FORECASTING DOMESTIC TOURISM:
APPLICATION TO JOHOR TOURISM DATA
NURUL FARHANA BINTI ZOLKIPLI
UNIVERSITI TEKNOLOGI MALAYSIA
FORECASTING DOMESTIC TOURISM:
APPLICATION TO JOHOR TOURISM DATA
NURUL FARHANA BINTI ZOLKIPLI
A thesis submitted in fulfilment of the
requirements for the award of the degree of
Master of Science (Mathematics)
Faculty of Science
Universiti Teknologi Malaysia
NOVEMBER 2009
iii
To my beloved Abah, Mak, Yaya and Ail
iv
ACKNOWLEDGEMENT
Assalamualaikum warahmatullahi wabarakatuh…
Firstly, I would like to thank Allah because of His blessings; I would be able to
successfully complete this thesis.
Besides that, I also want to convey my deepest appreciation to my supervisor,
Prof. Dr. Zuhaimy bin Hj. Ismail, for guiding and advising me all the way through.
Not forgotten, thousand of thanks to my beloved parents for giving me their
strongest and continuous support every time I drowned in troubles during the process of
doing this thesis.
Profuse thanks also to all my fellow friends for lending me their hands every
time I asked their favour.
I am very grateful and thankful because finally and
successfully, I manage to complete this thesis after trudging along the difficult road.
Thanks a lot.
v
ABSTRACT
Serious tourism forecasting spans over three decades. A number of tourism
forecasters have noted that complex forecasting models are seldom more accurate in
predicting the future than simple ones. Hence, in this study, it shows through a Johor
tourism forecasting comparison using the naïve model, moving average and exponential
smoothing. An approach to selecting the method appropriate to Johor tourism data is
based on criteria which are Mean Absolute Percentage Error (MAPE) and Mean Square
Error (MSE).
The naïve model, moving average and exponential smoothing are
relatively simple to apply, requiring no more than a data series and a computer
spreadsheet program. This study also explores the need of Tabu Search method in
implying forecast. Tabu Search will be used to ensure the pattern of the studied data by
searching the value of weighted constants  ,  and  . The confirmation is performed
by Automated Computerized Forecasting (ACF) System.
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ABSTRAK
Ramalan dalam industri pelancongan telah dijalankan secara serius menjangkau
lebih daripada tiga puluh tahun. Beberapa peramal menyedari bahawa model ramalan
yang kompleks tidak selalunya memberi keputusan yang lebih tepat daripada model
yang ringkas dalam meramal masa depan.
Oleh itu, kajian ini menunjukkan
perbandingan ramalan industri pelancongan negeri Johor menggunakan model naif,
Purata Bergerak dan Pelancar Eksponen.
Pendekatan untuk memilih kaedah yang
sesuai bagi data pelancongan Johor dengan berpandukan kepada kriteria yang mana
ianya adalah Min Peratusan Ralat Mutlak (MAPE) dan Min Kuasa Dua Ralat (MSE).
Model naif, Purata Bergerak dan Pelancar Eksponen agak mudah diaplikasi,
memerlukan hanya siri data dan boleh menggunakan program komputer berlembaran.
Kajian ini juga meneroka perlunya kaedah Carian Tabu dalam pelaksanaan ramalan.
Carian Tabu akan digunakan sebagai pengesahan kepada corak data yang dikaji dengan
mencari nilai pemberat bagi pemalar  ,  dan  . Pengesahan ini dilakukan melalui
sistem Pengkomputeran Automatik Ramalan (ACF).
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
x
LIST OF FIGURES
xi
LIST OF SYMBOLS
xiii
LIST OF APPENDICES
xiv
INTRODUCTION
1
1.1
Introduction
1
1.2
Background of the Problem
2
1.3
Statement of the Problem
4
1.4
Objectives of the Study
5
1.5
Scope of the Study
6
1.6
Significance of the Study
6
1.7
Dissertation Organization
7
viii
2
3
LITERATURE REVIEW
8
2.1
Introduction
8
2.2
Tourism Forecasting
8
2.3
History of Johor
10
2.4
Time Series Forecasting
12
2.5
Types of Forecasting Methods
14
2.6
The Forecasting Programme
15
2.7
The Naïve Forecasting Method
16
2.8
Moving Average Method
18
2.9
Single Exponential Smoothing Method
19
2.10
Tabu Search
21
2.11
Automated Computerized Forecasting System (ACF)
21
2.12
Summary
22
RESEARCH METHODOLOGY
23
3.1
Introduction
23
3.2
Forecast Accuracy
23
3.3
Exponential Smoothing
26
3.4
Holt’s Method
31
3.4.1 Applications of Holt’s Method
33
Holt-Winter’s Method
33
3.5.1 Applications of Holt-Winter’s
35
Tabu Search Method
36
3.6.1 A Basic Tabu Search Procedure
38
3.6.2 Tabu Search Application
39
Summary
40
3.5
3.6
3.7
ix
4
EXPERIMENTATION
41
4.1
Introduction
41
4.2
The Naïve Model
42
4.2.1 The Naïve 1 Model
42
4.2.2 The Naïve 2 Model
44
4.2.3 The Seasonal Naïve Model
46
4.3
Moving Average
48
4.4
Exponential Smoothing
51
4.4.1 Implementation of Exponential Smoothing
52
4.5
Holt’s Model
55
4.6
Tabu Search Application
58
4.6.1 Forecasting Graph Using Tabu Search
59
Summary
61
4.7
5
CONCLUSION AND RECOMMENDATION
62
5.1
Introduction
62
5.2
Conclusion
62
5.3
Recommendation
65
REFERENCES
66
Appendices A-G
72-93
x
LIST OF TABLES
TABLE NO.
TITLE
PAGE
1.1
Number of Visitors in Johor
72
4.1
Value of MAPE and MSE when m=4 and m=12
47
4.2
Value of MAPE and MSE Using 3-month Moving Average
50
And 6-month Moving Average
4.3
Value of  , MSE and MAPE for Exponential Smoothing
53
4.4
Calculation for Johor Tourism Data using Naïve 1 Model
76
4.5
Calculation for Johor Tourism Data using Naïve 2 Model
79
4.6
Calculation for Johor Tourism Data using Seasonal Naïve Model
82
4.7
Calculation for Johor Tourism Data using 3-month Moving
85
Average
4.8
Calculation for Johor Tourism Data using Exponential Smoothing
88
(   0.37 )
4.9
Calculation for Johor Tourism Data using Holt’s Model
91
(   0.36,   0 and a  1236616, b  6712.383 )
5.1
Comparison of the Selected Models to Forecast Johor Tourism
For 120 periods
64
xi
LIST OF FIGURES
FIGURE NO.
4.1
TITLE
Number of Visitors to Johor for 120 periods (January 1999
PAGE
42
-December 2008)
4.2
Actual and Forecast Value Using Naïve 1 Model for
44
Johor Tourism
4.3
Actual and Forecast Value Using Naïve 2 Model
46
for Johor Tourism
4.4
Actual and Forecast Value Using 3-month Moving Average
49
for Johor Tourism
4.5
Actual and Forecast Value Using 6-month Moving Average
50
for Johor Tourism
4.6
Actual and Forecast Value Using 3-month Moving Average
And 6-month Moving Average for Johor Tourism
51
xii
4.7
Actual and Forecast Value Using Exponential Smoothing
54
with   0.05 ,   0.20 and   0.37
4.8
The Forecasting Graph Using the Tabu Search in Finding MSE
59
4.9
The Forecasting Graph Using the Tabu Search in Finding MAPE
60
xiii
LIST OF SYMBOLS
MAPE
-
Mean Absolute Percentage Error
MSE
-
Mean Square Error

-
Weighted Constant (between 0 and 1)

-
Weighted Constant (between 0 and 1)

-
Weighted Constant (between 0 and 1)
AFC
-
Automated Computerized Forecasting
ADLM
-
Autoregressive Distributed Lag Model
ECM
-
Error Correction Model
VAR
-
Vector Autoregressive
TVP
-
Time Varying Parameter
AIDS
-
Almost Ideal Demand System
BSM
-
Basic Structural Model
SES
-
Single Exponential Smoothing
ARIMA
-
Autoregressive Moving Average
N ( s)
-
Neighbourhood
ST
-
Smoothed Estimate or Smoothed Statistic

