REGIONAL FREQUENCY ANALYSIS OF MAXIMUM DAILY RAINFALLS
USING TL-MOMENT APPROACH
NORATIQAH BINTI MOHD ARIFF
A dissertation submitted in partial fulfillment of the requirements for the award of the
degree of Master of Science (Mathematics)
Faculty of Science
Universiti Teknologi Malaysia
OCTOBER 2009
iii
To those whose moral supports and love had helped me countless of time to overcome
each and every obstacle
Father, Mohd Ariff bin Omar
Mother, Hajar binti Hawari
My lovely sisters,
Norlina binti Mohd Ariff
Norasyiqin binti Mohd Ariff
I thank God for blessing me with all your presence.
iv
ACKNOWLEDGEMENT
First of all, I am grateful to the Almighty Allah S.W.T. because with His
blessings, I am able to finish my Master’s dissertation in the allocated time given. I
thank God again for giving me good health in order for me to successfully complete this
thesis.
I wish to express my sincere gratitude to my supervisor, Dr. Ani Shabri for his
guidance in helping me throughout this thesis and not forgotten to Encik Abu Salim
from “Jabatan Pengairan dan Saliran, Malaysia” who made it easy for me to collect the
data for my thesis.
I am very much indebted to my beloved family members who had helped me in
each and every step of the way. All your patience, understanding, love and kindness
have blessed my life in more ways than one.
Last but not least, I am grateful to all my friends and all those who had helped
me to accomplish this project. May Allah repay all the kindness that you have given me
thus far.
v
ABSTRACT
Analyzing rainfalls data are important in order to obtain the probability
distribution of flood. The main aim of the study is to perform regional frequency
analysis of maximum daily rainfalls measured over stations in Selangor and Kuala
Lumpur by using the TL-moment method with t = 0, t = 1 and t = 2. Initially, the
maximum of each daily rainfall for each year were obtained. Then, parameters of every
distributions considered including the normal (N), logistic (LOG), generalized logistic
(GLO), extreme value type I (EV), generalized extreme value (GEV) and generalized
Pareto (GPA) distribution were estimated using TL-moment approach. TL-moments
with t = 0 are known as L-moments while TL-moments with
and
imply TL-
moments that are symmetrically trimmed by one and two conceptual sample values
respectively. The most suitable distribution were determined according to the mean
absolute deviation index (MADI), mean square deviation index (MSDI) and correlation,
r. L-moment and TL-moment ratio diagrams provided visual proofs of the results. The
L-moment method showed that the generalized logistic (GLO) distribution is the best
distribution whilst TL-moment method with t = 1 and t = 2 concluded that the extreme
value type I (EV) and generalized extreme value (GEV) distributions are the most
suitable distributions to fit the data of maximum daily rainfalls for stations in Selangor
and Kuala Lumpur.
.
vi
ABSTRAK
Penganalisaan taburan hujan adalah penting untuk mendapatkan taburan
kebarangkalian banjir. Objektif utama kajian ini adalah untuk menjalankan analisis
frekuensi rantau terhadap data hujan harian maksimum yang diukur pada stesen-stesen
hujan di Selangor dan Kuala Lumpur menggunakan pendekatan TL-momen dengan t =0,
t = 1 dan t = 2. Pada mulanya, jumlah maksimum hujan bagi setiap tahun dikenalpasti.
Kemudian, parameter untuk setiap taburan yang diambil kira termasuk taburan normal
(N), logistik (LOG), logistic teritlak (GLO), nilai ekstrim Jenis I (EV), nilai ekstrim
teritlak (GEV) dan Pareto teritlak (GPA) dikira melalui kaedah TL-momen. TL-momen
dengan t = 0 merupakan kaedah L-momen manakala TL-momen dengan
dan
menunjukkan TL-momen yang ditrim secara simetri oleh satu dan dua nilai
sampel masing-masing. Taburan yang paling sesuai untuk mewakili data stesen-stesen
ini dikenalpasti melalui sisihan indeks min mutlak (MADI), sisihan indeks min kuasa
dua (MSDI) dan korelasi, r. Rajah nisbah L-momen dan TL-momen digunakan sebagai
bukti dapatan kajian. Hasil dari kajian ini didapati bahawa apabila menggunakan kaedah
L-momen, taburan logistik teritlak (GLO) adalah taburan terbaik manakala penggunaan
kaedah TL-momen dengan
dan
menunjukkan taburan nilai ekstrim Jenis I
(EV) dan nilai ekstrim teritlak (GEV) adalah kaedah paling sesuai bagi mewakili data
taburan hujan harian maksimum di stesen-stesen Selangor dan Kuala Lumpur.
vii
TABLE OF CONTENTS
CHAPTER
1
TITLE
PAGE
DECLARATION
ii
DEDICATION
iii
ACKNOWLEDGEMENTS
iv
ABSTRACT
v
ABSTRAK
vi
TABLE OF CONTENTS
vii
LIST OF TABLES
xiii
LIST OF FIGURES
xvii
LIST OF SYMBOLS
xviii
LIST OF APPENDICES
xix
INTRODUCTION
1
1.1
Flood in Malaysia
1
1.2
Introduction to Flood Frequency Analysis
2
1.3
Introduction to L-Moment and TL-Moment
4
1.4
Objectives of the Study
6
1.5
Scope of the Study
6
1.6
Significance of the Study
7
1.7
Chapters’ Overview
7
viii
2
3
LITERATURE REVIEW
9
2.1
Introduction
9
2.2
Frequency Analysis
9
2.3
Parameter Estimations
13
2.4
Selection of Distributions
14
2.5
The Method of L-Moment
16
2.6
The Method of TL-Moment
20
METHODOLOGY
22
3.1
The Method of L-Moments
22
3.1.1
L-Moments Distributions
22
3.1.2
L-Moments Sample Estimates
24
3.2
3.3
The Method of TL-Moments
25
3.2.1 TL-Moments Distributions
25
3.2.2 TL-Moments Sample Estimates
27
Normal Distribution
29
3.3.1
Probability Density Function
29
3.3.2
Distribution Function
30
3.3.3
Quantile Function
31
3.3.4
L-Moments and L-Moments Ratios
31
3.3.5
Parameter Estimates using the L-Moment Method 32
3.3.6 TL-Moments at t = 1
32
3.3.7
33
Parameter Estimates using the TL-Moment
Method at t = 1
3.3.8 TL-Moments at t = 2
33
3.3.9
34
Parameter Estimates using the TL-Moment
Method at t = 2
3.4
Logistic Distribution (LOG)
34
3.4.1
Probability Density Function
34
3.4.2
Distribution Function
35
3.4.3
Quantile Function
35
ix
35
3.4.4
L-Moments and L-Moments Ratios
3.4.5
Parameter Estimates using the L-Moment Method 35
3.4.6
TL-Moments at t = 1
36
3.4.7
Parameter Estimates using the TL-Moment
36
Method at t = 1
3.4.8
TL-Moments at t = 2
36
3.4.9
Parameter Estimates using the TL-Moment
37
Method at t = 2
3.5
Generalized Logistic Distribution (GLO)
37
3.5.1
Probability Density Function
38
3.5.2
Distribution Function
38
3.5.3
Quantile Function
38
3.5.4
L-Moments and L-Moments Ratios
39
3.5.5
Parameter Estimates using the L-Moment Method 39
3.5.6
TL-Moments at t = 1
39
3.5.7
Parameter Estimates using the TL-Moment
40
Method at t = 1
3.5.8
TL-Moments at t = 2
40
3.5.9
Parameter Estimates using the TL-Moment
41
Method at t = 2
3.6
Extreme Value Type I Distribution (EV)
42
3.6.1
Probability Density Function
42
3.6.2
Distribution Function
42
3.6.3
Quantile Function
43
3.6.4
L-Moments and L-Moments Ratios
43
3.6.5
Parameter Estimates using the L-Moment Method 43
3.6.6
TL-Moments at t = 1
43
3.6.7
Parameter Estimates using the TL-Moment
44
Method at t = 1
3.6.8
TL-Moments at t = 2
44
x
3.6.9
Parameter Estimates using the TL-Moment
45
Method at t = 2
3.7
Generalized Extreme Value Distribution (GEV)
45
3.7.1
Probability Density Function
46
3.7.2
Distribution Function
47
3.7.3
Quantile Function
47
3.7.4
L-Moments and L-Moments Ratios
47
3.7.5
Parameter Estimates using the L-Moment Method 48
3.7.6
TL-Moments at t = 1
48
3.7.7
Parameter Estimates using the TL-Moment
49
Method at t = 1
3.7.8
TL-Moments at t = 2
50
3.7.9
Parameter Estimates using the TL-Moment
51
Method at t = 2
3.8
Generalized Pareto Distribution (GPA)
51
3.8.1
Probability Density Function
52
3.8.2
Distribution Function
52
3.8.3
Quantile Function
52
3.8.4
L-Moments and L-Moments Ratios
52
3.8.5
Parameter Estimates using the L-Moment Method 53
3.8.6
TL-Moments at t = 1
53
3.8.7
Parameter Estimates using the TL-Moment
54
Method at t = 1
3.8.8
TL-Moments at t = 2
54
3.8.9
Parameter Estimates using the TL-Moment
55
Method at t = 2
3.9
Goodness of Fit Criteria for Comparison of Probability 55
Distributions
3.9.1
Mean Absolute Deviation Index (MADI) and
55
Mean Square Deviation Index (MSDI)
3.9.2
Correlation (r)
57
xi
3.10
4
5
L-moment and TL-moment Ratio Diagrams
58
DATA ANALYSIS
59
4.1
Selangor
59
4.2
Kuala Lumpur
60
4.3
Flood in Selangor and Kuala Lumpur
61
4.4
Data Collection
62
4.5
Descriptive Statistics
66
4.6
L-Moments and L-Moments Ratios
66
4.7
TL-Moments and TL-Moments Ratios
69
RESULTS
72
5.1
Introduction
72
5.2
Mean Absolute Deviation Index (MADI)
73
5.2.1 Results for TL-Moment with t = 0 (L-Moment)
73
5.2.2 Discussions on Mean Absolute Deviation Index
75
(MADI) for TL-Moment with t = 0 (L-Moment)
5.2.3
Results for TL-Moment with t = 1
5.2.4 Discussions on Mean Absolute Deviation Index
77
78
(MADI) for TL-Moment with t = 1
5.2.5 Results for TL-Moment with t = 2
78
5.2.6 Discussions on Mean Absolute Deviation Index
80
(MADI) for TL-Moment with t = 2
5.3
Mean Square Deviation Index (MSDI)
5.3.1
Results for TL-Moment with t = 0 (L-Moment)
5.3.2 Discussions on Mean Square Deviation Index
81
81
83
(MSDI) for TL-Moment with t = 0 (L-Moment)
5.3.3
Results for TL-Moment with t = 1
5.3.4 Discussions on Mean Square Deviation Index
84
86
(MSDI) for TL-Moment with t = 1
5.3.5 Results for TL-Moment with t = 2
88
xii
5.3.6 Discussions on Mean Square Deviation Index
89
(MSDI) for TL-Moment with t = 2
5.4
Correlation (r)
89
Results for TL-Moment with t = 0 (L-Moment)
90
5.4.2 Discussions on Correlation (r) for TL-Moment
92
5.4.1
with t = 0 (L-Moment)
5.4.3
Results for TL-Moment with t = 1
5.4.4 Discussions on Correlation (r) for TL-Moment
94
95
with t = 1
5.4.5 Results for TL-Moment with t = 2
95
5.4.6 Discussions on Correlation (r) for TL-Moment
97
with t = 2
5.5
Summary on the Case of TL-Moment with t = 0
98
(L-Moment)
6
5.6
Summary on the Case of TL-Moment with t = 1
100
5.7
Summary on the Case of TL-Moment with t = 2
102
5.8
Conclusions
104
CONCLUSIONS AND RECOMMENDATIONS
105
6.1
Conclusions
105
6.2
Recommendations
109
REFERENCES
Appendices A – C
110
121-147
xiii
LIST OF TABLES
TABLE NO.
4.1
TITLE
PAGE
Accumulated hourly rainfall (mm) within 24 hours period from
Meteorological Stations in Petaling Jaya, Subang and KLIA on
10 June 2007
4.2
Name and information on all the stations in Selangor and
Kuala Lumpur
4.3
62
65
Descriptive Statistics on the maximum daily rainfalls for
stations in Selangor and Kuala Lumpur
67
4.4
L-Moments and L-Moments Ratios for all the stations
68
4.5
TL-Moments and TL-Moments Ratios for all the stations
(t = 1)
4.6
TL-Moments and TL-Moments Ratios for all the stations
(t = 2)
5.1
71
Mean Absolute Deviation Index (MADI) for stations in
Selangor and Kuala Lumpur (L-moment method, t = 0)
5.2
70
74
Ranks of Mean Absolute Deviation Index (MADI) for each
distribution with 55 stations (L-moment method, t = 0)
75
xiv
5.3
Ranks of Mean Absolute Deviation Index (MADI) for each
distribution with 39 stations excluding the 16 stations (Lmoment method, t = 0)
5.4
Mean Absolute Deviation Index (MADI) for stations in
Selangor and Kuala Lumpur (TL-moment method with t = 1)
5.5
76
Ranks of Mean Absolute Deviation Index (MADI) for each
distribution with 55 stations (TL-moment with t = 1)
5.6
75
77
Ranks of Mean Absolute Deviation Index (MADI) for each
distribution with 39 stations excluding the 16 stations (TLmoment with t = 1)
5.7
Mean Absolute Deviation Index (MADI) for stations in
Selangor and Kuala Lumpur (TL-moment method with t = 2)
5.8
79
Ranks of Mean Absolute Deviation Index (MADI) for each
distribution with 55 stations (TL-moment with t = 2)
5.9
77
80
Ranks of Mean Absolute Deviation Index (MADI) for each
distribution with 39 stations excluding the 16 stations (TLmoment with t = 2)
5.10
Mean Square Deviation Index (MSDI) for stations in Selangor
and Kuala Lumpur (L-moment method, t = 0)
5.11
80
82
Ranks of Mean Square Deviation Index (MSDI) for each
distribution with 55 stations (L-moment method, t = 0)
83
xv
5.12
Ranks of Mean Square Deviation Index (MSDI) for each
distribution with 39 stations excluding the 16 stations (Lmoment method, t = 0)
5.13
Mean Square Deviation Index (MSDI) for stations in Selangor
and Kuala Lumpur (TL-moment method with t = 1)
5.14
85
Ranks of Mean Square Deviation Index (MSDI) for each
distribution with 55 stations (TL-moment with t = 1)
5.15
83
86
Ranks of Mean Square Deviation Index (MSDI) for each
distribution with 39 stations excluding the 16 stations (TLmoment with t = 1)
5.16
Mean Square Deviation Index (MSDI) for stations in Selangor
and Kuala Lumpur (TL-moment method with t = 2)
5.17
87
Ranks of Mean Square Deviation Index (MSDI) for each
distribution with 55 stations (TL-moment with t = 2)
5.18
86
88
Ranks of Mean Square Deviation Index (MSDI) for each
distribution with 39 stations excluding the 16 stations (TLmoment with t = 2)
5.19
Correlation, r, for stations in Selangor and Kuala Lumpur (Lmoment method, t = 0)
5.20
88
91
Ranks of correlation, r, for each distribution with 55 stations
(L-moment method, t = 0)
92
xvi
5.21
Ranks of correlation, r, for each distribution with 39 stations
excluding the 16 stations (L-moment method, t = 0)
5.22
Correlation, r, for stations in Selangor and Kuala Lumpur (TLmoment method with t = 1)
5.23
96
Ranks of correlation, r, for each distribution with 55 stations
(TL-moment with t = 2)
5.27
94
Correlation, r, for stations in Selangor and Kuala Lumpur (TLmoment method with t = 2)
5.26
94
Ranks of correlation, r, for each distribution with 39 stations
excluding the 16 stations (TL-moment with t = 1)
5.25
93
Ranks of correlation, r, for each distribution with 55 stations
(TL-moment with t = 1)
5.24
92
97
Ranks of correlation, r, for each distribution with 39 stations
excluding the 16 stations (TL-moment with t = 2)
97
xvii
LIST OF FIGURES
FIGURE NO.
4.1
TITLE
PAGE
Location Map of Rainfall Gauge Stations in Selangor and
Kuala Lumpur
61
5.1
L-Moment Ratio Diagram (a)
99
5.2
L-Moment Ratio Diagram (b)
99
5.3
TL-Moment Ratio Diagram with t = 1 (a)
101
5.4
TL-Moment Ratio Diagram with t = 1 (b)
101
5.5
TL-Moment Ratio Diagram with t = 2 (a)
103
5.6
TL-Moment Ratio Diagram with t = 2 (b)
103
xviii
LIST OF SYMBOLS
K
-
shape parameter
r
-
correlation
sxx2
-
sample variance for observed flows
szz2
-
sample variance for predicted flows
sxz 2
-
sample covariance
xi
-
observed flows
zi
-
predicted flows
D
-
scale parameter
P
-
mean of the x series
V
-
standard deviation of the x series
[
-
location parameter
W3
-
L-moment skewness
W4
-
L-moment kurtosis
W 13
-
TL-moment skewness with t = 1
W 14
-
TL-moment kurtosis with t = 1
W 32
-
TL-moment skewness with t = 2
W 42
-
TL-moment kurtosis with t = 2
)
-
standard normal distribution function
) 1 (F )
-
the inverse of standard normal distribution function
xix
LIST OF APPENDICES
APPENDIX
TITLE
A
MathCAD Program using the L-Moment Method
B
MathCAD Program using the TL-Moment Method
with t = 1
C
PAGE
121
130
MathCAD Program using the TL-Moment Method
with t = 2
139
CHAPTER 1
INTRODUCTION
1.1
Flood in Malaysia
Human society faces great problems due to extreme environmental events. For
example, floods, rainstorms, droughts and high winds that cause tornadoes and such
destroy almost anything that is in their vicinity at the moment of occurrences. Flood,
also known as deluge, is a natural disaster that could diminish properties, infrastructures,
animals, plants and even human lives.
In terms of the number of population affected, frequency, area extent, duration
and social economic damage, flooding is the most natural hazard in Malaysia (Ministry
of Natural Resources and Environment, Malaysia, June 2007). According to the Ministry
of Natural Resources and Environment, Malaysia in June 2007, Malaysia has
experienced major floods since 1920 especially in the years 1926, 1963, 1965, 1967,
1969, 1971, 1973, 1979, 1983, 1988, 1993, 1998, 2005, December 2006 and January
2007. These flood events occurred in various states including Selangor and the capital
city of Malaysia, Kuala Lumpur.
The basic cause of river flooding is the incidence of heavy rainfall (monsoon or
convective) and the resultant large concentration of runoff, which exceeds river capacity
(Ministry of Natural Resources and Environment, Malaysia, June 2007). Flood had
resulted in a loss of millions in Malaysia. For example, the 1971 flood that hit Kuala
Lumpur and many other states had caused more than RM200 million losses and 61
deaths. Furthermore, the massive floods due to a few abnormally heavy rains in 2006
and 2007 cost RM 1.5 billion and hence deemed as the most expensive flood events ever
to occur in Malaysian history. This includes the cost of damage in infrastructures,
bridges, roads, agriculture and private commercial and residential properties. During this
flood event, 18 people unfortunately died and around 110,000 people were evacuated
from their homes and were sheltered in relief centers.
1.2
Introduction to Flood Frequency Analysis
Analyzing rainfalls and stream flows data are important in order to obtain the
probability distribution of flood and other phenomenon related to them. By knowing the
probability distribution, prediction of flood events and their characteristics can be
determined. With this, prevention acts and measures can be taken and flash flood
warning models can be built easily.
The study of water related characteristics and modeling throughout the Earth
such as the movement, distribution, resources, hydrologic cycle and quality of water is
called hydrology. By knowing and analyzing statistical properties of hydrologic records
and data like rainfall or river flow, hydrologists are able to estimate future hydrologic
phenomena. A very active area of investigation in Statistical Hydrology is the frequency
of floods (Rao et al., 2000).
3
As stated earlier, flood is the most costly natural hazard in Malaysia. It is also
one of the oldest natural hazards in the world. Hence, its characteristics and the
magnitude-recurrence interval relationship are important for hydrologist to design or
plan hydrological projects. In order to be able to plan and design these projects such as
hydraulic or water resources projects, continuous hydrological data, for example,
rainfalls data or river flow data is necessary. With the help of the data, flow pattern or
trend can be determined to make sure the design and planning can be done accordingly.
Hydraulic structures such as weirs, barrages, dams, spillways and bridges can be
modeled and damages can be minimized with a reliable and good estimation of
magnitude and frequency of occurrence of such extreme events. Many aspects of water
resources engineering and hydraulic studies need to estimate region or for a group of
sites (Rao et al., 2000). However, to select a reliable design quantile, which has affect on
design, operation, management and maintenance of hydraulic structure depends on
statistical methods used in parameter estimation belonging to probability distribution
(Hosking and Wallis, 1993).
Estimating flood and designing water related structure, erosion and agricultural
considerably need knowledge related to distributions of extreme rainfall depths.
Probability for future events can be predicted by fitting past observations to selected
probability distributions. The primary objective is to relate the magnitude of these
extreme events to their frequency occurrence through the use of probability distributions
(Chow et al., 1988).
However, extreme events are usually too short and too rare for a reliable
estimation to be obtained. This also includes the difficulties of identifying the
appropriate statistical distribution to describe the data and estimating the parameters of
the selected distribution. Hence, regional frequency analysis which was developed by
Hosking and Wallis (1991) is used since it can resolve this problem by trading space for
time.
4
With this method, this problem will be resolved. According to Cunnane (1989),
regional analysis is based on the concept of regional homogeneity which assumes that
annual maximum flow populations at several sites in a region are similar in statistical
characteristics and are not dependent on catchments size.
1.3
Introduction to L-Moment and TL-Moment
Extreme events such as flood are rare and often occur in a short amount of time.
Hence, it is difficult to analyze the characteristics of its statistical probability
distributions. By replacing space for time, frequency analysis is used to obtain the
probability distributions for extreme events. Outliers are common to be found in data
related to flood which is an extreme natural hazard.
Recently, the most popularized method in frequency analysis is the L-moment
approach introduced by Hosking in 1990 (Rao et al., 2000). The main role of the Lmoments is for estimating parameters for probability distributions. L-moments’
estimates are superior to standard moment-based estimates generally and especially for
small samples. They are also relatively insensitive to outliers compared to conventional
moments. Their small sample bias tends to be very small. L-moments are also preferable
when maximum likelihood estimates are unavailable, difficult to compute or have
undesirable properties.
Probability distributions are used to analyze data in many disciplines and are
often complicated by certain characteristics such as large range, variation or skewness.
Hence, outliers or highly influential values are common (Asquith, 2007). Outliers can
have undue influence on standard estimation methods (Elamir and Seheult, 2003).
According to Elamir and Seheult, if there is a concern about extreme observations
having undue influence, a robust method of estimation which is developed to reduce the
said influence of outliers on the final estimates should be preferable. TL-moments are
5
derived by Elamir and Seheult in 2003 from L-moments and might have additional
robust properties compared to L-moments. In other words, TL-moments are claimed to
be more robust than the L-moment. Hence, for extreme data, TL-moments are also
considered for estimating the parameters of the selected probability distributions.
Thus, this study focused on identifying a suitable probability distribution,
including normal (N), logistic (LOG), generalized logistic (GLO), extreme value type I
(EV), generalized extreme value type I (GEV) and generalized Pareto (GPA) by using
TL-moments technique for maximum daily rainfalls selected for each year among daily
rainfalls measured over the regions in Selangor and Kuala Lumpur, Malaysia. The TLmoments for all the said distributions were derived in order to be able to fit the rainfall
data to the probability distributions.
In the case of TL-moments which are symmetrically trimmed by one conceptual
sample value, i.e
, for normal (N) and logistic (LOG) distributions, the
TL-moments and their parameter estimates were computed and checked with those
obtained by Elamir and Seheult in 2003. Meanwhile, the TL-moments and their
parameter estimates for generalized logistic (GLO), extreme value type I (EV),
generalized extreme value type I (GEV) and generalized Pareto (GPA) distributions
were derived since none had been done before. However, for TL-moments which are
symmetrically trimmed by two conceptual sample values, i.e
all six
distributions’ TL-moments and their respective parameter estimates were all derived in
this study. The results from both cases (
and
) were then compared with those
obtained using the method of L-moments similar to the previous study by Shabri and
Ariff (2009).
6
1.4
Objectives of the Study
The objectives of this study are:
i. To derive the TL-moments for the selected distributions to be considered. Not all
distributions’ TL-moments have been derived thus far. New derivation will
be done for generalized logistic (GLO), extreme value type I (EV),
generalized extreme value type I (GEV) and generalized Pareto (GPA)
distributions.
ii. To obtain the respective parameter estimates for each distribution.
iii. To find the most suitable distribution to fit the maximum daily rainfalls data by
using the goodness-of-fit tests.
iv. To compare the results with the ones obtained from using the method of Lmoments in the previous study.
1.5
Scope of the Study
The scope of this study consisted of the TL-moment approach on maximum daily
rainfalls through regional frequency analysis. The data of maximum daily rainfalls were
selected each year and measured over stations in Selangor and Kuala Lumpur. The aims
of this study were to derive new TL-moments population and to determine the best
probability distribution among the selected distributions whose parameters were
estimated using the method of TL-moment. Furthermore, this study included the
comparison between the results achieved and those that have been obtained through the
L-moment approach.
7
1.6
Significance of the Study
The results of this study give benefits to statistical and hydrological studies. The
direct beneficiaries of the study are the statisticians, applied mathematicians, engineers
and hydrologists working in the research areas of applications from the result of
specifying the probability distribution of extreme events which in this case is flood.
Thus, this also helps our country from unnecessary cost and economic losses as well as
preventing possible danger due to overflow of water in the country. This study widened
the scope of TL-moments to distributions that have not been considered before which
were the generalized logistic (GLO), extreme value type I (EV), generalized extreme
value type I (GEV) and generalized Pareto (GPA) distributions and used all these TLmoments in estimating the probability distribution of rainfalls data. The comparison of
the TL-moment method and the L-moment method is also useful in helping statisticians
and mathematicians to determine the most suitable method for different situations.
1.7
Chapters’ Overview
Chapter 1 highlights introductions to flood occurences in Malaysia, flood
frequency analysis and the TL-moment method. It also covers the objectives, scope and
significance of the study.
Chapter 2 explains on frequency analysis, parameter estimations and selection of
distributions. It also includes a brief history and introductions on the statistical
distributions considered in this study which consists of normal, logistic, generalized
logistic, extreme value, generalized extreme value and the generalized Pareto
distribution.
8
Chapter 3 covers the probability density functions, distribution functions,
quantile functions, L-moments, L-moment ratios and parameter estimates using the Lmoment method for each statistical distributions that are being considered which are
normal (N), logistic (LOG), generalized logistic (GLO), extreme value type I (EV),
generalized extreme value type I (GEV) and generalized Pareto (GPA) distribution. This
chapter also includes the derivation of the TL-moments and TL-moment ratios in the
case of
and
for all the six distributions. In addition, it also covers the
goodness of fit test.
Chapter 4 includes a brief overview of Selangor and Kuala Lumpur, the data
collection process, the name and information on all the 55 stations considered and the
descriptive statistics of the data. It also presents the L-moments, L-moment ratios, TLmoments and TL-moment ratios for both
and
cases.
Chapter 5 discusses the analysis of the data using the TL-moments and Lmoment methods and their ratios that had been given in Chapter 3. It also covers the
results of the data using the mean absolute deviation index (MADI), mean square
deviation index (MSDI) and correlation, r.
Chapter
6 presents the conclusions made
from the analyzed
Recommendations for future research are also given in this section.
data.
CHAPTER 2
LITERATURE REVIEW
2.1
Introduction
The primary objective of frequency analysis is to relate the magnitude of extreme
events to their frequency of occurrence through the use of probability distributions
(Chow et al., 1988). Flood frequency analysis analyses data observed over an extended
period of time in a river or rainfall system. These data are assumed to be independent
and identically distributed. Furthermore, flood data are considered to be stochastic.
