Rainfall Estimation Using SPHONN Model
Ming Zhang*
Jessica Crane
Department of Physics, Computer Science and Engineering
Christopher Newport University,1 University Place Newport News, VA 23606, USA
* This research is supported by Christopher Newport University Applied Research Center.
ABSTRACT
Sigmoid Polynomial Higher Order Neural Network (SPHONN) model has been developed in this
paper. The SPHONN model for XOR and estimating heavy rainfall from satellite data has been
tested as well. For heavy rainfall estimation, the SPHONN model has 2.89% to 17.24% more
accuracy than M-HONN (Multiple Polynomial Higher Order Neural Network model) and PHONN
(Polynomial Higher Order Neural Network) Models.
KEY WORDS
Sigmoid Polynomial, Higher Order Neural Network, XOR Simulation, Rainfall Estimation.
1 INTRODUCTION
Currently, satellite derived precipitation estimates [1] and 3 hour precipitation outlooks for extratropical cyclones, and tropical cyclones are computed on the NOAA/NESDIS Interactive Flash
Flood Analyzer (IFFA) system and transmitted to National Weather Service Forecast Offices, and
River Forecast Centers. However, this system permits the computation of rainfall estimates is very
time consuming. This is due to the considerable time needed for image processing, interpretation,
and the computation involved in the estimation of rainfall. If there were several storms occurring, an
estimation technique would be useful in providing rainfall estimates for the entire country.
Automatic algorithms such as the Auto-Estimator (Vicente, Scofield, and Menzel, 1998 [2]) have
been accepted for operational implementation (for convection only) into National Weather Service
flash flood operations. However, initial research into Artificial Intelligence (AI) (Zhang and
Scofield, 1994, [8]) have shown promise as yet another way to help solve this all important flash
rainfall estimation problem.
Some artificial intelligence system for weather forecasting are designed to be objective and
automated, some are designed to augment human skill. In the Knowledge Augmented Severe Storms
Predictor (KASSPr), “knowledge” was elicited in a series of interviews and exchanges of
documentation between the developer and an expert in severe weather forecasting. But so far no AI
system can solve these problems very well.
Artificial Neural Network (ANN) computing is an area that is receiving increased research interest.
Since Richard Lippmann's [3] tutorial article "An Introduction to Computing with Neural
Networks", Lippmann's article becomes one of the most widely referenced papers in the neural
network literature. In Lippmann's [3] article, Multi-Layer Perceptron (MLP) neural network has
been introduced. So far, the MLP is still the most widely used neural network in the world. But so
results for rainfall estimation have not been very good using simple neural network models.
Artificial neuron network-based models are not yet sufficiently powerful to characterize complex
systems. Moreover, a gap exists in the research literature between complex systems and general
systems. To characterize complex systems, neuron-based and neural network-based group models
are studied. Lie Groups were used in Tsao’s [4] group theory approach to the computer simulation of
3D rigid motion. So far there have been limited studies with emphasis on setting a few free
parameters in the activation function. Vecci et al [5] studied the properties of a Feed-forward Neural
Network (FNN) which is able to adapt its activation function. In Chen and Chang’s [6] paper, real
variables a (gain) and b (slope) in the generalized sigmoid activation function are adjusted during the
learning process. In Campolucci et al [7], a neuron-adaptive activation function built as a piecewise
approximation can have arbitrary shape. And this enables reduction of the overall size of the neural
networks, trading connection complexity with activation function complexity. Neural networks with
a neuron-adaptive activation function seem to provide better fitting properties than classical
architectures with fixed activation function neurons.
Assessment of global climate change is a very important research area for the future of humans and
their environment. Rainfall estimation is a key parameter in this research. During the past 20 years,
there has been a great increase in our understanding of how satellite data can be used to estimate
rainfall. But, even with the use of interactive computer systems, the time needed to prepare
estimates of rainfall is very time consuming (about a half-hour). Verification results show that the
average error for an event is about 30% ([8], [9], and [10]). As mentioned previously, an automatic
estimation algorithm (Auto Estimator) has been developed for convective systems but there is still a
need to increase the speed of computation and improve the accuracy of the satellite derived rainfall
estimates.