-
Random Component
xiv
LIST OF APPENDICES
APPENDIX
TITLE
PAGE
A
Johor Tourism Data
69
B
Calculation for Johor Tourism Data Using Naïve 1 Model
73
C
Calculation for Johor Tourism Data Using Naïve 2 Model
76
D
Calculation for Johor Tourism Data Using Seasonal Naïve
79
(when m=4)
E
Calculation for Johor Tourism Data Using 3-month Moving
82
Average
F
Calculation for Johor Tourism Data Using Exponential
85
Smoothing (   0.37 )
G
Calculation for Johor Tourism Data Using Holt’s Model
(   0.36,   0 and a  1236616, b  6712.383 )
88
CHAPTER 1
INTRODUCTION
1.1
Introduction
Predictions of future events and conditions are called forecasts and the act of
making such predictions is called forecasting. Forecasting can be broadly considered as
a method or a technique for estimating many future aspects of a business or other
operation. Planning for the future is a critical aspect of managing any organization and
small business enterprise are no exception.
Indeed, their typically modest capital
resources make such planning particularly important. In fact, the long-term success of
both small and large organizations is closely tied to how well the management of the
organization is able to foresee its future and to develop appropriate strategies to deal
with likely future scenarios. Intuition, good judgment and an awareness of how well the
industry and national economy is going may give the manager of a firm a sense of
future market and economic trends.
Nevertheless, it is not easy to convert a feeling about the future into a precise
and useful number. Perfect accuracy (in forecasting) is not obtainable (Brealy and
Myers, 1988). If it were, the need for planning would be much less. Still the firm must
2
do the best it can. Forecasting cannot be reduced to a mechanical exercise. Naïve
extrapolation or fitting trends to past data is off limited value. It is because the future is
not likely to resemble the past that planning is needed. Forecasters rely on a variety of
data sources and forecasting methods. In other cases, the forecasters may employ
statistical techniques for analyzing and projecting time series. A time series is simply a
set of observations measured at successive points in time or over successive periods of
time.
As far as forecasting method are concerned, they can generally be classified into
quantitative and qualitative (Archer, 1980; Uysal & Crompton, 1985). The quantitative
approach gives more accurate forecasts than judgment forecasts. These methods are
further divided into causal models and time series approaches. Their main distinction is
that causal models attempt to identify and measure both economic and non-economic
variables affecting other variables such as price and quantity while time series
approaches identify stochastic components in each time series. All forecasting methods
can be divided into two broad categories: qualitative and quantitative. Division of
forecasting methods into qualitative and quantitative categories is based on the
availability of historical time series data.
Clearly, forecasting is very important in many type of organizations since
predictions of the future must be incorporated into decision-making process.
In
particular, any organizations must be able to make forecasts in order to make intelligent
decisions.
1.2
Background of the Problem
Some say that travel and tourism is the ‘world’s largest industry and generator
of quality jobs’ (World Travel and Tourism Council, 1995: 1). They estimate that
3
tourism directly and indirectly contributes nearly 11 percent of the gross world product,
the most comprehensive measure of the total value of the goods and services in the
world’s economies produce. The World Travel and Tourism Council estimated that in
2008, gross world product both directly and indirectly related to travel and tourism
would total about US$944 billion, up from US$ 857 billion in 2007.
This activity is
buttressed by more than US$8 trillion invested in world plan, equipment and
infrastructure related to travel and tourism. While these estimates may be controversial,
there is no doubt that tourism activities, encompassing travel away from home for
business or pleasure, comprise a substantial part of lifestyles of the world’s residents, or
that a very large industry has grown up to serve these travellers.
Futurist John Naisbitt (1944), in his bestselling book Global Paradox,
subscribes to the concept that tourism will be one of three industries that will drive the
world of economy into twenty-first century. However, the rate of growth has varied
immensely from year to year, and while most countries have enjoyed steady increase in
their tourist arrivals, some experienced declines in numbers at certain times. Planning
under these circumstances is exceptionally difficult and important. But in attempt to
plan successfully, accurate forecasts of tourism is required. This point has been noted
by several authors in tourism field. For example, Wandner and Van Erden (1980, p.
381) point out that since governments and private industry must plan for expected
tourism demand and provide tourism investment goods and infrastructure, the
availability of accurate estimates of international tourism demand has important
economic consequences. Archer (1987, p. 77) emphasizes the particular necessity for
accuracy in tourism forecasting.
The tourism industries and those interested in their success in contributing into
to the social and economic welfare of citizenry, need to reduce the risk of decisions
which is reduce the chances that a decision will fail to achieve desired objectives.
Future events important to tourism operations are somewhat predictable and somewhat
changeable. Future time, on the other hand, includes time that has passed for which we
4
do not yet have reasonably complete and accurate data, as well as time not yet
encountered. Some time series of interest to tourism forecasters may run three or six
months behind actual time, so that we may not know what happened to tourism industry
in two or three years ahead.
Planners and others in public agencies use tourism forecasts to predict the
economic, social/ cultural or environmental consequences of visitors. In short, sound
tourism forecasts can reduce the risks of decisions and the costs of attracting and
serving the travelling public. It is difficult to imagine the tourism industry without
forecasting to deal with the risks inherent in saving and investing.
1.3
Statement of the Problem
In this research, the tourism forecasting will be done by using basic
extrapolative models (naïve and simple moving average). These are seldom used on
their own to produce forecasts of tourism, but rather are the foundation for developing
and applying more sophisticated models. The naïve models are especially useful as
benchmarks for evaluating more complex forecasting techniques.
Then, we can test a number of alternative quantitative forecasting models,
including extrapolative methods. We look for the model that best simulates the data,
using criterion of error magnitude such as Mean Square Error (MSE) and Mean
Absolute Percentage Error (MAPE). This should be the best candidate for forecasting
future periods. Once we have determined this model, we can produce the forecasts we
need.
We will also examine intermediate extrapolative forecasting methods,
specifically single exponential smoothing, double exponential smoothing and triple
5
exponential smoothing. These methods are intermediate in their complexity among
time series forecasting methods. Single exponential smoothing can be used to forecast
from stationary time series while double exponential smoothing is designed for series
showing a linear trend or a trend with seasonality removed.
Triple exponential
smoothing can be used for series including both trend and seasonal components.
Intermediate time series methods are somewhat more complex than their basic brethren,
but can still be handled in modern computer spreadsheets. However, as complexity
arises, more data are required in time series models to obtain satisfactory results.
Finally, the use of Tabu Search will be applied to verify the pattern of the data
by searching the value of  ,  and  . This is because Tabu Search technique is a metaheuristic that guides a local heuristic search procedure to the solution space beyond
optimality. It uses adaptive memory (memory-based strategies) and takes advantage of
historical information of past solutions to form memories which are useful to help the
search to explore other regions via diversification and intensification whenever
necessary.
1.4
Objectives of the Study
Objectives of this study are:
i.
to forecast tourism using basic and intermediate extrapolative methods in
Microsoft Excel spreadsheets.
ii.
to use Tabu Search for the confirmation of data pattern based on the
value of  ,  and  found.
iii.
to compare the obtained results from those methods and select the best
model in forecasting Johor tourism..
6
1.5
Scope of the Study
In this study, the basic and intermediate extrapolative methods will be used in
determining the best model to forecast tourism. The naïve method will be introduced as
the basic of this research, followed by the moving average and exponential smoothing
method. At the mean time, the Tabu Search technique will be used to ensure the data
pattern based on the value of  ,  and  found using ACF system.
This study
concentrates only on univariate model where the time series forecasting based on the
past value. Measurement of error used is the MSE and MAPE. The data that will be
used in this study is the Johor tourism data which is taken from Majlis Tindakan
Pelancongan Negeri Johor (MTPNJ). The data covers the period for ten years from
January 1999 to December 2008. The data are listed in the Appendix A.
1.6
Significance of the Study
The need for quick, reliable and simple forecasts of various time series is often
encountered in economic and business environments. This study is an introduction of
scientific approach in making decision. The result from this study is practical for other
forecasters as a guide in forecasting especially in using extrapolative methods. A tourist
forecaster or policy maker can at least make a good estimate of visitor arrivals in the
absence of structural data by just using the available time series data. Relatively simple
methodologies can be used on a spreadsheet and can give reasonable estimates albeit a
short period time into the future. Moreover, these models are such that can be easily
updated as time goes on; hence a constant track can be kept of the expected number of
visitor arrivals. These models also can be extended to cover more regions or even
predict the total expected number of visitor arrivals. Further works need to be done to
generalize the findings to other tourist receiving destinations. That is, future studies can
employ the same forecasting models but with different data series, for forecasting
7
accuracy testing. Moreover, additional forecasting techniques can be incorporated into
future studies.
1.7
Dissertation Organization
Chapter 1 is the framework of the study that includes brief introduction,
problem background, problem statement, objectives of the study, scope of the study,
significance of the study and research outline.
Chapter 2 looks at the literature review involving tourism forecasting, history
of Johor, time series forecasting, the types of forecasting methods, the naïve methods,
the moving average methods, the single exponential smoothing method, the double
exponential smoothing method, the triple exponential smoothing method, the Tabu
Search technique and the Automated Computerized Forecasting (ACF) system.
Chapter 3 explores more about those methods mentioned above. The forecast
errors which are MSE and MAPE will be discussed in forecast accuracy. This chapter
will elaborate about the approach of the Tabu Search.
Chapter 4 presents the implementation of forecasting methods. Here, it shows
the calculation using each basic and intermediate extrapolative method on Johor tourism
data.
Chapter 5 discusses the comparison of each forecasting method. The best
model among these methods will be selected in order to forecast.
CHAPTER 2
LITERATURE REVIEW
2.1
Introduction
This chapter briefly discusses time series forecasting which includes the
component of time series and its applications. Explanation of naïve method, moving
average and single exponential smoothing as types of forecasting methods will also be
discussed. This is followed by the introduction of Tabu Search as a method to ensure
the pattern of the data.
2.2
Tourism Forecasting
Forecasting is a process designed to predict future events. In the realm of
tourism, an event may be the number of visitors to a destination, the number of roomnights sold in a hotel or a group of hotels, the number of passengers flying between two
points or the number of brochures requested by potential visitors. A valid forecast
9
event has two characteristics: a specific time and a specific outcome. These outcomes
are often precise volumes of demand. However, they may be stated as ranges or even
qualitative conditions, such as visitor demand is forecast to be greater next year than
this year.
Tourism forecasting has become an important component in tourism research
and different approaches have been used to generate forecasts of tourism. By the end of
the 1950s, the serious study of international tourism had begun. By virtue of the
accumulating record of international level at this time, researchers began to study
international tourism demand, one area of which has been forecasts. Over the last three
to four decades, a substantial number of such study have been undertaken. A large
portion of these have yielded empirical information and a considerable majority has
reported the accuracy of forecasts.
Witt and Witt (1955) provided a comprehensive review of the early tourism
forecasting literature. Together with the rapid development of modern econometrics,
many researches have now applied these recent developments in forecasting tourism
demand in various settings. Li, Song and Witt (2005) reviewed eighty-four-post-1990
empirical studies of international tourism demand modeling and forecasting and gave an
extensive and detailed view on issues such as data types and frequencies, independent
and dependant variables, estimation methods and reported diagnostic test statistics.
Their review suggested that the most frequently used forecasting methods in tourism are
the static regression model, the autoregressive distributed lag model (ADLM), the error
correction model (ECM), the vector autoregressive (VAR) models, time varying
parameter (TVP) model, almost ideal demand system (AIDS) and basic structural model
(BSM).
Song, Witt and Li (2003) used the general-to-specific modeling approach to
obtain ex ante forecasts of the demand for Thai tourism. Song and Witt (2006) used the
VAR modeling technique to forecast the demand for Macau tourism over the period
10
2003-2008.
Kulendran and King (1997) considered four time series models and
econometric model when predicting quarterly tourist flows into Australia from four
major tourist markets.
Song, Witt and Jensen (2003) compared the forecasting
performance of the ECM, ADLM, TVP and VAR models with those generated by two
univariate time series models in forecasting the demand for Denmark tourism and found
that the TVP model generates the most accurate one-year-ahead forecasts. Li, Song and
Witt (2006) reported the forecasts of tourist expenditure by United Kingdom (UK)
residents in a number of Western European countries using TVP and constant parameter
linear AIDS models.
2.3
History of Johor
Johor, the southern gateway and third largest state in Peninsula Malaysia is the
keeper of many national and natural treasures untold. It is a delightful mixture of the
traditional and modern, and natural and ultra-modern urban jungles. Johor gets its name
from the Arabic word Jauhar meaning 'precious stones'. Prior to that, it was known as
Ujung Tanah which mean 'land's end' in Malay, due to its location at the end of the
Malay Peninsula. It is also an apt name, as the southern-most tip of Asia is located at
Tanjung Piai, Johor. Johor has a rich and illustrious ancient history - its sultanate was
established in the early 16th century, commanding the southern Peninsula and Riau
islands. Upon Malacca's defeat to the Portuguese in 1511, the son of the last Sultan of
Malacca, Sultan Mahmud Shah, had established a monarchy in Johor and had posed a
constant threat to the Portuguese. Johor is the only state, apart from Melaka, that grew
into an empire. During its peak, the whole of Pahang and the present day Indonesian
territories of Riau Archipelago and part of Sumatra Island were under Johor's rule.
(Source: http://tourismjohor.com/about/)
11
With its 3.17 million populations, Johor is the most populous state in Malaysia.
Besides the cultural expressions practiced by the local Malays, Chinese and Indians,
culture in Johor is also influenced by visitors and traders throughout history. Johor is
divided into eight districts: Batu Pahat, Johor Bahru, Kluang, Kota Tinggi, Mersing,
Muar, Pontian and Segamat, the largest of these being Johor Bahru which is the state
capital.
Visitors commonly head for Johor Bahru's royal palaces and lively night
markets, considered as one of the state’s best attractions. Johor also has eight small
islands located off its coast: Tinggim, Rawa, Sibu, Tengah, Pemanggil, Aur, Besar dan
Lima. Most of these islands are sparsely populated, with Tinggi being the most
significance island and a popular tourist attraction which can be visited by boat from the
mainland.
(Source: http://www.hoteltravel.com/malaysia/johor/guides/overview.htm)
Johor is increasingly popular for eco tourism as it is endowed with rich natural
heritage. The Endau-Rompin Park, one of the world’s oldest tropical rain forests –
offers great natural beauty for the adventurous traveler. Furthermore, Johor’s beautiful
and natural forest parks also being developed into centers for eco tourism. The beacon
of new growth, Iskandar Malaysia will spur economic developments that actuate
Malaysia's global potential to greater heights. Sprawling 2,217 sq. km, it is 3 times the
size of Singapore and is comprised of most of Johor's major hubs including Johor
Bahru, Danga Bay, world-class ports Tanjung Pelepas and Pasir Gudang, amongst
others.
(Source: http://tourismjohor.com/about/)
12
2.4
Time Series Forecasting
A time series is a chronological sequence of observations on a particular
variable. In forecasting, we are trying to estimate how the sequence of observations
will continue into the future. Forecasting the future value of an observed time series is
an important problem in many areas, including economics, production planning, sales
forecasting and stock control. Forecast can be based entirely on past observations in a
given time series, by fitting a model to the data and extrapolating, which is called
univariate (Chatfield, 1984). Forecasting is done at all levels and events. Education,
job decision and long-life partner are made on personal prediction of future happenings.
In formal level, economists and politicians are continually forecasting economics and
politic trends. Many different forecasting techniques are used with varying degrees of
success (Runyon et. al, 1982).
Types of forecasting procedure can be classified into three categories (Chatfield,
1984):
a) Subjective- forecast can be made on subjective basis using judgment,
intuition, commercial knowledge and any other relevant information. For
example is Delphi method.
b) Univariate- forecast can be based entirely on past observations in a given
time series, by fitting a model to the data and extrapolating. For example,
forecast of future sales of a product based entirely on past sales.
c) Multivariate- forecast can be made by taking observations on other variables
into account. For example, sales may depend on stocks.
Although there are many different forecasting techniques, this study focuses on
the most commonly used forecasting technique, the univariate time series model. In
this model, past data are analyzed over a given time period and patterns are estimated.
13
A key assumption in the univariate time series model is that patterns which have
occurred in the past will continue in the future (Runyon et. al, 1982).
An important in selecting an appropriate forecasting method is to consider the
type of data patterns, so that the methods most appropriate to those patterns can be
utilized. Four types of time series data patterns can be distinguished (Makridakis,
Wheelwright, Hyndman, 1998): seasonal, cyclical, trend and irregularity.
a) Seasonal- exists when a series is influenced by seasonal factors (e.g., the
quarter of the year, the month or day of the week). Sales of product such as
soft drinks, ice creams and household electricity consumption all exhibit this
type of pattern. Seasonal series are sometimes called “periodic” although
they do not exactly repeat themselves over each period.
b) Cyclical- exists when the data exhibit rises and falls that are not of a fixed
period. For economic series, these are usually due to economic fluctuations
such as those associated with the business cycle. The sales of products such
as automobiles, steel and major appliances exhibit this type of pattern. The
major distinction between a seasonal and cyclical pattern is that the seasonal
is of a constant length and recurs on a regular periodic basis while the
cyclical is varies in length. Moreover, the average length of cycle is usually
longer than seasonal and the magnitude of a cycle is usually more variable
than seasonal.
c) Trend- exists when there is a long-term increase or decrease in the data. The
sales of companies, the gross national product (GNP) and many other
business or economic indicators follow a trend pattern in their movement over
time.
d) Irregularity- exhibit erratic movements in a time series that follow no
recognizable or regular pattern. Such movements represent what is ‘left over”
in a time series after seasonal, cyclical, trend have been accounted for. Many
irregularities in time series are caused by “unusual” events that cannot be
14
forecasted such as earthquake, accidents, hurricanes, wars, etc. This factor is
the source of error in forecasting.
Many data series include combinations of the preceding patterns. For example,
the sales of product shows trend, seasonality and cyclical behaviour. One of the things
that make forecasting interesting and challenging is the huge variety of patterns that
commonly occur in the real time series data. Forecasting methods that are capable of
distinguishing each of the patterns must be employed if a separation of the component
patterns is needed. Similarly, alternative methods of forecasting can be used to identify
the pattern and to best fit the data, so that future values can be forecasted.
2.5
Types of Forecasting Methods
Forecasting methods fall into two major categories: quantitative and qualitative.
Quantitative methods organize past information about a phenomenon by
mathematical rules. These rules take advantage of underlying patterns and relationships
in the data of interest to the forecaster. Objective numerical measurements consistent
over some historical period are required in these methods. These methods also assume
that at least some elements of past patterns will continue into the future (Frechtling,
2001).
There are two major subcategories of quantitative methods: extrapolative and
causal. Extrapolative methods also called ‘time series methods’, assume that the
variable’s past course is the key to predicting its future. Patterns in the data during the
past are used to project or extrapolate future values. Causal relationships are ignored.
15
The other subcategory of quantitative forecasting methods is causal methods.
These attempt to mathematically simulate cause-and-effect relationships. Determining
the causal variables (better called ‘explanatory variables’) that affect the forecast
variable and the appropriate mathematical expression of this relationship is the central
objective.
These methods have the advantage over time series methods of explicitly
portraying cause-and-effect relationships.
This is crucial in certain forecasting
situations, such as when management wants to know how much impact on demand an
increased advertising budget will have. Likewise, tourism policy forecasting requires
causal models.
However these methods are more costly and time-consuming to
construct than time series models and are often considerably less accurate.
Qualitative methods are also called ‘judgmental methods’. Past information
about the forecast variable is organized by experts using their judgment rather than
mathematical rules.
These are not necessarily cheaper or easier to apply than
quantitative methods, but they have the advantage of not requiring historical data series.
(Frechtling, 2001).
2.6
The Forecasting Programme
The objective of the forecasting programme is to establish a system for
periodically producing forecasts required by management, termed a tourism demand
forecasting programme. There are four major phases of developing this forecasting
programme, focusing on building a system that will be used repeatedly over a year or
16
more to produce forecasts: design phase, specification phase, implementation phase and
evaluation phase.
The design phase guides the forecaster in choosing the appropriate forecasting
method to employ.
This phase examines the problem, the resources and the
relationships that help determine a preliminary choice of method.
The specification phase includes determining the relationships that will
comprise the appropriate forecasting model and selecting an appropriate model.
The implementation phase comprises employing the selected model to
generate forecasts and preparing these forecasts for presentation to management.
The evaluation phase covers monitoring the forecasts over time to determine if
adjustments should be made in the forecasting model and making the appropriate
adjustments to secure the most accurate series of forecasts.
Following these steps ensures that the forecaster systematically develops a valid
strategy for solving his or her forecasting challenge. This help ensures that we do not
waste time and money in determining the shapes of the tourism features we are
interested in.
2.7
The Naïve Forecasting Method
The naïve forecasting method simply states the value for the period to be
forecast is equal to the actual value of the last period available. More formally,
17
Naïve 1 model: Ft  At 1
(2.7.1)
where F = forecast value
A = actual value
t = some time period.
This is the simplest forecasting model. As such, it is frequently used as a
benchmark to compare other forecast models against.
It is not unusual for more
elaborate models to produce higher MAPEs than the naïve model and are thus not worth
the time and money to operate. Witt and Witt (1992: 99-123) present a number of these
situations among international tourism demand series and conclude that more complex
forecasting models are less accurate than the naïve model for many series.
There are two other versions of the naïve concept that are sometimes used as a
benchmark forecast. The Naïve 2 model defines the forecast value as the current value
multiplied by the growth rate between the current value and the previous value. This
might be a useful benchmark for a series that trends upward or downward.
Naïve 2 model:
Ft  At 1 *
At 1
At  2
where F = forecast value
A = actual value
t = some time period
The ‘seasonal’ naïve can be used with a seasonal data and postulates that the
next period’s value is equal to the value of the same period in the previous year.
Seasonal naïve model: Ft  At  m
18
where F = forecast value
A = actual value
t = some time period
m = number of periods in a year (for example, four quarters, twelve
months)
2.8
Moving Average Method
The moving average method is the second in simplicity only to the naïve
method. We can average any number of periods to produce a forecast through the
moving average model. The general equation for the model is:
Ft 
At 1  At  2  At  n
n
where F = forecast value
A = actual value
t = some time period
n = number of periods
The moving average method allows some past values to determine forecast
values and all have the same influence on the forecast value. The more past values
included in the model, the smoother it becomes. This is because the more values (the
higher n is), the less influence any one values has on the average; rather, the values tend
to offset each other to provide a smooth forecast series.
19
If a time series shows wide variations around a trend, then the longer the moving
average, the better it will pick up the trend. However, long moving averages are slow to
pick up recent changes in trend because so many past values are affecting it. The
moving average method is more accurate in forecasting a series with very little variation
around its trend than one with significant seasonality or volatility.
2.9
Single Exponential Smoothing Method
Smoothing method was first developed in the late 1950s by operation
researches. It is unclear whether Holt (1957) or Brown (1956) was the first to introduce
exponential smoothing, or perhaps it was Magee (1958).
Most of the important
development work on exponential smoothing was completed in the 1950s and published
by the early 1960s. This work included that done by Brown (1956) and Holt (1957) and
subsequent work by Magee (1958), Brown (1959), Hot et al. (1960), Winters (1960),
Brown and Meyer (1961), and Brown (1963).
Since that time, the concept of
exponential smoothing has grown and become a practical method with wide application,
mainly in the forecasting of inventories.
Single exponential smoothing (SES) is a forecasting technique that attempts to
“track” changes in a time series by using newly observed time series values to “update”
the estimates of parameters describing the time series. It is also known as simple
exponential smoothing. Simple smoothing is used for short-range forecasting, usually
just one month into the future. The model assumes that the data fluctuates around a
reasonably stable mean (no trend or consistent pattern of growth). SES allows us to
vary the important of recent values to the forecast and includes all of the information
past values can provide us.
20
The logic of the SES is the evident in its general equation:
Ft  Ft 1   ( At 1  Ft 1 )
where F = forecast value
 = smoothing constant between 0 and 1
A = actual value
t = some time period
Another way of writing the above equation is:
Ft   At 1  (1   ) Ft 1
According to Witt, Newbould and Watkins (1992:38), exponential smoothing
tourism forecasting models ‘tend to perform well, with accuracy levels comparable to
more complex and statistically sophisticated forecasting methods which require
considerable user understanding to employ them successfully’. They further maintain
that while this method is not prominent in forecasting literature, in actual practice it,
along with the moving average method, is the most popular.
The major disadvantage of this method is that it cannot take into account factors
affecting the series other than its past values. It does not explain the relationships
between such factors and the series of interest. Should events occur that can radically
change tourism behaviour, such as pestilence, terrorism, entertainment mega-events or
natural disasters, time series methods fail. However, they can indicate the values that
should have been achieved in the absence of these catastrophes and thus measure the
magnitude of their impact on tourism.
21
2.10
Tabu Search
Tabu Search has its antecedents in methods designed to cross boundaries of
feasibility or local optimality normally treated as barriers and systematically to impose
and release constraints to permit exploration of otherwise forbidden regions. Early
examples of such procedures include heuristic based on surrogate constraint methods
and cutting plane approaches that systematically violate feasibility conditions. The
modern form of Tabu Search derives from F. Glover. Seminal ideas of the method are
also developed by P. Hansen in a steepest ascent/mildest descent formulation.
Additional contributions are shaping the evolution of the method and are responsible for
its growing body of successful applications.
Webster’s Dictionary defines tabu or taboo as ‘set apart as charged with a
dangerous supernatural power and forbidden to profane use or contact…’ or ‘banned on
grounds of morality or taste or as constituting a risk…’. Tabu Search scarcely involves
reference to supernatural or moral considerations, but instead is concerned with
imposing restrictions to a guide a search process to negotiate otherwise difficult regions.
These restrictions operate in several forms, both by direct exclusion of certain search
alternatives classed as ‘forbidden’ and also by translation into modified evaluations and
probabilities of selection.
2.11
Automated Computerized Forecasting System (ACF)
The ACF system has been developed by Mohd Nizam (2003) specifically as an
interactive system for solving Winter’s method of forecasting. It is built in order to
search for the smoothing parameter ( ,  and  ) within the range 0 to 1 using Genetic
Algorithm (and Tabu Search). The results are output in the form of Graph that show the
22
actual value and the forecast value using the constant parameter ( ,  and  ) found
earlier.
This software is written using Borland Delphi 5 Enterprise. There are two parts
in this system which are the integration of Genetic Algorithm into Winter’s method and
the Tabu Search as a refinement from Genetic Algorithm search result. In this study,
we only focus on the second part.
The Tabu Search properties Windows consist of Neighbourhood Size, Tabu List
Size and Number of Iteration. For neighbourhood size, it allows the user to specify the
number of neighbour. A larger number of neighbourhoods would lead to a slower
processing time (a more time is required to calculate the MSE or MAPE).
As for Tabu List Size, user can specify the size of interest. Tabu list is used to
prevent the search going back to the point visited previously. Lastly, in Convergence, it
allows the user to choose the number of iteration for the Tabu Search procedure.
2.12
Summary
The literature review provides an in-depth understanding of the area of
forecasting. The tourism industry, time series forecasting and the types of forecasting
methods have been discussed thoroughly. Tabu Search technique described here is
chosen to obtain the value of  ,  and  . The ACF also has been explained before its
implementation in chapter 4.
CHAPTER 3
RESEARCH METHODOLOGY
3.1
Introduction
In the previous chapter, literature review was given on tourism forecasting, time
series forecasting, types of forecasting methods, Tabu Search and the Automated
Computerized Forecasting System. This chapter will discuss the research methodology
which consists of forecast accuracy and application on Exponential Smoothing, Holt’s
method, Holt-Winter’s method and Tabu Search. All equations involved are given in
this chapter.
3.2
Forecast Accuracy
More accurate forecasts reduce the risks of decisions more than do less accurate
ones. Accuracy is the most important forecast evaluation criterion (Frechtling, 2001).
The word “accuracy” refers to “goodness of fit,” which in turn refers to how well the
24
forecasting model is able to reproduce the data that are already known.
To the
consumer of forecasts, it is the accuracy of the future forecast that is most important
(Makridakis, Wheelwright, Hyndman, 1998).
The most familiar concept of forecasting accuracy is called ‘error magnitude
accuracy’ and relates to forecast error associated with a particular forecasting model.
This is defined as:
et  At  Ft
where t = some time period, such as month, quarter or year
e = forecast error
A = actual value of the variable being forecast
F = forecast value
If the actual value is greater than the forecast value at time, t, then the
forecasting error is positive. If less than the forecast value, then the forecasting error is
negative. There are quite a few ways to summarize the error magnitude accuracy of a
forecasting model. Some of these compute measures of absolute error and are thus
subject to the units and time period over which the model is tested. They are often
difficult to interpret and compare across different models.
Other error magnitude measures compute percentage errors relative to the values
in the historical series. These allow comparing several different models across different
time periods. One of the most useful of these, due to its simplicity and intuitive clarity,
is the MAPE:
MAPE 
1 n et