They may even be assumed to be space and time independent since the floods are also
assumed not to have been affected by natural or manmade changes in the hydrological
regime in the system.
2.2
Frequency Analysis
Several summaries and discussions of flood frequency analysis had been done
along with many discussions on the general aspects of frequency analysis. Earlier works
of flood frequency analysis covered a wide range including a number of researches
related to parameter estimation, different probability distributions, and regionalization
methods which have been completed during the last two decades.
10
Based on the book written by A. Ramachandra Rao and Khaled H. Hamed in
2000, the history done on frequency analysis started as early as 1958 with the
development of computing the flood discharge probability based on the rational method,
the time concentration and unit hydrograph interpretation. The 1950s also found that
flood frequencies are second in importance to drainage area and hence investigating the
effects of channel slope on flood frequencies were done. Meanwhile, the next decade
discussed more on the problem of estimating the relationship between the magnitude and
frequency of rare floods, the use of moment ratio diagram, the random occurrence of
rare floods and the use of Poisson distribution in flood frequency analysis.
Discussions on the relationship between flood data and watershed characteristics
of small basins only started in 1970 by White and Reich. Then, frequency analysis was
developed further by the end of the 1970s by incorporating previous water levels at a site
into probability analysis.
The 1980s showed a great development in the history of frequency analysis with
the emphasized of the importance of visual interpretation of observed flood series.
Different characteristics of flood frequency analysis were also explored. For example,
Crippen (1982) developed envelope curves for extreme flood events in the U.S. and
Kuczera (1982) introduced the concept of robust flood frequency models. He also found
that regionalized estimates were preferable to estimates based on short record lengths
and estimates which combined both site and regional information were preferable for
large record lengths. Furthermore, he was able to develop an empirical Bayes approach
to combine site-specific and regional information.
Other developments included the serial dependence investigation on the
reliability of the T-year event, the procedures to estimate recurrence intervals of large
floods and the use of cox regression model for flood frequency analysis. These models
allowed incorporation of time varying exogenous information into flood frequency
analysis.
11
In 1990, a non-parametric flood frequency analysis method which can also use
historical information was developed (Adamowski and Feluch, 1990). A year later, flood
frequency groups were looked at in the context of regional flood frequency analysis.
Then, flood flow frequency model selection was explored along with the appraisal of
regional and index flood quantile estimators.
Escalante-Sandoval (1998) proposed models of multivariate extreme value
distributions with mixed Gumbel marginals and concluded that the proposed models are
suitable options to be considered when performing flood frequency analysis. By the end
of the millennium, floods in permeable drainage basins were proven to be able to
estimate (Faulkner and Robson, 1999).
Meanwhile, the millennium stated off with Daviau et al. (2000) using GIS, Lmoment and geostatistical methods in their regional frequency analysis. Various
methods were then used in flood frequency analysis. Stochastic and deterministic
methods were also combined in order to make frequency predictions and to build an
integrated simulation method for flash-flood risk assessment (Rulli and Rosso, 2002).
These extended to the use of the index-flood method in conducting the regional flood
frequency analysis (Kjeldsen et al., 2002).
The developments of different methods were continued for consequent years
with the investigation of the generalized probability weighted moments, generalized
moments and maximum likelihood fitting methods in the two-parameter log-logistic
model to extreme hydrologic data (Ashkar and Mahdi, 2003). The results obtained from
the investigations claimed that the log-logistic distribution has a good fitting potential
for fitting flood data. In the same year, developments concerning the comparison and
analyzing design floods were extended with the derivation through LH-moment methods
and the application of appropriate distribution on six Korean watersheds (Lee and
Maeng, 2003). The result presented the appropriate order of LH-moments that derived
appropriate design floods. Meanwhile, Cunderlik and Burn (2003) explored the non-
12
stationary pooled flood frequency analysis whilst Fowler and Kilsby (2003) did a
regional frequency analysis of United Kingdom extreme rainfall from 1961 to 2000.
Next, models were proposed for applications of flood frequency analysis using
the extended three-parameter Burr XII system (Shao et al., 2004). This was
demonstrated using data from China which then the Gan-Ming River basin in China was
used to conduct a regional flood frequency analysis. Regional rainfall frequency analysis
for the state of Michigan and historical as well as pooled flood frequency analysis for the
River Tay at Perth, Scotland were also explored (Jingyi and Hall (2004), Trefry et al.,
(2005) and Macdonald et al. (2006)).
A year later, regional analysis was used to improve estimates of the probabilities
of extreme events in Czech Republic by Kysely and Picek (2007a). Kysely and Picek
(2007b) also explored the probability estimates of heavy precipitation events in a floodprone central-European region with enhanced of Mediterranean cyclones. Meanwhile,
exploration of the regional Bayesian POT model for flood frequency analysis, usage of
the region-of-influence approach to a frequency analysis for heavy precipitation data and
application of the regional frequency analysis of extreme precipitation and flash flood
were also discussed thoroughly in the same year (Ribatet et al. (2007), Gaal et al. (2007)
and Norbiato et al. (2007)).
Recently, multivariate extension of the logistic model with the applications of
generalized extreme value (GEV) marginals was investigated especially the trivariate
generalized extreme value distribution in flood frequency analysis in order to provide a
regional at-site flood estimate (Sandoval, 2008). In the mean time, Srinivas et al. (2008)
combined self-organizing feature map and fuzzy clustering to regional flood frequency
analysis and Gaal et al. (2008) used the regional frequency analysis of heavy
precipitation totals in the High Tatras region in Slovakia for flood estimation while
Meshgi and Khalili (2009) did a comprehensive evaluation of regional flood frequency
analysis by L-moments and LH-moments.
13
2.3
Parameter Estimations
There are a few methods that can be used for parameter estimation which include
the method of moments (MOM), the maximum likelihood method (MLM), the
probability weighted moments method (PWM), the least squares method (LS),
maximum entropy (ENT), mixed moments (MIX), the generalized method of moments
(GMM), and incomplete means method (ICM) (Rao et al., 2000).
The maximum likelihood method (MLM) is often regarded as the most efficient
method. This is because it provides the smallest sampling variance of the estimated
parameters which leads to the smallest sampling variance of the estimated quantiles
compared to other methods. MLM has disadvantages in some particular cases, such as
the Pearson type III distribution where the optimality of the MLM is only asymptotic
and small sample estimates may lead to estimates of inferior quality (Bobeé et al., 1993).
Another disadvantage is that MLM often gives biased estimates. However, these biased
estimates can be corrected. Furthermore, MLM might be hard to compute especially if
the number of parameters is large. This will in turn make it hard and might also be
impossible to obtain ML estimates of small samples.
Another method is the method of moments (MOM) which is a relatively easy
parameter estimation method. Unfortunately, MOM estimates are usually inferior in
quality and generally not as efficient as the ML estimates especially in the case where
the distributions have a large number of parameters. This is due to the fact that higher
order moments are more likely to be highly biased in relatively small samples.
The probability weighted moments (PWM) method (Greenwood et al., 1979;
Hosking, 1986) gives parameter estimates comparable to the ML estimates. In fact, in
some cases the estimation procedures are much less complex and thus less complicated
since the computations are simpler than that of ML estimates. Landwehr et al., (1978)
stated that the parameter estimates from small samples using PWM are sometimes more
14
accurate than the ML estimates. Moreover, explicit expressions for the parameters in
some cases can be obtained by using PWM such as the symmetric lambda and Weibull
distributions. This is not the case with the ML or MOM methods.
2.4
Selection of Distributions
Flood frequency analysis requires selecting distributions to be considered in the
analysis. The choice of distributions has been a debatable topic for a long time. Rao et
al. (2000) had given a brief history of selection of distribution in flood frequency
analysis since the year 1914 with the selection of distribution in the context of storage
design for municipal water supply. Probability distributions that best fit distributions of
annual precipitation and runoff series were analyzed in the middle of 1960s before a
method based on the coefficient of determination for selection of a distribution which
best fits the original data was proposed and differences between these distributions were
detected using a nonparametric test.
Selection of distribution relies heavily on the determination of evaluation criteria.
Hence, extensive discussions were done on the criteria to evaluate the usefulness of
hydrologic analysis and a classification system for categorizing the available procedures
of flood frequency analysis was presented along with a literature review and
recommendation on reporting flood frequency analysis procedures. A presentation of
statistical terms used in flood frequency analysis and an interpretation of these terms
were also looked at around this point of time.
Next, different distributions were compared in various application of the
frequency analysis with various methods that are deemed suitable. The Type I extreme
value (EV), two-parameter lognormal (LN(2)), three-parameter lognormal (LN(3)) and
log Pearson type III (LP(3)) distributions were compared for fitting flood data from
Oregon and it was concluded that the LP (3) distribution was the best (Campbell and
15
Sidel, 1984). The Akaike’s information criterion and a new probability plot correlation
test were also proposed for the choice of distributions (Turkman, 1985). The probability
plot correlation coefficient test for the normal and lognormal distribution hypotheses
were discussed with the EV distribution by Vogel (1986).
The next decade started off with the development of a probability plot correlation
coefficient hypothesis test for the Pearson type III (LP(3)) distribution which was then
presented with a new estimator of the skewness coefficients (Vogel and McMartin,
1991). Nine distributions with data from 45 unregulated streams in Turkey were used to
conclude that LN(2) and EV distributions were superior to other distributions (Haktanir,
1992).
Commonly used procedures for flood frequency estimation were also reviewed
and reasons for the confusions concerning comparative studies were determined. In
correspondence to this, broad lines of a comparison strategy were presented. Next, in
response to the earlier conclusion that the LN(2) and EV distributions were superior to
other distributions, a total of 1819 site-years of data from 19 stations in the world were
analyzed and seven distributions were used to claim that the Generalized Extreme Value
(GEV) distribution was superior to other distributions instead (Onoz and Bayazit, 1995).
Hence, it was also concluded that most of the methods available for selection of
distributions from small samples are not sensitive enough to discriminate among
distributions.
Some statistics useful in regional frequency analysis were given by Hosking and
Wallis (1993) while Vogel et al. (1993) explored the flood-flow frequency model
selection in Southwestern United States. Later, the regional distributions for flood
frequency analysis of southern Africa were also identified (Mkhandi et al., 2000).
In 2007, the regional frequency distributions of floods in West Mediterranean
river basins in Turkey concluded that it may be more appropriate to use the Log-Pearson
16
type III distribution instead of the widely used Gumbel distribution for probability
distribution modeling of extreme values especially in West Mediterranean river basins
(Saf et al., 2007).
In the mean time, Ellouze and Abida (2008) looked at probability distribution of
flood flows in Tunisia and concluded that Northern Tunisia was shown to be represented
by the generalized normal distribution while both the generalized normal and
generalized extreme value distributions gave the best fit in the central and Southern
Tunisia.
2.5
The Method of L-Moment
L-moments are summary statistics for probability distributions and data samples.
They are analogous to ordinary moments and they provide measures of location,
dispersion, skewness, kurtosis, and other aspects of the shape of probability distributions
or data samples. However, they are computed from linear combinations of the ordered
data values and hence they are given the prefix L.
According to Hosking (1990), L-moments have various theoretical advantages
over ordinary moments. For example, for L-moments of a probability distribution to be
meaningful, only the distribution has to have finite mean and no higher-order moments
need be finite. Similarly, in order for the standard errors of L-moments to be finite, only
the distribution is required to have finite variance and no higher-order moments need be
finite. Although moment ratios can be arbitrarily large, sample moment ratios have
algebraic bounds but sample L-moment ratios can take any values that the corresponding
population quantities can (Hosking, 1990).
In addition, L-moments have properties that hold in a wide range of practical
situations. L-moments also give asymptotic approximations to sampling distributions
17
better than ordinary moments and provide better identification of the parent distribution
which generated a particular data sample. Furthermore, L-moments are less sensitive to
outlying data values (Vogel and Fennessey, 1993).
L-moments can also be used as the basis of a unified approach to the statistical
analysis of univariate probability distributions. L-moments can be defined for any
random variable whose mean exists. This forms the basis of a general theory that covers
the summarization and description of theoretical probability distributions and observed
data samples, estimation of parameters and quantiles of probability distributions and
hypothesis testing of probability distributions.
L-moment is a common method used in flood frequency analysis. In 1993, Vogel
et al. analyzed flood data from 383 sites in the southwestern U.S. in order to explore the
suitability of different distributions to model flood frequencies. L-moment ratio
diagrams were used by them for selection of distributions (Rao et al., 2000).
In the same year, Vogel and Fennessey also explored the advantages of Lmoments compared to product moments while previously, Gingras and Adamowski
(1992) combined L-moments and nonparametric methods to underlying distributions
which are nonunimodal.
The L-moment approach introduced by Hosking (1990) is widely popularized to
frequency analysis especially in the case of parameter estimation belonging to statistical
distributions. The method of L-moments has been used increasingly by hydrologists
(Chen et al., 2006).
For example, the L-moment method was used for analyzing the
regional flood frequency in New Zealand and for finding the value of regional
information to flood frequency analysis (Pearson, 1991). The L-moment method was
also applied in a regional frequency analysis concerning the Wabash River flood data
and the regional frequency analysis in Southern Ontario (Rao et al. (1997) and Glaves et
al. (1997)).
18
The L-moment method is very popular and commonly used in analyzing rivers
and basins in India. The method of L-moments was exploited in the regional frequency
analysis for Mahi-Sabarmati basin in India and it was found out that the three-parameter
lognormal distribution (LN(3)) is a suitable probability distribution for modeling floods
for the basin (Parida et al., 1998). Meanwhile, investigation and comparison of sampling
properties of L-moments and conventional moments were done by Sankarasubramanian
and Srinivasan in 1999. Nest, the development of regional flood formulae for gauged
and ungauged cathchments of South Bihar, Jharkhand and North Brahmaputra river
system were also done by using L-moments (Kumar et al. (2002) and Kumar et al.
(2003a)). Another frequency analysis done with the help of L-moments and L-moment
ratio diagrams in India was for the Middle Ganga Plains sub-zone where it was
concluded that the generalized extreme value distribution (GEV) is the robust
distribution for that particular area (Kumar et al., 2003b).
The search for L-moment estimation using annual maximum and peak over
threshold series in regional analysis of flood frequencies started off in the early
millennium by Gottschalk and Krasovskaia (2002). Around the same time, sampling
variance of flood quantiles from the generalized logistic distribution was also estimated
with the help of the L-moment method (Kjeldsen and Jones, 2004).
In Malaysia, the regional frequency analysis was also done using an index flood
estimation procedure based on L-moments where Lim and Lye (2003) found that the
generalized extreme value (GEV) and generalized logistic distributions (GLO) were
appropriate for the distribution of extreme flood events in the Sarawak region of
Malaysia. As in previous researches, the regional flood frequency was also done using
the method of L-moment to other rivers and basins around the globe. In Canada, Lmoment was utilized in the determination of regional probability distributions of
Canadian flood flows (Yue and Wang, 2004) which was then followed with a research
done by Chebana and Ouarda (2007) where the multivariate L-moment homogeneity test
was explored. Then, the use of L-moment helped in analyzing the maximum monthly
19
rainfall for an arid region in Isfahan Province, Iran and also in the regional flood
frequency analysis in Sicily, Italy (Eslamian and Feizi (2007) and Noto et al. (2008)).
In the mean time, assessment of the regional floods in the case of the River Nile
using the L-moments approach found that the generalized logistic distribution represent
the hydrologically homogeneous region formed by eight sites of the River Nile (Atiem
and Harmancioglu, 2006). Next, the Halil-River basin regional flood frequency analysis
based on L-moment approach was surveyed and the results obtained showed that the
generalized Pareto distribution was ana appropriate distribution for fitting the observed
for region A of the Halil-River while for Region B, the suitable distributions were the
generalized extreme value, Pearson type III, lognormal, generalized logistic and
generalized Pareto distributions (Rahnama and Rostami, 2007).
Recently, Hussain and Pasha (2009) used L-moments for regional flood
frequency analysis of the seven sites of Punjab, Pakistan and the results obtained with
the help of L-moment ratio diagram showed that the generalized normal, the generalized
Pareto and the generalized extreme value distributions were suitable candidates for
regional distribution. Meanwhile, Saf (2009a) performed the regional flood frequency
analysis using L-moments for the West Mediterranean region of Turkey. The Pearson
type III distribution was identified as the best-fit distribution for the Antalya and LowerWest Mediterranean subregions and the generalized logistic was the best for the UpperWest Mediterranean subregion. Saf (2009b) did another regional flood frequency
analysis using L-moments for the buyuk and kucuk Menderes river basins of Turkey and
concluded that the generalized normal extreme value distribution was the best-fit
distribution for both the upper- and lower-Menderes subregions.
In a previous study by Shabri and Ariff, 2009, the generalized logistic
distribution (GLO) was found to be the most suitable distribution to fit the data of
maximum daily rainfalls for stations in Selangor and Kuala Lumpur.
20
2.6
The Method of TL-Moment
TL-moments were introduced by Elamir and Seheult in 2003 as an alternative to
LQ-moments which are a robust version of L-moments. TL-moments censored or
trimmed a predetermined percentage of the extreme values from the sample before
estimating the mean and standard deviation from the un-trimmed sample values (Elamir
and Seheult, 2003). In other words, TL-moments are an extension of L-moments. This is
obvious since TL-moments are derived from L-moments (Asquith, 2007).
TL-moments, as stated earlier, are more robust than L-moments and they exist
even if the particular distribution does not have a mean. For example, TL-moments exist
for Cauchy distribution which is known to have no mean (Abdul Moniem, 2009).
Considering that TL-moments are extensions of L-moments, TL-moments are deemed to
be able to estimate the probability distributions of extreme events. In fact, L-moments
are a special case of TL-moments where
which means both sides are
symmetrically untrimmed. Thus, this study used the method of TL-moments to estimate
parameters for probability distributions to fit maximum rainfalls data for flood events.
Since it has been introduced, several studies had been done using the method of
TL-moment in order to estimate the parameters of the sample TL-moments for all kinds
of selected probability distributions and to determine the most suitable probability
distribution to fit the original data. L-moments and TL-moments of the generalized
lambda distribution had been derived and compared by Asquith in 2007. Meanwhile, the
methods of L-moments and TL-moments estimators have also been applied to estimate
the parameters of exponential distribution (Abdul-Moniem, 2007).
The most recent thus far, is the application of both these methods in estimating
parameters for the generalized Pareto distribution (GPA) (Abdul-Moniem, 2009).
However, the function used in estimating parameters for the generalized Pareto
21
distribution by Abdul-Moniem is different than the ones used by Rao et.al. in 2000 and
in the previous study (Shabri and Ariff, 2009).
Hence, this study derived new TL-moments with
and
for
distributions such as the normal, logistic (LOG), generalized logistic (GLO), extreme
value type I (EV), generalized extreme value type I (GEV) and generalized Pareto
(GPA) distributions. The TL-moments were used in regional frequency analysis to find
the best probability distributions to fit the original maximum rainfall data. Finally,
comparison of both L-moments and TL-moments in regional frequency analysis were
compared and summarized.
CHAPTER 3
METHODOLOGY
3.1
The Method of L-Moment
3.1.1 L-Moment Distributions
As defined by Hosking (1990), L-moments are linear combinations of probability
weighted moments (PWM).
The theory of PWM are summarized and defined by
Greenwood et al. (1979) as
Er
1
³ x F F
r
dF
where Er is the r th order PWM.
0
Hosking (1986 and 1990) introduced the L-moments, which are linear functions
of PWMs. The L-moments are more convenient than PWMs because they can be
directly interpreted as measures of scale and shape of probability distributions. In this
respect they are analogous to conventional moments.
23
L-moments are defined by Hosking in terms of the PWMs ȕ as:
O1
O2
O3
O4
E0 ,
2E 1 E 0 ,
6 E 2 6E 1 E 0 ,
20E 3 30E 2 12E 1 E 0
Hosking (1990) defines the L-moment ratios as:
W2
W3
W4
with
Wr
O2
,
O1
O3
,
O2
O4
O2
Or
,r t 3
O2
where O1 is a measure of location, W is a measure of scale and dispersion (LCv), W 3 is a
measure of skewness (LCs), and W 4 is a measure of kurtosis (LCk).
It can be shown (Hosking, 1990) that for r greater than or equal to 3, the absolute
value of W r is less than one. Furthermore, if x t 0 almost surely, then W , the LCv of x
satisfies 0 W 1. This boundedness of L-moment ratios is an advantage (Hosking,
1990) because it is easier to interpret a measure such as W 3 , which is constrained to lie
within the interval (–1,1), than conventional skewness coefficient which can take
arbitrarily large values.
L-moment and nonparametric methods were coupled to
underlying distributions which are nonunimodal by Gingras and Adamowski (1992).
The advantages of L-moments compared to product moments are brought out by Vogel
and Fennessey (1993).
24
3.1.2 L-Moment Sample Estimates
Hosking and Wallis (1997) defined unbiased sample estimators of PWMs as (bi)
and obtained unbiased sample estimators of the first four L-moments by PWM sample
estimators.
Unbiased sample estimates of the PWM for any distribution can be
computed using
br
1
n
n
j 1 j 2... j r ¦ n 1n 2...n r x
j r 1
j
where x j is an ordered set of observations x1 d x2 d x3 d ... d xn . For any distribution
the first four L-moments are easily computed from PWM using :
l1
l2
b0 ,
2b1 b0 ,
l3
6b2 6b1 b0 ,
l4
20b3 30b2 12b1 b0
Sample estimates for L-moments ratios are given by:
t2
t3
t4
with
tr
l2
,
l1
l3
,
l2
l4
l2
lr
,r t3
l2
25
3.2
The Method of TL-Moment
3.2.1 TL-Moment Distributions
TL-moments have been derived from the L-moments method. Let Y1, Y 2, …, Yr
be a conceptual random sample of size r from a continuous function with quantile
and let
function
statistics.
The
th
r
Y 1:r ”Y 2:r ”…” Yr:r denote the corresponding order
L-moment
In TL-moments, the
defined
is replaced by
by
Hosking
(1990)
is
. Hence, the
conceptual sample size of r is increased from r to r+t1+t2 . This study used only the
expectations of the r order statistics
. This is done by
trimming the t1 smallest and t2 largest from the conceptual sample. Elamir and Seheult
(2003) defined the r th TL-moment as
The relationship between L-moments and TL-moments is that TL-moments is
reduced to L-moments when t 1=t2=0. For symmetric cases, the r th TL-moment is written
as
26
The first four TL-moments when t=1 (TL-moments which are symmetrically
trimmed by one conceptual sample value) are :
However,
can be written as
Thus, equation for
can be re-expressed as
Therefore the alternative expressions for the first four TL-moments when t=1 are
27
Using the same expression of
for the first four TL-moments when t=2, the
following equations were obtained:
The population TL-skewness,
and TL-kurtosis,
are defined as
and
3.2.2 TL-Moment Sample Estimates
Given the linear combinations of order statistics X1:n ” X2:n ” … ” Xn:n of a random
sample X1, X2,…, Xn of size n from the population. Elamir and Seheult (2003) defined the
th
r sample TL-moment,
is a linear combination of X1:n , X2:n , … , Xn:n that is an unbiased
This shows that
estimator of
, as
.
The unbiased estimator of
can be written in the form:
28
Hence, an alternative of equation for
is made by a simple re-arrangement as follows
The first four sample TL-moments when t=1 are
Similarly, the first four sample TL-moments when t=2 are
29
Hence, estimation for TL-skewness,
and TL-kurtosis,
are
and
3.3
Normal Distribution
The normal distribution or also known as the Gaussian distribution is a
continuous probability distribution in probability theory and statistics. The normal
distribution describes data that clusters around an average or a mean. The probability
density function is well known for its shape which is the bell-shape. The shape has a
peak at the mean and is called the Gaussian function or the bell curve.
3.3.1
Probability Density Function
The probability density function of a normally distributed variable x is given by
f ( x)
1
1
V 2S
e
2V 2
x P 2
where ȝ and ı are the parameters of the distribution. The variable x can take any value
in the range (-’,’). The standard normal variate u is a normal variable with a mean
equal to zero and standard deviation equal to one.
The probability density function of u is given by
f (u )
1
2S
e
u 2
2
which can be numerically approximated by (Abramowitz and Stegun, 1965)
30
f (u )
(b0 b2 u 2 b4u 4 b8 u 8 b10u 10 ) 1 H (u )
where 0 d u f and
b0
2.5052367b6
0.1306469
b2
1.2831204b8
0.0202490
b4
0.2264718b10
0.0039132
The error H (u) is less than 2.3 u 104 . f (u) is an even function so that f (u)
3.3.2
f ( u) .
Distribution Function
The cumulative distribution F (u ) which is the area under the probability density
function is given by
u
F (u )
³
f
1
2S
e
t 2
2
dt
The distribution function can be numerically approximated (Abramowitz and Stegun,
1965) as
F (u ) 1 f (u)(b1 q b2 q b3 q b4 q b5 q ) H (u )
2
where q
p
1
1 pu
0.2316419 and 0 d u f
3
4
5
31
b1
0.319381530
b2
0.356563782
b3
b4
1.781477937
1.821255978
b5
1.330274429
The error H (u) is less than 7.5 u 108 and F ( u) 1 F (u) .
3.3.3
Quantile Function
Quantile function for normal distribution is written as:
P V) 1 [u]
Q(u)
) 1[u ] is the inverse standard normal distribution function.
For normal distribution, an approximation of the quantile function was given by Bayazit
and Onoz (2002) since the analytical expression is sometimes hard to compute. The
approximation is written as:
3.3.4 L-Moments and L-Moment Ratios
Since the normal distribution function cannot be explicitly expressed in terms of
x, the evaluation of the probability weighted moments becomes a complicated
procedure. However, the resulting expressions are simple. Hosking (1990) gives the
following properties of the normal distribution as
O1
E0
P
O2
2E1 E0
V
S
32
3.3.5
W3
O3
O2
0
W4
O4
O2
30
tan 1 ( 2 9)
S
Parameter Estimates using the L-Moment Method
PÖ
VÖ
3.3.6
Hence
0.1226
l1
S (l 2 )
TL-Moments at t = 1
33
3.3.7
Parameter Estimates using the TL-Moment Method at t = 1
3.3.8 TL-Moments at t = 2
Hence
34
3.3.9 Parameter Estimates using the TL-Moment Method at t = 2
3.4 Logistic Distribution (LOG)
Similar to the normal distribution, the logistic distribution is a continuous
probability distribution in probability theory and statistics. However, its cumulative
distribution function is that of the logistic function which is often seen in logistic
regression and feedforward neural networks. The shape resembles the normal
distribution’s shape which is the bell shape but it has heavier tails which mean that the
logistic distribution has higher kurtosis compared to that of the normal distribution.
3.4.1
Probability Density Function
The probability density function of x is given by
f ( x)
§ x [ ·
¸
D ¹
1 ¨©
e
D
§ x [ ·
ª
¸º
¨
«1 e © D ¹ »
«¬
»¼
2
The variable x takes the values in the range -’ < x < ’.
3.4.2
Distribution Function
F( x)
§ x [ · º
ª
¸
¨
«1 e © D ¹ »
¼»
¬«
1
35
3.4.3 Quantile Function
ª u º
[ D ln «
»
¬1 u ¼
Q (u )
3.4.4 L-moments and L-Moment Ratios
O1
O2
W3
W4
[
D
0
1
6
3.4.5 Parameter Estimates using the L-Moment Method
[Ö l1
DÖ l 2
3.4.6 TL-moments at
36
Hence
3.4.7 Parameter Estimates using the TL-Moment Method at
3.4.8
Hence
TL-moments at
37
3.4.9 Parameter Estimates using the TL-Moment Method at
3.5 Generalized Logistic Distribution (GLO)
The generalized logistic distribution is equivalent to the log-logistic distribution
(Ahmad et al., 1988).