Zhang, Murugesan, and Sadeghi (1995) developed a Polynomial Higher Order Neural Network
(PHONN)[12] for data simulation. Ming Zhang, Jing Chun Zhang, and Steve Keen [13] developed a
Trigonometric Polynomial Higher Order Neural Network (THONN) system for higher frequency
non-linear data simulation and prediction. Lu and Zhang [14] developed a model, called
Polynomial and Trigonometric polynomial Higher Order Neural Network (PT-HONN). Hui Qi,
Ming Zhang, and Roderick Scofield, [15] used multiple Polynomial Higher Order Neural Network
(M-PHONN) Model for Rainfall Estimation. The special feature of M-PHONN model would be it
could simulate multiple polynomial functions.
This paper will develop a new higher order neural network model called Sigmoid Polynomial Higher
Order Neural Network (SPHONN) model. Sigmoid functions have been used as neuron activity
functions. For testing SPHONN model, a SPHONN simulator has been built. XOR data has been
used to test SPHONN model convergence. The comparison experimental results between SPHONN,
PHONN, and M-HONN models for heavy rainfall estimation also will be presented here.
2 SPHONN MODEL
The network architecture of SPHONN is developed based on the characteristics of PHONN,
THONN, PT-HONN and M-PHONN models. The different part of SPHONN is, SPHONN used
sigmoid function for the neurons. It is a multi-layer that consists of an input layer with input-units,
and output layer with output-units, and two hidden layers consisting of intermediate processing
units. The specify definition of SPHONN is presented in the following:
Z
n
 (a
i , j 0
ij
((1  exp(  x)) i (1  exp(  y))  j )
The output layer weights are updated according to:
(2.8)
wkj (t  1)  wkj (t )   (E p / wkj )
o
o
o
where  = learning rate (positive & usually < 1)
j = input index (j=1…L means input-L of neuron-j)
k = kth output neuron
E = error
t = training time
o = output layer
p = pth training vector
w = weight
For a sigmoid output neuron, we have:
 E p / wkj  ( y pk  o pk ) f k ' (net pk )i pj
o
o
o
 ( y pk  o pk )o pk (1  o pk )i pj
For sigmoid output neurons, the weight update equations are formulated as follows:
ospk = (ypk - opk) fok'(netopk) = (ypk - opk)
opk(1 - opk)
wkj (t  1)  wkj (t )   (E p / wkj )
o
o
o
 wkj (t )   ( y pk  o pk )o pk (1  o pk )i pj
o
 wkj (t )   pk i pj
o
os
Based on derivatives of SPHONN Model, a leaning algorithm has been developed.
3 SPHONN SIMULATOR & XORTEST
The SPHONN simulator has been written in the C computing language, running under X window on
Sun workstation, based on the previous work did by Zhang and Fulcher [11]. A user-friendly GUI
(Graphical User Interface) system has also been incorporated. When you run the system, any step,
data or calculation can be reviewed and modified from different windows during processing. Hence,
changing data, network models and comparing results can be done very easily and efficiently. The
SPHONN Simulator can be seen in the following figure 1.
Figure 1 SPHONN Simulator
Figure2: SPHONN Weights for XOR Test
XOR test result also can be seen from the Figure 1. After 63796 Epochs, the difference between
desired result and actual result already reached to 9.426059e-15. The SPHONN structure and
weights can be seen from Figure 2.
4 Heavy Rainfall Estimating
For testing rainfall estimation data simulation, our expert knowledge of rainfall estimation based on
the satellite observed cloud top temperature and cloud growth was used [9]. For example, when the
cloud top temperature is between -58C and -60C, and the cloud growth is more than 2/3 latitude,
the half hour rainfall estimate is 0.94 inch according to the technique developed by Scofield [1].
Details of this expert knowledge are listed in the following Table 1 [8].