n t 1 At
25
where n = number of periods
e = forecast error
A = actual value of the variable being forecast
t = some time period
The MAPE is the sum of the absolute errors for each time period divided by the
actual value for the period; this sum is divided by the number of periods to obtain a
mean value. Then, by convention, this is multiplied by 100 to state it in percentage
terms. This is a simple measure permitting comparison across different forecasting
models with different time periods and numbers of observations and weighting all
percentage error magnitudes the same. Lower MAPE values are preferred to higher
ones because they indicate a forecasting model is producing smaller percentage errors.
Moreover, its interpretation is intuitive. The MAPE indicates, on the average,
the percentage error a given forecasting model produces for a specified period. One
author has suggested the following interpretation of MAPE values:

less than 10 percent is highly accurate forecasting

between 10 and 20 percent is good forecasting

between 20 and 50 percent is reasonable forecasting

greater than 50 percent is inaccurate forecasting
Such a standard can be quite misleading because it ignores the change characteristics of
the time series being forecast.
The MSE is calculated using this equation:
MSE 
1 n 2
 et
n t 1
26
where n = number of periods
t = some time period
e = forecast error
Here, the errors are made positive by squaring each one and then the squared errors are
averaged. The MSE has the advantage of being easier to handle mathematically and so
it is often used in statistical optimization.
There is no consensus among statisticians as to which measure is preferable.
The decision is which one to use depends on the make up of the data; if there are only
one or two large errors, these will be magnified by using MSE, thus another forecast
error such as MAPE will be used. But if all the error is similar in magnitude, the MSE
will be used.
3.3
Exponential Smoothing
Exponential smoothing is a procedure for continually revising a forecast in the
light of more recent experience.
Exponential smoothing assigns exponentially
decreasing weights as the observation get older. In other words, recent observations are
given relatively more weight in forecasting than the older observations.
To begin the exponential smoothing approach, let now suppose that at the end of
a particular time period, which we shall call time period T-1, we have obtained a set of
observations for the time series, which we shall denote A1, A2,..., AT 1 .
Given these
observations we wish to estimate   , the average level of the time series. As just stated,
27
the least squares estimate of   , which we now denote as b (T  1) to emphasize the fact
that the most recent observation in the time series corresponds to time period T-1, is
T 1
b (T  1)  A  
At
t 1
(T  1)
So, the estimate of   is simply the average of the observations in the time series
through period T-1. Given this estimate, the forecast for any future time period, say
period T  1   , where  is a positive integer, is b (T  1) . Thus the forecast for any
future period is the average of the observations in the time series through period T-1.
Now suppose that we obtain a new observation AT at the end of the next time
period, period T. We would like to incorporate this new observation into the estimate
of   . That is, we wish to obtain an updated estimate of   that is based on the new
observation AT as well as the old observations A1, A2,..., AT 1 . We shall call this new
estimate b (T ) to indicate that the most recent observation in the time series
corresponds to time period T. Regression may be used to obtain such an estimate. This
estimate is given by
T
b (T )  A  
t 1
At
T
So this new estimate can be found by simply recalculating A , which is now the average
of the observations in the time series through period T . This new estimate b (T ) is then
the new forecast for the average level of the time series in any future time period.
Another way to incorporate this new observation into the estimate of   is
known as simple exponential smoothing. This approach generates the new estimate
28
b (T ) in a different a different way. It seems intuitive to change the old estimate
b (T  1) by some fraction of the forecast error which resulted when the old estimate was
used to forecast the value of the time series for the present period. This forecast error is
given by
eT  AT  b (T  1)
and is the difference between the observed value in period T and the forecast made for
period T in period T-1. If the fraction we use is  , then the updated estimate is given by
b (T )  b (T  1)   [ AT  b (T  1)]
Then the new estimate is partly based on the old estimate b (T  1) . If the old estimate
yielded a forecast for period T that was too low, then the new estimate is higher. If the
old estimate yielded a forecast for period T that was too high, then the new estimate is
lower. The magnitude of the adjustment up or down is determined by the magnitude of
the forecast error. A large error leads to a large adjustment while a small error leads to
a small adjustment.
In order to simplify notation at this point, we shall define. So the equation we
use to update the estimate can be written as
ST  ST 1   ( AT  ST 1 )  ST 1   AT   ST 1
ST   AT  (1   ) ST 1
This equation defines the updating procedure called simple exponential smoothing. We
call ST the smoothed estimate or smoothed statistic. The fraction  is called the
smoothing constant.
29
Examining the smoothing equation
ST   AT  (1   ) ST 1
we see that the smoothed estimate is simply an estimate based on the observations
A1, A2,..., AT .
This is true since
ST 1
or
b (T  1) is the average of the
observations A1, A2,..., AT 1 . Now, let us change the time origin so that the initial estimate
of   is assumed to be generated in time period zero.
We shall call this initial
estimate S . In practice, the estimate S would be obtained by calculating the average of
an initial set of the observations in the time series. If such an initial set of observations
is not available, S is commonly set to the first observed value of the time series. Let
us also assume that the smoothing equation
ST   AT  (1   ) ST 1
has been used to update the estimates for each time period t from period 1 to period T,
the present period. In this situation, the smoothed estimate for period T, that is ST , can
be shown to be a linear combination of all past observations. To see this, we consider
the smoothed estimate
ST   AT  (1   ) ST 1
from which we can see that
ST 1   AT 1  (1   )ST  2
Substitution therefore gives us
ST   AT  (1   )[ AT 1  (1   ) ST 2
  AT   (1   ) AT 1  (1   ) 2 ST  2
30
Again, we can see that
ST  2   AT 2  (1   ) ST 3
and substituting again we have
ST   AT   (1   ) AT 1  (1   ) 2 [ AT 2  (1   ) ST 3 ]
  AT   (1   ) AT 1   (1   )2 AT  2  (1   )3 ST 3
Substituting recursively for ST 3 , ST  4,..., S 2 and S1 , we obtain
ST   AT   (1   ) AT 1   (1   ) 2 AT 2  ...   (1   )T 1 A1  (1   )T A
Thus we see that ST , the estimate of   in time period T, can be expressed in terms of
the observations A1, A2,..., AT and the initial estimate S .
The coefficients of the
observations,  ,  (1   ),  (1   )2 ,...,  (1   )T 1 measure the contributions that the
observations AT , AT 1, AT  2,..., A1 make the most recent estimate ST . It can be seen that
these coefficients decrease geometrically with the age of the observations.
For
example, if the smoothing constant  is 0.1, then these coefficients are 0.1, 0.09, 0.081
and so on. The updating procedure we have described is called simple exponential
smoothing because these coefficients decrease exponentially.
Since these coefficients are decreasing, the most recent observation AT makes
the largest contribution to the current estimate of   . Older observations make smaller
and smaller contributions to the current estimate of   at each successive time point.
Thus, remote observations are “dampened out” of the current estimate of   as time
advances. The rate at which remote observations are dampened out depends on the
31
smoothing constant  . For values of  near 1, remote observations are dampened out
quickly, while for values of  near 0, remote observations are dampened out more
slowly. For example, for   0.09 , we obtain coefficients 0.9,0.09,0.009,0.0009,…
while for   0.1 , we obtain coefficients 0.1,0.09,0.081,0.0081,… So, the choice of the
smoothing constant has a great bearing on the estimate ST . In general, when the time
series quite volatile, that is when the random component  t has a large variance, we
would select a small smoothing constant, so that the smoothed estimate ST will weight
ST 1 , the smoothed estimate for the previous period, to a greater degree than it weights
the observation AT . For more a stable time series, in which the random component
 t has a smaller variance, we would select a larger smoothing constant.
In making forecast, suppose we are in period T and the current estimate of   is
ST  b (T ) . We wish to forecast the time series for a future period T   . Since the
model is At      t , the forecast is simply FT  (T )  ST , the current estimate of   .
Here the use of the hat F indicates that FT  (T ) is a forecast value rather than an
observation of the time series. The notation of FT  (T ) is used to emphasize that this
forecast is being made for period T   and is being made in period T.
3.4
Holt’s Method
This method was developed to deal with time series showing a linear trend over
time. In essence, the Holt’s model computes a smoothed level and trend at each data
point. These values for the last point in the time series can be used to forecast one or
two points ahead in the future. The level is a smoothed estimate of the value of the data
32
at the end of each period. The trend is a smoothed estimate of average growth at the
end of each period. The equations for this model are:
Level
:  At  (1   )(at 1  bt 1 )
Trend
: bt   (at  at 1 )  (1   )bt 1
Forecast : Ft  m  at  mbt
where a = level of the series
 = level smoothing constant between 0 and 1
A= actual value
b= trend of the series
  trend smoothing constant between 0 and 1
t = some time period
m = number of time periods ahead to be forecast
We need to develop two initial estimates to begin the Holt’s process.
Makridakis, Wheelwright and Hyndman (1998: 159) suggest the following:
Level initialization : a1  A1
Trend initialization : b1  A2  A1
A variation on this Holt’s method is double exponential smoothing method (Makridakis,
Wheelwright and Hyndman, 1998:158). Here, the smoothing constant,  , in the level
equation is replaced by an independent smoothing constant,  , in the trend equation.
These two values are varied between zero and one to produce the equation with the
lowest MAPE or other measure of error.
33
The advantages of Holt’s method are that it is relatively simple, captures linear
trends up or down well and can forecast several periods ahead (unlike single
exponential smoothing). Its disadvantages are that it is cannot track non-linear trends
well, fails to simulate a stepped series well and cannot deal with seasonality. Finally, in
common with all time series methods, it does not incorporate any causal relationship
that may be important to management.
3.4.1 Applications of Holt’s Method
Sheldon (1993) tested eight models for forecasting international visitor
expenditures in six developed countries and found that Holt’s method was the second
most accurate model in terms of MAPE. The most accurate model was the Naïve 1
model of no change from the previous year.
Martin and Witt (1989) compared the accuracy of seven forecasting methods for
simulating visitor flows among twenty-four origin-destination pairs and, like Sheldon,
found that exponential smoothing was the second most accurate model in terms of
MAPE, after Naïve 1.
3.5
Holt-Winter’s Method
The Holt-Winter’s trend and seasonality method employs triple exponential
smoothing: one equation for the level, one for the trend and one for the seasonality. The
34
equations associated with each of these elements are as follows (Makridakis,
Wheelwright and Hyndman, 1998:165):
yt
 (1   )(at 1  bt 1 )
St  s
Level
: at  
Trend
: bt   (at  at 1 )  (1   )bt 1
Seasonality : St  
Forecast
At
 (1   ) St  s
at
: Ft  m  (at  mbt ) St  s  m
where a = level of the series
 = level smoothing constant between 0 and 1
A = actual value
s = number of seasonal periods in a year (for example, four quarters,
twelve months)
b = trend of the series
 = trend smoothing constant between 0 and 1
S = seasonal component
 = seasonal smoothing constant between 0 and 1
t = some time period
m = number of time periods ahead to be forecast
Initialization is more complex than the earlier Holt’s method. Values must be
sought for as , bs and S s , that is at the end of the first complete season.
One
recommended approach is the following (Makridakis, Wheelwright and Hyndman,
1998:168):
35
A1  A2  ...  As
s
Initial level
as 
Initial trend
A  As 
1  A  A A  A2
bs   s 1 1  s  2
 ...  s  s