In fact, the logistic distribution is a special case of this
distribution. Hence, these two distributions have very similar mathematical reasoning for
the construction of the probability plotting scales. The L-moments of the generalized
logistic distribution were given by Hosking (1986).
3.5.1 Probability Density Function
f ( x)
1
D
ª
§ x [ ·º
¸»
«1 K ¨
© D ¹¼
¬
(
1
1 )
K
1
ª
K
[
x
­
½
§
·
«1 1 K ¨
¸¾
« ®¯
© D ¹¿
«¬
The variable x takes values in the range
º
»
»
»¼
2
38
[
D
D
d x f for K 0 and f x d [ for K ! 0
K
K
3.5.2 Distribution Function
F( x)
1
ª
K
[
x
­
½
§
·
«1 1 K ¨
¸¾
« ®¯
© D ¹¿
¬«
º
»
»
¼»
1
3.5.3 Quantile Function
Q (u )
D
[
K
­° ª 1 u º K ½°
®1 «
» ¾
°̄ ¬ u ¼ °¿
3.5.4 L-Moments and L-Moment Ratios
39
D
^1 * (1 K )* (1 K )`
K
D* (1 K )* (1 K )
K
O1
[
O2
W3
(1 5W 3 )
6
3
W4
or W 4
0.16667 0.83333W 3
2
3.5.5 Parameter Estimates using the L-Moment Method
KÖ
DÖ
[Ö
t 3
l2
*(1 K )*(1 K )
(l DÖ )
l1 2
KÖ
3.5.6 TL-Moments at
40
Hence
3.5.7 Parameter Estimates using the TL-Moment Method at
3.5.8
TL-moments at
41
Hence
3.5.9 Parameter Estimates using the TL-Moment Method at
3.6 Extreme Value Type I (EV) Distribution
The extreme value Type I distribution is a special case of the generalized
extreme value distribution. Of all the three families in the generalized extreme value
distribution (Type I, Type II and Type III), Type I is by far the one most commonly
referred to in discussions of ‘extreme value’ distributions. The extreme value type I
distribution has two forms where one is based on the smallest extreme and the other is
42
based on the largest extreme. These forms are the minimum and maximum cases
respectively. The extreme value Type I distribution is also known as the Gumbel
distribution.
3.6.1 Probability Density Function
The probability density function of extreme value Type I distribution is given by
f ( x)
ª § x [ · §¨ x[ ·¸ º
1
D
exp« ¨
¸ e © ¹»
D
D
¹
«¬ ©
»¼
The variable x takes values in the range -’ < x < ’.
3.6.2 Distribution Function
The distribution function of x is given by
F( x)
3.6.3
ª §¨ x[ ·¸ º
exp« e © D ¹ »
«¬
»¼
Quantile Function
Q (u )
[ D ln( ln(u ))
3.6.4 L-Moments and L-Moment Ratios
43
O1
O2
W3
W4
3.6.5
[ 0.5772D
D ln 2
0.1669
0.1504
Parameter Estimates using the L-Moment Method
DÖ
l2
ln 2
[Ö
l1 0.5772DÖ
3.6.6 TL-moments at
where
Hence
44
3.6.7 Parameter Estimates using the TL-Moment Method at
3.6.8
where
Hence
TL-moments at
45
3.6.9 Parameter Estimates using the TL-Moment Method at
3.7
Generalized Extreme Value (GEV) Distribution
The generalized extreme value distribution in probability theory and statistics is a
family of continuous probability distributions. Extreme value distributions are generally
considered to comprise three families which is Type I, Type II and Type III (Rao et. al.,
2000). It is developed within extreme value theory to combine all three Type I, II and III
extreme value distributions which are respectively known as the Gumbel, Fréchet and
Weibull families. According to the extreme value theorem, the generalized extreme
value distribution is the limiting distribution of properly normalized maxima of a
sequence of independent and identically distributed random variables. Hence, the name
‘extreme value’ is attached to these distributions because they can be obtained as
limiting distributions (as n goes to infinity) of the greatest value among n independent
random variables each having the same continuous distribution.
Although the
distributions are labeled ‘extreme value’, it is to be borne in mind that they do not
represent distributions of all kinds of ‘extreme values’ and they can be used empirically
in the same way as other distributions (Johnson et. al., 1970).
3.7.1 Probability Density Function
46
The
f ( x)
probability
1
D
density
function
1
1)
K
ª
§ x [ · º K
«1 K ¨
¸»
© D ¹¼
¬
ª
§ x [ ·º
«1 K ¨ D ¸ »
©
¹¼
¬
(
of
the
GEV
is
of
the
form
1
e
The range of the variable x depends on the sign of the parameter K. The extreme
value type I distribution is a special case of generalized extreme value distribution in
which the shape parameter K is equal to zero. When K is negative, it becomes the Type
II extreme value distribution where the variable x can take values in the range u + Į/K <
x < ’ which makes it suitable for flood frequency analysis. Meanwhile, when K is
positive, it turns into the Type III extreme value distribution where the variable x
becomes upper bounded and takes values in the range -’ < x < u + Į/K which may not
be acceptable for analyzing floods unless there is sufficient evidence that such an upper
bound does exist. (Rao et. al., 2000)
3.7.2 Distribution Function
The generalized distribution function is of the form (Jenkinson, 1955)
F (x )
1
½
­
§ x [ ·º K °
° ª
exp® «1 K ¨
¸» ¾
© D ¹¼ °
°̄ ¬
¿
47
3.7.3 Quantile Function
Q (u )
[
D
^1 ( ln(u)) K `
K
3.7.4 L-Moments and L-moment Ratios
D
^1 * (1 K )`
K
O1
[
O2
D
(1 2 K )*(1 K )
K
W3
2(1 3 )
3
(1 2 K )
W4
0.10701 0.11090W 3 0.84838W 3 0.06669W 3 0.00567W 3 0.04208W 3 0.03763W 3
K
2
3
3.7.5 Parameter Estimates using the L-Moment Method
KÖ
7.8590C 2.9554C 2 where C
DÖ
l 2 KÖ
KÖ
*(1 KÖ )(1 2 )
2
ln 2
3 t 3 ln 3
4
5
6
48
DÖ
[Ö l1 [*(1 KÖ ) 1]
KÖ
3.7.6 TL-moments at
Hence
49
3.7.7 Parameter Estimates using the TL-Moment Method at
For the estimate of K, the method of regression was used. The polynomial where both
the values of MAE and RMSE start to be less than 0.005 is used as the estimate for K.
3.7.8
TL-moments at
50
Hence
3.7.9 Parameter Estimates using the TL-Moment Method at
For the estimate of K, the method of regression was used. The polynomial where both
the values of MAE and RMSE start to be less than 0.005 is used as the estimate for K.
51
3.8
Generalized Pareto Distribution (GPA)
The Generalized Pareto (GPA) is a right-skewed distribution, parameterized with
a shape parameter, K, and a scale parameter, D (Rao et. al, 2000). The GPA distribution
is a generalization of both the exponential distribution
K
0 and the Pareto
distribution K 0 where those two distributions are included in a larger family. Thus,
a continuous range of shapes is possible. The generalized Pareto distribution is also a
special case of the Wakeby distribution.
3.8.1
Probability Density Function
The probability density function is written as
1
f ( x)
1
D
ª K
ºK
«1 D ( x [ )»
¬
¼
1
3.8.2 Distribution Function
The distribution function F = F(x) is explicitly written as in
52
1
ª K
ºK
F (x ) 1 «1 ( x [ )»
¬ D
¼
3.8.3
Quantile Function
Q (u )
3.8.4
[
D
^1 [1 u]K `
K
L-Moments and L-Moment Ratios
D
1 K
O1
[
O2
D
(1 K )(2 K )
W3
(1 K )
(3 K )
W4
W 3 (1 5W 3 )
(5 W 3 )
W4
0.20196W 3 0.95924W 3 0.20096W 3 0.04061W 3
or
2
3
3.8.5 Parameter Estimates using the L-Moment Method
KÖ
(1 3t3 )
(1 t 3 )
DÖ
l2 (1 KÖ )(2 KÖ )
4
53
[Ö
l1 l2 ( 2 KÖ )
3.8.6 TL-moments at
Hence
3.8.7 Parameter Estimates using the TL-Moment Method at
54
3.8.8
TL-moments at
Hence
3.8.9 Parameter Estimates using the TL-Moment Method at
55
3.9
Goodness of Fit Criteria for Comparison of Probability Distributions
3.9.1
Mean Absolute Deviation Index (MADI) and Mean Square Deviation Index
(MSDI)
For comparison among the probability distributions for fitting the data used in
the study, two indices (mean absolute deviation index and mean square deviation index),
which were proposed by Jain and Sing (1987), were taken into account to measure the
relative goodness of fit. The mean absolute deviation index (MADI) and mean square
deviation index (MSDI) can be calculated by :
MADI
MSDI
N
x i zi
xi
1
N
¦
1
N
§ xi z i
¨¨
¦
xi
i 1 ©
i 1
N
·
¸¸
¹
2
Where xi are observed flows whereas zi are predicted flows respectively for
successive values of empirical probability of exceedence given by Gringorten plotting
position formula. Jain and Singh (1987) claimed and believed that Gringorten formula
ensures to maintain unbiasedness for different distributions. Hence, they suggest this
plotting position formula for comparison of the probability distributions of fitting the
56
data.
The formula for T (return period) and F (cummulative probability of non-
exceedence) in the Gringorten plotting position is given by :
T
N 0.12
m 0.44
F
i 0.44
N 0.12
with i is the rank in ascending order, i
N m 1 ; m is the rank in descending order,
N i m ; and N is the number of observations.
and m
The smaller the value obtained for the mean absolute deviation index (MADI)
and mean square deviation index (MSDI) of a given distribution shows that the
distribution is more fitted for the actual data. Hence the distribution with the smallest
value implies that the particular distribution is the most fitted whereas the largest shows
that it is the least fitted to present the observed data.
3.9.2
Correlation (r)
Correlation relies on descriptive statistics that measure location, variation and
linear association. Let n be the length of data for each station,
x
mean of x
1 n
¦ x i where xi are observed flows and
ni1
z
mean of z
1 n
¦ zi where zi are predicted flows.
ni1
The measure spread is given by the sample variance which is defined as
s xx
2
1 n
( xi x ) 2 for the observed flows and
¦
n 1 i 1
57
s zz
2
1 n
2
( z i z ) for the predicted flows.
¦
n 1 i 1
Meanwhile, the sample covariance is written as
s xz
2
1 n
¦ ( x x )(zi z) .
n 1 i 1
Hence the sample correlation, r, is given as
r
s xz
.
s xx s zz
The value of the correlation, r, must be between -1 and +1 inclusive. If the
correlation, r, is near 0, then it means there is a lack of linear association between the
observed and predicted flows. Hence the distribution with the particular correlation is
not suitable to represent the data. Meanwhile, if the correlation, r, is near 1 or -1, it
implies a tendency for both observed and predicted flows to be large or small together
and this shows that the distribution is able to fit the actual data. Thus the distribution
with the nearest value to 1 or -1 is the best fitted distribution for the data and the one
nearest to 0 is the least fitted.
3.10
L-moment and TL-moment Ratio Diagrams
The L-moment and TL-moment ratio diagrams are based on relationships
between the L-moment and TL-moment ratios respectively. The ratio diagrams are
based on unbiased sample quantities and the sample L-moment or TL-moment ratios
plot as fairly well separated group. Thus, this permits better discrimination between the
distributions. Hence, the identification of a parent distribution can be achieved.
Relationships between W 3 and W 4 are used for all the distributions including
normal (N), logistic (LOG), generalized logistic (GLO), extreme value type I (EV),
generalized extreme value type I (GEV) and generalized Pareto (GPA). The sample Lmoment and TL-moment ratios for each distribution is taken for the range 1 d t3 d 1.
58
For this interval, the values for t4 are counted for all the distribution using their
relationships with t3. Then the average values for the sample L-moment and TL-moment
ratios were calculated as points in the diagram (( ,
,( ,
and ( ,
.
The distributions which have L-moment or TL-moment ratios that are nearest to
the average sample values of sample ratios are considered good distributions for fitting
the observed data. Otherwise, they are taken as unsuitable distribution to represent the
data.
CHAPTER 4
DATA ANALYSIS
4.1
Selangor
The State of Selangor is Malaysia's most populated and prosperous state
compared to the rest of the states in Malaysia. This is due to its advantageous geographic
position and rich natural resources. Since the KL International Airport and the largest
port in the country, Port Klang, are both located in this state, it is often referred to as the
Gateway to Malaysia. The capital of Selangor is Shah Alam while the other larger
towns are Petaling Jaya, Ampang and Klang. The country's capital city, Kuala Lumpur,
used to be part of Selangor 20 years ago before it was made a Federal Territory.
Similarly, Putrajaya, the country's administrative centre was previously part of Selangor
(Frederick Fernandez, 1995-2000).
Selangor covers an area of approximately 125,000 sq. km and extends along the
west coast of Peninsular Malaysia at the northern outlet of the Straits of Malacca. It is
known to be the most populated state in Malaysia, with about 3.7 million inhabitants. A
large proportion of Selangor's population lives around the Federal Territory of Kuala
Lumpur, though the balance is now shifting towards its new capital, Shah
60
Alam. Selangor is also the country's premier state with its huge resources, well
developed communications network, industrial estates, and skilled manpower (Asian
Vacation Inc., 2000-2007).
4.2
Kuala Lumpur
Kuala Lumpur (KL) is the capital city of Malaysia. The name Kuala Lumpur,
when translated into English, literally means “muddy confluence”. That is because the
city obtained its name from its location which is at the confluence of the Klang and
Gombak rivers. Situated midway along the west coast of Peninsular Malaysia, Kuala
Lumpur functions as the commercial, business capital and principle center of
entertainment and international activities of the country. Until the year 1999, Kuala
Lumpur was also the administrative capital, which has since been moved to Putrajaya
(Elizabeth Ng, 1995-2008).
Kuala Lumpur is approximately 35km from the coast and sits at the centre of the
Peninsula's extensive and modern transportation network. Kuala Lumpur is easily the
largest city in the nation, possessing a population of 1.4 million citizens as of the year
2000 statistics drawn from all of Malaysia's many ethnic groups.
61
Figure 4.1: Location Map of Rainfall Gauge Stations in Selangor and Kuala Lumpur
4.3
Flood in Selangor and Kuala Lumpur
Flash floods are not a rare occurrence in Selangor and Kuala Lumpur. In fact,
they occurred several times in Selangor and Kuala Lumpur. One of the most recent ones
will be the flash flood occurrence in June 2007. During the late evening and night on
the 10th of June 2007, severe thunderstorms and heavy rains accompanied by strong
winds occurred in Kuala Lumpur, Putrajaya and Selangor (mainly in Klang Valley
areas). The areas affected by the floods were Jalan Masjid India, Jalan Ipoh, Kampung
Baru, Kampung Chubadak and Sentul. The areas surrounding Sultan Abdul Samad
62
Complex and underground car park of the Merdeka Square were also submerged All this
had been stated and recorded by the Research Division of Malaysian Meteorological
th
Department, Ministry of Science, Technology and Innovations, Malaysia on the 13
June 2007. Details of the hourly rainfall starting from 6 pm to 11 pm are shown in Table
4.1 below.
Table 4.1: Accumulated hourly rainfall (mm) within 24 hours period from
Meteorological Stations in Petaling Jaya, Subang and KLIA on 10 June 2007
Time
6.00 pm
7.00 pm
8.00 pm
9.00 pm
10.00 pm
11.00 pm
Accumulated
rainfall (24hour)
4.4
Petaling Jaya (mm)
0.0
0.0
1.2
28.8
6.4
0.2
Subang (mm)
0.0
0.0
0.0
13.4
6.6
1.2
KLIA, Sepang (mm)
0.0
1.2
0.2
3.2
3.8
0.2
37.4
24.8
8.6
Data Collection
The data of daily rainfalls for stations in Selangor and Kuala Lumpur was
collected and taken from “Jabatan Pengairan dan Saliran Malaysia”. The data of daily
rainfalls for 55 stations were sent by email. The data contains measurements of daily
rainfalls in millimeters from the year 1971 until 2007. The data is listed in Table 4.2
including informations on the data.
The maximum rainfalls of each month were
identified followed by the maximum of each year (1971-2007). This is done to all the
55 stations in Selangor and Kuala Lumpur.
63
Example:
The maximum of daily rainfalls for five consecutive years, 1971-1975, for the station in
Subang was obtained as follows:
DAILY RAINFALLS (mm)
SUBANG (1971-1975)
200
180
160
140
120
100
80
60
40
20
0
-20
1971
1972
1973
1974
1975
DAYS
The first two digit of the station number represent the latitude, the next two
represent the longitude and the rest are the code numbers for the stations in “Jabatan
Pengairan dan Saliran, Malaysia”.
Example:
For the station number 2615131,
26
Latitude
15
131
Longitude
Station’s code
The data from the 55 stations in Selangor and Kuala Lumpur have the latitude
that ranges from 26o up to 38o while a longitude from 8o to 18o. The maximum of data
from each station were checked for randomness and homogeneity using the run’s test for
randomness with the mean as the point of reference and Mann-Whitney U test for
homogeneity. A majority of the stations are random and homogeneous. However there
64
are stations which are either not random, not homogeneous or have a sample size less
than 30 where their randomness cannot be tested (16 stations) but for this study, they are
still analyzed and their distributions were taken into account.
The maximum data of daily rainfalls for each year were then analyzed for all the
55 stations using MathCAD program. A MathCAD program was created to find the Lmoments, L-moment ratios, TL-moment, TL-moment ratios with
and
and
parameter estimations using both L-moment and TL-moment for six probability
statistical distributions which were the normal (N), logistic (LOG), generalized logistic
(GLO), extreme value type I (EV), generalized extreme value (GEV) and generalized
Pareto (GPA) distribution. Meanwhile, Microsoft Excel was used to obtain the
descriptive statistics of each station’s data of maximum daily rainfalls every year.
65
Table 4.2:
Name and information on all the stations in Selangor and Kuala Lumpur
NAME OF STATION
LDG. BATU UNTONG
LDG. TELOK MERBAU
LDG. SEPANG
LDG. BUTE
PEJABAT JPS. SG. MANGG
LDG. BROOKLANDS
SMK. BDR TASIK KESUMA
P.KWLN P.S TELOK GONG
LDG. WEST
JPS. PULAU LUMUT
LDG. BKT. CHEEDING
PEJABAT JPS. KLANG
LDG. DOMINION
LDG. BUKIT KERAYONG
LDG. SG. KAPAR
LDG. NORTH HUMMOCK
LDG. HARPENDEN
LDG. ELMINA
SG. BULOH
LDG. EDINBURGH SITE 2
JPS AMPANG
PEMASOKAN AMPANG
SEK.KEB.KG.LUI
LDG. BRAUNSTON
LDG. BKT. CHERAKAH
LDG. TUAN MEE
LDG. BKT. IJOK
KG. SG. TUA
KEPONG (SEMAIAN)
IBU BEKALAN KM. 16
EMPANGAN GENTING KLANG
IBU BEKALAN KM. 11
STN. JENALETRIK LLN.
LDG. BKT. BELIMBING
JLN. KELANG
LDG. BKT. TALANG
LDG. KUALA SELANGOR
LDG. SG. BULOH
RMH PAM JPS JAYA SETIA
LDG. SG. GAPI
AIR TERJUN SG BATU
GENTING SEMPAH
PARIT 1 SG. BURONG
IBU BEKALAN SG. TENGKI
LDG. RAJA MUSA
LDG. SG. TINGGI
LDG. HOPEFUL
FDC. SEKICHAN
PARIT 1 SG. BESAR
SG. NIPAH
LDG. SG. GUMUT
RMH PAM JPS BGN TERAP
PARIT 6 SG. BESAR
PARIT SALIRAN SG. AIR TAWAR
LDG SG. BERNAM
STATION NUMBER
2615131
2616135
2617134
2717114
2815001
2815115
2818110
2913001
2913121
2913122
2915116
3014084
3018107
3113059
3113087
3114085
3114086
3115053
3115079
3116006
3117070
3118069
3118102
3213057
3213058
3214054
3214055
3216001
3216002
3217001
3217002
3217003
3218101
3312042
3312045
3313040
3313043
3313060
3314001
3316028
3317001
3317004
3411016
3412001
3412041
3414029
3414030
3510001
3609012
3610014
3615002
3710006
3710011
3808001
3809009
n
37
37
35
37
37
35
34
33
37
37
38
36
38
37
37
36
27
38
38
31
37
22
37
34
38
36
35
36
7
36
36
9
37
36
37
35
37
38
36
35
23
34
36
33
37
36
35
33
36
33
7
37
37
33
37
RANDOMNESS
random
random
random
random
random
random
not random
random
random
random
random
random
random
random
not random
not random
random
random
random
random
random
random
random
random
random
not random
random
random
not random
random
random
random
random
random
random
random
random
random
random
random
random
not random
random
random
random
not random
random
random
random
random
random
random
random
random
random
HOMOGENEITY
homogeneous
homogeneous
not homogeneous
homogeneous
homogeneous
not homogeneous
not homogeneous
homogeneous
homogeneous
not homogeneous
homogeneous
homogeneous
homogeneous
homogeneous
homogeneous
homogeneous
homogeneous
homogeneous
homogeneous
not homogeneous
homogeneous
homogeneous
not homogeneous
homogeneous
homogeneous
homogeneous
homogeneous
homogeneous
homogeneous
homogeneous
homogeneous
homogeneous
homogeneous
homogeneous
homogeneous
homogeneous
homogeneous
homogeneous
homogeneous
homogeneous
homogeneous
homogeneous
homogeneous
homogeneous
homogeneous
not homogeneous
homogeneous
homogeneous
homogeneous
homogeneous
homogeneous
homogeneous
homogeneous
homogeneous
homogeneous
66
4.5
Descriptive Statistics
As can be seen from Table 4.3, the means for the maximum daily rainfalls for the
55 stations in Selangor and Kuala Lumpur range from 70.3571 mm (Kepong, Semaian)
to 153.0528 mm (Ldg. North Hummock). Meanwhile, their standard deviations are
from 17.1225 mm (Ldg. Sg. Gumut) to 207.288 mm (Ldg. North Hummock). Hence
they are a few stations with values that vary greatly. This is the case since they are
several years where Selangor and Kuala Lumpur experience flash floods due to a sudden
large amount of rainfalls. The kurtosis of the data are from -0.9613 (Kepong, Semaian)
to 31.6294 (Ldg. Bukit Kerayong) and the skewness are from -0.71717 (Kepong,
Semaian ) up to 5.4500 (Ldg. Bukit Kerayong).
4.6
L-Moments and L-Moments Ratios
L-Moments and L-Moments Ratios for all the stations in Selangor and Kuala
Lumpur were calculated using the MathCAD program. They were then tabulated as in
Table 4.4. These values were used in the calculation of quantile function for each
distribution using the L-Moment method.
67
Table 4.3:
Descriptive Statistics on the maximum daily rainfalls for stations in
Selangor and Kuala Lumpur
NAME OF STATION
LDG. BATU UNTONG
LDG. TELOK MERBAU
LDG. SEPANG
LDG. BUTE
PEJABAT JPS. SG. MANGG
LDG. BROOKLANDS
SMK. BDR TASIK KESUMA
P.KWLN P.S TELOK GONG
LDG. WEST
JPS. PULAU LUMUT
LDG. BKT. CHEEDING
PEJABAT JPS. KLANG
LDG. DOMINION
LDG. BUKIT KERAYONG
LDG. SG. KAPAR
LDG. NORTH HUMMOCK
LDG. HARPENDEN
LDG. ELMINA
SG. BULOH
LDG. EDINBURGH SITE 2
JPS AMPANG
PEMASOKAN AMPANG
SEK.KEB.KG.LUI
LDG. BRAUNSTON
LDG. BKT. CHERAKAH
LDG. TUAN MEE
LDG. BKT. IJOK
KG. SG. TUA
KEPONG (SEMAIAN)
IBU BEKALAN KM. 16
EMPANGAN GENTING KLANG
IBU BEKALAN KM. 11
STN. JENALETRIK LLN.
LDG. BKT. BELIMBING
JLN. KELANG
LDG. BKT. TALANG
LDG. KUALA SELANGOR
LDG. SG. BULOH
RMH PAM JPS JAYA SETIA
LDG. SG. GAPI
AIR TERJUN SG BATU
GENTING SEMPAH
PARIT 1 SG. BURONG
IBU BEKALAN SG. TENGKI
LDG. RAJA MUSA
LDG. SG. TINGGI
LDG. HOPEFUL
FDC. SEKICHAN
PARIT 1 SG. BESAR
SG. NIPAH
LDG. SG. GUMUT
RMH PAM JPS BGN TERAP
PARIT 6 SG. BESAR
PARIT SALIRAN SG. AIR TAWAR
LDG SG. BERNAM
MEAN (X)
132.7595
105.9000
103.8629
95.9081
88.6568
88.1429
117.1706
119.7879
108.3270
98.0297
90.2605
86.8139
96.5395
108.5865
105.1135
153.0528
93.0667
108.7237
94.0974
95.3258
106.7270
103.0136
114.6541
91.8324
96.9816
83.9556
106.9714
98.2250
70.3571
97.2583
100.3056
114.4444
108.2568
97.8278
96.5595
98.6457
100.5973
93.2421
108.4000
113.8914
97.9565
105.9912
102.8389
91.2667
93.0486
98.9556
109.2543
87.9848
93.6278
83.9061
109.4429
86.5270
88.0243
89.9242
91.6541
STANDARD DEVIATION (X)
35.6394
34.9830
31.9488
26.6090
26.7828
24.5133
56.9465
76.6036
39.8844
34.0849
29.2144
26.3272
26.4200
99.0321
30.0823
207.2880
26.2533
56.6307
29.1214
22.5880
25.7755
35.8925
61.2085
33.2971
69.3055
25.1605
46.2387
29.7754
38.8122
21.4893
38.4195
28.8145
56.6823
40.1725
32.3812
48.7262
40.4633
31.4104
75.2407
30.9925
25.6563
125.5961
31.9110
29.3514
40.9093
58.1997
36.9129
30.2408
23.9500
33.3141
17.1225
22.6775
23.4848
39.6707
30.0644
KURTOSIS
-0.5398
2.1444
-0.0723
-0.2296
1.6465
2.7466
0.9720
21.3283
1.2915
0.9015
3.1395
2.3136
4.6805
31.6293
-0.3152
11.1780
-0.0479
6.8762
0.1748
-0.6577
0.4752
0.9882
2.4264
0.0264
27.0971
0.7242
1.6965
0.8072
-0.9613
1.3390
11.2649
-0.3041
2.9205
3.7216
7.1436
6.7642
0.5613
1.4370
21.7805
1.4904
3.4352
30.7065
0.5561
0.4574
4.9140
9.4544
0.1453
2.2465
-0.1085
0.0486
-0.5096
0.0444
2.9111
2.6719
1.4684
SKEWNESS
0.2823
1.0247
0.8805
0.2745
0.7936
1.0308
1.5466
4.2461
0.8167
0.8815
0.8682
1.5118
0.8955
5.4500
0.5022
3.4223
0.5590
2.2829
0.4141
0.4324
0.4342
0.3136
1.5993
0.7691
4.8298
0.8109
1.4857
1.2593
-0.7172
0.4007
2.7163
0.5501
1.6358
1.9177
2.2008
2.4333
0.8433
0.9866
4.2549
1.2819
1.5418
5.4288
0.4396
0.9606
2.0198
2.2788
0.9395
1.3535
0.5586
-0.2284
-0.2980
0.3636
1.1031
1.5380
1.1681
68
Table 4.4:
NAME OF STATION
LDG. BATU UNTONG
LDG. TELOK MERBAU
LDG. SEPANG
LDG. BUTE
PEJABAT JPS. SG. MANGG
LDG. BROOKLANDS
SMK. BDR TASIK KESUMA
P.KWLN P.S TELOK GONG
LDG. WEST
JPS. PULAU LUMUT
LDG. BKT. CHEEDING
PEJABAT JPS. KLANG
LDG. DOMINION
LDG. BUKIT KERAYONG
LDG. SG. KAPAR
LDG. NORTH HUMMOCK
LDG. HARPENDEN
LDG. ELMINA
SG. BULOH
LDG. EDINBURGH SITE 2
JPS AMPANG
PEMASOKAN AMPANG
SEK.KEB.KG.LUI
LDG. BRAUNSTON
LDG. BKT. CHERAKAH
LDG. TUAN MEE
LDG. BKT. IJOK
KG. SG. TUA
KEPONG (SEMAIAN)
IBU BEKALAN KM. 16
EMPANGAN GENTING KLANG
IBU BEKALAN KM. 11
STN. JENALETRIK LLN.
LDG. BKT. BELIMBING
JLN. KELANG
LDG. BKT. TALANG
LDG. KUALA SELANGOR
LDG. SG. BULOH
RMH PAM JPS JAYA SETIA
LDG. SG. GAPI
AIR TERJUN SG BATU
GENTING SEMPAH
PARIT 1 SG. BURONG
IBU BEKALAN SG. TENGKI
LDG. RAJA MUSA
LDG. SG. TINGGI
LDG. HOPEFUL
FDC. SEKICHAN
PARIT 1 SG. BESAR
SG. NIPAH
LDG. SG. GUMUT
RMH PAM JPS BGN TERAP
PARIT 6 SG. BESAR
PARIT SALIRAN SG. AIR TAWAR
LDG SG. BERNAM
L-Moments and L-Moments Ratios for all the stations
l1
132.759
105.9
103.863
95.908
88.657
88.143
117.171
119.788
108.327
98.03
90.261
86.814
96.539
108.586
105.114
153.053
93.067
108.724
94.097
95.326
106.727
103.014
114.654
91.832
96.982
83.956
106.971
98.225
70.357
97.258
100.306
114.444
108.257
97.828
96.559
98.646
100.597
93.242
108.4
113.891
97.957
105.991
102.839
91.267
93.049
98.956
109.254
87.985
93.628
83.906
109.443
86.527
88.024
89.924
91.654
l2
20.554
19.262
17.775
15.272
14.894
13.199
27.498
29.634
21.884
18.973
15.308
13.577
13.646
28.954
17.081
67.881
14.927
27.402
16.474
13.011
14.423
19.372
31.008
18.727
24.858
13.959
23.862
15.763
22.819
11.837
18.314
17.125
28.904
19.803
16.059
22.927
22.196
16.852
28.285
16.677
13.577
34.68
18.04
16.291
20.421
27.461
20.463
16.285
13.269
18.884
10.41
12.854
12.583
20.779
16.405
l3
1.416
2.527
3.982
0.709
1.276
1.288
12.498
13.258
3.367
2.972
1.399
4.325
0.591
17.779
2.137
50.52
1.797
10.381
1.705
1.417
0.876
2.739
11.306
3.355
9.851
2.033
8.117
4.779
-4.68
0.914
4.581
2.593
8.379
7.59
4.74
8.56
3.946
3.452
13.769
4.482
3.153
23.323
1.135
3.449
6.767
4.863
4.785
4.081
2.129
-0.216
-0.846
1.326
1.55
5.84
3.643
l4
1.971
2.41
1.841
1.465
2.123
2.932
6.531
11.323
4.631
2.977
4.674
3.36
4.005
16.914
2.186
42.454
2.32
7.715
2.015
1.064
2.681
5.406
8.113
2.144
10.724
2.709
5.367
2.968
1.371
2.527
4.663
1.976
7.81
5.583
4.297
6.797
4.577
3.821
13.237
2.956
3.473
21.575
2.232
1.933
4.886
10.751
2.275
2.872
1.815
2.992
1.329
1.696
2.975
4.713
2.571
t2
0.155
0.182
0.171
0.159
0.168
0.15
0.235
0.247
0.202
0.194
0.17
0.156
0.141
0.267
0.163
0.444
0.16
0.252
0.175
0.136
0.135
0.188
0.27
0.204
0.256
0.166
0.223
0.16
0.324
0.122
0.183
0.15
0.267
0.202
0.166
0.232
0.221
0.181
0.261
0.146
0.139
0.327
0.175
0.178
0.219
0.278
0.187
0.185
0.142
0.225
0.095
0.149
0.143
0.231
0.179
t3
0.069
0.131
0.224
0.046
0.086
0.098
0.455
0.447
0.154
0.157
0.091
0.319
0.043
0.614
0.125
0.744
0.12
0.379
0.103
0.109
0.061
0.141
0.365
0.179
0.396
0.146
0.34
0.303
-0.205
0.077
0.25
0.151
0.29
0.383
0.295
0.373
0.178
0.205
0.487
0.269
0.232
0.673
0.063
0.212
0.331
0.177
0.234
0.251
0.16
-0.011
-0.081
0.103
0.123
0.281
0.222
t4
0.096
0.125
0.104
0.096
0.143
0.222
0.238
0.382
0.212
0.157
0.305
0.247
0.294
0.584
0.128
0.625
0.155
0.282
0.122
0.082
0.186
0.279
0.262
0.114
0.431
0.194
0.225
0.188
0.06
0.213
0.255
0.115
0.27
0.282
0.268
0.296
0.206
0.227
0.468
0.177
0.256
0.622
0.124
0.119
0.239
0.391
0.111
0.176
0.137
0.158
0.128
0.132
0.236
0.227
0.157
69
4.7
TL-Moments and TL-Moments Ratios
Similarly, TL-Moments and TL-Moments Ratios for all the stations in Selangor
and Kuala Lumpur were calculated using the MathCAD program for t =1 and t = 2.
They were then tabulated as in Table 4.5 and Table 4.6 respectively. These values were
used in the calculation of quantile function for each distribution using the TL-Moment
method.
70
Table 4.5:
TL-Moments and TL-Moments Ratios for all the stations ( t = 1 )
NAME OF STATION
LDG. BATU UNTONG
LDG. TELOK MERBAU
LDG. SEPANG
LDG. BUTE
PEJABAT JPS. SG. MANGG
LDG. BROOKLANDS
SMK. BDR TASIK KESUMA
P.KWLN P.S TELOK GONG
LDG. WEST
JPS. PULAU LUMUT
LDG. BKT. CHEEDING
PEJABAT JPS. KLANG
LDG. DOMINION
LDG. BUKIT KERAYONG
LDG. SG. KAPAR
LDG. NORTH HUMMOCK
LDG. HARPENDEN
LDG. ELMINA
SG. BULOH
LDG. EDINBURGH SITE 2
JPS AMPANG
PEMASOKAN AMPANG
SEK.KEB.KG.LUI
LDG. BRAUNSTON
LDG. BKT. CHERAKAH
LDG. TUAN MEE
LDG. BKT. IJOK
KG. SG. TUA
KEPONG (SEMAIAN)
IBU BEKALAN KM. 16
EMPANGAN GENTING KLANG
IBU BEKALAN KM. 11
STN. JENALETRIK LLN.
LDG. BKT. BELIMBING
JLN. KELANG
LDG. BKT. TALANG
LDG. KUALA SELANGOR
LDG. SG. BULOH
RMH PAM JPS JAYA SETIA
LDG. SG. GAPI
AIR TERJUN SG BATU
GENTING SEMPAH
PARIT 1 SG. BURONG
IBU BEKALAN SG. TENGKI
LDG. RAJA MUSA
LDG. SG. TINGGI
LDG. HOPEFUL
FDC. SEKICHAN
PARIT 1 SG. BESAR
SG. NIPAH
LDG. SG. GUMUT
RMH PAM JPS BGN TERAP
PARIT 6 SG. BESAR
PARIT SALIRAN SG. AIR TAWAR
LDG SG. BERNAM
l1
131.344
103.373
99.88
95.199
87.38
86.855
104.673
106.53
104.96
95.058
88.862
82.489
95.949
90.808
102.976
102.533
91.27
98.342
92.392
93.908
105.851
100.274
103.348
88.478
87.131
81.922
98.855
93.446
75.037
96.344
95.724
111.851
99.878
90.237
91.819
90.085
96.651
89.791
94.631
109.41
94.804
82.668
101.704
87.818
86.282
94.092
104.469
83.904
91.499
84.122
110.289
85.201
86.474
84.084
88.011
l2
11.15
10.112
9.561
8.284
7.663
6.161
12.58
10.986
10.352
9.598
6.381
6.13
5.784
7.224
8.937
15.257
7.565
11.812
8.675
7.169
7.045
8.379
13.737
9.95
8.481
6.75
11.097
7.677
12.869
5.586
8.191
9.089
12.657
8.532
7.057
9.678
10.571
7.819
9.029
8.233
6.062
7.863
9.485
8.614
9.321
10.026
10.912
8.048
6.872
9.535
5.449
6.695
5.765
9.64
8.301
l3
0.705
0.487
2.008
0.094
0.121
-0.114
6.024
1.399
1.266
0.662
0.205
1.657
-0.301
1.748
1.264
9.224
0.735
3.156
1
0.748
-0.051
2.587
4.879
1.421
0.653
0.548
3.219
2.11
-6.102
0.357
0.151
1.488
2.746
2.44
0.957
1.786
1.77
1.352
2.891
1.702
0.395
1.763
0.032
1.279
1.695
-0.222
2.26
1.149
1.648
0.731
0.416
0.799
0.19
1.846
1.123
l4
0.687
0.380
0.81
0.112
0.454
0.73
3.649
0.786
1.246
0.874
0.961
1.14
0.562
0.96
0.877
8.936
1.049
1.973
0.281
0.635
0.77
1.755
2.78
0.492
0.586
1.005
1.836
1.506
9.536
0.375
0.298
-0.826
2.224
1.645
0.867
1.297
1.741
1.018
1.806
0.919
0.742
1.033
0.062
0.498
1.061
2.591
0.656
0.597
0.567
0.968
2.911
0.311
0.625
1.27
0.607
t3
0.063
0.048
0.21
0.011
0.016
-0.018
0.479
0.127
0.122
0.069
0.032
0.27
-0.052
0.242
0.141
0.605
0.097
0.267
0.115
0.104
-0.00731
0.309
0.355
0.143
0.077
0.081
0.29
0.275
-0.474
0.064
0.018
0.164
0.217
0.286
0.136
0.185
0.167
0.173
0.32
0.207
0.065
0.224
0.00338
0.148
0.182
-0.022
0.207
0.143
0.24
0.077
0.076
0.119
0.033
0.192
0.135
t4
0.062
0.038
0.085
0.013
0.059
0.119
0.29
0.072
0.12
0.091
0.151
0.186
0.097
0.133
0.098
0.586
0.139
0.167
0.032
0.089
0.109
0.209
0.202
0.049
0.069
0.149
0.165
0.196
0.741
0.067
0.036
-0.091
0.176
0.193
0.123
0.134
0.165
0.13
0.2
0.112
0.122
0.131
0.00657
0.058
0.114
0.258
0.06
0.074
0.082
0.102
0.534
0.047
0.108
0.132
0.073
71
Table 4.6:
TL-Moments and TL-Moments Ratios for all the stations ( t = 2 )
NAME OF STATION (t=2)
LDG. BATU UNTONG
LDG. TELOK MERBAU
LDG. SEPANG
LDG. BUTE
PEJABAT JPS. SG. MANGG
LDG. BROOKLANDS
SMK. BDR TASIK KESUMA
P.KWLN P.S TELOK GONG
LDG. WEST
JPS. PULAU LUMUT
LDG. BKT. CHEEDING
PEJABAT JPS. KLANG
LDG. DOMINION
LDG. BUKIT KERAYONG
LDG. SG. KAPAR
LDG. NORTH HUMMOCK
LDG. HARPENDEN
LDG. ELMINA
SG. BULOH
LDG. EDINBURGH SITE 2
JPS AMPANG
PEMASOKAN AMPANG
SEK.KEB.KG.LUI
LDG. BRAUNSTON
LDG. BKT. CHERAKAH
LDG. TUAN MEE
LDG. BKT. IJOK
KG. SG. TUA
KEPONG (SEMAIAN)
IBU BEKALAN KM. 16
EMPANGAN GENTING KLANG
IBU BEKALAN KM. 11
STN. JENALETRIK LLN.
LDG. BKT. BELIMBING
JLN. KELANG
LDG. BKT. TALANG
LDG. KUALA SELANGOR
LDG. SG. BULOH
RMH PAM JPS JAYA SETIA
LDG. SG. GAPI
AIR TERJUN SG BATU
GENTING SEMPAH
PARIT 1 SG. BURONG
IBU BEKALAN SG. TENGKI
LDG. RAJA MUSA
LDG. SG. TINGGI
LDG. HOPEFUL
FDC. SEKICHAN
PARIT 1 SG. BESAR
SG. NIPAH
LDG. SG. GUMUT
RMH PAM JPS BGN TERAP
PARIT 6 SG. BESAR
PARIT SALIRAN SG. AIR TAWAR
LDG SG. BERNAM
l1
130.709
102.935
98.073
95.114
87.272
86.957
99.251
105.271
103.821
94.462
88.678
80.997
96.22
89.235
101.838
94.231
90.609
95.502
91.492
93.236
105.898
97.946
98.957
87.199
86.543
81.429
95.958
91.547
80.529
96.023
95.588
110.512
97.407
88.041
90.958
88.477
95.058
88.574
92.029
107.878
94.448
81.082
101.675
86.666
84.756
94.292
102.435
82.87
90.015
83.464
109.914
84.482
86.304
82.423
87
l2
7.572
7.005
6.366
5.853
5.214
3.983
6.9
7.398
6.682
6.357
4.008
3.727
3.811
4.615
5.883
5.791
4.804
7.31
6.036
4.758
4.592
4.983
8.224
6.826
5.723
4.247
6.877
4.623
3.743
3.775
5.681
6.964
7.77
5.154
4.545
6.172
6.556
5.003
5.417
5.356
3.906
5.026
6.739
5.869
6.052
5.681
7.42
5.407
4.585
6.258
2.229
4.604
3.76
6.16
5.582
l3
0.406
0.468
1.26
0.041
0.122
-0.00065
2.926
0.606
0.592
0.198
0.108
0.755
-0.226
0.818
0.888
1.707
0.386
1.498
0.678
0.438
-0.328
1.451
2.57
0.727
0.495
0.166
1.35
1.066
0.033
0.132
-0.071
1.505
1.345
0.976
0.272
0.549
0.759
0.463
1.488
0.956
0.025
0.796
-0.101
0.65
0.706
-0.067
1.218
0.471
1.333
0.639
1.067
0.468
0.142
0.846
0.502
l4
0.249
0.233
0.511
-0.067
0.269
0.335
2.071
0.358
0.637
0.319
0.235
0.572
0.196
0.31
0.395
1.448
0.467
1.09
0.093
0.388
0.306
0.715
1.441
-0.081
0.253
0.426
0.644
0.9
0.035
0.024
-2.25
0.571
0.633
0.313
0.224
0.681
0.338
1.038
0.35
0.244
0.425
-0.095
0.1
0.347
1.141
0.096
0.183
0.416
0.345
0.09
0.209
0.363
0.156
t3
0.054
0.067
0.198
0.00703
0.023
-0.00016
0.424
0.082
0.089
0.031
0.027
0.203
-0.059
0.177
0.151
0.295
0.08
0.205
0.112
0.092
-0.071
0.291
0.313
0.107
0.086
0.039
0.196
0.231
0.0089
0.035
-0.013
0.216
0.173
0.189
0.06
0.089
0.116
0.093
0.275
0.178
0.00648
0.158
-0.015
0.111
0.117
-0.012
0.164
0.087
0.291
0.102
0.479
0.102
0.038
0.137
0.09
t4
0.033
0.033
0.08
-0.011
0.052
0.084
0.292
0.048
0.095
0.05
0.059
0.153
0.051
0.067
0.067
0.25
0.097
0.149
0.015
0.082
0.067
0.143
0.175
-0.012
0.044
0.1
0.094
0.195
0.00939
0.00417
-0.323
0.074
0.123
0.069
0.036
0.104
0.068
0.192
0.065
0.062
0.085
-0.014
0.017
0.057
0.201
0.013
0.034
0.091
0.055
0.019
0.056
0.059
0.028
CHAPTER 5
RESULTS
5.1
Introduction
All the maximum values of daily rainfalls for each year for the 55 stations were
analyzed using MathCAD. Three MathCAD programs were built and constructed for
each 55 stations. One for t = 0, t = 1 and t = 2 respectively. The case of t = 0 are actually
the L-moment method. Meanwhile, t = 1 referred to TL-moment which was
symmetrically trimmed for one conceptual sample value and t = 2 referred to TLmoment which was symmetrically trimmed for two conceptual sample values. Then,
their distributions for each case were compared using mean absolute deviation index
(MADI), mean square deviation index (MSDI) and their correlation, r. For better view,
the ratio diagrams were constructed for each case.
Each MADI, MSDI and correlation, r, for all the 55 stations were calculated for
all the distributions which includes normal (N), logistic (LOG), generalized logistic
(GLO), extreme value type I (EV), generalized extreme value type I (GEV) and
generalized Pareto (GPA). Then, the distributions were ranked according to their MADI,
MSDI and correlation, r, from the best distribution that fits the data to the least. The
number of times each distribution obtains a given rank were then calculated and
tabulated.
73
The ranking process was repeated for 39 stations excluding 16 stations that are
either nonrandom, nonhomogeneous or those that have their n values less than 30 (their
randomness cannot be tested).
5.2
Mean Absolute Deviation Index (MADI)
The maximum daily rainfalls for all the 55 stations were analyzed using
MathCAD to obtain their mean absolute deviation vectors for all three cases (t = 0, t = 1
and t = 2). Hence the results obtained and tabulated.
For each station, their MADI were then ranked with the smallest value as the
distribution which best fit the data and so on. From all the 55 stations, the number of
times the distribution obtained a given rank was summed up and the totals for each rank
were also put into a table.
Another set of rankings were done on the 39 stations excluding the 16 stations
that are not random, not homogeneous or have n less than 30. The results were then
listed and tabulated. All these are repeated three times for all three cases.
5.2.1 Results for TL-Moment with t = 0 (L-Moment)
Table 5.1 showed the MADI obtained for all six distributions in the case of using
the L-moment. The sum of each rank for all the distributions with the 55 and 39 stations
were given in Table 5.2 and Table 5.3 respectively.
74
Table 5.1:
Mean Absolute Deviation Index (MADI) for stations in Selangor and
Kuala Lumpur (L-moment method, t = 0)
NAME OF STATION
LDG. BATU UNTONG
LDG. TELOK MERBAU
LDG. SEPANG
LDG. BUTE
PEJABAT JPS. SG. MANGG
LDG. BROOKLANDS
SMK. BDR TASIK KESUMA
P.KWLN P.S TELOK GONG
LDG. WEST
JPS. PULAU LUMUT
LDG. BKT. CHEEDING
PEJABAT JPS. KLANG
LDG. DOMINION
LDG. BUKIT KERAYONG
LDG. SG. KAPAR
LDG. NORTH HUMMOCK
LDG. HARPENDEN
LDG. ELMINA
SG. BULOH
LDG. EDINBURGH SITE 2
JPS AMPANG
PEMASOKAN AMPANG
SEK.KEB.KG.LUI
LDG. BRAUNSTON
LDG. BKT. CHERAKAH
LDG. TUAN MEE
LDG. BKT. IJOK
KG. SG. TUA
KEPONG (SEMAIAN)
IBU BEKALAN KM. 16
EMPANGAN GENTING KLANG
IBU BEKALAN KM. 11
STN. JENALETRIK LLN.
LDG. BKT. BELIMBING
JLN. KELANG
LDG. BKT. TALANG
LDG. KUALA SELANGOR
LDG. SG. BULOH
RMH PAM JPS JAYA SETIA
LDG. SG. GAPI
AIR TERJUN SG BATU
GENTING SEMPAH
PARIT 1 SG. BURONG
IBU BEKALAN SG. TENGKI
LDG. RAJA MUSA
LDG. SG. TINGGI
LDG. HOPEFUL
FDC. SEKICHAN
PARIT 1 SG. BESAR
SG. NIPAH
LDG. SG. GUMUT
RMH PAM JPS BGN TERAP
PARIT 6 SG. BESAR
PARIT SALIRAN SG. AIR TAWAR
LDG SG. BERNAM
Normal
0.030
0.053
0.082
0.030
0.036
0.045
0.237
0.185
0.063
0.061
0.067
0.089
0.052
0.289
0.049
0.791
0.043
0.175
0.043
0.036
0.030
0.117
0.214
0.080
0.193
0.048
0.157
0.095
0.191
0.030
0.082
0.042
0.175
0.141
0.086
0.158
0.093
0.077
0.232
0.068
0.057
0.405
0.036
0.075
0.129
0.189
0.090
0.085
0.061
0.102
0.017
0.232
0.042
0.126
0.078
EV
0.032
0.039
0.040
0.039
0.037
0.044
0.166
0.120
0.047
0.028
0.08
0.052
0.072
0.216
0.031
0.623
0.031
0.093
0.041
0.026
0.036
0.129
0.116
0.038
0.125
0.027
0.087
0.053
0.309
0.035
0.046
0.023
0.085
0.081
0.042
0.091
0.053
0.055
0.157
0.030
0.028
0.312
0.045
0.035
0.067
0.205
0.043
0.031
0.052
0.221
0.035
0.157
0.038
0.055
0.027
GEV
0.018
0.038
0.032
0.024
0.029
0.042
0.091
0.050
0.047
0.028
0.073
0.037
0.059
0.067
0.029
0.112
0.027
0.047
0.035
0.023
0.028
0.125
0.085
0.038
0.09
0.027
0.052
0.039
0.273
0.029
0.040
0.023
0.069
0.036
0.027
0.035
0.054
0.054
0.105
0.022
0.025
0.064
0.032
0.033
0.026
0.208
0.036
0.020
0.051
0.104
0.017
0.105
0.034
0.033
0.020
LOG
0.038
0.062
0.084
0.037
0.042
0.043
0.237
0.185
0.066
0.061
0.051
0.091
0.039
0.286
0.050
0.779
0.045
0.180
0.048
0.043
0.030
0.103
0.220
0.081
0.190
0.047
0.159
0.097
0.218
0.025
0.083
0.042
0.173
0.140
0.085
0.157
0.095
0.075
0.235
0.068
0.055
0.398
0.043
0.077
0.129
0.193
0.091
0.087
0.061
0.073
0.019
0.235
0.039
0.125
0.079
GLO
0.029
0.044
0.038
0.034
0.034
0.036
0.094
0.046
0.038
0.030
0.057
0.037
0.044
0.066
0.033
0.113
0.031
0.048
0.037
0.029
0.024
0.115
0.084
0.044
0.082
0.025
0.053
0.040
0.236
0.022
0.039
0.026
0.061
0.036
0.025
0.030
0.043
0.048
0.103
0.026
0.022
0.063
0.032
0.039
0.027
0.173
0.045
0.024
0.050
0.073
0.018
0.103
0.028
0.032
0.026
GPA
0.030
0.046
0.032
0.037
0.045
0.064
0.091
0.068
0.087
0.045
0.111
0.042
0.092
0.083
0.041
0.123
0.040
0.056
0.055
0.023
0.050
0.159
0.102
0.044
0.120
0.053
0.059
0.044
0.283
0.053
0.059
0.032
0.102
0.045
0.044
0.057
0.090
0.075
0.125
0.023
0.042
0.080
0.051
0.030
0.041
0.292
0.031
0.031
0.059
0.206
0.027
0.125
0.056
0.057
0.030
75
Table 5.2: Ranks of Mean Absolute Deviation Index (MADI) for each distribution with
55 stations (L-moment method, t = 0)
Distribution
Normal
EV
GEV
LOG
GLO
GPA
1
2
3
25
4
27
5
Number of times a distribution had the ranking
2
3
4
5
2
8
3
29
6
14
21
7
22
2
5
1
2
3
6
0
14
11
3
0
1
15
3
18
6
11
4
0
20
0
13
Table 5.3: Ranks of Mean Absolute Deviation Index (MADI) for each distribution with
39 stations excluding the 16 stations (L-moment method, t = 0)
Distribution
Normal
EV
GEV
LOG
GLO
GPA
1
0
1
17
3
21
3
Number of times a distribution had the ranking
2
3
4
5
1
7
3
20
3
13
14
6
17
2
3
0
1
1
4
14
11
4
3
0
1
10
1
15
6
8
2
0
16
0
9
5.2.2 Discussions on Mean Absolute Deviation Index (MADI) for TL-Moment
with t = 0 (L-Moment)
From both Table 5.2 and Table 5.3, it was obvious that the generalized logistic
(GLO) distribution ranked first most of the time compared to the other distributions.
This was followed closely by the generalized extreme value (GEV) distribution and
hence it was no surprise that for the second rank, the generalized extreme value (GEV)
distribution was the most frequent. Next, the Generalized Pareto (GPA) distribution
obtained the third rank the most. Meanwhile, extreme value type I (EV) distribution
ranked fourth the most for the calculations involving all the 55 and also the 39 stations.
The normal distribution was frequently ranked fifth for both tables. Lastly, the logistic
(LOG) distribution was the most often to rank last.
76
Table 5.4:
Mean Absolute Deviation Index (MADI) for stations in Selangor and
Kuala Lumpur (TL-moment method with t = 1)
NAME OF STATION
LDG. BATU UNTONG
LDG. TELOK MERBAU
LDG. SEPANG
LDG. BUTE
PEJABAT JPS. SG. MANGG
LDG. BROOKLANDS
SMK. BDR TASIK KESUMA
P.KWLN P.S TELOK GONG
LDG. WEST
JPS. PULAU LUMUT
LDG. BKT. CHEEDING
PEJABAT JPS. KLANG
LDG. DOMINION
LDG. BUKIT KERAYONG
LDG. SG. KAPAR
LDG. NORTH HUMMOCK
LDG. HARPENDEN
LDG. ELMINA
SG. BULOH
LDG. EDINBURGH SITE 2
JPS AMPANG
PEMASOKAN AMPANG
SEK.KEB.KG.LUI
LDG. BRAUNSTON
LDG. BKT. CHERAKAH
LDG. TUAN MEE
LDG. BKT. IJOK
KG. SG. TUA
KEPONG (SEMAIAN)
IBU BEKALAN KM. 16
EMPANGAN GENTING KLANG
IBU BEKALAN KM. 11
STN. JENALETRIK LLN.