s
s
s
s

Seasonal indices for the first year
S1 
A
A1
A
, S 2  2 ,..., S s  s
as
as
as
The level is initialized by the average of the first season of values. The trend is
initialized by the average of each of the s estimates of trend over the first season.
Finally, the seasonal indices are initially set as the ratio of the first year’s values to the
mean of the first year, as .
3.5.1 Applications of Holt-Winter’s Method
Chu (1998) tested six forecasting methods in forecasting monthly visitor arrivals
in ten Asia-Pacific nations over the 1975-94 period: Naïve 1, Naïve 2 time trend
regression, sine wave regression, autoregressive/ moving average (ARIMA) and HoltWinters. Using the MAPE as his measure in eighteen-month ex post forecasts, he found
Holt-Winter’s models were the second only to ARIMA in producing superior forecasts
for nine of the ten countries and led ARIMA in forecasting New Zealand visitor
arrivals.
Turner, Kulendran and Fernando (1997) compared six different methods for
forecasting quarterly tourism flows to each of Japan, Australia and New Zealand from
seven originating countries over the 1978-95 period. They grouped the quarters into
four ‘periodic series’ for each origin-destination pair and applied Holt’s exponential
36
smoothing method and the autoregressive method to the series to develop ex post
forecasts. They tested these against Winter’s exponential smoothing method and the
Box-Jenkins ARIMA for the straight seasonalized series.
They found the models
applied to the periodic series generally proved less accurate in ex post forecasting than
the ARIMA model or the naïve model applied to the seasonalized series. The Winter’s
models proved generally less accurate than the Naïve 1 model for the seasonalized
series.
3.6
Tabu Search
Tabu Search is a meta-heuristic approach designed to find a near-optimal
solution for combinatorial optimization problems. Like other traditional methods, Tabu
Search needs strong domain knowledge in order to utilize the procedure. Below are
some terminologies used in implementing Tabu Search:

A move - A transition from a current solution to its neighbouring (or
another) solution.

Tabu list – A list of moves that are currently tabu (a list of forbidden
exchanges to avoid cycling between the same solutions endlessly)

Tabu list size – The number of iterations for which a recently accepted
move is not allowed to be reserved, Ts

Termination – Tabu Search stops when there is no more move available
or until it reaches a preset maximum iteration.

Parameter settings – Since Tabu Search is problem-oriented, it requires
fine-tuning of its parameters such as searching strategy, sopping rules,
initial solution, length of tabu list and maximum iteration.
37
Tabu Search can be thought of as an iterative method. An initial solution is
randomly generated and a neighbourhood around that solution is examined. If a new
solution in the neighbourhood that is preferred in the original, then the new solution
replaces the old and the process repeats. If no new solution is found that improves upon
the old function evaluation, then unlike the gradient descent procedure which would
stop at that point (a local minimum), the Tabu Search algorithm may continue by
accepting a new value that is worse than the old value. Therefore, a collection of
solution in a given neighbourhood is generated and the final solution would be based on
the best solution so far for that particular neighbourhood.
To keep from cycling, an additional step is included that prohibits solutions
from recurring for a user defined number of solutions. This Tabu list is generated by
adding the last solution to the beginning of the list and discarding the oldest solution
from the list. During this procedure, the best solution found so far is retained. From the
subset of acceptable neighbourhoods the best solution is chosen. In a problem in which
there is no known solution, a given value for a maximum number of iteration can be
used to terminate the process. When the number of iterations has taken place with no
improvement over the best solution, the algorithm will terminate.
According to Hertz and De-Werra (1990), the easiest way to introduce Tabu
Search is to think of an iterative descent method used for finding in a (finite or infinite)
set X of feasible solutions, s which minimizes a real valued objective function, f.
A neighbourhood N ( s) should be defined for each solution s in X; the procedure
would then start from an initial feasible solution s. Whenever a feasible solution s has
been reached, the neighbourhood N(s) is examined. If a solution s’ in N(s) with f(s’)
<f(s) has been found, one move to s’ and repeats the step. If no such s’ is found, the
descent procedure stops. In such a case, the method has found a local minimum with
respect to the topology chosen on X for defining the neighbourhood N.
38
In contrast to this descent method, Tabu Search may try to continue from there:
in order to avoid being trapped in local minima, a movement from an s to an s’ in N(s)
even if f(s’)  f ( s) is allowed. We therefore generate, when we are at s, a collection of
solution si in N(s); we then move to the best si generated. Call it s ; this may, of course
introduce cycling in the algorithm.
In order to prevent cycling, we do not allow going back to a solution visited in
the last k iterations (for a given k). More precisely, we introduced a so-called Tabu list
T of length T =k (fixed or variable) which is used as a queue as follows: whenever a
move s  s ' is made. We should introduce s at the end of T and we should remove the
oldest solution introduced into T (in the case where T is fixed). Thus, all moves back
to s are now forbidden for the next T iterations, s is a Tabu Solution. Any moving
back to s is a Tabu move.
3.6.1 A Basic Tabu Search Procedure
1.
Initialization

Generate an initial solution, say x (which is not necessarily feasible)

Set the best current solution xbest  x

Evaluate all the moves in the neighbourhood N ( x)

Set values for tabu list size and the data structure for candidate list of
solutions.

Set counters: iteration=0 (current iteration) and best iteration=0
(iteration for which the xbest is found)
39
2.
Candidate list of solutions

Determine strategically, by the use of a special data structure, the
candidate list of the best moves in the neighbourhood N ' ( x )  N ( x)

3.
4.
Update the data structure if necessary
Selection strategy: Forbidding strategy and freeing strategy

Choose the best admissible solution x '  N ' ( x )

Set x  x ' and iteration = iteration +1

If F ( x ' )  F ( xbest ) , set xbest  x ' and best iteration = iteration

Update the tabu list
Stopping criterion
If the stopping criterion is met, go to step 5, otherwise go to step 2.
5.
Diversification (optional)
Apply some forms of diversification on well-defined solutions and go to
step 2, or else stop.
3.6.2 Tabu Search Application
Tabu Search techniques have enjoyed success in a variety of problem settings
such as scheduling, transportation, layout and circuit design, electronic circuit design,
graphs, probabilistic logic and expert systems, neural networks and others (Reeves,
1993). Tabu Search has the ability to employ any procedure to arrive at a best solution.
Furthermore, it has the ability to escape from a local optimum, thereby increasing the
chances of reaching global optimum.
40
3.7
Summary
This chapter discussed about the forecast accuracy used in forecasting which are
MAPE and MSE. Each method also has been elaborated and all equations involved are
given in this chapter. Their applications also have been included to give more clearly
understanding.
CHAPTER 4
EXPERIMENTATION
4.1
Introduction
This chapter presents calculation and implementation of forecasting methods
which have been discussed in the previous chapter. The calculation begins with Naïve
model, followed by Moving Average, Exponential Smoothing and Holt’s model. Tabu
Search will be the last to be implemented.
Figure 4.1 below shows the number of visitors (in million) to Johor for ten years
(120 periods) which covers from January 1999 until December 2008. The graph shows
a series with irregular patterns. It follows no recognizable or regular pattern.
42
4,000,000
3,500,000
3,000,000
visitors
2,500,000
2,000,000
1,500,000
1,000,000
500,000
0
1
8
15
22
29
36
43
50
57
64
71
78
85
92
99 106 113 120
t
Figure 4.1
Number of visitors to Johor for 120 periods (January 1999-December
2008)
4.2
The Naïve Model
4.2.1 The Naïve 1 Model
For the Naïve 1 model, the equation that is being used is:
Ft  At 1
where F = forecast value
A = actual value
43
t = some time period
To use this equation, the last value is the best forecast of the next value. For example,
the value of F2 is the value of the previous actual value, A1 . Thus,
F2  A1
F3  A2

F120  A119
which are
F2  A1  1213874
F3  A 2  1107735

F120  A119  2217748
From this model, we obtained MAPE = 0.213690271 and MSE = 2.44245138814E11.
44
4,000,000
3,500,000
3,000,000
visitors
2,500,000
2,000,000
1,500,000
1,000,000
500,000
0
1
8
15
22
29 36
43
50 57
64 71
78
85 92
99 106 113 120
t
Actual Data At
Figure 4.2
Forecast value Ft
Actual and Forecast Value Using Naïve 1 Model for Johor Tourism
4.2.2 The Naïve 2 Model
This model uses an equation:
Ft  At 1 
At 1
At 2
where F = forecast value
A = actual value
45
t = some time period
The forecast value is obtained by multiplying the current value by the growth rate
between the current value and the previous value. For example,
F3  A2 
A2
A1
F4  A3 
A3
 1151772
A2
F5  A4 
A4
 852033
A3
Thus,

F120  A119 
A119
 2079311
A118
MAPE and MSE for this model are 0.421844828 and 1.172248109134E12 respectively.
46
9,000,000
8,000,000
7,000,000
visitors
6,000,000
5,000,000
4,000,000
3,000,000
2,000,000
1,000,000
0
1
8
15 22 29 36 43 50 57 64 71 78 85 92 99 106 113 120
t
Actual Data At
Figure 4.3
Forecast value Ft
Actual and Forecast Value Using Naïve 2 Model for Johor Tourism
4.2.3 The Seasonal Naïve Model
We used an equation which is:
Ft  At  m
where F = forecast value
A = actual value
t = some time period
47
m = number of periods in a year
For this model, we used m  4 and m  12 which represent 4 quarters and 12
months respectively. For example, when m  4 , the calculation becomes:
F5  A1  1213874
F6  A2  1107735

F120  A116  2225317
while for m  12 ,
F13  A1  1213874
F14  A2  1107735

F120  A108  2145665
Table 4.1
Value of MAPE and MSE when m  4 and m  12
m
MAPE
MSE
m=4
0.287878460
3.60123345545E11
m=12
0.254703814
3.18035348453E11
From the table above, we can see that the seasonal model for m=12 gives much smaller
MAPE and MSE than when m=4.
48
4.3
Moving Average
We can average any number of periods to produce a forecast through moving
average model. The general equation we used is:
Ft 
At 1  At  2  At  n
n
where F = forecast value
A = actual value
t = some time period
n = number of past period
For this model, we used the average of the previous three values and previous six values
to serve as our forecast for the next period. Example for these models is as follows:
Calculation example for 3-month moving average model
A1  A2  A3 1213874  1107735  1129539

 1150383
3
3
A  A3  A4 1107735  1129539  981022
F5  2

 1072765
3
3

A  A118  A119 1866076  2365402  2217748
F120  117

 2149742
3
3
F4 
49
4,000,000
3,500,000
3,000,000
visitors
2,500,000
2,000,000
1,500,000
1,000,000
500,000
0
1
9
17
25
33
41
49
57
65
73
81
89
97 105 113
t
Actual Data At
Figure 4.4
Forecast value Ft
Actual and Forecast Value Using 3-month Moving Average for Johor
Tourism
Calculation example for 6-month moving average model
A1  A2    A6 1213874  1107735    1240579

 1146404
6
6
A  A3    A7 1107735  1129539    1334745
F8  2

 1166549
6
6

A  A115    A119 1770749  1914976   2217748
F120  114

 2060045
6
6
F7 
50
4,000,000
3,500,000
3,000,000
visitors
2,500,000
2,000,000
1,500,000
1,000,000
500,000
0
1
8
15 22 29 36 43 50 57 64 71 78 85 92 99 106 113 120
t
Actual Data At
6-month moving average
Figure 4.5
Actual and Forecast Value Using 6-month Moving Average for Johor
Tourism
Table 4.2
Value of MAPE and MSE using 3-month moving average and 6-month
moving average
Moving average
MAPE
MSE
3-month moving average
0.202699029
1.89307879049E11
6-month moving average
0.220517655
2.13774356448E11
The smallest MAPE and MSE among these two models is produced by 3-month moving
average with MAPE=0.202699029 and MSE=1.89307879049E11.
51
4,000,000
3,500,000
3,000,000
visitors
2,500,000
2,000,000
1,500,000
1,000,000
500,000
0
1
8
15 22 29 36 43 50 57 64 71 78 85 92 99 106 113 120
t
Actual Data At
Figure 4.6
4.4
3-month moving average
6-month moving average
Actual and Forecast Value of 3-month Moving Average and 6-month
Moving Average for Johor Tourism
Exponential Smoothing
Exponential smoothing is a technique that used all the preceding observation to
determine a smoothed value for a particular time period. Single exponential smoothing
that is being used is
Ft   At 1  (1   ) Ft 1
where F = forecast value
A = actual value
t = some time period
 = smoothing constant between 0 and 1
52
The one-step ahead forecast Ft represents a weighted average of all past
observations. The starting value for the exponentially weighted average is the number
of visitors in the first month of year 1999 (January) which is 1213874. Thus,
F1  A1
Then,
F2   A1  (1   ) F1
F3   A2  (1   ) F2

Fn   An 1  (1   ) Fn 1
for n = 1,2,…,120
For example, when  =0.35,
F1  A1  1213874
F2   A1  (1   ) F1  0.35(1213874)  (1  0.35)(1213874)  1213874
F3   A2  (1   ) F2  0.35(1107735)  (1  0.35)(1213874)  1176725