LDG. BKT. BELIMBING
JLN. KELANG
LDG. BKT. TALANG
LDG. KUALA SELANGOR
LDG. SG. BULOH
RMH PAM JPS JAYA SETIA
LDG. SG. GAPI
AIR TERJUN SG BATU
GENTING SEMPAH
PARIT 1 SG. BURONG
IBU BEKALAN SG. TENGKI
LDG. RAJA MUSA
LDG. SG. TINGGI
LDG. HOPEFUL
FDC. SEKICHAN
PARIT 1 SG. BESAR
SG. NIPAH
LDG. SG. GUMUT
RMH PAM JPS BGN TERAP
PARIT 6 SG. BESAR
PARIT SALIRAN SG. AIR TAWAR
LDG SG. BERNAM
Normal
0.030
0.052
0.084
0.030
0.033
0.037
0.205
0.082
0.054
0.054
0.065
0.076
0.050
0.082
0.051
0.273
0.040
0.134
0.044
0.038
0.027
0.123
0.060
0.083
0.057
0.038
0.128
0.089
0.220
0.029
0.049
0.047
0.116
0.104
0.061
0.094
0.072
0.062
0.120
0.063
0.037
0.107
0.034
0.074
0.094
0.205
0.088
0.070
0.063
0.120
0.021
0.035
0.032
0.100
0.066
EV
0.027
0.039
0.046
0.036
0.038
0.046
0.153
0.044
0.052
0.028
0.088
0.049
0.073
0.059
0.031
0.220
0.031
0.076
0.038
0.026
0.043
0.139
0.017
0.042
0.053
0.030
0.075
0.055
0.370
0.032
0.037
0.026
0.065
0.062
0.034
0.051
0.055
0.049
0.111
0.032
0.025
0.068
0.043
0.040
0.048
0.296
0.048
0.031
0.052
0.231
0.034
0.030
0.042
0.044
0.027
GEV
0.019
0.042
0.039
0.026
0.030
0.037
0.115
0.040
0.053
0.031
0.075
0.037
0.045
0.047
0.039
0.099
0.030
0.045
0.040
0.026
0.029
0.179
0.079
0.038
0.047
0.028
0.055
0.048
0.955
0.031
0.044
0.032
0.084
0.039
0.029
0.033
0.070
0.054
0.104
0.023
0.027
0.044
0.031
0.034
0.028
0.197
0.038
0.022
0.069
0.204
0.030
0.031
0.035
0.039
0.021
LOG
0.038
0.060
0.087
0.040
0.041
0.037
0.210
0.088
0.057
0.060
0.052
0.078
0.041
0.088
0.057
0.280
0.043
0.143
0.054
0.044
0.030
0.107
0.100
0.091
0.066
0.041
0.135
0.090
0.232
0.023
0.055
0.049
0.131
0.108
0.064
0.101
0.083
0.058
0.114
0.066
0.038
0.112
0.045
0.080
0.100
0.170
0.094
0.076
0.065
0.071
0.021
0.040
0.032
0.108
0.071
GLO
0.028
0.048
0.042
0.037
0.037
0.039
0.116
0.045
0.048
0.032
0.061
0.038
0.031
0.046
0.039
0.103
0.030
0.047
0.042
0.030
0.031
0.181
0.085
0.043
0.042
0.025
0.057
0.049
0.875
0.026
0.051
0.031
0.082
0.040
0.028
0.031
0.067
0.052
0.105
0.024
0.026
0.043
0.044
0.040
0.028
0.174
0.042
0.025
0.072
0.172
0.030
0.033
0.030
0.038
0.028
GPA
0.030
0.037
0.037
0.031
0.036
0.050
0.114
0.036
0.081
0.046
0.105
0.041
0.072
0.056
0.045
0.101
0.045
0.057
0.049
0.025
0.049
0.182
0.094
0.044
0.070
0.051
0.057
0.049
0.891
0.045
0.036
0.035
0.105
0.043
0.044
0.047
0.093
0.067
0.113
0.026
0.038
0.053
0.042
0.029
0.037
0.301
0.034
0.030
0.067
0.286
0.031
0.034
0.053
0.055
0.030
77
5.2.3 Results for TL-Moment with t = 1
Table 5.4 presented the MADI obtained in the case of TL-moment symmetrically
trimmed for one conceptual sample value (t = 1) for all the 55 stations considered. Table
5.5 and Table 5.6 gave the sum of each ranking for each distribution for all the 55 and
39 stations respectively.
Table 5.5: Ranks of Mean Absolute Deviation Index (MADI) for each distribution with
55 stations (TL-moment with t = 1)
Distribution
Normal
EV
GEV
LOG
GLO
GPA
1
4
11
18
7
12
8
Number of times a distribution had the ranking
2
3
4
5
8
2
11
29
7
10
8
17
18
12
5
1
3
2
1
7
17
8
4
13
3
12
12
6
6
1
2
1
35
1
14
Table 5.6: Ranks of Mean Absolute Deviation Index (MADI) for each distribution with
39 stations excluding the 16 stations (TL-moment with t = 1)
Distribution
Normal
EV
GEV
LOG
GLO
GPA
1
1
6
15
3
9
6
Number of times a distribution had the ranking
2
3
4
5
6
2
7
22
7
6
13
6
12
7
4
1
2
2
1
4
13
8
5
3
2
11
9
2
6
1
1
0
27
1
9
78
5.2.4 Discussions on Mean Absolute Deviation Index (MADI) for TL-Moment
with t = 1
Table 5.5 and Table 5.6 showed that the generalized extreme value (GEV)
distribution was the distribution with the most number of times to be ranked first
compared to the other distribution. For the second rank, the generalized extreme value
(GEV) distribution remained the most frequently ranked second for all 55 stations.
However, the generalized logistic (GLO) distribution had only one value difference from
the GEV distribution. For the 39 stations excluding the 16 stations which were either not
random, not homogeneous or had their n values less 30, this situation was switched with
the generalized logistic (GLO) distribution ranked second the most and GEV distribution
had only one value difference compared to GLO distribution. The third rank also gave
different distributions for the 55 stations and 39 stations. The generalized logistic (GLO)
distribution was ranked third the most for the 55 stations but the generalized Pareto
(GPA) distribution was the most frequent for the 39 stations. Meanwhile, the extreme
value type I (EV) distribution was ranked fourth the most, the normal distribution was
the most to be ranked fifth and the last ranked was usually the logistic distribution for
both the 55 and 39 stations.
5.2.5 Results for TL-Moment with t = 2
Table 5.7 gave the MADI computed for the 55 stations in Selangor and Kuala
Lumpur in the case of TL-moment symmetrically trimmed for two conceptual sample
values (t = 2). Meanwhile, Table 5.8 and Table 5.9 showed the total number of rankings
for each distribution for all the 55 and 39 stations respectively.
79
Table 5.7:
Mean Absolute Deviation Index (MADI) for stations in Selangor and
Kuala Lumpur (TL-moment method with t = 2)
NAME OF STATION
LDG. BATU UNTONG
LDG. TELOK MERBAU
LDG. SEPANG
LDG. BUTE
PEJABAT JPS. SG. MANGG
LDG. BROOKLANDS
SMK. BDR TASIK KESUMA
P.KWLN P.S TELOK GONG
LDG. WEST
JPS. PULAU LUMUT
LDG. BKT. CHEEDING
PEJABAT JPS. KLANG
LDG. DOMINION
LDG. BUKIT KERAYONG
LDG. SG. KAPAR
LDG. NORTH HUMMOCK
LDG. HARPENDEN
LDG. ELMINA
SG. BULOH
LDG. EDINBURGH SITE 2
JPS AMPANG
PEMASOKAN AMPANG
SEK.KEB.KG.LUI
LDG. BRAUNSTON
LDG. BKT. CHERAKAH
LDG. TUAN MEE
LDG. BKT. IJOK
KG. SG. TUA
KEPONG (SEMAIAN)
IBU BEKALAN KM. 16
EMPANGAN GENTING KLANG
IBU BEKALAN KM. 11
STN. JENALETRIK LLN.
LDG. BKT. BELIMBING
JLN. KELANG
LDG. BKT. TALANG
LDG. KUALA SELANGOR
LDG. SG. BULOH
RMH PAM JPS JAYA SETIA
LDG. SG. GAPI
AIR TERJUN SG BATU
GENTING SEMPAH
PARIT 1 SG. BURONG
IBU BEKALAN SG. TENGKI
LDG. RAJA MUSA
LDG. SG. TINGGI
LDG. HOPEFUL
FDC. SEKICHAN
PARIT 1 SG. BESAR
SG. NIPAH
LDG. SG. GUMUT
RMH PAM JPS BGN TERAP
PARIT 6 SG. BESAR
PARIT SALIRAN SG. AIR TAWAR
LDG SG. BERNAM
Normal
0.031
0.056
0.084
0.034
0.034
0.038
0.174
0.083
0.056
0.053
0.072
0.069
0.053
0.081
0.050
0.146
0.040
0.127
0.051
0.038
0.030
0.138
0.156
0.089
0.058
0.035
0.114
0.082
0.769
0.029
0.052
0.062
0.101
0.094
0.058
0.089
0.054
0.061
0.121
0.065
0.035
0.103
0.039
0.077
0.091
0.273
0.093
0.072
0.068
0.128
0.049
0.038
0.032
0.095
0.066
EV
0.027
0.041
0.046
0.037
0.039
0.049
0.133
0.044
0.058
0.029
0.093
0.048
0.076
0.060
0.031
0.115
0.036
0.075
0.041
0.025
0.047
0.153
0.107
0.046
0.054
0.036
0.068
0.052
0.824
0.032
0.037
0.050
0.073
0.058
0.034
0.047
0.061
0.050
0.114
0.033
0.028
0.067
0.046
0.043
0.046
0.348
0.051
0.032
0.053
0.239
0.050
0.031
0.043
0.044
0.028
GEV
0.019
0.041
0.057
0.029
0.029
0.039
0.188
0.044
0.059
0.037
0.082
0.040
0.043
0.047
0.058
0.098
0.036
0.050
0.050
0.027
0.046
0.212
0.143
0.040
0.057
0.034
0.053
0.060
0.780
0.031
0.054
0.074
0.095
0.038
0.038
0.043
0.074
0.049
0.109
0.030
0.033
0.044
0.044
0.037
0.031
0.271
0.044
0.028
0.127
0.267
0.108
0.035
0.039
0.040
0.024
LOG
0.040
0.067
0.090
0.047
0.044
0.037
0.180
0.091
0.057
0.061
0.058
0.072
0.043
0.088
0.057
0.153
0.043
0.138
0.064
0.045
0.029
0.120
0.174
0.101
0.070
0.038
0.123
0.085
0.754
0.022
0.060
0.065
0.118
0.099
0.062
0.098
0.069
0.055
0.110
0.069
0.037
0.110
0.054
0.084
0.099
0.233
0.102
0.078
0.071
0.072
0.047
0.045
0.032
0.105
0.073
GLO
0.029
0.046
0.061
0.045
0.035
0.037
0.178
0.048
0.051
0.041
0.068
0.041
0.027
0.046
0.062
0.100
0.034
0.052
0.054
0.031
0.060
0.213
0.145
0.049
0.053
0.028
0.056
0.062
0.761
0.024
0.066
0.075
0.096
0.039
0.037
0.044
0.070
0.048
0.111
0.034
0.035
0.043
0.063
0.045
0.031
0.215
0.050
0.029
0.127
0.255
0.107
0.039
0.031
0.040
0.031
GPA
0.032
0.045
0.052
0.031
0.042
0.056
0.183
0.037
0.092
0.049
0.111
0.043
0.073
0.057
0.057
0.105
0.053
0.062
0.053
0.029
0.049
0.209
0.139
0.043
0.078
0.059
0.056
0.056
0.825
0.046
0.037
0.071
0.112
0.047
0.049
0.048
0.099
0.072
0.117
0.030
0.040
0.055
0.044
0.029
0.039
0.372
0.036
0.034
0.117
0.327
0.107
0.035
0.057
0.059
0.032
80
Table 5.8: Ranks of Mean Absolute Deviation Index (MADI) for each distribution with
55 stations (TL-moment with t = 2)
Distribution
Normal
EV
GEV
LOG
GLO
GPA
1
2
16
17
8
13
5
Number of times a distribution had the ranking
2
3
4
5
11
9
7
25
6
10
16
6
16
7
6
7
4
5
2
6
11
9
12
4
8
11
7
10
6
1
1
2
30
6
14
Table 5.9: Ranks of Mean Absolute Deviation Index (MADI) for each distribution with
39 stations excluding the 16 stations (TL-moment with t = 2)
Distribution
Normal
EV
GEV
LOG
GLO
GPA
1
2
9
16
4
8
5
Number of times a distribution had the ranking
2
3
4
5
6
5
5
20
5
8
13
3
11
4
4
4
3
3
1
3
9
6
9
4
6
7
7
6
6
1
1
0
25
3
8
5.2.6 Discussions on Mean Absolute Deviation Index (MADI) for TL-Moment
with t = 2
Both Table 5.8 and Table 5.9 showed that for all the 55 and 39 stations, the
generalized extreme value (GEV) distribution was the most to be ranked first and
second. The extreme value type I (EV) distribution ranked third the most for the
calculations involving only the 39 stations but tied with the generalized Pareto (GPA)
distribution for calculations of all the 55 stations. For the fourth, fifth and last rank, both
the tables gave the same results with the extreme value type I (EV) distribution remained
the most in the fourth rank. Similar to the case of L-moment and TL-moment with t = 1,
81
the normal distribution ranked fifth and the logistic (LOG) distribution ranked last the
most often.
5.3
Mean Square Deviation Index (MSDI)
Using similar method through the MathCAD program, the results obtained for
the mean square deviation index (MSDI) of all the 55 stations in Selangor and Kuala
Lumpur were tabulated.
Following the same procedure as with the mean absolute deviation index
(MADI), the mean square deviation index (MSDI) for each distribution were ranked
with the smallest value as the first and the largest as the last for all the 55 stations. All
the number of times a given distribution obtained a given rank was then summed up and
tabulated. The same procedure was used for the 39 stations which exclude the 16
stations that are either not random, not homogeneous or have their n values less than 30.
The sums of all the ranks of each distribution were again listed in a table. These were
done three times for all the three considered cases.
5.3.1 Results for TL-Moment with t = 0 (L-Moment)
Table 5.10 showed the MSDI obtained for all six distributions in the case of
using the L-moment method on all the stations in Selangor and Kuala Lumpur. The
number of times each distribution obtained a given rank for all 55 and 39 stations was
listed in Table 5.11 and Table 5.12 respectively.
82
Table 5.10:
Mean Square Deviation Index (MSDI) for stations in Selangor and Kuala
Lumpur (L-moment method, t = 0)
NAME OF STATION
LDG. BATU UNTONG
LDG. TELOK MERBAU
LDG. SEPANG
LDG. BUTE
PEJABAT JPS. SG. MANGG
LDG. BROOKLANDS
SMK. BDR TASIK KESUMA
P.KWLN P.S TELOK GONG
LDG. WEST
JPS. PULAU LUMUT
LDG. BKT. CHEEDING
PEJABAT JPS. KLANG
LDG. DOMINION
LDG. BUKIT KERAYONG
LDG. SG. KAPAR
LDG. NORTH HUMMOCK
LDG. HARPENDEN
LDG. ELMINA
SG. BULOH
LDG. EDINBURGH SITE 2
JPS AMPANG
PEMASOKAN AMPANG
SEK.KEB.KG.LUI
LDG. BRAUNSTON
LDG. BKT. CHERAKAH
LDG. TUAN MEE
LDG. BKT. IJOK
KG. SG. TUA
KEPONG (SEMAIAN)
IBU BEKALAN KM. 16
EMPANGAN GENTING KLANG
IBU BEKALAN KM. 11
STN. JENALETRIK LLN.
LDG. BKT. BELIMBING
JLN. KELANG
LDG. BKT. TALANG
LDG. KUALA SELANGOR
LDG. SG. BULOH
RMH PAM JPS JAYA SETIA
LDG. SG. GAPI
AIR TERJUN SG BATU
GENTING SEMPAH
PARIT 1 SG. BURONG
IBU BEKALAN SG. TENGKI
LDG. RAJA MUSA
LDG. SG. TINGGI
LDG. HOPEFUL
FDC. SEKICHAN
PARIT 1 SG. BESAR
SG. NIPAH
LDG. SG. GUMUT
RMH PAM JPS BGN TERAP
PARIT 6 SG. BESAR
PARIT SALIRAN SG. AIR TAWAR
LDG SG. BERNAM
Normal
0.00212
0.00885
0.01100
0.00215
0.00414
0.00437
0.07800
0.06200
0.00621
0.01000
0.00781
0.01200
0.00993
0.12300
0.00426
0.90200
0.00357
0.06100
0.00352
0.00287
0.00127
0.03900
0.08800
0.01400
0.06700
0.00491
0.03700
0.01400
0.10700
0.00192
0.01500
0.00282
0.06200
0.03100
0.01300
0.04400
0.01300
0.00855
0.09300
0.00822
0.00567
0.23800
0.00265
0.01200
0.03200
0.10100
0.01500
0.01400
0.00523
0.05300
0.00046
0.00209
0.00305
0.03100
0.01100
EV
0.00206
0.00265
0.00250
0.00284
0.00236
0.00369
0.03700
0.02600
0.01700
0.00142
0.03000
0.00392
0.02800
0.06500
0.00191
0.53100
0.00146
0.01500
0.00953
0.00143
0.00265
0.11200
0.01900
0.00210
0.02600
0.00136
0.00990
0.00440
0.15600
0.00522
0.00520
0.00907
0.00969
0.00961
0.00335
0.01400
0.00757
0.01400
0.03700
0.00158
0.00146
0.13300
0.00656
0.00214
0.00831
0.11800
0.00281
0.00190
0.00402
0.46500
0.00162
0.00479
0.00245
0.00517
0.00136
GEV
0.00071
0.00281
0.00209
0.00110
0.00174
0.00315
0.01500
0.00733
0.01500
0.00149
0.01900
0.00296
0.01500
0.01400
0.00144
0.03000
0.00109
0.00506
0.00532
0.00110
0.00112
0.09800
0.02700
0.00203
0.02000
0.00123
0.00526
0.00289
0.19700
0.00309
0.00391
0.00906
0.00959
0.00294
0.00136
0.00321
0.00826
0.01700
0.07300
0.00082
0.00103
0.01400
0.00227
0.00172
0.00179
0.12500
0.00221
0.00059
0.00390
0.05700
0.00037
0.00275
0.00205
0.00203
0.00740
LOG
0.00393
0.00404
0.01400
0.00393
0.00636
0.00532
0.08200
0.06700
0.00859
0.01400
0.00401
0.01300
0.00541
0.13000
0.00614
0.92600
0.00489
0.07300
0.00431
0.00419
0.00155
0.02600
0.11800
0.01900
0.07900
0.00662
0.04300
0.01600
0.13400
0.00108
0.01800
0.00279
0.08400
0.03400
0.01400
0.04900
0.02100
0.00848
0.11600
0.00939
0.00592
0.24700
0.00418
0.01400
0.03700
0.17900
0.01800
0.01600
0.00588
0.02200
0.00047
0.00223
0.00340
0.03700
0.01400
GLO
0.00186
0.01200
0.00302
0.00231
0.00259
0.00299
0.01600
0.00664
0.00755
0.00251
0.01000
0.00291
0.00835
0.01400
0.00206
0.03000
0.00154
0.00467
0.00297
0.00203
0.00081
0.07700
0.02400
0.00339
0.01700
0.00133
0.00530
0.00335
0.21200
0.00164
0.00339
0.00103
0.00695
0.00304
0.00111
0.00305
0.00442
0.01300
0.07100
0.00114
0.00083
0.01400
0.00181
0.00266
0.00200
0.09700
0.00323
0.00096
0.00369
0.02000
0.00037
0.00173
0.00130
0.00196
0.00124
GPA
0.00189
0.00320
0.00145
0.00275
0.00348
0.00590
0.01500
0.01000
0.04200
0.00289
0.04500
0.00451
0.03500
0.01900
0.00310
0.03100
0.00267
0.01000
0.01700
0.00810
0.00450
0.16000
0.05000
0.00278
0.03200
0.00390
0.00812
0.00270
0.20300
0.00828
0.00671
0.00145
0.02600
0.00452
0.00333
0.00550
0.02800
0.03300
0.09700
0.00098
0.00256
0.01700
0.01000
0.00141
0.00296
0.25800
0.00187
0.00154
0.00625
0.38800
0.00093
0.00833
0.00587
0.00565
0.00155
83
Table 5.11: Ranks of Mean Square Deviation Index (MSDI) for each distribution with
55 stations (L-moment method, t = 0)
Distribution
Normal
EV
GEV
LOG
GLO
GPA
1
2
4
18
5
26
5
Number of times a distribution had the ranking
2
3
4
5
7
6
7
33
7
15
16
10
21
3
11
2
2
4
4
6
11
12
4
0
2
13
4
15
6
0
3
0
34
2
16
Table 5.12: Ranks of Mean Square Deviation Index (MSDI) for each distribution with
39 stations excluding the 16 stations (L-moment method, t = 0)
Distribution
Normal
EV
GEV
LOG
GLO
GPA
1
1
2
11
4
19
3
Number of times a distribution had the ranking
2
3
4
5
5
5
5
23
5
9
13
9
17
2
8
1
1
3
3
4
9
7
3
0
1
13
7
2
6
0
1
0
24
1
13
5.3.2 Discussions on Mean Square Deviation Index (MSDI) for TL-Moment
with t = 0 (L-Moment)
From both Table 5.11 and Table 5.12, the results obtained were basically the
same except for the third rank where for the calculations from all the 55 stations, both
the extreme value type I (EV) and generalized Pareto (GPA) distributions were tied with
the most number of times to be ranked third. Meanwhile, for the calculations excluding
the 16 nonrandom, nonhomogeneous and small sample size stations (n less than 30),
only the generalized Pareto (GPA) distribution was the most often ranked third. All the
other ranks gave the same distributions for both tables which were also similar to the
results obtained using the mean absolute deviation index (MADI). The generalized
84
logistic (GLO) distribution was the most frequent to be ranked first compared to the
other distributions, the generalized extreme value (GEV) distribution was the most to be
ranked second, the extreme value type I (EV) distribution ranked fourth the most often,
the normal distribution ranked fifth most of the time and lastly, the logistic (LOG)
distribution was the one with the most number of times ranking last.
5.3.3 Results for TL-Moment with t = 1
Table 5.13 presented the MSDI obtained in the case of TL-moment
symmetrically trimmed for one conceptual sample value (t = 1) for all the 55 stations
considered. Table 5.14 and Table 5.15 listed the sum of each ranking for each
distribution for all the 55 and 39 stations respectively.
85
Table 5.13:
Mean Square Deviation Index (MSDI) for stations in Selangor and Kuala
Lumpur (TL-moment method with t = 1)
NAME OF STATION
LDG. BATU UNTONG
LDG. TELOK MERBAU
LDG. SEPANG
LDG. BUTE
PEJABAT JPS. SG. MANGG
LDG. BROOKLANDS
SMK. BDR TASIK KESUMA
P.KWLN P.S TELOK GONG
LDG. WEST
JPS. PULAU LUMUT
LDG. BKT. CHEEDING
PEJABAT JPS. KLANG
LDG. DOMINION
LDG. BUKIT KERAYONG
LDG. SG. KAPAR
LDG. NORTH HUMMOCK
LDG. HARPENDEN
LDG. ELMINA
SG. BULOH
LDG. EDINBURGH SITE 2
JPS AMPANG
PEMASOKAN AMPANG
SEK.KEB.KG.LUI
LDG. BRAUNSTON
LDG. BKT. CHERAKAH
LDG. TUAN MEE
LDG. BKT. IJOK
KG. SG. TUA
KEPONG (SEMAIAN)
IBU BEKALAN KM. 16
EMPANGAN GENTING KLANG
IBU BEKALAN KM. 11
STN. JENALETRIK LLN.
LDG. BKT. BELIMBING
JLN. KELANG
LDG. BKT. TALANG
LDG. KUALA SELANGOR
LDG. SG. BULOH
RMH PAM JPS JAYA SETIA
LDG. SG. GAPI
AIR TERJUN SG BATU
GENTING SEMPAH
PARIT 1 SG. BURONG
IBU BEKALAN SG. TENGKI
LDG. RAJA MUSA
LDG. SG. TINGGI
LDG. HOPEFUL
FDC. SEKICHAN
PARIT 1 SG. BESAR
SG. NIPAH
LDG. SG. GUMUT
RMH PAM JPS BGN TERAP
PARIT 6 SG. BESAR
PARIT SALIRAN SG. AIR TAWAR
LDG SG. BERNAM
Normal
0.00321
0.01000
0.01500
0.00304
0.00411
0.00395
0.06500
0.02800
0.00544
0.01000
0.02400
0.01000
0.02200
0.02500
0.00504
0.14300
0.00352
0.04000
0.00422
0.00424
0.00121
0.07900
0.06000
0.01900
0.01600
0.00403
0.03200
0.01400
0.13500
0.00274
0.01100
0.00390
0.03400
0.02200
0.00955
0.02800
0.00872
0.00836
0.03000
0.00845
0.00465
0.03600
0.00301
0.01500
0.02600
0.19400
0.02000
0.01400
0.00556
0.09500
0.00521
0.00233
0.00273
0.02400
0.01200
EV
0.00219
0.00278
0.00363
0.00299
0.00269
0.00542
0.03200
0.01300
0.02500
0.00144
0.05700
0.00410
0.04500
0.01900
0.00192
0.08700
0.00158
0.01100
0.00922
0.00181
0.00375
0.17300
0.01700
0.00280
0.01500
0.00191
0.00896
0.00444
0.27500
0.00714
0.00442
0.00104
0.00835
0.00799
0.00318
0.00927
0.01200
0.01900
0.05100
0.00170
0.00180
0.02300
0.00695
0.00288
0.00683
0.49800
0.00423
0.00199
0.00398
0.63200
0.00200
0.00484
0.00410
0.00410
0.00151
GEV
0.00087
0.00430
0.00629
0.00181
0.00263
0.00448
0.09200
0.01100
0.03000
0.00225
0.03600
0.00608
0.01600
0.01700
0.00364
0.02500
0.00138
0.00754
0.01000
0.00177
0.00148
0.33400
0.07900
0.00285
0.01300
0.00158
0.01300
0.00860
4.02600
0.00528
0.00843
0.00156
0.02400
0.00474
0.00237
0.00414
0.02600
0.03000
0.11200
0.00159
0.00254
0.01700
0.00249
0.00232
0.00211
0.17400
0.00633
0.00075
0.01400
0.47600
0.00146
0.00581
0.00289
0.00280
0.00832
LOG
0.00735
0.01700
0.02100
0.00718
0.00815
0.00471
0.07600
0.03600
0.00736
0.01800
0.01200
0.01200
0.01300
0.02700
0.00913
0.16600
0.00615
0.05800
0.00852
0.00722
0.00146
0.04900
0.10000
0.03100
0.02100
0.00667
0.04300
0.01700
0.14800
0.00127
0.01400
0.00479
0.06100
0.02800
0.01200
0.03700
0.02000
0.00770
0.02800
0.01100
0.00523
0.04200
0.00738
0.02100
0.03500
0.14400
0.02900
0.02000
0.00742
0.02000
0.00539
0.00376
0.00281
0.03600
0.01800
GLO
0.00256
0.00785
0.00850
0.00563
0.00592
0.00600
0.08900
0.01100
0.02100
0.00441
0.02100
0.00694
0.00508
0.01600
0.00454
0.02600
0.00166
0.00783
0.00788
0.00329
0.00178
0.33800
0.08500
0.00479
0.01100
0.00132
0.01500
0.01000
3.11900
0.00310
0.01100
0.00168
0.02200
0.00544
0.00171
0.00364
0.02200
0.02700
0.11500
0.00239
0.00217
0.01600
0.00670
0.00390
0.00214
0.12900
0.00877
0.00120
0.01600
0.27800
0.00139
0.00505
0.00181
0.00260
0.00157
GPA
0.00284
0.00340
0.00312
0.00327
0.00410
0.00741
0.08100
0.01200
0.06900
0.00403
0.07800
0.00637
0.04700
0.02200
0.00573
0.02500
0.00419
0.01400
0.02600
0.00108
0.00609
0.37600
0.09400
0.00369
0.02600
0.00611
0.01300
0.00535
3.72900
0.01200
0.00803
0.00177
0.04700
0.00592
0.00548
0.00751
0.05500
0.04900
0.13700
0.00123
0.00464
0.01900
0.01100
0.00143
0.00407
0.48100
0.00347
0.00218
0.01300
1.14800
0.00181
0.01300
0.00830
0.00834
0.00193
86
Table 5.14: Ranks of Mean Square Deviation Index (MSDI) for each distribution with
55 stations (TL-moment with t = 1)
Distribution
Normal
EV
GEV
LOG
GLO
GPA
1
7
15
11
6
12
6
Number of times a distribution had the ranking
2
3
4
5
10
2
9
27
7
15
11
6
18
11
7
6
6
6
0
2
9
9
18
6
4
11
11
7
6
0
1
2
35
1
16
Table 5.15: Ranks of Mean Square Deviation Index (MSDI) for each distribution with
39 stations excluding the 16 stations (TL-moment with t = 1)
Distribution
Normal
EV
GEV
LOG
GLO
GPA
1
5
9
9
5
8
4
Number of times a distribution had the ranking
2
3
4
5
7
2
5
20
5
7
6
12
11
7
6
6
4
4
0
2
8
5
14
3
4
8
8
1
6
0
0
0
24
1
14
5.3.4 Discussions on Mean Square Deviation Index (MSDI) for TL-Moment
with t = 1
The extreme value type I (EV) distribution was the most often ranked as first in
the calculations involving all the 55 stations but it was tied with the generalized extreme
value (GEV) distribution for the 39 stations. However, for the rest of the rankings from
the second to the sixth (the last rank), both Table 5.14 and Table 5.15 showed the same
results. The generalized extreme value (GEV) distribution was the most often to be
ranked second. The extreme value type I (EV) distribution also ranked third the most
although it was also mostly ranked first. The most frequent to rank fourth was the
generalized logistic (GLO) distribution, followed by the normal distribution in the fifth
rank and logistic (LOG) distribution in the last rank.