F120   A119  (1   ) F119  0.35(2217748)  (1  0.35)(2108095)  2146474
4.4.1 Implementation of Exponential Smoothing
Different value of  has been used to find the best value of  to minimize the
value of MAPE and MSE by using the exponential smoothing method. Possible values
for the smoothing constant,  in 0.05 increments in finding the one that minimizes the
MSE and MAPE are shown in the table below.
53
Table 4.3
Values of  , MSE and MAPE for Exponential Smoothing
Smoothing
Constant,
MSE
MAPE
0.05
2.19519581800E11
0.227142
0.10
1.98058992400E11
0.218336
0.15
1.89277642000E11
0.213353
0.20
1.85128160100E11
0.209261
0.25
1.82917659200E11
0.205530
0.30
1.81694205300E11
0.201835
0.35
1.81156299600E11
0.199749
0.40
1.81244894900E11
0.197842
0.45
1.81983353400E11
0.197005
0.50
1.83416091700 E11
0.196779
0.55
1.85587696400 E11
0.197236
0.60
1.88538524400 E11
0.198176
0.65
1.92306673000 E11
0.199154
0.70
1.96932085000 E11
0.200304
0.75
2.02461112300 E11
0.201745
0.80
2.08951002000 E11
0.203417
0.85
2.16474284900 E11
0.205170
0.90
2.25123289500 E11
0.207034
0.95
2.35015131700 E11
0.210168
1.00
2.46297619000 E11
0.215486

54
From Table 4.3, we can see that the value of MSE decreases until  =0.35 and
starts to increase when  =0.40. For MAPE, the value also decreases until  =0.50 and
increases when  =0.55.
The smallest MSE and MAPE from this table is when
 =0.35 and  =0.50 respectively. The search then continues until it found that the best
value for MSE is 1.81116159000E11 when  =0.37. While for MAPE, the best value
is 0.19673 when  =0.48.
4,000,000
3,500,000
visitors
3,000,000
2,500,000
2,000,000
1,500,000
1,000,000
500,000
0
1
8
15 22 29
36 43 50
57 64
71 78 85
92 99 106 113 120
t
Actual Data At
Figure 4.7
alpha=0.05
alpha=0.20
alpha=0.37
Actual and Forecast Value Using Exponential Smoothing With
  0.05 ,   0.20 and   0.37
55
4.5
Holt’s Model
When a time series contains a trend component, exponential smoothing involves
two updating steps, one for the smoothed estimate and one for a trend estimate. The
updating formula for the smoothed estimate is as follows:
at   At  (1   )(at 1  bt 1 )
where at = the smoothed estimate for the current period t
at 1 = the smoothed estimate for the preceding period t-1
At = the actual data point for the current period t
 = smoothing constant between 0 and 1
While the updating formula for the trend estimate is:
bt   (at  at 1 )  (1   )bt 1
where bt = the trend estimate for the current period t
bt 1 = the trend estimate for the preceding period t-1
 = the smoothing constant between 0 and 1
In Holt’s method, we begin by calculating the initial value, a and b . Equations below
are used to calculate the initial trend value, b and overall smoothed value, a
respectively.
b 
Am  A1
(m  1) L
where L = the period
56
m = the number of year
A1 = the average of data for year 1
Am = the average of data for year m
b is calculated as follows:
b 
Am  A1
A10  A1 2001828  1276890