87
Table 5.16:
Mean Square Deviation Index (MSDI) for stations in Selangor and Kuala
Lumpur (TL-moment method with t = 2)
NAME OF STATION
LDG. BATU UNTONG
LDG. TELOK MERBAU
LDG. SEPANG
LDG. BUTE
PEJABAT JPS. SG. MANGG
LDG. BROOKLANDS
SMK. BDR TASIK KESUMA
P.KWLN P.S TELOK GONG
LDG. WEST
JPS. PULAU LUMUT
LDG. BKT. CHEEDING
PEJABAT JPS. KLANG
LDG. DOMINION
LDG. BUKIT KERAYONG
LDG. SG. KAPAR
LDG. NORTH HUMMOCK
LDG. HARPENDEN
LDG. ELMINA
SG. BULOH
LDG. EDINBURGH SITE 2
JPS AMPANG
PEMASOKAN AMPANG
SEK.KEB.KG.LUI
LDG. BRAUNSTON
LDG. BKT. CHERAKAH
LDG. TUAN MEE
LDG. BKT. IJOK
KG. SG. TUA
KEPONG (SEMAIAN)
IBU BEKALAN KM. 16
EMPANGAN GENTING KLANG
IBU BEKALAN KM. 11
STN. JENALETRIK LLN.
LDG. BKT. BELIMBING
JLN. KELANG
LDG. BKT. TALANG
LDG. KUALA SELANGOR
LDG. SG. BULOH
RMH PAM JPS JAYA SETIA
LDG. SG. GAPI
AIR TERJUN SG BATU
GENTING SEMPAH
PARIT 1 SG. BURONG
IBU BEKALAN SG. TENGKI
LDG. RAJA MUSA
LDG. SG. TINGGI
LDG. HOPEFUL
FDC. SEKICHAN
PARIT 1 SG. BESAR
SG. NIPAH
LDG. SG. GUMUT
RMH PAM JPS BGN TERAP
PARIT 6 SG. BESAR
PARIT SALIRAN SG. AIR TAWAR
LDG SG. BERNAM
Normal
0.00355
0.01200
0.01700
0.00448
0.00425
0.00437
0.05100
0.03000
0.00658
0.00961
0.03200
0.01000
0.02500
0.02500
0.00502
0.06900
0.00302
0.03500
0.00599
0.00416
0.00146
0.10800
0.04500
0.02300
0.01600
0.00373
0.02900
0.01300
2.05100
0.00275
0.01100
0.00935
0.02500
0.02100
0.00946
0.02600
0.00692
0.00964
0.03600
0.00879
0.00470
0.03600
0.00470
0.01700
0.02600
0.40800
0.02300
0.01500
0.00635
0.12200
0.00446
0.00296
0.00303
0.02200
0.01300
EV
0.00231
0.00310
0.00395
0.00333
0.00294
0.00693
0.02800
0.01300
0.03000
0.00156
0.06800
0.00524
0.04900
0.02000
0.00193
0.05700
0.00219
0.01100
0.00807
0.00171
0.00476
0.20300
0.02000
0.00360
0.01500
0.00321
0.00922
0.00469
2.42200
0.00715
0.00438
0.00352
0.01200
0.00895
0.00352
0.00920
0.01800
0.02100
0.06300
0.00186
0.00218
0.02300
0.00629
0.00339
0.00681
0.78200
0.00535
0.00215
0.00394
0.69500
0.00589
0.00448
0.00497
0.00428
0.00162
GEV
0.00098
0.00315
0.02000
0.00287
0.00212
0.00476
0.40300
0.01200
0.03300
0.00384
0.04700
0.00720
0.01400
0.01700
0.01100
0.03800
0.00224
0.00983
0.01500
0.00232
0.00477
0.42500
0.15100
0.00353
0.01500
0.00260
0.01100
0.01700
2.12100
0.00463
0.01300
0.01500
0.03600
0.00422
0.00452
0.00801
0.03000
0.02400
0.13200
0.00415
0.00457
0.01700
0.00619
0.00284
0.00296
0.39400
0.01100
0.00160
0.10000
0.87400
0.02200
0.00703
0.00370
0.00315
0.00114
LOG
0.00878
0.02100
0.02400
0.01100
0.00930
0.00460
0.06000
0.03900
0.00661
0.01800
0.01700
0.01200
0.01400
0.02800
0.00958
0.07500
0.00520
0.05100
0.01400
0.00759
0.00133
0.07000
0.07600
0.03900
0.02300
0.00557
0.04000
0.01500
1.96600
0.00123
0.01600
0.01200
0.04600
0.02500
0.01200
0.03500
0.01400
0.00780
0.02800
0.01200
0.00514
0.04200
0.01300
0.02500
0.03500
0.29100
0.03500
0.02200
0.00883
0.02100
0.00405
0.00564
0.00278
0.03400
0.02000
GLO
0.00296
0.00532
0.02600
0.00900
0.00467
0.00462
0.32800
0.01300
0.02400
0.00746
0.02900
0.00858
0.00359
0.01600
0.01500
0.03900
0.00222
0.01100
0.01600
0.00438
0.01400
0.43500
0.15400
0.00674
0.01200
0.00190
0.01300
0.02000
2.01400
0.00226
0.02100
0.01500
0.03600
0.00472
0.00371
0.00747
0.02600
0.01900
0.13800
0.00608
0.00459
0.01600
0.02100
0.00522
0.00274
0.24700
0.01600
0.00228
0.09900
0.73900
0.02200
0.00767
0.00191
0.00297
0.00213
GPA
0.00371
0.00406
0.00950
0.00339
0.00545
0.01100
0.36700
0.01300
0.07800
0.00560
0.09500
0.00716
0.05100
0.02300
0.01000
0.04200
0.00674
0.01700
0.03000
0.00168
0.00673
0.44300
0.14200
0.00369
0.03100
0.00908
0.01200
0.00898
2.39100
0.01300
0.00900
0.01200
0.06200
0.00729
0.00809
0.01100
0.06500
0.04900
0.15100
0.00199
0.00663
0.02000
0.01000
0.00147
0.00579
0.84400
0.00440
0.00316
0.07100
1.56000
0.02200
0.01400
0.01100
0.01100
0.00250
88
5.3.5 Results for TL-Moment with t = 2
Table 5.16 presented the MSDI computed for the 55 stations in Selangor and
Kuala Lumpur in the case of TL-moment symmetrically trimmed for two conceptual
sample values (t = 2). Meanwhile, Table 5.17 and Table 5.18 showed the total number of
rankings for each distribution for all the 55 and 39 stations respectively.
Table 5.17: Ranks of Mean Square Deviation Index (MSDI) for each distribution with
55 stations (TL-moment with t = 2)
Distribution
Normal
EV
GEV
LOG
GLO
GPA
1
6
17
10
9
10
3
Number of times a distribution had the ranking
2
3
4
5
12
8
9
20
13
6
6
12
14
8
13
8
6
7
1
8
9
8
7
13
6
10
12
6
6
0
1
2
24
8
18
Table 5.18: Ranks of Mean Square Deviation Index (MSDI) for each distribution with
39 stations excluding the 16 stations (TL-moment with t = 2)
Distribution
Normal
EV
GEV
LOG
GLO
GPA
1
5
9
9
6
7
3
Number of times a distribution had the ranking
2
3
4
5
6
6
5
17
13
9
4
4
10
6
8
5
4
3
1
4
6
6
10
6
5
6
9
3
6
0
0
1
21
4
13
89
5.3.6 Discussions on Mean Absolute Deviation Index (MADI) for TL-Moment
with t = 2
The first rank was mostly taken by the extreme value type I (EV) distribution for
the total of 55 stations. However, for the 39 stations excluding the 16 stations which
were not random, not homogeneous and had small sample sizes (n less than 30), both the
extreme value type I and generalized extreme value distribution are usually the ones in
the first rank. The generalized extreme value (GEV) distribution was the most often
ranked second for the computations including all 55 stations but the extreme value type I
(EV) distribution was the most often for the computations with only the 39 stations.
However, there was only one value difference between the generalized extreme value
(GEV) and extreme value type I (EV) distributions for the calculations of the 55 stations.
The third rank was mostly filled by the extreme value type I (EV) distribution for both
Table 5.17 and Table 5.18. The generalized logistic (GLO) distribution was the most
frequently ranked fourth for both table but it was tied with the extreme value type I
distribution for all the 55 stations. Similar to the results from mean absolute deviation
index (MADI) and mean square deviation index (MSDI) for all the three cases of t = 0,
t = 1 and t = 2, the fifth ranked was mostly the normal distribution and the logistic
distribution showed the highest total number of times to be ranked last.
5.4
Correlation (r)
The maximum daily rainfalls for all the 55 stations were again analyzed using
MathCAD to find the correlation, r, between the actual flows and predicted flows of
rainfalls. The results obtained were given in a table.
The correlation for each distribution was then ranked with the value closest to
one as the best rank to the value furthest to one as the least. This was done for all the 55
stations. The number of times each distribution obtained each ranking was then summed
90
up and the totals were put into table. The same methods were applied to the 39 stations
without considering the 16 nonrandom, nonhomogeneous and small sample sizes
stations (n less than 30). Then, the sums of each ranking for all the distributions were
tabulated. Similar to the procedures using mean absolute deviation index (MADI) and
mean square deviation index (MSDI), these steps were done three times for all the three
cases considered (TL-moment with t = 0, t = 1 and t = 2).
5.4.1 Results for TL-Moment with t = 0 (L-Moment)
Table 5.19 listed the correlation, r, obtained for all the six distributions
considered in this study for the case of using the TL-moment with t = 0 or also known as
the L-moment method on all the stations in Selangor and Kuala Lumpur. The number of
times each distribution obtained a given rank for all 55 and 39 stations was listed in
Table 5.20 and Table 5.21 respectively.
91
Table 5.19:
Correlation, r, for stations in Selangor and Kuala Lumpur (L-moment
method, t = 0)
NAME OF STATION
LDG. BATU UNTONG
LDG. TELOK MERBAU
LDG. SEPANG
LDG. BUTE
PEJABAT JPS. SG. MANGG
LDG. BROOKLANDS
SMK. BDR TASIK KESUMA
P.KWLN P.S TELOK GONG
LDG. WEST
JPS. PULAU LUMUT
LDG. BKT. CHEEDING
PEJABAT JPS. KLANG
LDG. DOMINION
LDG. BUKIT KERAYONG
LDG. SG. KAPAR
LDG. NORTH HUMMOCK
LDG. HARPENDEN
LDG. ELMINA
SG. BULOH
LDG. EDINBURGH SITE 2
JPS AMPANG
PEMASOKAN AMPANG
SEK.KEB.KG.LUI
LDG. BRAUNSTON
LDG. BKT. CHERAKAH
LDG. TUAN MEE
LDG. BKT. IJOK
KG. SG. TUA
KEPONG (SEMAIAN)
IBU BEKALAN KM. 16
EMPANGAN GENTING KLANG
IBU BEKALAN KM. 11
STN. JENALETRIK LLN.
LDG. BKT. BELIMBING
JLN. KELANG
LDG. BKT. TALANG
LDG. KUALA SELANGOR
LDG. SG. BULOH
RMH PAM JPS JAYA SETIA
LDG. SG. GAPI
AIR TERJUN SG BATU
GENTING SEMPAH
PARIT 1 SG. BURONG
IBU BEKALAN SG. TENGKI
LDG. RAJA MUSA
LDG. SG. TINGGI
LDG. HOPEFUL
FDC. SEKICHAN
PARIT 1 SG. BESAR
SG. NIPAH
LDG. SG. GUMUT
RMH PAM JPS BGN TERAP
PARIT 6 SG. BESAR
PARIT SALIRAN SG. AIR TAWAR
LDG SG. BERNAM
Normal
0.987
0.958
0.953
0.988
0.970
0.952
0.847
0.725
0.972
0.970
0.951
0.915
0.940
0.584
0.978
0.639
0.974
0.876
0.983
0.973
0.979
0.945
0.905
0.968
0.689
0.970
0.911
0.921
0.923
0.974
0.863
0.905
0.912
0.884
0.893
0.851
0.963
0.952
0.721
0.937
0.923
0.554
0.984
0.955
0.891
0.875
0.953
0.943
0.959
0.984
0.784
0.983
0.954
0.926
0.952
EV
0.977
0.980
0.979
0.979
0.980
0.972
0.914
0.821
0.990
0.993
0.966
0.969
0.949
0.698
0.981
0.764
0.980
0.953
0.983
0.971
0.977
0.947
0.967
0.986
0.779
0.988
0.966
0.966
0.885
0.975
0.926
0.915
0.964
0.956
0.958
0.927
0.981
0.978
0.819
0.981
0.967
0.671
0.983
0.983
0.959
0.924
0.980
0.988
0.967
0.952
0.728
0.981
0.977
0.976
0.989
GEV
0.990
0.978
0.975
0.990
0.979
0.967
0.906
0.936
0.990
0.993
0.960
0.977
0.940
0.925
0.986
0.935
0.983
0.994
0.989
0.979
0.982
0.949
0.980
0.986
0.886
0.988
0.971
0.962
0.937
0.977
0.948
0.915
0.974
0.981
0.984
0.968
0.981
0.978
0.950
0.983
0.975
0.925
0.988
0.982
0.984
0.925
0.976
0.994
0.968
0.985
0.751
0.988
0.973
0.985
0.991
LOG
0.981
0.959
0.948
0.985
0.971
0.959
0.848
0.744
0.977
0.970
0.965
0.921
0.956
0.612
0.974
0.661
0.972
0.887
0.983
0.965
0.981
0.951
0.912
0.964
0.712
0.971
0.914
0.920
0.920
0.981
0.875
0.905
0.919
0.892
0.904
0.861
0.965
0.957
0.744
0.938
0.930
0.581
0.985
0.951
0.898
0.893
0.948
0.946
0.958
0.984
0.747
0.983
0.962
0.931
0.953
GLO
0.983
0.980
0.968
0.987
0.982
0.975
0.902
0.939
0.993
0.992
0.975
0.974
0.960
0.925
0.980
0.935
0.980
0.994
0.987
0.969
0.985
0.952
0.977
0.979
0.894
0.988
0.966
0.956
0.932
0.985
0.957
0.914
0.972
0.979
0.987
0.968
0.978
0.977
0.952
0.979
0.978
0.925
0.990
0.976
0.983
0.942
0.968
0.993
0.964
0.984
0.750
0.985
0.982
0.984
0.989
GPA
0.987
0.964
0.986
0.981
0.960
0.936
0.921
0.919
0.972
0.983
0.915
0.979
0.887
0.918
0.984
0.937
0.977
0.988
0.978
0.984
0.959
0.932
0.985
0.989
0.855
0.976
0.979
0.974
0.942
0.947
0.917
0.914
0.970
0.984
0.967
0.964
0.975
0.968
0.936
0.986
0.959
0.922
0.970
0.986
0.979
0.876
0.987
0.989
0.966
0.964
0.747
0.978
0.942
0.982
0.987
92
Table 5.11: Ranks of correlation, r, for each distribution with 55 stations (L-moment
method, t = 0)
Distribution
Normal
EV
GEV
LOG
GLO
GPA
1
1
6
22
0
23
14
Number of times a distribution had the ranking
2
3
4
5
3
3
4
17
14
8
19
3
20
11
2
0
4
5
7
30
10
11
10
1
2
10
4
14
6
27
5
0
9
0
11
Table 5.12: Ranks of correlation, r, for each distribution with 39 stations excluding the
16 stations (L-moment method, t = 0)
Distribution
Normal
EV
GEV
LOG
GLO
GPA
1
0
3
16
0
17
9
Number of times a distribution had the ranking
2
3
4
5
3
3
3
10
10
5
3
15
14
7
2
0
3
5
4
21
8
6
7
1
1
12
5
3
6
20
3
0
6
0
9
5.4.2 Discussions on Correlation, r, for TL-Moment with t = 0 (L-Moment)
Both Table 5.20 and Table 5.21 showed the same results for all the six ranks.
The first rank was mostly the generalized logistic (GLO) distribution but in both tables,
the generalized extreme value (GEV) distribution had only one value difference from the
GLO distribution. Hence, as expected the generalized extreme value (GEV) distribution
ranked second the most for both the 55 and 39 stations’ calculations. The third rank was
mostly taken by the generalized Pareto (GPA) distribution while the fourth was filled
mostly by the extreme value type I (EV) distribution. Contrary to the results from the
mean absolute deviation index (MADI) and mean square deviation index (MSDI), the
fifth and sixth ranks were switched with the fifth rank being monopolized by the logistic
(LOG) distribution and the last rank monopolized by the normal distribution.
93
Table 5.22:
Correlation, r, for stations in Selangor and Kuala Lumpur (TL-moment
method with t = 1)
NAME OF STATION
LDG. BATU UNTONG
LDG. TELOK MERBAU
LDG. SEPANG
LDG. BUTE
PEJABAT JPS. SG. MANGG
LDG. BROOKLANDS
SMK. BDR TASIK KESUMA
P.KWLN P.S TELOK GONG
LDG. WEST
JPS. PULAU LUMUT
LDG. BKT. CHEEDING
PEJABAT JPS. KLANG
LDG. DOMINION
LDG. BUKIT KERAYONG
LDG. SG. KAPAR
LDG. NORTH HUMMOCK
LDG. HARPENDEN
LDG. ELMINA
SG. BULOH
LDG. EDINBURGH SITE 2
JPS AMPANG
PEMASOKAN AMPANG
SEK.KEB.KG.LUI
LDG. BRAUNSTON
LDG. BKT. CHERAKAH
LDG. TUAN MEE
LDG. BKT. IJOK
KG. SG. TUA
KEPONG (SEMAIAN)
IBU BEKALAN KM. 16
EMPANGAN GENTING KLANG
IBU BEKALAN KM. 11
STN. JENALETRIK LLN.
LDG. BKT. BELIMBING
JLN. KELANG
LDG. BKT. TALANG
LDG. KUALA SELANGOR
LDG. SG. BULOH
RMH PAM JPS JAYA SETIA
LDG. SG. GAPI
AIR TERJUN SG BATU
GENTING SEMPAH
PARIT 1 SG. BURONG
IBU BEKALAN SG. TENGKI
LDG. RAJA MUSA
LDG. SG. TINGGI
LDG. HOPEFUL
FDC. SEKICHAN
PARIT 1 SG. BESAR
SG. NIPAH
LDG. SG. GUMUT
RMH PAM JPS BGN TERAP
PARIT 6 SG. BESAR
PARIT SALIRAN SG. AIR TAWAR
LDG SG. BERNAM
Normal
0.987
0.958
0.953
0.988
0.970
0.952
0.847
0.725
0.972
0.970
0.951
0.915
0.940
0.584
0.978
0.639
0.974
0.876
0.983
0.973
0.979
0.945
0.905
0.968
0.689
0.970
0.911
0.921
0.923
0.974
0.863
0.905
0.912
0.884
0.893
0.851
0.963
0.952
0.721
0.937
0.923
0.554
0.984
0.955
0.891
0.875
0.953
0.943
0.959
0.984
0.748
0.983
0.954
0.926
0.952
EV
0.977
0.980
0.979
0.979
0.980
0.972
0.914
0.821
0.990
0.993
0.966
0.969
0.949
0.698
0.981
0.764
0.980
0.953
0.983
0.971
0.977
0.947
0.967
0.986
0.779
0.988
0.966
0.966
0.885
0.975
0.926
0.915
0.964
0.956
0.958
0.927
0.981
0.978
0.819
0.981
0.967
0.671
0.983
0.983
0.959
0.924
0.980
0.988
0.967
0.952
0.728
0.981
0.977
0.976
0.989
GEV
0.988
0.972
0.948
0.989
0.972
0.940
0.765
0.841
0.990
0.990
0.954
0.954
0.917
0.847
0.970
0.896
0.981
0.993
0.981
0.971
0.975
0.855
0.921
0.979
0.751
0.988
0.937
0.920
0.838
0.978
0.868
0.909
0.968
0.968
0.971
0.962
0.968
0.970
0.963
0.974
0.952
0.808
0.982
0.979
0.982
0.851
0.951
0.994
0.905
0.966
0.735
0.977
0.961
0.984
0.991
LOG
0.981
0.959
0.948
0.985
0.971
0.959
0.848
0.744
0.977
0.970
0.965
0.921
0.956
0.612
0.974
0.661
0.972
0.887
0.983
0.965
0.981
0.951
0.912
0.964
0.712
0.971
0.914
0.920
0.920
0.981
0.875
0.905
0.919
0.892
0.904
0.861
0.965
0.957
0.744
0.938
0.930
0.581
0.985
0.951
0.898
0.893
0.948
0.946
0.958
0.984
0.747
0.983
0.962
0.931
0.953
GLO
0.979
0.976
0.935
0.986
0.976
0.950
0.769
0.867
0.989
0.990
0.972
0.946
0.939
0.865
0.958
0.901
0.976
0.991
0.974
0.960
0.980
0.849
0.916
0.970
0.788
0.988
0.929
0.910
0.840
0.984
0.890
0.908
0.962
0.962
0.981
0.965
0.958
0.963
0.966
0.966
0.962
0.830
0.986
0.971
0.983
0.878
0.938
0.993
0.893
0.959
0.735
0.969
0.974
0.980
0.989
GPA
0.988
0.953
0.973
0.978
0.951
0.909
0.779
0.777
0.974
0.973
0.904
0.971
0.866
0.786
0.981
0.898
0.978
0.991
0.979
0.983
0.949
0.870
0.941
0.989
0.676
0.971
0.960
0.948
0.856
0.948
0.823
0.910
0.971
0.981
0.936
0.940
0.975
0.971
0.942
0.985
0.921
0.742
0.963
0.986
0.968
0.794
0.976
0.983
0.929
0.960
0.735
0.978
0.924
0.982
0.984
94
5.4.3 Results for TL-Moment with t = 1
Table 5.22 provide the correlation, r, obtained for all the 55 stations considered
in the case of TL-moment with t = 1 which implied that the TL-moment was
symmetrically trimmed for one conceptual sample value. Table 5.23 and Table 5.24
gave the sum of each ranking of the six distributions for all the 55 and 39 stations
respectively.
Table 5.23: Ranks of correlation, r, for each distribution with 55 stations (TL-moment
with t = 1)
Distribution
Normal
EV
GEV
LOG
GLO
GPA
1
5
22
9
6
13
9
Number of times a distribution had the ranking
2
3
4
5
3
11
1
14
10
7
1
12
10
18
7
9
7
4
12
23
10
6
12
4
11
7
12
2
6
21
3
2
3
10
14
Table 5.24: Ranks of correlation, r, for each distribution with 39 stations excluding the
16 stations (TL-moment with t = 1)
Distribution
Normal
EV
GEV
LOG
GLO
GPA
1
3
12
7
5
10
8
Number of times a distribution had the ranking
2
3
4
5
3
6
0
10
8
6
10
1
8
13
6
5
2
4
8
18
8
4
8
2
7
4
7
2
6
17
2
0
2
7
11
95
5.4.4 Discussions on Correlation, r, for TL-Moment with t = 1
The extreme value type I (EV) distribution was rank first the most for both Table
5.23 and Table 5.24. As for the second rank, the generalized Pareto (GPA) distribution
was the most frequent for the calculations involving all the 55 stations in Selangor and
Kuala Lumpur. This was followed closely by the extreme value type I (EV), generalized
extreme value (GEV) and generalized logistic (GLO) distributions with a difference of
only one value from the generalized Pareto (GPA) distribution. However, this situation
was switched in the calculations of only the 39 stations. The three distributions (EV,
GEV and GLO distribution) were tied for the most to rank second followed by the
generalized Pareto (GPA) distribution with just one value less. The third rank was solely
monopolized by the generalized extreme value (GEV) distribution. The extreme value
type I (EV) distribution was the most frequent to be ranked fourth for the 39 stations’
computations but it was tied with the logistic (LOG), generalized logistic (GLO) and
generalized Pareto (GPA) distributions for the 55 stations’ computation. According to
both the tables, the logistic distribution mostly ranked fifth compared to the other
distributions and the normal distribution mostly ranked last.
5.3.5 Results for TL-Moment with t = 2
The correlation, r, for the 55 stations in Selangor and Kuala Lumpur in the case
of TL-moment symmetrically trimmed for two conceptual sample values (t = 2) was
shown in Table 5.25. Meanwhile, Table 5.26 and Table 5.27 presented the total number
of rankings for each distribution for all the 55 and 39 stations respectively.
96
Table 5.25:
Correlation, r, for stations in Selangor and Kuala Lumpur (TL-moment
method with t = 2)
NAME OF STATION
LDG. BATU UNTONG
LDG. TELOK MERBAU
LDG. SEPANG
LDG. BUTE
PEJABAT JPS. SG. MANGG
LDG. BROOKLANDS
SMK. BDR TASIK KESUMA
P.KWLN P.S TELOK GONG
LDG. WEST
JPS. PULAU LUMUT
LDG. BKT. CHEEDING
PEJABAT JPS. KLANG
LDG. DOMINION
LDG. BUKIT KERAYONG
LDG. SG. KAPAR
LDG. NORTH HUMMOCK
LDG. HARPENDEN
LDG. ELMINA
SG. BULOH
LDG. EDINBURGH SITE 2
JPS AMPANG
PEMASOKAN AMPANG
SEK.KEB.KG.LUI
LDG. BRAUNSTON
LDG. BKT. CHERAKAH
LDG. TUAN MEE
LDG. BKT. IJOK
KG. SG. TUA
KEPONG (SEMAIAN)
IBU BEKALAN KM. 16
EMPANGAN GENTING KLANG
IBU BEKALAN KM. 11
STN. JENALETRIK LLN.
LDG. BKT. BELIMBING
JLN. KELANG
LDG. BKT. TALANG
LDG. KUALA SELANGOR
LDG. SG. BULOH
RMH PAM JPS JAYA SETIA
LDG. SG. GAPI
AIR TERJUN SG BATU
GENTING SEMPAH
PARIT 1 SG. BURONG
IBU BEKALAN SG. TENGKI
LDG. RAJA MUSA
LDG. SG. TINGGI
LDG. HOPEFUL
FDC. SEKICHAN
PARIT 1 SG. BESAR
SG. NIPAH
LDG. SG. GUMUT
RMH PAM JPS BGN TERAP
PARIT 6 SG. BESAR
PARIT SALIRAN SG. AIR TAWAR
LDG SG. BERNAM
Normal
0.987
0.958
0.953
0.988
0.970
0.952
0.847
0.725
0.972
0.970
0.951
0.915
0.940
0.584
0.978
0.639
0.974
0.876
0.983
0.973
0.979
0.945
0.905
0.968
0.689
0.970
0.911
0.921
0.923
0.974
0.863
0.905
0.912
0.884
0.893
0.851
0.963
0.952
0.721
0.937
0.923
0.554
0.984
0.955
0.891
0.875
0.953
0.943
0.959
0.984
0.748
0.983
0.954
0.926
0.952
EV
0.977
0.980
0.979
0.979
0.980
0.972
0.914
0.821
0.990
0.993
0.966
0.969
0.949
0.698
0.981
0.764
0.980
0.953
0.983
0.971
0.977
0.947
0.967
0.986
0.779
0.988
0.966
0.966
0.885
0.975
0.926
0.915
0.964
0.956
0.958
0.927
0.981
0.978
0.819
0.981
0.967
0.671
0.983
0.983
0.959
0.924
0.980
0.988
0.967
0.952
0.728
0.981
0.977
0.976
0.989
GEV
0.987
0.979
0.902
0.988
0.975
0.946
0.677
0.825
0.990
0.983
0.955
0.947
0.906
0.852
0.931
0.935
0.980
0.989
0.967
0.965
0.941
0.781
0.860
0.979
0.789
0.983
0.945
0.884
0.921
0.977
0.839
0.872
0.961
0.974
0.943
0.935
0.971
0.978
0.977
0.957
0.922
0.804
0.976
0.978
0.978
0.853
0.939
0.991
0.757
0.932
0.537
0.970
0.966
0.984
0.991
LOG
0.981
0.959
0.948
0.985
0.971
0.959
0.848
0.744
0.977
0.970
0.965
0.921
0.956
0.612
0.974
0.661
0.972
0.887
0.983
0.965
0.981
0.951
0.912
0.964
0.712
0.971
0.914
0.920
0.920
0.981
0.875
0.905
0.919
0.892
0.904
0.861
0.965
0.957
0.744
0.938
0.930
0.581
0.985
0.951
0.898
0.893
0.948
0.946
0.958
0.984
0.747
0.983
0.962
0.931
0.953
GLO
0.976
0.981
0.888
0.986
0.980
0.958
0.691
0.858
0.988
0.986
0.973
0.937
0.925
0.873
0.912
0.934
0.972
0.986
0.954
0.949
0.937
0.780
0.862
0.966
0.828
0.986
0.934
0.874
0.917
0.985
0.860
0.870
0.952
0.967
0.962
0.949
0.959
0.974
0.977
0.945
0.935
0.832
0.979
0.968
0.982
0.881
0.922
0.993
0.757
0.915
0.537
0.957
0.979
0.979
0.989
GPA
0.988
0.962
0.941
0.978
0.954
0.915
0.682
0.757
0.973
0.964
0.905
0.970
0.860
0.781
0.961
0.939
0.978
0.992
0.975
0.981
0.926
0.802
0.884
0.989
0.710
0.962
0.970
0.923
0.922
0.945
0.803
0.878
0.971
0.984
0.899
0.895
0.976
0.964
0.966
0.979
0.890
0.724
0.958
0.986
0.955
0.798
0.972
0.973
0.788
0.943
0.537
0.976
0.929
0.980
0.980
97
Table 5.26: Ranks of correlation, r, for each distribution with 55 stations (TL-moment
with t = 2)
Distribution
Normal
EV
GEV
LOG
GLO
GPA
1
6
21
7
7
14
10
Number of times a distribution had the ranking
2
3
4
5
7
8
6
7
12
11
8
2
11
15
4
16
8
5
14
19
4
9
10
6
6
4
4
17
6
21
1
2
2
12
14
Table 5.27: Ranks of correlation, r, for each distribution with 39 stations excluding the
16 stations (TL-moment with t = 2)
Distribution
Normal
EV
GEV
LOG
GLO
GPA
1
4
10
6
6
13
9
Number of times a distribution had the ranking
2
3
4
5
4
5
1
6
11
10
6
2
9
12
3
9
2
2
11
16
1
6
8
3
5
2
11
3
6
19
0
0
2
8
9
5.4.6 Discussions on Correlation, r, for TL-Moment with t = 2
Table 5.26 showed that the extreme value type I (EV) distribution ranked first
most of the time compared with the other five distributions for the 55 stations’ analysis.
Meanwhile, Table 5.27 showed that the generalized logistic (GLO) distribution ranked
first most of the time instead for the 39 stations’ analysis. However, both analyses
obtained the extreme value type I (EV) distribution as the most to rank second and the
generalized extreme value (GEV) distribution as the most to rank third. The generalized
Pareto (GPA) distribution ranked fourth the most for both calculations although it was
tied with the logistic distribution for the calculations involving only the 39 stations. In
accordance to the correlation obtained from the L-moment and TL-moment with t = 1
98
cases, the logistic distribution ranked fifth the most and the most usual to rank last was
the normal distribution.
5.5
Summary on the Case of TL-Moment with t = 0 (L-Moment)
The results obtained from using the L-moment method were quite precise. It
gave almost the same results in all the methods of goodness-of-fit test used in this study
which were the mean absolute deviation index (MADI), mean square deviation index
(MSDI) and correlation, r. It was obvious that the generalized logistic (GLO)
distribution was the best distribution to fir the whole data whether the analysis was done
on the whole 55 stations or only the 39 random and homogeneous stations. The
generalized extreme value distribution was also deemed suitable. This can be seen
clearly in the L-moment ratio diagrams (Figure 5.1 and Figure 5.2) since both the Lmoment ratios of the generalized logistic (GLO) and generalized extreme value (GEV)
distributions were the closest to the average of the sample L-moment ratios,
and
for
the 55 and 39 stations’ analyses. Meanwhile, the normal and logistic distributions were
both the least suitable and therefore each of them was not a good distribution to
represent the actual data. This was also seen in the L-moment ratio diagram where both
points of average sample L-moment ratios for the 55 and 39 stations’ analyses were the
furthest from the L-moment ratios of the logistic and normal distributions.
The average values for the sample L-moment ratios,
stations were as follows:
= 0.221236
= 0.227782
and
for all the 55
99
Meanwhile the average values for the sample L-moment ratios,
stations which were random and homogeneous:
= 0.222923
= 0.222231
Figure 5.1: L-Moment Ratio Diagram (a)
Figure 5.2: L-Moment Ratio Diagram (b)
and
for only 39
100
5.6
Summary on the Case of TL-Moment with t = 1
From all the six tables concerning the use of TL-moment method with t = 1, the
extreme value type I (EV), the generalized extreme value (GEV) and generalized logistic
(GLO) distributions were seen to be a good fit for the actual data. However, the extreme
value type I (EV) and the generalized extreme value (GEV) distributions were the ones
to monopolize the first ranks. As stated in Chapter 2 and Chapter 3, the extreme value
type I (EV) distribution is a special case of the generalized extreme value (GEV)
distribution. Hence, both distributions are almost the same. Similar to the case of using
the L-moment method, the normal and logistic distributions were deemed not able to fit
the actual data properly or as good compared to all the other distributions considered in
this study. The TL-moment ratio diagrams (Figure 5.3 and Figure 5.4) constructed
proved the results of the data analysis since the average of the sample TL-moment ratios,
and
, for both the calculations of the 55 and 39 stations were nearest to those TL-
moment ratios of the extreme value type I (EV), generalized extreme value (GEV) and
generalized logistic (GLO) distributions. Meanwhile, the furthest were those from the
normal and logistic distributions.
The average values for the sample TL-moment ratios,
and
for all the 55
stations of Selangor and Kuala Lumpur were as follows:
= 0.141019
= 0.139319
Meanwhile the average values for the sample TL-moment ratios,
and
for only 39
stations excluding the 16 stations which were not random, not homogeneous and had
small sample sizes (n less than 30):
= 0.137438
= 0.102604
101
Figure 5.3: TL-Moment Ratio Diagram with t = 1 (a)
Figure 5.4: TL-Moment Ratio Diagram with t = 1 (b)
102
5.7
Summary on the Case of TL-Moment with t = 2
The results for the TL-moment method with t = 2 were more widely spread.
However, the extreme value type I (EV), generalized extreme value (GEV) and
generalized logistic (GLO) distributions were still deduced as the distributions that were
the most able to fit the rainfalls data for stations in Selangor and Kuala Lumpur. As in
the case of TL-moment with t = 1, the extreme value type I (EV) and the generalized
extreme value (GEV) distributions were the ones to monopolize the first ranks in five
out of six tables concerning the use of the TL-moment method with t = 2 method. As
mentioned earlier, the extreme value type I (EV) distribution is a special case of the
generalized extreme value (GEV) distribution. The normal and logistic distributions
were both concluded as not suitable distributions since both always ranked fifth and last.
This was also shown in the TL-moment ratio diagrams (Figure 5.5 and Figure 5.6). The
average values for the sample TL-moment ratios,
and
, for the computations
involving the whole 55 stations and the average for the sample TL-moment ratios for the
computations with only the 39 random and homogeneous stations were nearest to the
TL-moment ratios of the extreme value type I (EV), generalized extreme value (GEV)
and generalized logistic (GLO) distributions while both the average points were furthest
from the TL-moment ratios of the normal and logistic distributions.
The average values for the sample TL-moment ratios,
and
for all the 55
stations were as follows:
= 0.122277
= 0.067937
The average values for the sample TL-moment ratios,
and
for only 39 stations
excluding the 16 stations which were nonrandom, nonhomogeneous and had small
sample sizes (n less than 30):
= 0.102206
= 0.060040
103
Figure 5.5: TL-Moment Ratio Diagram with t = 2 (a)
Figure 5.6: TL-Moment Ratio Diagram with t = 2 (b)
104
5.8
Conclusions
Overall, the extreme value type I (EV), generalized extreme value (GEV) and
generalized logistic (GLO) distributions were good distributions to represent the actual
maximum daily rainfalls of stations in Selangor and Kuala Lumpur. The L-moment
method gave a more precise result and showed that the generalized logistic (GLO)
distribution was the best distribution to fit the data independent on any goodness-of-fit
test used (mean absolute deviation index (MADI), mean square deviation index (MSDI)
and correlation, r) and in both analyses of the 55 and 39 stations. Meanwhile, the TLmoment, method with either t = 1 or t = 2, had a wider spread answer and showed that
the extreme value type I (EV) and generalized extreme value (GEV) distributions were
the most suitable distributions. Extreme value type I (EV) distribution is a special case
of the generalized extreme value (GEV) distribution. Hence, both distributions are
similar.
However, bear in mind that the TL-moment method with t t = 1 and t = 2)
had trimmed the actual data symmetrically by one and two conceptual sample values.
Thus, the results obtained from using this method did not represent the whole observed
data but only those that remained after trimming. Meanwhile, the L-moment method is a
special case of the TL-moment method with t = 0 which implies no trimming is done on
the actual data. In other words, the results from each value of t were distributions for
different sets of sample data. However, in accordance with most flood frequency
analysis, the extreme value type I (EV), generalized extreme value (GEV) and
generalized logistic (GLO) distributions were proven as good distributions to fit the
maximum daily rainfalls data.
Normal and logistic (LOG) distributions were also shown that both were not
suitable distributions to present the actual data in all the goodness-of- fit test used in this
research.
CHAPTER 6
CONCLUSIONS AND RECOMMENDATIONS
6.1
Conclusions
Flood or also called a deluge is a natural disaster that could destroy properties,
infrastructures, animals, plants and even human lives. Flooding is the most natural
hazard and the most costly disastrous phenomenon in Malaysia. It is also one of the
oldest natural hazards in the world.
Analyzing rainfalls and stream flows data are important in order to obtain the
probability distribution of flood and other phenomenon related to them. By knowing the
probability distribution, prediction of flood events and their characteristic can be
determined. With this, prevention acts and measures can be taken and flash flood
warning models can be built easily.
In order to be able to plan and design these projects such as hydraulic or water
resources projects, continuous hydrological data, for example, rainfalls data or river flow
data is necessary. With the help of the data, flow pattern or trend can be determined to
make sure the design and planning can be done accordingly. However, to select a
reliable design quantile, which affects design, operation, management and maintenance
of hydraulic structure depends on statistical methods used in parameter estimation
belonging to probability distribution (Hosking and Wallis, 1993).
106
Extreme events are usually too short and too rare for a reliable estimation to be
obtained and this creates difficulties in identifying the appropriate statistical distribution
to describe the data and estimating the parameters of the selected distribution. Hence,
regional frequency analysis which was developed by Hosking and Wallis (1991) is used
since it can resolve this problem by trading space for time.
Recently, the most popularized method in frequency analysis is the L-moment
approach introduced by Hosking in 1990 (Rao et al., 2000). The main role of the Lmoments is for estimating parameters for probability distributions. Probability
distributions are used to analyze data in many disciplines and are often complicated by
certain characteristics such as large range, variation or skewness. Hence, outliers or
highly influential values are common (Asquith, 2007). TL-moments are derived by
Elamir and Seheult in 2003 from L-moments and might have additional robust
properties compared to L-moments. In other words, TL-moments are claimed to be more
robust than the L-moment. Hence, TL-moments are also considered for estimating the
parameters of the selected probability distributions.
This study focused on identifying a suitable probability distribution, including
normal (N), logistic (LOG), generalized logistic (GLO), extreme value type I (EV),
generalized extreme value type I (GEV) and generalized Pareto (GPA) by using TLmoments technique for maximum daily rainfalls selected for each year among daily
rainfalls measured over the regions in Selangor and Kuala Lumpur, Malaysia. The TLmoments for all the said distributions were derived for
and
which implies
TL-moments that are symmetrically trimmed by one and two conceptual sample values
respectively in order to be able to fit the rainfall data to the probability distributions. The
results from both cases (
and
) were then compared with those obtained using
the method of L-moments similar to a previous study by Shabri and Ariff (2009).
107
The data of daily rainfalls for stations in Selangor and Kuala Lumpur was
collected and taken from “Jabatan Pengairan dan Saliran Malaysia”. The data of daily
rainfalls for 55 stations were sent by email. The data contains measurements of daily
rainfalls in millimeters from the year 1971 until 2007. The maximum rainfalls of each
month were identified followed by the maximum of each year (1971-2007). This is
done to all the 55 stations in Selangor and Kuala Lumpur.
The maximum data of daily rainfalls for each year were then analyzed for all the
55 stations using MathCAD program. A MathCAD program was created to find the Lmoments, L-moment ratios, TL-moment, TL-moment ratios with
and
and
parameter estimations using both L-moment and TL-moment for six probability
statistical distributions which were the normal (N), logistic (LOG), generalized logistic
(GLO), extreme value type I (EV), generalized extreme value (GEV) and generalized
Pareto (GPA) distribution.
Three MathCAD programs were built and constructed for each 55 stations. One
for t = 0, t = 1 and t = 2 respectively. The case of t = 0 are actually the L-moment
method. Meanwhile, t = 1 referred to TL-moment which was symmetrically trimmed for
one conceptual sample value and t = 2 referred to TL-moment which was symmetrically
trimmed for two conceptual sample values. Then, their distributions for each case were
compared using mean absolute deviation index (MADI), mean square deviation index
(MSDI) and their correlation, r. For better view, the ratio diagrams were constructed for
each case.
Each MADI, MSDI and correlation, r, for all the 55 stations were calculated for
all the distributions which includes normal (N), logistic (LOG), generalized logistic
(GLO), extreme value type I (EV), generalized extreme value type I (GEV) and
generalized Pareto (GPA). Then, the distributions were ranked according to their MADI,
MSDI and correlation, r, from the best distribution that fits the data to the least. The
108
number of times each distribution obtains a given rank were then calculated and
tabulated.
The ranking process was repeated for 39 stations excluding 16 stations that are
either nonrandom, nonhomogeneous or those that have their n values less than 30 (their
randomness cannot be tested).
The L-moment method gave a more precise result. From the use of mean
absolute deviation index (MADI), mean square deviation index (MSDI) and correlation,
r, the results showed that the generalized logistic (GLO) distribution was the best
distribution to fit the data. Meanwhile, the TL-moment, method with either t = 1 or t = 2,
were more widely spread and the results obtained were that the extreme value type I
(EV) and generalized extreme value (GEV) distributions were the most suitable
distributions. Since extreme value type I (EV) distribution is a special case of the
generalized extreme value (GEV) distribution, both distributions are regarded as similar.
TL-moment method with t  t = 1 and t = 2) had trimmed the actual data
symmetrically by one and two conceptual sample values and hence the results did not
represent the whole observed data. Meanwhile, the L-moment method is a special case
of the TL-moment method with t = 0 which implies no trimming is done on the actual
data. Therefore, the results from each value of t were distributions for different sets of
sample data. All in all, the extreme value type I (EV), generalized extreme value (GEV)
and generalized logistic (GLO) distributions were good distributions to represent the
actual maximum daily rainfalls of stations in Selangor and Kuala Lumpur.
On the other hand, normal and logistic (LOG) distributions were shown to be not
suitable to present the actual data in all the goodness-of- fit test used in this study. The
results were proven visually through the use of ratio diagrams.
109
6.2
Recommendations
A few recommendations and suggestions are given for future research and
developments of this study. The following is a list of some suggestions to bear in mind
for future researches:
1) Research on regional frequency analysis of maximum daily rainfalls using TLmoment approach over regions in the whole Malaysia.
2) Study on the TL-moment approach of regional frequency analysis for other
extreme events such as draught and compare the results to the ones obtained
from L-moment.
3) A research of flood frequency analysis by considering other distributions such as
Gamma distribution using the TL-moment method and derive the TL-moment
for those distributions.
4) Compare the TL-moment method with other estimation methods such as the
maximum likelihood method or the method of moments.
REFERENCES
Abdul-Moniem, I.B. (2007). “L-moments and TL-moments estimation for the
exponential distribution”, Far East J. Theo. Stat., 23 (1) (2007), pp. 51-61.
Abdul-Moniem, I.B. (2009). “TL-moments and L-moments estimation for the
generalized Pareto distribution”, Applied Mathematical Sciences, Vol. 3 (2009),
no. 1, pp. 43-52.
Abramowitz, M., and Stegun, I.A. (1965). Handbook of Mathematical Functions.
Dover Publications, New York.
Adamowski, K., and Feluch, W. (1990). “Non Parametric Flood-Frequency Analysis
with Historical Information”, Journal of Hydraulic Engineering, ASCE, pp. 10351047.
Ahmad, M.I., Sinclair, C.I., and Werritty, A. (1988). “Log-Logistic Flood Frequency
Analysis”, Journal of Hydraulics, Vol. 98, pp. 205-224.
Ashkar, F., and Mahdi, S. (2003). “Comparisan of two fitting methods for the loglogistic distribution”, Water Resources Research, 39 (8), pp. SWC71-SWC78.
Asquith, W. H. (2007). “L-moments and TL-moments of the generalized lambda
distribution”, Computational Statistics & Data Analysis, 51 (2007), pp. 4484-4496.
111
Atiem, I. A., and Harmancioglu, N.B. (2006). “Assessment of regional floods using Lmoments approach: The case of the River Nile”, Water Resources Management, 20
(5), pp. 723-747.
Bayazit, M., and Onoz, B. (2002). “LL-moments for estimating low flow quantiles”,
Hydrological Sciences, 47, pp. 707-720.
Bobée, B., Caradias, G., Ashkar, F., Bernier, J., and Rasmussen, P. (1993). “Towards a
Systematic Approach to Comparing Distributions used in Flood Frequency
Analysis”, Journal of Hydraulics, Vol. 142, pp. 121-136.
Campbell, A.J., and Sidel, R.C. (1984). “Prediction of Peak Flows on Small Watersheds
in Oregon for Use in Culvert Design”, Water Resources Bulletin , Vol. 20, No. 1,
pp. 9-14.
Chebana, F., and Ouarda, T.B.M.J. (2007). “Multivariate L-moment homogeneity test”,
Water Resources Research, 43 (8), art. No. W08406.
Chen, Y.D., Huang, G., Shao, Q., and Xu, C.Y. (2006). “Regional analysis of low flow
using L-moments for Dongjiang basin, South China.” Hydrological Sciences–
Journal–des Sciences Hydrologiques, 51(6).
Chow, V.T., Maidment, D.R. and Mays, L.W. (1988). Applied Hydrology, McGrawHill, New York, NY.
Crippen, J.R. (1982). “Envelope Curves for Extreme Flood Events”, Journal
of the Hydraulic Division, ASCE, pp. 1208-1212.
112
Cunderlik, J.M., and Burn, D.H. (2003). “Non-stationary pooled flood frequency
analysis”, Journal of Hydrology, 276 (1-4), pp. 210-223.
Cunnane, C. (1989). “Statistical Distributions for Flood Frequency Analysis”. Word
Meteorological Organization Operational Hydrology, Report No. 33, WMO-No.
718, Geneva, Switzerland.
Daviau, J. L., Adamowski, K., and Patry, G.G. (2000). “Regional flood frequency
analysis using GIS, L-moment and geostatistical methods”, Hydrological
Processes, 14 (15), pp. 2731-2753.
Elamir, E.A., and Seheult, A.H. (2003). “Trimmed L-moments”, Computational
Statistics & Data Analysis, 43 (2003), pp. 299-314.
Ellouze, M., and Abida, H. (2008). “Regional flood frequency analysis in Tunisia:
Identification of regional distributions”, Water Resources Management, 22 (8), pp.
943-957.
Escalante-Sandoval, C. (1998). “Multivariate extreme value distribution with mixed
Gumbel marginals”, Journal of the American Water Resources Association, 34 (2),
pp. 321-333.
Eslamian, S.S., and Feizi, H. (2007). “Maximum monthly rainfall analysis using Lmoments for an arid region in Isfahan Province, Iran”, Journal of Applied
Meteorology and Climatology, 46 (4), pp. 494-503.
Faulkner, D.S., and Robson, A.J. (1999). “Estimating floods in permeable drainage
basins”, IAHS-AISH Publication, (255), pp. 245-250.
113
Fowler, H.J., and Kilsby, C.J. (2003). “A regional frequency analysis of United
Kingdom extreme rainfall from 1961 to 2000”, International Journal of
Climatology, 23 (11), pp. 1313-1334.
Gaal, L., Kysely, J., and Szolgay, J. (2007). “Region-of-influence approach as to a
frequency analysis of heavy precipitation in Slovakia”, Hydrology and Earth
System Sciences Discussions, 4 (4), pp. 2361-2401.
Gaal, L., Szolgay, J., and Lapin, M. (2008). “Regional frequency analysis of heavy
precipitation totals in the High Tatras region in Slovakia for flood risk estimation”,
Contributions to geophysics and Geodesy, 38 (3), pp. 327-355.
Gingras, D., and Adamowski, K. (1992). “Coupling of Nonparametric Frequency and LMoment Analyses for Mixed Distribution Identification”, Water Resources
Bulletin, Vol. 28, No. 2, pp. 263-372.
Glaves, R., and Waylen, P.R. (1997). “Regional flood frequency analysis in Southern
Ontario using L-moments”, Canadian Geographer, 41 (2), pp. 178-193.
Gottschalk, L., and Krasovskaia, I. (2002). “L-moment estimation using annual
maximum (AM) and peak over threshold (POT) series in regional analysis of flood
frequencies”, Norsk Geografisk Tidsskrift, 56 (2), pp. 179-187.
Greenwood, J.A., Landwehr, J.M., Matalas, N.C., and Wallis, J.R. (1979). “Probability
Weighted Moments: Definition and Relation to Parameters of Several
Distributions Expressible in Inverse Form”, Water Resources Research, Vol. 15,
No. 5, pp. 1049-1054.
114
Haktanir, J. (1992). “Comparison of Various Flood Frequency Distributions using
Annual Flood Peaks Data of Rivers in Anatolia”, Journal of Hydraulics, Vol.
136, pp. 1-31.
Hosking, J.R.M. (1986). “The Theory of Probability Weighted Moments”, Res. Rep. RC
12210, IBM Research Division, Yorktown Heights, NY. 10598.
Hosking, J.R.M. (1990). “L-Moments: Analysis and Estimation of Distributions using
Linear Combinations of Order Statistics”, Journal of Royal Statistical Society B,
Vol 52, pp. 105-124.
Hosking, J.R.M., and Wallis, J.R. (1991). “Some Statistics Useful in Regional
Frequency Analysis”, Res. Rep. RC 17096, IBM Research Division, Yorktown
Heights, NY 10598.
Hosking, J.R.M., and Wallis, J.R. (1993). “Some Statistics Useful in Regional
Frequency Analysis”, Water Resources Research, Vol. 29, No. 2, pp. 271-281.
Hussain, Z., and Pasha, G.R. (2009). “Regional flood frequency analysis of the seven
sites of Punjab, Pakistan, using L-moments”, Water Resources Management, 23
(10), pp. 1917-1933.
Jain, D., and Singh, V.P. (1986). “A comparison of Transformation Methods for Flood
Frequency Analysis”, Water Resources Bulletin, Vol.22, No.6, pp. 903-912.
Jingyi, Z., and Hall, M.J. (2004). “Regional flood frequency analysis for the Gan-Ming
River basin in China”, Journal of Hydrology, 296 (1-4), pp. 98-117.
115
Johnson, N. L., and Kotz, S. (1970). “Distributions in Statistics: Continuous Univariate
Distributions”, Wiley Series in Probability and Mathematical Statistics.
Kjeldsen, T.R., Smithers, J.C. and Schulze, R.E. (2002). “Regional flood frequency
analysis in the kwazulu-natal province, South Africa, using the index-flood
method”, Journal of Hydrology, 255 (1-4), pp. 194-211.
Kjeldsen, T.R. and Jones, D.A. (2004). “Sampling variance of flood quantiles from the
generalized logistic distribution estimated using the method of L-moments”,
Hydrology and Earth System Sciences, 8 (2), pp. 183-190.
Kuczera, G. (1982). “Robust Flood Frequency Models”, Water Resources Research,
Vol. 18, No. 2, pp. 315-324.
Kumar, R., Chatterjee, C., Kumar, S., and Upadhyay, P. (2002). “Development of
regional flood formulae using L-moments for gauged and ungauged catchments
of South Bihar and Jharkhand”, Water and Energy International, 59 (2), pp. 52–
59.
Kumar, R., Chatterjee, C., Panigrihy, N., Patwary, B.C., and Singh, R.D. (2003a).
“Development of regional flood formulae using L-moments for gauged and
ungauged catchments of North Brahmaputra river system”, Journal of the
Institution of Engineers (India): Civil Engineering Division, 84 (1), pp. 57–63.
Kumar, R., Chatterjee, C., and Kumar, S. (2003b). “Regional flood formulas using Lmoments for small watersheds of sone subzone of India”, Applied Engineering in
Agriculture, 19 (1), pp. 47–53.
116
Kysely, J., and Picek, J. (2007a). “Regional growth curves and improved design value
estimates of extreme precipitation events in the Czech Republic”, Climate
Research, 33 (3), pp. 243–255.
Kysely, J., and Picek, J. (2007b). “Probability estimates of heavy precipitation events in
a flood-prone central-European region with enhanced influence of Mediterranean
cyclones”, Advances in Geosciences, 12, pp. 43–50.
Landwehr, J.M., Matalas, N.C., and Wallis, J.R. (1978). “Some Comparisons of Flood
Statistics in Real and Log Space”, Water Resources Research, Vol. 14, No. 5, pp.
902-920.
Lee, S.H., and Maeng, S.J. (2003). “Comparison and analysis of design floods by the
change in the order of LH-moment methods”, Irrigation and Drainage, 52 (3),
pp. 231–245.
Lim, Y.H. and Lye, L.M. (2003). “Regional flood estimation for ungauged basins in
Sarawak, Malaysia”, Hydrological Sciences Journal, 48(1), pp. 79–94.
Macdonald, N., Werritty, A., Black, A.R., and McEwen, L.J. (2006). “Historical and
pooled flood frequency analysis for the River Tay at Perth, Scotland”, Area, 38
(1), pp. 34–46.
Meshgi, A., and Khalili, D. (2009). “Comprehensive evaluation of regional flood
frequency analysis by L- and LH-moments”, Stochastic Environmental Research
and Risk Assessment, 23 (1), pp. 137–152.
117
Mkhandi, S.H., Kachroo, R.K., and Gunasekara, T.A.G. (2000). “Flood frequency
analysis of
southern
Africa:
Identification
of
regional
distributions”,
Hydrological Sciences Journal, 45 (3), pp. 449–464.
Norbiato, D., Borga, M., Sangati, M., and Zanon, F. (2007). “Regional frequency
analysis of extreme precipitation in the eastern Italian Alps and the August 29,
2003 flash flood”, Journal of Hydrology, 345 (3-4), pp. 149–166.
Noto, L.V., and La Loggia, G. (2008). “Use of L-moments approach for regional
frequency analysis in Sicily, Italy”, Water Resources Management, pp. 1–23.