 6712.383
(m  1) L (10  1)12
108
Next, calculate the initial overall smoothed value, a :
a  A1 
L
12
b  1276890  (6712.383)  1236616
2
2
Using the initial value a =1236616 and b =6712.383, the update estimates, a1 and b1 as
shown below:
at   At  (1   )(at 1  bt 1 )
a1  0.36 *(1213874)  (1  0.36) *(1236616  6712.383)
a1  1232724.8
bt   (at  at 1 )  (1   )bt 1
b1  0 * (1232724.8  1236616)  (1  0) * 6712.383
b1  6712.383
Then, the forecast made in F2 is
F2  a1  b1
F2  1239437
57
Next, we can obtain the updated estimates, a2 and b2 using the same way we did to a1
and b1 which is as follows:
at   At  (1   )(at 1  bt 1 )
a2  0.36 * (1107735)  (1  0.36)* (1232724.8  6712.383)
a2  1192024.4
bt   (at  at 1 )  (1   )bt 1
b2  0 *(1192024.4  1232724.8)  (1  0) * (6712.383)
b2  6712.383
Since we have calculated a2 and b2 , the forecast F3 is
F3  a2  b2
F3  11920.24  6712.383
F3  1198737
The procedure continuous through the entire 120 periods of historical data.
The
forecast value for time period t+1 is the current smoothed values plus the current
smoothed trend value. An equation below is used to forecast m number of time periods
into the future.
Ft  m  at  mbt
(m=1, 2, 3…)
Hence, the forecast for one period ahead is, for F121 :
F1201  a120  (1) * (b120 )
F121  1989607.3  (1) * (6712.383)
F121  1996320
58
For F122 :
F120  2  a120  (2)* (b120 )
F122  1989607.3  (2)* (6712.383)
F122  2003032
and so on. Referring to Appendix G , we knew that by using   0.36 and   0.00 ,
the value of MSE and MAPE are 1.79321039826E11 and 0.201822197 respectively.
The search continues and found that the best value of  and  are   0.48 and
  0.00 for MSE. For MAPE,   0.36 and   0.00 are the best value found.
However, the value of  from the results indicates that the data does not show a linear
trend. Thus, Holt’s method is not an appropriate method for the data.
4.6
Tabu Search Application
The following characteristics will be examined to get the best value of  , 
and  to minimize MSE and MAPE:
i. an initial solution
ii. the tabu list size, Ts
iii. the neighbourhood size
In this study, the neighbourhood size used is 1 and the tabu list size is 100. The
Automated Computerized Forecasting System has been run to obtain the best value of
 ,  and  to minimize MSE and MAPE.
59
4.6.1 Forecasting Graph Using Tabu Search
Figure 4.8
The Forecasting Graph Using Tabu Search in Finding MSE
Figure 4.7 shows the parameter values of  ,  and  found using Tabu Search
which are  =0.49,  =0.00 and  =0.00 with MSE=1.52074940277814E11. Since the
values of  and  are zeroes, it clearly shows that Holt-Winter’s method is not suitable
to be applied into Johor tourism data.
60
Figure 4.9
The Forecasting Graph Using Tabu Search in Finding MAPE
The same also goes for MAPE. The parameter values of  =0.69,  =0.00 and
 =0.00 with MAPE=0.182282 are found which is shown in Figure 4.9 above. Thus,
the result strictly declines that the data cannot be applied using Holt-Winter’s method
because the data pattern does not show any linear trend and seasonality.
61
4.7
Summary
In this chapter, the forecast has been done using the naïve model, moving
average and single exponential smoothing. While from the result of Holt’s method and
Tabu Search, we can ensure that the data shows irregular patterns (no linear trend and
seasonality).
CHAPTER 5
CONCLUSION AND RECOMMENDATION
5.1
Introduction
This chapter summarized the results and findings obtained in Chapter 4. From
these observations, conclusion is drawn. Finally, some recommendations for future
research in this area which might be pursued are offered.
5.2
Conclusion
In this study, we have examined four time series forecast which are the naïve
model, moving average, single exponential smoothing and Holt’s method. Tabu Search
also has been applied in order to gain the objectives of this study. Each method has
advantages and disadvantages in dealing with particular time series.
63
From Table 5.1 below, the minimum MAPE and MSE for Naïve model equal to
0.213690271 and 2.44245138814E11 (both belong to Naïve 1). For seasonal Naïve, it
produces the smallest MAPE and MSE when m=12 i.e. MAPE=0.254703814 and
MSE=3.18035348453E11.
While
for
moving
average,
the
minimum
MAPE=0.202699029 and MSE=1.89307879049, which come from 3-month moving
average. For exponential smoothing, when  =0.48, it shows the smallest MAPE and
when  =0.37, it shows the smallest MSE with the value are 0.196703000 and
1.81116159000E11 respectively.
For Holt’s method, its initial overall value, a  1236616 and trend,
b  6712.383 . From this method, we obtained the smallest MAPE=0.197288000 when
 =0.48 and  =0.00 while the smallest MSE=1.79328268900 when  =0.36 and
 =0.00. Nevertheless, the Holt’s method cannot be used as the forecast method for
this data since the value of  =0.00 which means the data has no linear trend. Lastly,
using Tabu Search, we get the smallest MAPE=0.182282000 when  =0.69,  =0.00
and  =0.00 and when  =0.49,  =0.00 and  =0.00, it gives the smallest
MSE=1.52074940277. From the value of  and  , it certainly confirmed that the data
has no linear trend and seasonality. Hence, Holt-Winter’s is also not an appropriate
method to be applied.
Comparison of the MAPE and MSE among the naïve model, moving average
and single exponential smoothing clearly indicates that single exponential smoothing
gives the best fit while the Naïve 2 model gives the poorest fit.
64
Table 5.1
Comparison of the Selected Model to Forecast Johor Tourism for 120
periods
Method
Characteristics
MAPE
MSE
1. Naïve 1
0.213690271
2.44245138814E11
2. Naïve 2
0.421844828
1.172248109134E12
m=4
0.287878460
3.60123345545E11
m=12
0.254703814
3.18035348453E11
3-month
0.202699099
1.89307879049E11
0.220517655
2.13774356448E11
 =0.35
0.199749000
1.81156299600E11
Exponential
 =0.37
0.198978000
1.81116159000E11
Smoothing
 =0.48
0.196703000
1.82756737700E11
 =0.50
0.196779000
1.83416091700E11
3. Seasonal Naïve
moving average
4. Moving Average
6-month
moving average
5. Single
65
5.3
Recommendation
In this study, it focuses on forecasting technique i.e. the Naïve, moving average,
single exponential smoothing and Holt’s method. For further studies, these models can
be extended to cover more regions. Better method should also be used as a comparison
to these models such as Box-Jenkins, ARIMA, Neural Network and others.
Besides that, combining quantitative and qualitative forecasting technique could
be an especially effective way of achieving convergent validity. Related to this concept
is the value of combining forecasts from different models into a single forecast. Current
and previous values of other related factors which could provide additional useful
information, such as economic variables, also could be incorporated into forecast of
visitors in future studies.
This study also focuses on Tabu Search which helps to ensure the pattern of the
data.
Other heuristic method could be involved in the future such as Simulated
Annealing, Neural Network or Evolutionary Algorithm.
66
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72
APPENDIX A
Johor Tourism Data
(Data is taken from Majlis Tindakan Pelancongan Negeri Johor)
Table 1.1
Period
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
Number of Visitors in Johor
Year Month
Jan
Feb
Mar
Apr
May
1999 June
July
Aug
Sept
Oct
Nov
Dec
Jan
Feb
Mar
Apr
May
2000 June
July
Aug
Sept
Oct
Nov
Dec
Jan
Feb
No. of
Visitors
1,213,874
1,107,735
1,129,539
981,022
1,205,673
1,240,579
1,334,745
1,514,357
1,213,478
1,474,035
1,514,357
1,393,287
1,088,378
1,080,564
1,109,080
1,049,116
1,013,572
1,205,808
1,109,254
2,146,602
953,087
952,753
930,598
1,193,784
1,080,693
944,095
73
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
2001
2002
2003
2004
Mar
Apr
May
June
July
Aug
Sept
Oct
Nov
Dec
Jan
Feb
Mar
Apr
May
June
July
Aug
Sept
Oct
Nov
Dec
Jan
Feb
Mar
Apr
May
June
July
Aug
Sept
Oct
Nov
Dec
Jan
Feb
Mar
Apr
May
June
1,305,077
1,217,528
1,190,167
1,345,568
1,120,333
1,393,845
981,729
969,556
962,730
1,239,244
1,234,351
1,148,229
1,333,280
985,363
914,062
1,035,614
1,270,599
1,911,344
1,505,878
2,590,348
1,747,456
3,450,757
1,057,455
1,171,256
1,087,495
1,083,396
1,277,491
1,274,229
1,133,624
1,103,639
2,103,175
2,188,585
1,148,499
1,545,169
1,333,513
1,145,025
1,270,344
889,574
919,763
1,129,101
74
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
2005
2006
2007
July
Aug
Sept
Oct
Nov
Dec
Jan
Feb
Mar
Apr
May
June
July
Aug
Sept
Oct
Nov
Dec
Jan
Feb
Mar
Apr
May
June
July
Aug
Sept
Oct
Nov
Dec
Jan
Feb
Mar
Apr
May
June
July
Aug
Sept
Oct
870,424
923,640
994,139
2,846,747
906,309
1,068,855
1,492,337
840,330
1,016,850
887,480
1,003,125
907,830
979,130
859,112
794,946
600,394
814,820
1,073,763
949,066
1,173,758
1,212,861
1,649,596
2,204,547
2,528,855
2,136,125
1,750,787
1,452,891
975,078
1,302,898
1,364,959
1,544,982
1,863,259
1,806,980
1,806,074
2,229,541
2,002,932
2,147,831
2,164,777
2,057,634
2,224,213
75
107
108
109
110
111
112
113
114
115
116
117
118
119
120
2008
Nov
Dec
Jan
Feb
Mar
Apr
May
June
July
Aug
Sept
Oct
Nov
Dec
1,979,464
2,145,665
2,345,687
1,609,402
2,087,459
1,927,879
2,019,837
1,770,749
1,914,976
2,225,317
1,866,076
2,365,402
2,217,748
1,671,398
76
APPENDIX B
Table 4.4
Calculation for Johor Tourism Data using Naïve 1 Model
(Assume F2  A1 )
Period
Actual
Data
Forecast
value
Error
t
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
At
1,213,874
1,107,735
1,129,539
981,022
1,205,673
1,240,579
1,334,745
1,514,357
1,213,478
1,474,035
1,514,357
1,393,287
1,088,378
1,080,564
1,109,080
1,049,116
1,013,572
1,205,808
1,109,254
2,146,602
953,087
952,753
930,598
1,193,784
1,080,693
944,095
1,305,077
1,217,528
1,190,167
1,345,568
1,120,333
1,393,845
981,729
Ft
et
Square Error
Absolute
Percentage
Error
1,213,874 -106,139
11265487321 0.095816238
1,107,735
21,804
475414416
0.01930345
1,129,539 -148,517
22057299289 0.151390081
981,022
224,651
50468071801
0.1863283
1,205,673
34,906
1218428836
0.028136862
1,240,579
94,166
8867235556
0.070549805
1,334,745
179,612
32260470544 0.118606115
1,514,357 -300,879
90528172641 0.247947635
1,213,478
260,557
67889950249 0.176764459
1,474,035
40,322
1625863684
0.026626482
1,514,357 -121,070
14657944900 0.086895234
1,393,287 -304,909
92969498281 0.280149911
1,088,378
-7,814
61058596
0.007231409
1,080,564
28,516
813162256
0.0257114
1,109,080
-59,964
3595681296
0.057156692
1,049,116
-35,544
1263375936
0.035068056
1,013,572
192,236
36954679696 0.159425049
1,205,808
-96,554
9322674916
0.087044085
1,109,254 1,037,348 1076090873104 0.483251204
2,146,602 -1,193,515 1424478055225 1.252262385
953,087
-334
111556
0.000350563
952,753
-22,155
490844025
0.023807272
930,598
263,186
69266870596 0.220463668
1,193,784 -113,091
12789574281 0.104646741
1,080,693 -136,598
18659013604 0.144686711
944,095
360,982
130308004324 0.276598239
1,305,077
-87,549
7664827401
0.071907176
1,217,528
-27,361
748624321
0.022989211
1,190,167
155,401
24149470801 0.115491005
1,345,568 -225,235
50730805225 0.201042904
1,120,333
273,512
74808814144 0.196228419
1,393,845 -412,116
169839597456 0.419785908
77
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
969,556
962,730
1,239,244
1,234,351
1,148,229
1,333,280
985,363
914,062
1,035,614
1,270,599
1,911,344
1,505,878
2,590,348
1,747,456
3,450,757
1,057,455
1,171,256
1,087,495
1,083,396
1,277,491
1,274,229
1,133,624
1,103,639
2,103,175
2,188,585
1,148,499
1,545,169
1,333,513
1,145,025
1,270,344
889,574
919,763
1,129,101
870,424
923,640
994,139
2,846,747
906,309
1,068,855
1,492,337
840,330
1,016,850
887,480
1,003,125
907,830
979,130
981,729
-12,173
148181929
0.012555231
969,556
-6,826
46594276
0.007090254
962,730
276,514
76459992196 0.223131199
1,239,244
-4,893
23941449
0.003964026
1,234,351
-86,122
7416998884
0.075004202
1,148,229
185,051
34243872601 0.138793802
1,333,280 -347,917
121046238889 0.353085107
985,363
-71,301
5083832601
0.078004555
914,062
121,552
14774888704 0.117371917
1,035,614
234,985
55217950225 0.184940331
1,270,599
640,745
410554155025 0.335232695
1,911,344 -405,466
164402677156 0.269255544
1,505,878 1,084,470 1176075180900 0.418658034
2,590,348 -842,892
710466923664 0.482353776
1,747,456 1,703,301 2901234296601 0.493602128
3,450,757 -2,393,302 5727894463204 2.263266049
1,057,455
113,801
12950667601 0.097161509
1,171,256
-83,761
7015905121
0.077021963
1,087,495
-4,099
16801801
0.003783473
1,083,396
194,095
37672869025 0.151934534
1,277,491
-3,262
10640644
0.002559979
1,274,229 -140,605
19769766025 0.124031425
1,133,624
-29,985
899100225
0.02716921
1,103,639
999,536
999072215296 0.47525099
2,103,175
85,410
7294868100
0.039025215
2,188,585 -1,040,086 1081778887396 0.90560462
1,148,499
396,670
157347088900 0.256716256
1,545,169 -211,656
44798262336 0.158720612
1,333,513 -188,488
35527726144 0.164614746
1,145,025
125,319
15704851761 0.098649657
1,270,344 -380,770
144985792900 0.428036341
889,574
30,189
911375721
0.032822586
919,763
209,338
43822398244 0.185402369
1,129,101 -258,677
66913790329
0.29718505
870,424
53,216
2831942656
0.057615521
923,640
70,499
4970109001
0.070914631
994,139
1,852,608 3432156401664 0.650780698
2,846,747 -1,940,438 3765299631844 2.141033577
906,309
162,546
26421202116 0.152074884
1,068,855
423,482
179337004324 0.283771025
1,492,337 -652,007
425113128049 0.775893994
840,330
176,520
31159310400 0.173594926
1,016,850 -129,370
16736596900 0.145772299
887,480
115,645
13373766025 0.115284735
1,003,125
-95,295
9081137025
0.104970094
907,830
71,300
5083690000
0.072819748
78
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
859,112
794,946
600,394
814,820
1,073,763
949,066
1,173,758
1,212,861
1,649,596
2,204,547
2,528,855
2,136,125
1,750,787
1,452,891
975,078
1,302,898
1,364,959
1,544,982
1,863,259
1,806,980
1,806,074
2,229,541
2,002,932
2,147,831
2,164,777
2,057,634
2,224,213
1,979,464
2,145,665
2,345,687
1,609,402
2,087,459
1,927,879
2,019,837
1,770,749
1,914,976
2,225,317
1,866,076
2,365,402
2,217,748
1,671,398
979,130
859,112
794,946
600,394
814,820
1,073,763
949,066
1,173,758
1,212,861
1,649,596
2,204,547
2,528,855
2,136,125
1,750,787
1,452,891
975,078
1,302,898
1,364,959
1,544,982
1,863,259
1,806,980
1,806,074
2,229,541
2,002,932
2,147,831
2,164,777
2,057,634
2,224,213
1,979,464
2,145,665
2,345,687
1,609,402
2,087,459
1,927,879
2,019,837
1,770,749
1,914,976
2,225,317
1,866,076
2,365,402
2,217,748
-120,018
-64,166
-194,552
214,426
258,943
-124,697
224,692
39,103
436,735
554,951
324,308
-392,730
-385,338
-297,896
-477,813
327,820
62,061
180,023
318,277
-56,279
-906
423,467
-226,609
144,899
16,946
-107,143
166,579
-244,749
166,201
200,022
-736,285
478,057
-159,580
91,958
-249,088
144,227
310,341
-359,241
499,326
-147,654
-546,350
14404320324
4117275556
37850480704
45978509476
67051477249
15549341809
50486494864
1529044609
190737460225
307970612401
105175678864
154236852900
148485374244
88742026816
228305262969
107465952400
3851567721
32408280529
101300248729
3167325841
820836
179324300089
51351638881
20995720201
287166916
11479622449
27748563241
59902073001
27622772401
40008800484
542115601225
228538495249
25465776400
8456273764
62044831744
20801427529
96311536281
129054096081
249326454276
21801703716
298498322500
0.139700062
0.080717432
0.324040547
0.263157507
0.241154705
0.131389176
0.191429579
0.032240298
0.264752703
0.251730174
0.128243019
0.1838516
0.220094163
0.205036716
0.490025413
0.251608338
0.0454673
0.116521099
0.170817369
0.031145336
0.000501641
0.18993461
0.113138639
0.067462943
0.007828058
0.052070971
0.074893457
0.123644077
0.07745897
0.085272246
0.457489801
0.229013839
0.082774904
0.045527436
0.140668158
0.075315304
0.139459232
0.192511452
0.211095619
0.066578349
0.326882047
79
APPENDIX C
Table 4.5
Calculation for Johor Tourism Data using Naïve 2 Model
Period
Actual
Data
Forecast
value
Error
t
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
At
1,213,874
1,107,735
1,129,539
981,022
1,205,673
1,240,579
1,334,745
1,514,357
1,213,478
1,474,035
1,514,357
1,393,287
1,088,378
1,080,564
1,109,080
1,049,116
1,013,572
1,205,808
1,109,254
2,146,602
953,087
952,753
930,598
1,193,784
1,080,693
944,095
1,305,077
1,217,528
1,190,167
1,345,568
1,120,333
1,393,845
981,729
969,556
962,730
Ft
et
1010877
1151772
852033
1481768
1276496
1436059
1718139
972379
1790539
1555782
1281896
850196
1072806
1138349
992394
979232
1434504
1020431
4154053