Onoz, B., and Baryazit, M. (1995). “Best-fit Distributions of Largest Available Flood
Samples”, Journal of Hydrology, Vol. 167, pp. 195–208.
Parida, B.P., Kachroo, R.K., and Shrestha, D.B. (1998). “Regional flood frequency
analysis of Mahi-Sabarmati Basin (Subzone 3-a) using index flood procedure
with L-moments”, Water Resources Management, 12, pp. 1–12.
Pearson, C.P. (1991). “Regional flood frequency analysis for small New Zealand: Flood
frequency groups”, Journal of Hydrology (New Zealand), 30 (2), pp. 77–92.
Rahnama, M.B., and Rostami, R. (2007). “Halil-River basin regional flood frequency
analysis based on L-moment approach”, International Journal of Agricultural
Research, 2 (3), pp. 261–267.
Rao, A.R., and Hamed, K.H. (1997). “Regional frequency analysis of Wabash River
flood data by L-moments”, Journal of Hydrologic Engineering, 2 (4), pp. 169–
179.
118
Rao, A.R., and Hamed, K.H. (2000). “Flood Frequency Analysis”, CRC Press, Boca
Raton, London, New York, Washigton, D.C.
Ribatet, M., Sauquet, E., Gresillon, J.M., and Ouarda, T.B.M.J. (2007). “A regional
Bayesian POT model for flood frequency analysis”, Stochastic Environmental
Research and Risk Assessment, 21 (4), pp. 327–339.
Rulli, M.C., and Rosso, R. (2002). “An integrated simulation method for flash-flood risk
assessment: Frequency predictions in the Bisagno River by combining stochastic
and deterministic methods”, Hydrology and Earth System Sciences, 6 (2), pp.
267–283.
Saf, B., Dikbas, F., and Yasar, M. (2007). “Determination of regional frequency
distributions of floods in West Mediterranean river basins in Turkey”, Fresenius
Environmental Bulletin, 16 (10), pp. 1300–1308.
Saf, B. (2009a). “Regional flood frequency analysis using L-moments for the West
Mediterranean region of Turkey”, Water Resources Management, 23 (3), pp.
531–551.
Saf, B. (2009b). “Regional flood frequency analysis using L-moments for the buyuk and
kucuk menderes river basins of Turkey”, Journal of Hydrologic Engineering, 14
(8), pp. 783–794.
Sandoval, C.E., and Raynal-Villasenor, J. (2008). “Trivariate generalized extreme value
distribution in flood frequency analysis”, Hydrological Sciences Journal, 53 (3),
pp. 550–567.
119
Sankarasubramaniam, A., and Srinivasan, K. (1999). “Investigation and comparison of
sampling properties of L-moments and conventional moments”, Journal of
Hydrology, 218, pp. 13–34.
Shabri, A., and Ariff, N.M. (2009). “Frequency analysis of maximum daily rainfalls via
L-moment approach”, Sains Malaysiana, 38 (2), pp. 149–158.
Shao, Q., Wong, H., Xia, J., and Ip, W.C. (2004). “Models for extremes using the
extended three-parameter Burr XII system with application to flood frequency
analysis”, Hydrological Sciences Journal, 49 (4), pp. 685–701.
Srinivas, V.V., Tripathi, S., Rao, A.R., and Govindaraju, R.S. (2008). “Regional flood
frequency analysis by combining self-organizing feature map and fuzzy
clustering”, Journal of Hydrology, 348 (1-2), pp. 148–166.
Stedinger, J.R., Vogel, R.M., and Foufoula-Georgiou, E. (1993). Frequency Analysis of
Extreme Events, in “Handbook of Hydrology”, ed. D.R. Maidment, McGrawHill, New York, NY, pp. 18.1–18.66.
Trefry, C.M., Watkins Jr., D.W., and Johnson, D. (2005). “Regional rainfall frequency
analysis for the state of Michigan”, Journal of Hydrologic Engineering, 10 (6),
pp. 437–449.
Turkman, K.F. (1985). “The Choice of Extremal Models by Akaike’s Information
Criterion”, Journal of Hydraulic, Vol. 82, pp. 307-315.
Vogel, R.M. (1986). “The Probability Plot Correlation Coefficient Test for the Normal,
Lognormal and Gumbel Distributional Hypotheses”, Water Resources Research,
vol. 22, No. 4, pp. 587-590.
120
Vogel, R.M., and McMartin, D.E. (1991). “Probability Plot Goodness of Fit and
Skewness Estimation Procedures for the Pearson Type 3 Distribution”, Water
Resources Research, Vol. 27, No. 12, pp. 3149-3158.
Vogel, R.M., and Fennessey, N.M. (1993). “L-Moment Diagrams Should Replace
Product Moment Diagrams”, Water Resources Research, Vol. 29, No. 6, pp.
1745-1752.
Vogel, R.M., Thomas, Jr., W.O., and McMahon, T.A. (1993). “Flood Flow Frequency
Model Selection in Southwestern United States”, Journal Water Research
Planning and Management., ASCE, pp. 353-366.
White, E.L. and Reich, B.M. (1970). “Behavior of Annual Floods in Limestone Basins
in Pennsylvania”, Journal of Hydraulics, 10, pp. 193-198.
Yue, S., and Wang, C. (2004). “Determination of regional probability distributions of
Canadian flood flows using L-moments”, Journal of Hydrology New Zealand, 43
(1), pp. 59-73.
APPENDIX A
MathCAD Program for L-Moment
A
0
B sort ( A)
n length ( A)
i 1 n
x B
i
i1
A
b 0
b 1
1
n
n
˜
¦
i
1
˜
n
2
n
¦
2
n
n ¦
1
˜
i
b 3
3
n
n ¦
1
˜
i
b 4
i
1
i
b x
1
˜
n
4
n
¦
i
5
ª (i 1) xº
« n 1 i»
¬
¼
ª (i 1) ˜( i 2) xº
«
»
¬ (n 1) ˜( n 2) i¼
ª ( i 1) ˜ (i 2) ˜ (i 3) xº
«
»
¬ (n 1)˜ ( n 2)˜ ( n 3) i¼
ª ( i 1)˜ ( i 2) ˜ ( i 3) ˜ (i 4) xº
»
¬ (n 1)˜ ( n 2) ˜ ( n 3) ˜ (n 4) i¼
0
83.8
1
82.5
2
190.4
3
86.4
4
129.5
5
170
6
160
7
126
8
121
9
115
10
115
11
117
12
160
13
120
14
205
15
...
122
l b
1
0
l 2˜ b b
2
1
0
l 6˜ b 6˜ b b
3
2
1
0
l 20˜ b 30˜ b 12˜ b b
4
3
2
t 3
1
l
3
l
2
l
t 4
l
F
0
i
0.015
0.042
4
2
F
i
i 0.44
n 0.12
0.069
0.096
0.123
0.15
§ 0 ·
¨
¸
132.759
¨
¸
l ¨ 20.554 ¸
¨ 1.416 ¸
¨
¸
© 1.971 ¹
§ 0 ·
¨
¸
0
¨
¸
t2 ¨ 0 ¸
¨ 0.069 ¸
¨
¸
© 0.096 ¹
0.177
0.204
0.231
0.258
0.284
0.311
0.338
i g ln ln F
i
0.365
0.392
...
123
normal
P l
1
P
1
V V
S˜l
1
2
132.759
1
1
36.431
y
y
1 i
P V ˜ qnorm F 0 1
1
1
i
1 i
53.784
69.82
78.713
250
85.208
90.468
94.968
98.954
102.573
105.915
200
xi
y 1 i
150
100
109.045
112.009
114.842
117.57
120.215
122.796
...
50
2
0
2
gi
4
6
124
ev
l
D 1
2
D
ln( 2)
[ l 0.5772D
˜
1
y
2 i
1
1
1
29.653
[
1
115.644
i [ D ˜ ln ln F
1
1
0
y
2 i
0
73.132
1
81.437
2
86.476
3
90.379
4
93.688
5
96.633
6
99.336
7
101.869
8
104.279
9
106.601
10
108.859
11
111.074
12
113.262
13
115.436
14
117.609
15
...
250
200
xi
y 2 i
150
100
50
2
0
2
gi
4
6
125
gev
2
C
3t
ln( 2)
C
ln( 3)
3
2
K 7.8590C
˜ 2.9554C
˜
K
2
a * 1 K
a
0.165
2
2
0.928
D 2
l ˜K
2
2
K2
a 12
[ l 2
y
0.021
1
3 i
y
D
K
[ 2
2
D
˜ (a 1)
[
2
D
ª
K ¬
2
2
33.796
2
118.048
K 2º
i ˜ 1 ln F
2
¼
3 i
63.404
75.116
250
81.964
200
87.136
91.436
95.197
98.595
101.733
xi
y 3 i
150
100
104.678
107.478
110.168
112.773
115.315
117.811
120.275
...
50
2
0
2
gi
4
6
126
logistic
[ l
3
D l
3
y
4 i
[
1
3
D
2
3
132.759
20.554
§ Fi ·¸
¨ 1 Fi ¸
©
¹
[ D ˜ ln ¨
3
3
y
250
4 i
46.869
200
68.497
79.264
86.645
92.355
xi
150
y 4 i
100
97.072
101.133
50
104.733
107.994
110.997
113.802
116.45
118.976
121.405
123.759
...
0
2
0
2
gi
4
6
127
glo
K t
4
K
3
4
0.069
l
D 4
* 1 K * 1K
[ l 4
2
1
4
l D
2
K
4
4
D
20.394
4
4
[
4
130.436
K 4º
ª
« §¨ 1 Fi ¸· »
y [ ˜ 1
»
5 i
4
K «
¨ Fi
4 ¬
©
¹̧ ¼
D
4
y
5 i
56.163
71.53
79.894
250
85.923
200
90.759
94.87
98.497
101.781
104.813
xi
y 5 i
150
100
107.654
110.35
112.934
115.434
117.872
120.266
...
50
2
0
2
gi
4
6
128
gpa
K 5
1 3˜ t
1t
3
K
D l ˜ 1 K ˜ 2 K
5
2
5
[ l l ˜ 2 K
5
y
6 i
1
2
[ 5
0.742
5
3
D
ª
K ¬
5
5
5
D
[
5
˜ 1 1F
i
y
5
98.196
76.396
5
K 5º
¼
6 i
83.295
85.621
250
87.969
90.339
92.732
95.148
97.59
200
xi
y 6 i
150
100.057
102.552
100
105.075
107.628
110.212
112.828
115.48
118.168
...
50
2
0
2
gi
4
6
129
j2 1 6
MADI j
MSDI j
n
1
˜
n
¦
i
1
˜
n
1
n
¦
i
1
xy
i
j i
meanx mean (x)
n
i
§¨ xi yj i ¸·
¨ xi ¸
©
¹
n
˜
¦
i
2
1
Sx ˜
n
y
j i
1
n
2
¦ xi meanx
i
1
Sy j
n
2
y meany ¦
j
i
j
n
1
˜
i
j
j
meany j
x
Sxy r 1
1
n
1
n
˜
¦ ª¬xi meanx ˜ y ji meany jº¼
i
1
Sxy
j
Sx˜Sy
j
§
·
¨
¸
0.03
¨
¸
¨ 0.032 ¸
MADI ¨ 0.018 ¸
¨
¸
¨ 0.038 ¸
¨ 0.029 ¸
¨
¸
© 0.03 ¹
0
0
§
·
¨
¸
3
¨ 2.119u 10 ¸
¨
¸
¨ 2.057u 10 3 ¸
¨
¸
4
MSDI ¨ 7.086u 10 ¸
¨
3¸
¨ 3.927u 10 ¸
¨
3¸
¨ 1.857u 10 ¸
¨
3¸
© 1.887u 10 ¹
meanx
§ 0 ·
¨
¸
¨ 0.987 ¸
¨ 0.977 ¸
¨ 0.99 ¸
r
¨
¸
¨ 0.981 ¸
¨ 0.983 ¸
¨
¸
© 0.987 ¹
stdev (x)
129.266
40.681
130
APPENDIX B
MathCAD Program for TL-Moment with t = 1
0
A
A
B sort ( A)
n length ( A)
i 1 n
x B
i
i1
0
83.8
1
82.5
2
190.4
3
86.4
4
129.5
5
170
6
160
7
126
8
121
9
115
10
115
11
117
12
160
13
120
14
205
15
...
131
n 1
1
i
n 1
¦
i
2
l 2
n 1
2
ª§ combin (i 1 2)˜ combin (n i 1) combin( i 11) ˜combin (n i 2) · ˜xº o 22.30003331062154591
¨
¸ »i
combin (n 4)
©
¹ ¼
1
˜ 22.3000333106215459
2
ª§ combin ( i 13) ˜combin ( n i1) 2 ˜combin( i 1 2) ˜combin( n i2 ) combin (i 1 1) ˜combin (n i3) · ˜x º o 2.114048731695790519
¨
¸ i»
combin( n 5)
¹ ¼
©
¦
i
ª§ combin( i 11) ˜ combin( n i 1) · ˜ xº
Ǭ
¸ i»
combin (n 3)
©
¹ ¼
¦
l 2
l 3
n1
¦ ª«¬§¨©
1
˜ 2.114048731695790519
3
combin
(i 14)˜combin
(n i1) 3combin
˜
(i 13)˜combin
(n i2) 3˜combin
(i 12)˜combin
( n i3) combin
( i 11)˜combin
(n i4) ·
i 2
l 4
1
4
˜ 2.747202320731732496
l
t 3
l
3
t 4
l
2
4
l
2
i 0.44
n 0.12
F
i
º
¸ ˜x»i o2.7472023207317
¹¼
combin
(n 6)
F
i
0.015
0.042
§ 0 ·
¨
¸
131.344
¨
¸
l ¨ 11.15 ¸
¨ 0.705 ¸
¨
¸
© 0.687 ¹
§ 0 ·
¨
¸
0
¨
¸
t ¨ 0 ¸
¨ 0.063 ¸
¨
¸
© 0.062 ¹
0.069
0.096
0.123
0.15
0.177
0.204
0.231
i g ln ln F
i
0.258
0.284
0.311
0.338
0.365
0.392
...
132
normal
P l
1
P
1
V 3.373˜ l
1
y
1 i
2
131.344
1
V
37.609
1
i P V ˜ qnorm F 01
1
1
0
y
1 i
0
49.814
1
66.369
2
75.549
3
82.255
4
87.684
5
92.33
6
96.445
7
100.181
8
103.631
9
106.862
10
109.922
11
112.847
12
115.663
13
118.394
14
121.058
15
...
250
200
xi
150
y 1 i
100
50
0
2
0
2
gi
4
6
133
l
ev
D 1
2
ln §¨
729 ·
¸
© 512 ¹
D
31.555
1
[ l 3˜ D ˜ ln( 2) 2˜ D ˜ ln( 3) J ˜ D
1
y
2 i
1
1
1
[
1
i [ D ˜ ln ln F
1
1
250
200
0
y
2 i
0
71.607
xi
1
80.445
y 2 i
2
85.807
3
89.96
4
93.482
5
96.616
6
99.492
7
102.187
8
104.752
9
107.223
10
109.626
11
111.983
12
114.311
13
116.625
14
118.937
15
...
150
100
50
2
0
2
gi
4
6
1
116.846
134
gev
2 0.403498762˜ t 3 3 0.707333631˜ t 3 4 1.728715237˜ t 3 5 4.076511188˜ t3 6 2.525801801˜ t 3 7 5.225208913˜ t 3 8 1.910928577˜ t 3 9 2.856823577˜ t3 10
K 2 0.291922291 2.89036313t
˜ 3 1.291839815˜ t 3
D 2
l
2
ª1
6˜ * K ˜ « ˜§¨
2 2
¬ ©
D
§
[ l 3˜¨
2
1 K
©
2
y
3 i
2
[ 2
D
ª
K ¬
2
1·
¸
4¹
K2
1
§¨ ·¸
© 3¹
K2
1 1
˜ §¨ ·¸
2 © 2¹
K2º
K
0.114
D
33.754
[
117.416
2
»
¼
2
K2
§1·
¸ ˜ D2 ˜ *K2 2˜ ¨ ¸ ˜ D2˜ * K2 2¹
©3¹
1·
i ˜ 1 ln F
2
K2
2
K 2º
¼
0
y
3 i
0
64.834
1
75.795
2
82.276
3
87.21
4
91.336
5
94.964
6
98.257
7
101.312
8
104.192
9
106.941
xi
10
109.592
11
112.169
y 3 i
12
114.693
13
117.18
14
119.645
15
...
250
200
150
100
50
2
0
2
gi
4
6
135
logistic
[ l
3
[
1
D 2˜ l
3
y
4 i
2
3
D
3
131.344
22.3
§ Fi ·¸
¨ 1 Fi ¸
©
¹
[ D ˜ ln¨
3
3
y
4 i
38.157
61.622
73.303
250
81.312
87.507
200
92.624
97.03
100.936
104.474
xi
y 4 i
150
100
107.733
110.776
50
113.649
116.389
119.024
121.578
...
0
2
0
2
gi
4
6
136
glo
9˜ t
K 4
3
K
5
D 4
2˜ l ˜ sin S ˜ K
2
4
D
4 2 1º¼
S˜K ˜ª K
4
4¬
[ l 4
1
4 2º¼
sin S ˜ K 4
S ˜ D ˜ ª1 K
4¬
ª
D
22.115
4
K
§ 1 Fi ·
4
¸
y [ ˜ 1¨
5 i
4
K
¨ F ¸
4 ¬
© i ¹
D
0.114
4
[
4
129.713
4
K4º
»
»
¼
y
5 i
56.163
71.53
250
79.894
85.923
200
90.759
94.87
98.497
101.781
104.813
xi
y 5 i
150
100
107.654
110.35
112.934
115.434
117.872
120.266
...
50
2
0
2
gi
4
6
137
gpa
5˜ 9˜ t 2
K 5
3
9˜ t 10
K
3
D 5
0.677
5
ª¬l 2˜ K5 2˜ K5 3˜ K5 4º¼
6
D
85.559
[
82.001
5
[ l 5
y
6 i
1
[ 5
3˜ D
5
2˜ D
5
K5 2 K5 3
D
ª
K ¬
5
5
˜ 1 1F
K5º
i
y
5
¼
6 i
83.295
85.621
250
87.969
90.339
92.732
95.148
97.59
100.057
102.552
200
xi
y 6 i
150
100
105.075
107.628
110.212
112.828
115.48
118.168
...
50
2
0
2
gi
4
6
138
j 1 6
MADI j
xy
n
1
˜
n
¦
i
1
˜
MSDI j
n
¦
1
j i
meanx mean (x)
j
§¨ xi yj i ¸·
¨ xi ¸
©
¹
2
Sx 1
˜
n
n
¦ i
x meanx
i
j
2
Sy 1
1
n
j
1
˜
n
n
˜
¦
i
y
j i
1
n
2
¦ yj i meanyj
i
1
n
˜
¦ ª¬ xi meanx˜ yj i meany jº¼
i
j
n
i
Sxy r 1
meany x
1
n
i
i
1
Sxy
j
Sx˜Sy
j
§ 0 ·
¨
¸
¨ 0.03 ¸
¨ 0.027 ¸
¨ 0.019 ¸
MADI
¨
¸
¨ 0.038 ¸
¨ 0.028 ¸
¨
¸
© 0.03 ¹
0
§
·
¨
¸
3
¨ 3.213 u 10 ¸
¨
¸
¨ 2.187 u 10 3 ¸
¨
4¸
MSDI ¨ 8.718 u 10 ¸
¨
3¸
¨ 7.352 u 10 ¸
¨
3 ¸
¨ 2.56 u 10 ¸
¨
3¸
© 2.839 u 10 ¹
§ 0 ·
¨
¸
¨ 0.987¸
¨ 0.977¸
¨ 0.988¸
r
¨
¸
¨ 0.981¸
¨ 0.979¸
¨
¸
© 0.988¹
meanx
129.266
stdev ( x)
40.681
139
APPENDIX C
MathCAD Program for TL-Moment with t = 2
0
A
0
A
B sort ( A)
n length ( A)
i 1 n
x B
i
i1
83.8
1
82.5
2
190.4
3
86.4
4
129.5
5
170
6
160
7
126
8
121
9
115
10
115
11
117
12
160
13
120
14
205
15
...
140
n 2
¦
l 1
i
n2
¦
i
3
l 2
n 2
¦
i
3
¦
3
1
2
˜ 15.14368027309203779
1
˜1.217647622177982709
3
3
i
ª§ combin( i 13) ˜ combin( n i 2) combin ( i 12) ˜combin ( n i 3) · ˜ xº o 15.14368027309203779
Ǭ
¸ i»
combin( n 6)
©
¹ ¼
ª§ combin( i 1 4 )˜ combin(n i2) 2˜combin( i 1 3) ˜combin( n i3) combin( i 1 2) ˜combin( n i4 ) · ˜x º o 1.21764762217798270
¨
¸ i»
combin( n 7)
¹ ¼
©
l n 2
3
ª§ combin (i 12)˜ combin (n i 2) · ˜ xº
¨
¸ i»
combin (n 5)
©
¹ ¼
ª§ combin ( i 15)˜ combin ( n i 2) 3˜ combin (i 1 4)˜ combin (n i 3) 3˜ combin (i 13) ˜ combin (n i 4) combin ( i 12) ˜combin ( n i 5) · ˜ xº o 0.9970891125729835407
¨
¸ »i
combin (n 8)
¹ ¼
©
1
˜ 0.997089112572983540
4
l 4
l
t 3
l
3
t 4
l
2
4
l
2
F
i
0.015
0.042
i 0.44
n 0.12
F
i
0.069
0.096
§ 0 ·
¨
¸
130.709
¨
¸
l ¨ 7.572 ¸
¨ 0.406 ¸
¨
¸
© 0.249 ¹
§ 0 ·
¨
¸
0
¨
¸
t ¨ 0 ¸
¨ 0.054 ¸
¨
¸
© 0.033 ¹
0.123
0.15
0.177
0.204
0.231
0.258
0.284
0.311
i g ln ln F
i
0.338
0.365
0.392
...
141
normal
V 4.9736l
˜
1
2
8
P l 1.473˜ ( 10)
1
˜V
1
V
y
1 i
P
1
1
130.709
37.659
1
i P V ˜ qnorm F 01
1
1
0
y
1 i
0
49.071
1
65.647
2
74.84
3
81.555
4
86.991
5
91.643
6
95.764
7
99.505
8
102.96
9
106.195
10
109.26
11
112.188
12
115.008
13
117.742
14
120.41
15
...
250
200
xi
150
y 1 i
100
50
0
2
0
2
gi
4
6
142
ev
D 4.216˜ l
1
2
D
31.923
1
[ l 6˜ D ˜ ln( 5) 30˜ D ˜ln (2) 10˜ D ˜ ln( 3) J ˜ D
1
y
2 i
1
1
1
1
[
1
i [ D ˜ ln ln F
1
1
300
250
0
y
2 i
0
1
71.358
80.299
2
85.724
3
89.925
4
93.488
5
96.659
6
99.569
7
102.295
8
104.89
9
107.389
10
109.821
11
112.205
12
114.56
13
116.901
14
119.24
15
...
xi
200
y 2 i
150
100
50
2
0
2
gi
4
6
1
117.125
143
gev
32 1.11298955˜ t33 1.326160015˜ t 3 4 0.578634686˜ t 35 1.462068119˜ t 36 1.103598046˜ t 3 7 1.366381534˜ t 38
K 0.300983183 3.911819242t
˜ 1.38875248˜ t
2
3
l
D [ l 2
y
2
K2
ª 1 1 K2
§ ·
§1·
30˜ *K ˜ « ˜ ¨ ¸ ¨ ¸ 2 ¬3 © 3 ¹
©4¹
2
1
3 i
D
10˜ §¨
2
K
¸
©3¹
2
[ 2
1·
D
ª
K ¬
2
2
K2
2
˜D ˜* K
2
i
˜ 1 ln F
§1·
¨ ¸
©5¹
15˜ §¨
K2
1·
1
3
2
˜D ˜* K
2
K
0.095
D
33.533
2
§1· »
¸
©6¹ ¼
˜¨
K2
¸
©4¹
K 2º
2
6˜ §¨
1·
K2
¸ ˜ D2˜ * K2 ©5¹
[
K 2º
¼
0
y
3 i
0
1
65.832
76.454
2
82.762
3
87.578
4
91.615
5
95.172
6
98.407
7
101.412
8
104.25
9
106.963
xi
10
109.583
11
112.134
y 3 i
12
114.635
13
117.104
14
119.554
15
...
250
200
150
100
50
2
0
2
gi
4
6
2
117.339
144
logistic
[ l
3
[
1
D 3˜ l
3
y
4 i
D
2
§ Fi ·
¸
¨ 1 Fi
©
¹̧
[ D ˜ ln¨
3
3
y
130.709
3
3
22.716
4 i
35.786
59.688
71.588
250
79.745
86.056
200
91.269
95.757
99.736
103.339
xi
y 4 i
150
100
106.659
109.758
50
112.685
115.476
118.161
120.762
...
0
2
0
2
gi
4
6
145
glo
18˜ t
K 4
3
K
7
D 4
4
2
S ˜ K ˜ ªK 5˜ K 4º
4¬ 4
4
¼
2
[ l 4
12˜ l ˜sin S ˜ K
1
0.138
4
4
D
4
S ˜D ˜ª K
4 4 5˜ K4 2 4º¼ D4
4 sin S ˜ K K
4
4
4¬
ª
§ 1 Fi ·
¸
y [ ˜ 1¨
5 i
4
F
K
¨
4 ¬
© i ¹̧
D
4
22.546
[
4
129.477
K4º
»
»
¼
y
5 i
57.859
72.21
80.17
300
85.968
250
90.654
94.663
98.218
101.451
104.447
xi
200
y 5 i
150
100
107.266
109.95
112.531
115.035
117.484
119.896
...
50
2
0
2
gi
4
6
146
gpa
7˜ 6˜ t 1
K 5
3
6˜ t 7
K
3
D 5
0.649
5
ª¬l 2˜ K5 3˜ K5 4˜ K5 5 ˜ K5 6º¼
60
D
80.382
[
84.392
5
[ l 5
y
6 i
1
[ 5
10˜ D
5
K 3
5
D
ª
K ¬
5
5
15˜ D
5
K 4
5
˜ 1 1F
i
y
6˜ D
5
K 5
5
5
K5º
¼
6 i
85.608
87.795
250
90.005
92.237
94.492
200
96.772
xi
99.078
y 6 i
101.41
103.77
150
100
106.159
108.579
111.031
113.518
116.04
118.599
...
50
2
0
2
gi
4
6
147
j 1 6
MADI j
xy
n
1
˜
n
¦
i
1
MSDI ˜
j
n
¦
1
j i
meanx mean (x)
1
meany j
x
n
i
1
n
i
i
n
¦
˜
i
§ xi yj i ·
¨
¸
¨ xi ¸
©
¹
2
Sx 1
˜
n
n
¦ i
x meanx
i
Sxy j
2
Sy 1
1
n
j
1
˜
n
y
j i
1
n
2
¦ yj i meanyj
i
1
n
˜
¦ ª¬ xi meanx˜ yj i meany jº¼
i
1
Sxy
r j
j
Sx˜Sy
j
§ 0 ·
¨
¸
¨ 0.031 ¸
¨ 0.027 ¸
MADI ¨ 0.019 ¸
¨
¸
¨ 0.04 ¸
¨ 0.029 ¸
¨
¸
© 0.032 ¹
0
§
·
¨
¸
3
¨ 3.548 u 10 ¸
¨
¸
3
¨ 2.31 u 10 ¸
¨
¸
4
MSDI ¨ 9.844 u 10 ¸
¨
3¸
¨ 8.776 u 10 ¸
¨
3¸
¨ 2.961 u 10 ¸
¨
3¸
© 3.708 u 10 ¹
§ 0 ·
¨
¸
¨ 0.987¸
¨ 0.977¸
r ¨ 0.987¸
¨
¸
¨ 0.981¸
¨ 0.976¸
¨
¸
© 0.988¹
meanx
129.266
stdev ( x)
40.681