423169
952419
908958
1531403
978315
824763
1804083
1135852
1163421
1521260
932800
1734131
691463
957534
118,662
-170,750
353,640
-241,189
58,249
78,298
-504,661
501,656
-276,182
-162,495
-193,518
230,368
36,274
-89,233
21,178
226,576
-325,250
1,126,171
-3,200,966
529,584
-21,821
284,826
-450,710
-34,220
480,314
-586,555
54,315
182,147
-400,927
461,045
-752,402
278,093
5,196
Square Error
Absolute
Percentage
Error
14080763556
29155622955
125061461434
58172318620
3392994948
6130630216
254682485885
251658801182
76276267973
26404625704
37449339895
53069535584
1315795782
7962445307
448506006
51336579994
105787462751
1268260050428
10246184717954
280459505242
476161149
81125744908
203139184120
1171040866
230701744584
344047061328
2950110054
33177576106
160742319330
212562354634
566108364505
77335882563
26999042
0.105053826
0.174053362
0.293313609
0.194416787
0.04364086
0.051704018
0.415879615
0.340328458
0.18237548
0.116627086
0.177804328
0.213192611
0.032706297
0.085054975
0.020894382
0.187903689
0.293214941
0.524629402
3.358524685
0.555846348
0.023448489
0.238590746
0.417056134
0.036246855
0.368035154
0.481759146
0.045636382
0.135368207
0.357863979
0.330771966
0.766404711
0.286825411
0.005397214
80
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
1,239,244
1,234,351
1,148,229
1,333,280
985,363
914,062
1,035,614
1,270,599
1,911,344
1,505,878
2,590,348
1,747,456
3,450,757
1,057,455
1,171,256
1,087,495
1,083,396
1,277,491
1,274,229
1,133,624
1,103,639
2,103,175
2,188,585
1,148,499
1,545,169
1,333,513
1,145,025
1,270,344
889,574
919,763
1,129,101
870,424
923,640
994,139
2,846,747
906,309
1,068,855
1,492,337
840,330
1,016,850
887,480
1,003,125
907,830
979,130
859,112
794,946
955952
1595178
1229477
1068116
1548154
728234
847920
1173330
1558903
2875208
1186426
4455808
1178839
6814320
324048
1297304
1009724
1079312
1506359
1270975
1008534
1074447
4007964
2277464
602695
2078841
1150849
983179
1409379
622935
950977
1386084
671010
980110
1070019
8151746
288538
1260554
2083603
473187
1230450
774569
1133839
821588
1056030
753805
283,292
-360,827
-81,248
265,164
-562,791
185,828
187,694
97,269
352,441
-1,369,330
1,403,922
-2,708,352
2,271,918
-5,756,865
847,208
-209,809
73,672
198,179
-232,130
-137,351
95,105
1,028,728
-1,819,379
-1,128,965
942,474
-745,328
-5,824
287,165
-519,805
296,828
178,124
-515,660
252,630
14,029
1,776,728
-7,245,437
780,317
231,783
-1,243,273
543,663
-342,970
228,556
-226,009
157,542
-196,918
41,141
80254324780
130196105636
6601289405
70312039817
316733944099
34531930111
35228906248
9461260375
124214626246
1875063521070
1970996490925
7335168835368
5161612756492
33141491354415
717761310469
44019822916
5427553309
39274737679
53884311153
18865387669
9044945636
1058281050439
3310138955302
1274560848286
888256623645
555514384263
33924424
82463619734
270196955531
88106785955
31728334634
265905560896
63821971598
196826284
3156762377670
52496355357759
608893898162
53723576042
1545728303887
295569427783
117628318049
52237727020
51080236111
24819530010
38776630180
1692553402
0.228600617
0.292321207
0.070759682
0.198881087
0.571151148
0.20329878
0.181239004
0.076553665
0.184394308
0.909323058
0.541981936
1.549882619
0.658382581
5.444075366
0.723332858
0.192928717
0.068000925
0.155131073
0.182172863
0.121161275
0.086173938
0.4891309
0.831303663
0.982991281
0.609948603
0.558920965
0.00508676
0.226052782
0.584329947
0.322722128
0.157757802
0.592424285
0.273515773
0.014112191
0.624125712
7.99444435
0.730049012
0.155315768
1.479505934
0.534654052
0.38645361
0.22784373
0.24895561
0.160900139
0.22921089
0.051752765
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
600,394
814,820
1,073,763
949,066
1,173,758
1,212,861
1,649,596
2,204,547
2,528,855
2,136,125
1,750,787
1,452,891
975,078
1,302,898
1,364,959
1,544,982
1,863,259
1,806,980
1,806,074
2,229,541
2,002,932
2,147,831
2,164,777
2,057,634
2,224,213
1,979,464
2,145,665
2,345,687
1,609,402
2,087,459
1,927,879
2,019,837
1,770,749
1,914,976
2,225,317
1,866,076
2,365,402
2,217,748
1,671,398
735572
453456
1105827
1414996
838850
1451646
1253267
2243593
2946193
2900872
1804386
1434961
1205682
654404
1740931
1429976
1748748
2247103
1752401
1805168
2752298
1799355
2303212
2181857
1955794
2404278
1761647
2325821
2564355
1104229
2707518
1780498
2116181
1552379
2070950
2585952
1564829
2998338
2079311
-135,178
361,364
-32,064
-465,930
334,908
-238,785
396,329
-39,046
-417,338
-764,747
-53,599
17,930
-230,604
648,494
-375,972
115,006
114,511
-440,123
53,673
424,373
-749,366
348,476
-138,435
-124,223
268,419
-424,814
384,018
19,866
-954,953
983,230
-779,639
239,339
-345,432
362,597
154,367
-719,876
800,573
-780,590
-407,913
18273220976
130584011537
1028071958
217090695652
112163256301
57018266125
157076920750
1524623188
174170645566
584837242782
2872829809
321500794
53178174824
420544982121
141354698618
13226344511
13112759223
193708490967
2880803284
180092057579
561548748573
121435256689
19164385657
15431279416
72048806674
180466650146
147469973329
394670957
911935882795
966741914227
607837156772
57282958783
119323480372
131476771546
23829094144
518221254889
640917803245
609320781494
166392944341
0.225149615
0.44348948
0.029860929
0.49093522
0.285329542
0.196877448
0.240258408
0.017711767
0.165030248
0.358006447
0.03061411
0.012341217
0.236497937
0.497732283
0.275445396
0.074438308
0.061457348
0.243568422
0.029718115
0.190340768
0.374134301
0.162245362
0.063949078
0.060371621
0.120680478
0.214610453
0.178973975
0.0084693
0.593359111
0.4710178
0.404402517
0.11849401
0.195076947
0.1893482
0.069368432
0.385769851
0.338451317
0.351974174
0.244054924
82
APPENDIX D
Table 4.6
Calculation for Johor Tourism Data using Seasonal Naïve Model
(when m=4)
Period
Actual
Data
Year
t
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
At
1,213,874
1,107,735
1,129,539
981,022
1,205,673
1,240,579
1,334,745
1,514,357
1,213,478
1,474,035
1,514,357
1,393,287
1,088,378
1,080,564
1,109,080
1,049,116
1,013,572
1,205,808
1,109,254
2,146,602
953,087
952,753
930,598
1,193,784
1,080,693
944,095
1,305,077
1,217,528
1,190,167
1,345,568
1,120,333
1,393,845
981,729
1
1
1
1
2
2
2
2
3
3
3
3
4
4
4
4
5
5
5
5
6
6
6
6
7
7
7
7
8
8
8
8
9
Forecast
value
Error
Ft
et
1,213,874
1,107,735
1,129,539
981,022
1,205,673
1,240,579
1,334,745
1,514,357
1,213,478
1,474,035
1,514,357
1,393,287
1,088,378
1,080,564
1,109,080
1,049,116
1,013,572
1,205,808
1,109,254
2,146,602
953,087
952,753
930,598
1,193,784
1,080,693
944,095
1,305,077
1,217,528
1,190,167
-8,201
132,844
205,206
533,335
7,805
233,456
179,612
-121,070
-125,100
-393,471
-405,277
-344,171
-74,806
125,244
174
1,097,486
-60,485
-253,055
-178,656
-952,818
127,606
-8,658
374,479
23,744
109,474
401,473
-184,744
176,317
-208,438
Square Error
Absolute
Percentage
Error
67256401
17647528336
42109502436
284446222225
60918025
54501703936
32260470544
14657944900
15650010000
154819427841
164249446729
118453677241
5595937636
15686059536
30276
1204475520196
3658435225
64036833025
31917966336
907862141124
16283291236
74960964
140234521441
563777536
11984556676
161180569729
34130345536
31087684489
43446399844
0.00680201
0.107082258
0.153741726
0.352185779
0.006431925
0.158378872
0.118606115
0.086895234
0.114941684
0.364134841
0.365417283
0.328058098
0.073804328
0.103867282
0.000156862
0.511266644
0.063462202
0.265603992
0.191979781
0.798149414
0.118077937
0.009170687
0.286940158
0.01950181
0.09198205
0.298366935
0.164900971
0.126496849
0.212317248
83
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
969,556
962,730
1,239,244
1,234,351
1,148,229
1,333,280
985,363
914,062
1,035,614
1,270,599
1,911,344
1,505,878
2,590,348
1,747,456
3,450,757
1,057,455
1,171,256
1,087,495
1,083,396
1,277,491
1,274,229
1,133,624
1,103,639
2,103,175
2,188,585
1,148,499
1,545,169
1,333,513
1,145,025
1,270,344
889,574
919,763
1,129,101
870,424
923,640
994,139
2,846,747
906,309
1,068,855
1,492,337
840,330
1,016,850
887,480
1,003,125
907,830
979,130
9
9
9
10
10
10
10
11
11
11
11
12
12
12
12
13
13
13
13
14
14
14
14
15
15
15
15
16
16
16
16
17
17
17
17
18
18
18
18
19
19
19
19
20
20
20
1,345,568
1,120,333
1,393,845
981,729
969,556
962,730
1,239,244
1,234,351
1,148,229
1,333,280
985,363
914,062
1,035,614
1,270,599
1,911,344
1,505,878
2,590,348
1,747,456
3,450,757
1,057,455
1,171,256
1,087,495
1,083,396
1,277,491
1,274,229
1,133,624
1,103,639
2,103,175
2,188,585
1,148,499
1,545,169
1,333,513
1,145,025
1,270,344
889,574
919,763
1,129,101
870,424
923,640
994,139
2,846,747
906,309
1,068,855
1,492,337
840,330
1,016,850
-376,012
-157,603
-154,601
252,622
178,673
370,550
-253,881
-320,289
-112,615
-62,681
925,981
591,816
1,554,734
476,857
1,539,413
-448,423
-1,419,092
-659,961
-2,367,361
220,036
102,973
46,129
20,243
825,684
914,356
14,875
441,530
-769,662
-1,043,560
121,845
-655,595
-413,750
-15,924
-399,920
34,066
74,376
1,717,646
35,885
145,215
498,198
-2,006,417
110,541
-181,375
-489,212
67,500
-37,720
141385024144
24838705609
23901469201
63817874884
31924040929
137307302500
64455562161
102585043521
12682138225
3928907761
857440812361
350246177856
2417197810756
227392598449
2369792384569
201083186929
2013822104464
435548521521
5604398104321
48415841296
10603438729
2127884641
409779049
681754067856
836046894736
221265625
194948740900
592379594244
1089017473600
14846204025
429804804025
171189062500
253573776
159936006400
1160492356
5531789376
2950307781316
1287733225
21087396225
248201247204
4025709177889
12219312681
32896890625
239328380944
4556250000
1422798400
0.387818754
0.163704258
0.124754286
0.204659777
0.155607462
0.277923617
0.257652256
0.350401833
0.108742253
0.049331851
0.484465905
0.393003949
0.600202753
0.272886413
0.446108781
0.424058707
1.211598489
0.60686348
2.185129906
0.172240744
0.080812005
0.040691623
0.018342048
0.3925893
0.417784093
0.012951687
0.285748679
0.577168727
0.911386214
0.095914965
0.73697635
0.449844145
0.014103256
0.459454243
0.036882335
0.074814488
0.603371497
0.039594664
0.135860337
0.333837464
2.38765366
0.108709249
0.204370803
0.487687975
0.074353128
0.038523996
84
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
859,112
794,946
600,394
814,820
1,073,763
949,066
1,173,758
1,212,861
1,649,596
2,204,547
2,528,855
2,136,125
1,750,787
1,452,891
975,078
1,302,898
1,364,959
1,544,982
1,863,259
1,806,980
1,806,074
2,229,541
2,002,932
2,147,831
2,164,777
2,057,634
2,224,213
1,979,464
2,145,665
2,345,687
1,609,402
2,087,459
1,927,879
2,019,837
1,770,749
1,914,976
2,225,317
1,866,076
2,365,402
2,217,748
1,671,398
20
21
21
21
21
22
22
22
22
23
23
23
23
24
24
24
24
25
25
25
25
26
26
26
26
27
27
27
27
28
28
28
28
29
29
29
29
30
30
30
30
887,480
1,003,125
907,830
979,130
859,112
794,946
600,394
814,820
1,073,763
949,066
1,173,758
1,212,861
1,649,596
2,204,547
2,528,855
2,136,125
1,750,787
1,452,891
975,078
1,302,898
1,364,959
1,544,982
1,863,259
1,806,980
1,806,074
2,229,541
2,002,932
2,147,831
2,164,777
2,057,634
2,224,213
1,979,464
2,145,665
2,345,687
1,609,402
2,087,459
1,927,879
2,019,837
1,770,749
1,914,976
2,225,317
-28,368
804743424
0.033020142
-208,179
43338496041 0.261878165
-307,436
94516894096 0.512057083
-164,310
26997776100 0.201651899
214,651
46075051801 0.199905379
154,120
23752974400 0.162391235
573,364
328746276496 0.488485701
398,041
158436637681 0.328183526
575,833
331583643889 0.349075167
1,255,481 1576232541361 0.569496137
1,355,097 1836287879409 0.535853973
923,264
852416413696 0.432214407
101,191
10239618481 0.057797436
-751,656
564986742336 0.517351956
-1,553,777 2414222965729 1.593489957
-833,227
694267233529 0.639518212
-385,828
148863245584 0.282666366
92,091
8480752281
0.05960652
888,181
788865488761 0.476681449
504,082
254098662724 0.278963796
441,115
194582443225 0.244239716
684,559
468621024481 0.307040328
139,673
19508546929
0.06973427
340,851
116179404201 0.158695447
358,703
128667842209 0.165699746
-171,907
29552016649 0.083545956
221,281
48965280961 0.099487324
-168,367
28347446689 0.085056864
-19,112
365268544
0.008907262
288,053
82974530809 0.122801124
-614,811
377992565721 0.382012077
107,995
11662920025 0.051735148
-217,786
47430741796 0.112966633
-325,850
106178222500 0.161324899
161,347
26032854409 0.091117939
-172,483
29750385289 0.090070581
297,438
88469363844 0.133660957
-153,761
23642445121 0.082398037
594,653
353612190409 0.251396169
302,772
91670883984 0.136522274
-553,919
306826258561 0.331410592
85
APPENDIX E
Table 4.7
Calculation for Johor Tourism Data using 3-month Moving Average
Period
Actual
Data
t
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
At
1,213,874
1,107,735
1,129,539
981,022
1,205,673
1,240,579
1,334,745
1,514,357
1,213,478
1,474,035
1,514,357
1,393,287
1,088,378
1,080,564
1,109,080
1,049,116
1,013,572
1,205,808
1,109,254
2,146,602
953,087
952,753
930,598
1,193,784
1,080,693
944,095
1,305,077
1,217,528
1,190,167
1,345,568
1,120,333
1,393,845
981,729
969,556
962,730
3-month
moving
average
Error
1150383
1072765
1105411
1142425
1260332
1363227
1354193
1400623
1400623
1460560
1332007
1187410
1092674
1079587
1057256
1089499
1109545
1487221
1402981
1350814
945479
1025712
1068358
1072857
1109955
1155567
1237591
1251088
1218689
1286582
1165302
1115043
-169,361
132,908
135,168
192,320
254,025
-149,749
119,842
113,734
-7,336
-372,182
-251,443
-78,330
-43,558
-66,015
148,552
19,755
1,037,057
-534,134
-450,228
-420,216
248,305
54,981
-124,263
232,220
107,573
34,600
107,977
-130,755
175,156
-304,853
-195,746
-152,313
Square Error
Absolute
Percentage
Error
28683035414
17664447859
18270298112
36987110613
64528531275
22424763001
14362025069
12935346933
53821787
138519193003
63223749878
6135536680
1897299364
4357936215
22067696704
390273195
1075487912620
285299486045
202705251984
176581486656
61655207488
3022947015
15441376011
53925973587
11571950329
1197183067
11659104514
17096782855
30679507565
92935351609
38316627013
23199351511
0.172636971
0.110235252
0.108955308
0.144087697
0.167744242
0.123404792
0.081301778
0.075103603
0.005265486
0.341959932
0.232696382
0.070625804
0.041518764
0.065130713
0.12319706
0.017809567
0.483115796
0.560425578
0.472554796
0.451554807
0.207997985
0.050875997
0.131621641
0.177935606
0.088353615
0.029071831
0.080246657
0.116710538
0.125663662
0.310526632
0.201892756
0.158209813
et
86
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
1,239,244
1,234,351
1,148,229
1,333,280
985,363
914,062
1,035,614
1,270,599
1,911,344
1,505,878
2,590,348
1,747,456
3,450,757
1,057,455
1,171,256
1,087,495
1,083,396
1,277,491
1,274,229
1,133,624
1,103,639
2,103,175
2,188,585
1,148,499
1,545,169
1,333,513
1,145,025
1,270,344
889,574
919,763
1,129,101
870,424
923,640
994,139
2,846,747
906,309
1,068,855
1,492,337
840,330
1,016,850
887,480
1,003,125
907,830
979,130
859,112
794,946
971338
1057177
1145442
1207275
1238620
1155624
1077568
978346
1073425
1405852
1562607
2002523
1947894
2596187
2085223
1893156
1105402
1114049
1149461
1211705
1228448
1170497
1446813
1798466
1813420
1627418
1342394
1341236
1249627
1101648
1026560
979479
973096
974388
929401
1588175
1582398
1607304
1155834
1133841
1116506
914887
969152
932812
963362
915357
267,906
177,174
2,787
126,005
-253,257
-241,562
-41,954
292,253
837,919
100,026
1,027,741
-255,067
1,502,863
-1,538,732
-913,967
-805,661
-22,006
163,442
124,768
-78,081
-124,809
932,678
741,772
-649,967
-268,251
-293,905
-197,369
-70,892
-360,053
-181,885
102,541
-109,055
-49,456
19,751
1,917,346
-681,866
-513,543
-114,967
-315,504
-116,991
-229,026
88,238
-61,322
46,318
-104,250
-120,411
71773446232
31390744392
7769227
15877344028
64139108049
58352199844
1760166085
85411621174
702108250561
10005133992
1056251563081
65059344534
2258597196769
2367696167824
835335067778
649089646921
484264036
26713287364
15567137003
6096694615
15577286481
869887629899
550226194499
422457534400
71958420167
86379953088
38954390582
5025628403
129638402844
33082031968
10514588320
11893065728
2445895936
390088834
3676215683716
464941696533
263726755211
13217334444
99542563680
13686816087
52452755992
7786003469
3760346803
2145388003
10867993000
14498889195
0.21618476
0.143536428
0.002427506
0.09450778
0.257018987
0.264273102
0.040511555
0.230011724
0.438392566
0.066423486
0.396757887
0.145964953
0.435516903
1.455127641
0.780330403
0.740841107
0.020312056
0.127939845
0.097916727
0.068877629
0.11308861
0.44346175
0.338927816
0.565927644
0.173606037
0.220398801
0.172370618
0.055805094
0.40474804
0.197751667
0.090816204
0.125289897
0.053544671
0.019867108
0.673521743
0.752355249
0.480461179
0.077038006
0.375452104
0.11505204
0.258062905
0.087963448
0.067547522
0.047305601
0.121345839
0.151471085
87
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
600,394
814,820
1,073,763
949,066
1,173,758
1,212,861
1,649,596
2,204,547
2,528,855
2,136,125
1,750,787
1,452,891
975,078
1,302,898
1,364,959
1,544,982
1,863,259
1,806,980
1,806,074
2,229,541
2,002,932
2,147,831
2,164,777
2,057,634
2,224,213
1,979,464
2,145,665
2,345,687
1,609,402
2,087,459
1,927,879
2,019,837
1,770,749
1,914,976
2,225,317
1,866,076
2,365,402
2,217,748
1,671,398
877729
751484
736720
829659
945883
1065529
1111895
1345405
1689001
2127666
2289842
2138589
1779934
1392919
1243622
1214312
1404280
1591067
1738407
1825438
1947532
2012849
2126768
2105180
2123414
2148875
2087104
2116447
2156939
2033585
2014183
1874913
2011725
1906155
1901854
1970347
2002123
2152265
2149742
-277,335
63,336
337,043
119,407
227,875
147,332
537,701
859,142
839,854
8,459
-539,055
-685,698
-804,856
-90,021
121,337
330,670
458,979
215,913
67,667
404,103
55,400
134,982
38,009
-47,546
100,799
-169,411
58,561
229,240
-547,537
53,874
-86,304
144,924
-240,976
8,821
323,463
-104,271
363,279
65,483
-478,344
76914887115
4011448896
113597983849
14258031649
51927015625
21706718224
289122365401
738124976164
705354181413
71554681
290580652395
470181747204
647793717307
8103720427
14722586678
109342869347
210662028427
46618567511
4578822889
163299504011
3069196933
18220140324
1444684081
2260622116
10160438401
28699973980
3429429762
52550824773
299796401344
2902443792
7448322880
21002869160
58069432576
77810041
104628312369
10872510955
131971631841
4288023289
228812982336
0.461922227
0.077730051
0.313889564
0.125815275
0.194141382
0.121474761
0.325959205
0.389713624
0.332108273
0.003959974
0.307893155
0.471954193
0.825427641
0.069092643
0.088894001
0.214028599
0.246331473
0.119488502
0.03746635
0.181249564
0.027659618
0.062845727
0.017557929
0.023107122
0.045318951
0.085584111
0.027292859
0.097728157
0.34021125
0.025808571
0.044766122
0.071750179
0.136087046
0.004606324
0.14535592
0.055877324
0.153580237
0.029526799
0.286193953
88
APPENDIX F
Table 4.8
Calculation for Johor Tourism Data using Exponential Smoothing
(   0.37 )
Period
Actual
Data
Forecast
Error
Square
t
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
At
1,213,874
1,107,735
1,129,539
981,022
1,205,673
1,240,579
1,334,745
1,514,357
1,213,478
1,474,035
1,514,357
1,393,287
1,088,378
1,080,564
1,109,080
1,049,116
1,013,572
1,205,808
1,109,254
2,146,602
953,087
952,753
930,598
1,193,784
1,080,693
944,095
1,305,077
1,217,528
1,190,167
1,345,568
1,120,333
1,393,845
981,729
Ft
1213874
1213874
1174603
1157929
1092473
1134357
1173659
1233261
1337267
1291465
1359016
1416492
1407906
1289681
1212308
1174113
1127864
1085576
1130062
1122363
1501331
1298481
1170562
1081775
1123218
1107484
1047030
1142507
1170265
1177629
1239766
1195576
1268936
et
0
-106,139
-45,064
-176,907
113,200
106,222
161,086
281,096
-123,789
182,570
155,341
-23,205
-319,528
-209,117
-103,228
-124,997
-114,292
120,232
-20,808
1,024,239
-548,244
-345,728
-239,964
112,009
-42,525
-163,389
258,047
75,021
19,902
167,939
-119,433
198,269
-287,207
Error
0
11265487321
2030725341
31296104021
12814140173
11283054270
25948597910
79014950841
15323600562
33331887084
24130901403
538472845
102098245755
43729811581
10655926491
15624338684
13062737489
14455692925
432970665
1049065515384
300571959658
119527845532
57582546575
12545995775
1808408549
26695963320
66588220073
5628085673
396087934
28203586172
14264309052
39310608973
82487579365
Absolute
Percentage
Error
0
0.095816238
0.039895541
0.180329339
0.093889105
0.085622699
0.120686487
0.185620684
0.102011352
0.123857456
0.10257901
0.016654873
0.293581973
0.193525549
0.093074933
0.119145409
0.112761929
0.099710592
0.018758505
0.477144339
0.575230209
0.362872637
0.257859608
0.093826781
0.039350109
0.173064145
0.197725448
0.06161712
0.016721988
0.124809176
0.106605164
0.142246112
0.292551722
89
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
969,556
962,730
1,239,244
1,234,351
1,148,229
1,333,280
985,363
914,062
1,035,614
1,270,599
1,911,344
1,505,878
2,590,348
1,747,456
3,450,757
1,057,455
1,171,256
1,087,495
1,083,396
1,277,491
1,274,229
1,133,624
1,103,639
2,103,175
2,188,585
1,148,499
1,545,169
1,333,513
1,145,025
1,270,344
889,574
919,763
1,129,101
870,424
923,640
994,139
2,846,747
906,309
1,068,855
1,492,337
840,330
1,016,850
887,480
1,003,125
907,830
979,130
1162669
1091217
1043677
1116037
1159813
1155527
1221296
1134001
1052623
1046330
1129309
1418662
1450932
1872516
1826244
2427314
1920466
1643258
1437626
1306561
1295805
1287822
1230769
1183731
1523925
1769849
1539950
1541881
1464785
1346474
1318306
1159675
1070908
1092439
1010294
978232
984117
1673290
1389507
1270866
1352810
1163193
1109046
1027066
1018208
977368
-193,113
-128,487
195,567
118,314
-11,584
177,753
-235,933
-219,939
-17,009
224,269
782,035
87,216
1,139,416
-125,060
1,624,513
-1,369,859
-749,210
-555,763
-354,230
-29,070
-21,576
-154,198
-127,130
919,444
664,660
-621,350
5,219
-208,368
-319,760
-76,130
-428,732
-239,912
58,193
-222,015
-86,654
15,907
1,862,630
-766,981
-320,652
221,471
-512,480
-146,343
-221,566
-23,941
-110,378
1,762
37292669885
16508974390
38246463263
13998256776
134189884
31596149442
55664179375
48372953234
289315216
50296656496
611578070736
7606592269
1298268687416
15639992831
2639043225101
1876512767313
561315564395
308872817725
125478795972
845054075
465523029
23776982835
16161949824
845377849773
441772807826
386076134514
27241529
43417146183
102246280601
5795719899
183810839845
57557742992
3386480241
49290710101
7508832661
253041618
3469388980634
588260406291
102817869713
49049438520
262635976584
21416138728
49091403546
573193207
12183328460
3103893
0.199176841
0.133461359
0.157811561
0.095851366
0.010088611
0.133320126
0.239437216
0.240616634
0.016424332
0.17650664
0.409154276
0.057916896
0.439869833
0.07156687
0.470770103
1.295429751
0.639663712
0.511049039
0.326962499
0.022755396
0.016932579
0.136022058
0.115191342
0.437169668
0.303693902
0.541010703
0.003377845
0.156254806
0.279260036
0.059928355
0.481951658
0.260841052
0.051539652
0.255065477
0.093817419
0.016001064
0.654301063
0.846269164
0.29999603
0.14840554
0.609855915
0.143917529
0.249657231
0.02386687
0.121584565
0.001799339
90
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
859,112
794,946
600,394
814,820
1,073,763
949,066
1,173,758
1,212,861
1,649,596
2,204,547
2,528,855
2,136,125
1,750,787
1,452,891
975,078
1,302,898
1,364,959
1,544,982
1,863,259
1,806,980
1,806,074
2,229,541
2,002,932
2,147,831
2,164,777
2,057,634
2,224,213
1,979,464
2,145,665
2,345,687
1,609,402
2,087,459
1,927,879
2,019,837
1,770,749
1,914,976
2,225,317
1,866,076
2,365,402
2,217,748
1,671,398
978020
934024
882565
778162
791725
896079
915684
1011172
1085797
1294402
1631156
1963305
2027248
1924958
1750293
1463463
1404054
1389589
1447084
1601069
1677256
1724919
1911629
1945411
2020306
2073761
2067794
2125669
2071573
2098987
2190266
1975346
2016828
1983917
1997207
1913418
1913994
2029184
1968834
2115564
2153372
-118,908
-139,078
-282,171
36,658
282,038
52,987
258,074
201,689
563,799
910,145
897,699
172,820
-276,461
-472,067
-775,215
-160,565
-39,095
155,393
416,175
205,911
128,818
504,622
91,303
202,420
144,471
-16,127
156,419
-146,205
74,092
246,700
-580,864
112,113
-88,949
35,920
-226,458
1,558
311,323
-163,108
396,568
102,184
-481,974
14139130106
19342714210
79620583086
1343819769
79545226341
2807591298
66601996683
40678607893
317869664605
828363132438
805863632974
29866897069
76430759890
222846795411
600958148937
25781244514
1528434334
24146992731
173201303336
42399340859
16594059459
254643662431
8336246389
40973823851
20871739629
260065741
24466988352
21375860867
5489615377
60860851201
337403044048
12569246019
7911930163
1290254324
51283375457
2428149
96921822677
26604121756
157266296867
10441556352
232299053920
0.138408117
0.174952873
0.469976706
0.044989258
0.262662834
0.055830373
0.219869536
0.16629225
0.341780237
0.412848792
0.354982424
0.080903701
0.157906779
0.324915301
0.795028608
0.123237115
0.028642029
0.100579183
0.223358431
0.113953116
0.071324836
0.226334612
0.045584696
0.094243877
0.06673692
0.007837426
0.070325671
0.073860833
0.034530991
0.105171714
0.36091918
0.053707713
0.046138285
0.017783668
0.127888442
0.000813719
0.139900382
0.087406783
0.167653595
0.046075539
0.28836586
91
APPENDIX G
Table 4.9
Calculation for Johor Tourism Data using Holt’s Model
(   0.36,   0 and a0  1236616, b  6712.383 )
Period
t
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
Actual Data
At
1,213,874
1,107,735
1,129,539
981,022
1,205,673
1,240,579
1,334,745
1,514,357
1,213,478
1,474,035
1,514,357
1,393,287
1,088,378
1,080,564
1,109,080
1,049,116
1,013,572
1,205,808
1,109,254
2,146,602
953,087
952,753
930,598
1,193,784
1,080,693
944,095
1,305,077
1,217,528
1,190,167
1,345,568
1,120,333
1,393,845
981,729
969,556
962,730
1,239,244
1,234,351
1,148,229
at
bt
Ft
1232724.8
1192024.4
1173825.6
1108712.2
1147914
1185569.3
1243568.5
1345348.3
1302170.9
1368337.9
1425200.7
1418007.7
1303636.9
1227626.6
1189245.7
1143095
1100762.6
1142874.9
1135067.3
1503515.7
1309657.3
1185467.7
1098010.5
1136784.9
1120887.7
1061538.3
1153508.1
1180851.2
1188500.8
1249340.9
1207194
1278684.3
1176076.3
1106024.9
1058734.7
1128014
1170591.2
1166836.7
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
1239437
1198737
1180538
1115425
1154626
1192282
1250281
1352061
1308883
1375050
1431913
1424720
1310349
1234339
1195958
1149807
1107475
1149587
1141780
1510228
1316370
1192180
1104723
1143497
1127600
1068251
1160221
1187564
1195213
1256053
1213906
1285397
1182789
1112737
1065447
1134726
1177304
Error
-131,702
-69,198
-199,516
90,248
85,953
142,463
264,076
-138,583
165,152
139,307
-38,626
-336,342
-229,785
-125,259
-146,842
-136,235
98,333
-40,333
1,004,822
-557,141
-363,617
-261,582
89,061
-62,804
-183,505
236,826
57,307
2,603
150,355
-135,720
179,939
-303,668
-213,233
-150,007
173,797
99,625
-29,075
Square Error
17345466356
4788333227
39806620040
8144773667
7387848221
20295785134
69736193943
19205156450
27275086702
19406359510
1491974710
113125994522
52801290510
15689813056
21562611941
18560070028
9669377772
1626772397
1009667907538
310406198773
132217092335
68425174259
7931879252
3944377588
33674129876
56086714744
3284146556
6777693
22606563826
18420004006
32377905225
92214053378
45468180661
22502191693
30205376876
9925072793
845332416
Absolute
Percentage Error
0.118893226
0.061261969
0.203375627
0.074853132
0.069284256
0.106734452
0.174381677
0.11420287
0.112040561
0.091990667
0.027722995
0.309030575
0.212653127
0.112939539
0.13996749
0.13441112
0.081549463
0.036360713
0.468099036
0.584564782
0.381648426
0.281090288
0.074604031
0.058114821
0.194371459
0.181465415
0.047068711
0.002187424
0.111740762
0.121142835
0.12909514
0.309319241
0.219928185
0.155814512
0.140244328
0.080710154
0.025321256
92
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
1,333,280
985,363
914,062
1,035,614
1,270,599
1,911,344
1,505,878
2,590,348
1,747,456
3,450,757
1,057,455
1,171,256
1,087,495
1,083,396
1,277,491
1,274,229
1,133,624
1,103,639
2,103,175
2,188,585
1,148,499
1,545,169
1,333,513
1,145,025
1,270,344
889,574
919,763
1,129,101
870,424
923,640
994,139
2,846,747
906,309
1,068,855
1,492,337
840,330
1,016,850
887,480
1,003,125
907,830
979,130
859,112
794,946
600,394
814,820
1,073,763
949,066
1,173,758
1,212,861
1,649,596
1231052.2
1146900
1067374.3
1060236.5
1140262.9
1422148
1456586.7
1869036.7
1829563.6
2417489.1
1932172.8
1662538.7
1459818.9
1328602.6
1314498.3
1304297.3
1247150.8
1199782.5
1529299.7
1770938.3
1551156.1
1553296.7
1478470.5
1362726
1333764.4
1178151.8
1089427.8
1108006
1026772.4
993940.68
998308
1668042
1398114
1283876.7
1363218.3
1179274.5
1125097.6
1043851.2
1033485.7
992545.56
992011.88
948463.85
897493.35
794833.51
806324.57
906898.33
926374.62
1019728.6
1093552.2
1298023.9
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
1173549
1237765
1153612
1074087
1066949
1146975
1428860
1463299
1875749
1836276
2424202
1938885
1669251
1466531
1335315
1321211
1311010
1253863
1206495
1536012
1777651
1557868
1560009
1485183
1369438
1340477
1184864
1096140
1114718
1033485
1000653
1005020
1674754
1404826
1290589
1369931
1185987
1131810
1050564
1040198
999258
998724
955176
904206
801546
813037
913611
933087
1026441
1100265
159,731
-252,402
-239,550
-38,473
203,650
764,369
77,018
1,127,049
-128,293
1,614,481
-1,366,747
-767,629
-581,756
-383,135
-57,824
-46,982
-177,386
-150,224
896,680
652,573
-629,152
-12,699
-226,496
-340,158
-99,094
-450,903
-265,101
32,961
-244,294
-109,845
-6,514
1,841,727
-768,445
-335,971
201,748
-529,601
-169,137
-244,330
-47,439
-132,368
-20,128
-139,612
-160,230
-303,812
13,274
260,726
35,455
240,671
186,420
549,331
25513951609
63706580121
57384405049
1480145073
41473370852
584259498803
5931707707
1270239152039
16459121341
2606548991151
1867996063071
589254525557
338440095523
146792621015
3343608502
2207280809
31465677905
22567314886
804035236093
425851381019
395831901014
161277073
51300463957
115707369482
9819702723
203313341751
70278635509
1086418421
59679769188
12065884275
42433071
3391956920686
590508256006
112876786341
40702221813
280476918853
28607271160
59697130122
2250416824
17521303409
405134035
19491584720
25673727573
92301568580
176201955
67978071271
1257077355
57922530474
34752437768
301765046001
0.119802946
0.256150905
0.262072401
0.037149607
0.160278828
0.399911629
0.051144635
0.435095543
0.073417074
0.467862857
1.292486701
0.655389735
0.534950547
0.353642852
0.04526368
0.036870694
0.156476641
0.136117168
0.426345938
0.298171144
0.547803465
0.008218836
0.169849156
0.297074614
0.078005968
0.506874984
0.288227706
0.029192129
0.280661415
0.118926009
0.006552471
0.646958305
0.847884496
0.31432833
0.135189248
0.630229453
0.166334112
0.275307569
0.047290775
0.145807101
0.020556966
0.162507642
0.201561154
0.5060206
0.016290848
0.242815264
0.037358083
0.205043118
0.153702739
0.333009691
93
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
2,204,547
2,528,855
2,136,125
1,750,787
1,452,891
975,078
1,302,898
1,364,959
1,544,982
1,863,259
1,806,980
1,806,074
2,229,541
2,002,932
2,147,831
2,164,777
2,057,634
2,224,213
1,979,464
2,145,665
2,345,687
1,609,402
2,087,459
1,927,879
2,019,837
1,770,749
1,914,976
2,225,317
1,866,076
2,365,402
2,217,748
1,671,398
1628668.1
1957031.3
2025801
1931091.9
1763235.5
1483794.7
1422967.8
1406380.6
1460573
1609835.9
1685103.7
1732948.9
1916018
1951603
2026541
2080601.9
2076629.4
2134055.4
2082698.4
2109662.3
2198927.1
1990994
2030017.3
1997543.5
2009865.1
1928079.2
1927658
2039111.1
1981114.4
2123753.9
2161887.7
1989607.3
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
6712.383
1304736
1635381
1963744
2032513
1937804
1769948
1490507
1429680
1413093
1467285
1616548
1691816
1739661
1922730
1958315
2033253
2087314
2083342
2140768
2089411
2116375
2205640
1997706
2036730
2004256
2016577
1934792
1934370
2045824
1987827
2130466
2168600
1996320
2003032
2009744
2016457
2023169
2029882
2036594
2043306
2050019
2056731
2063444
2070156
899,811
893,474
172,381
-281,726
-484,913
-794,870
-187,609
-64,721
131,889
395,974
190,432
114,258
489,880
80,202
189,516
131,524
-29,680
140,871
-161,304
56,254
229,312
-596,238
89,753
-108,851
15,581
-245,828
-19,816
290,947
-179,748
377,575
87,282
-497,202
m=1
m=2
m=3
m=4
m=5
m=6
m=7
m=8
m=9
m=10
m=11
m=12
809659381054
798296673764
29715310520
79369739013
235140860122
631818099206
35197172662
4188834251
17394720566
156795097210
36264241998
13054872554
239982107304
6432299273
35916182162
17298466289
880917671
19844709738
26018909925
3164534662
52584132297
355499167250
8055531214
11848477469
242772591
60431621761
392657385
84649949063
32309175077
142563029273
7618102658
247209895918
0.40816129
0.353311872
0.08069813
0.160914123
0.333757488
0.815185927
0.143993694
0.047416226
0.085366073
0.212516675
0.105386737
0.063263145
0.219722215
0.040042106
0.08823583
0.060756204
0.014424459
0.063335325
0.081488616
0.026217605
0.097759123
0.37047146
0.042996107
0.056461382
0.007714069
0.138827377
0.010347693
0.130743909
0.0963238
0.159624113
0.039356024
0.297476763