Cloud Top Cloud Growth Half Hour Cloud
Top Cloud Growth Half
Hour
Temperature Latitude
Rainfall
Temperature
Latitude
Rainfall
Degree
Inches
Degree
Inches
2/3
0.05
1/3
0.85
> -32 C
- 70 C
2/3
0.20
1/3
0.95
-36 C
< - 80  C
2/3
0.48
0
0.03
-46 C
> -32 C
2/3
0.79
0
0.06
-55 C
- 36 C
2/3
0.94
0
0.11
-60 C
- 46 C
1/3
0.05
0
0.22
> - 32 C
- 55 C
1/3
0.13
0
0.36
- 36 C
- 60 C
1/3
0.24
0
0.49
- 46 C
-70 C
1/3
0.43
0
0.55
- 55 C
< -80 C
1/3
0.65
- 60 C
Table 1 Cloud Top Temperature, Cloud growth, and Rainfall inches
The cloud top temperature and cloud growth have been used as input of SPHONN model. The
desired output values are the heavy rainfall estimation values (inches). The input can be seen from
the figure 3.
The simulation can be seen from Figure 4. In the figure 4, after 54695 epochs, the SPHONN is
convergence. The SPHONN simulator’s output can be seen from Figure 5. The SPHONN structure
and weights are in Figure 6.
Figure 3: Rainfall Estimation Input
Figure 5: SPHONN Running Results
Figure 4: Rainfall Estimation Simulating
Figure 6: SPHONN weights for rainfall Estimating
5 COMPARISONS
In this section, we will compare SPHONN with M-PHONN and PHONN with the error per pattern
and the average percent error. This comparison will be pay particular attention on the degree of the
accuracy of the error. The aim of this comparative analysis is to identify how the feature of
SPHONN is when it is compared with M-PHONN, and PHONN. Rules of the comparison
experiments - ‘Keep compared programs in the same environment’. There for the comparison will be
based on the following conditions: same groups of testing data and same parameters of experimental
environment such as learning rates and number of hidden layers and so on.
Cloud
Top
Temper
ature
Cloud
Growth
Latitude
Degree
> -32 2/3
C
-36 C 2/3
P
HO
NN
|Erro
|r%
3.22
MHO
NN
|Erro
r| %
8.78
SP
HONN
|Error|
%
Cloud
Top
Tempe
rature
Cloud
Growth
Latitude
Degree
9.88
9.92
6.52
6.05
22.5
8
14.2
1
0.12
5.20
- 70
C
< - 80
C
> -32
C
- 36
C
- 46
C
- 55
C
- 60
C
-70 C
< -80
C
Avera
ge
-46 C
2/3
-55 C
2/3
-60 C
2/3
25.1
9
14.6
3
3.66
> - 32 1/3
C
- 36 C 1/3
10.4
7
3.50
8.03
4.51
4.18
3.32
- 46 C
- 55 C
1/3
1/3
3.52
0.22
4.24
1.89
8.01
7.72
- 60 C
1/3
3.21
0.65
4.66
2.92
5.84
MHONN
|Error|
%
SP
HONN
|Error|
%
1/3
P
HO
NN
|Erro
|r%
9.01
5.12
6.20
1/3
3.89
1.32
0.73
0
9.81
7.24
11.27
0
2.98
3.25
7.73
0
5.69
5.67
3.01
0
5.28
4.03
5.10
0
3.32
1.43
2.19
0
0
0.77
2.50
2.78
1.12
2.43
3.23
6.36
5.42
5.263
Table 2 Rainfall data simulation using PHONN, M-PHONN, and SPHONN
Table 2 presents the rainfall data estimation results using MPHONN, PHONN and M-HONN model.
The average error of M-PHON is 5.42%. The average of error of PHONN is 6.32%. The average
error of SPHONN is 5.263%. This means the SPHONN model is about 2.89% better than M-HONN
model when using the rainfall estimate experimental database in Table 1. And this means the
SPHONN model is about 17.24% better than PHONN model when using the rainfall estimate
experimental database.
6 RESULTS AND CONCLUSION
SPHONN models are studied in this paper. Using SPHONN model, heavy rainfall estimation could
have better results than M-HONN and PHONN higher order neural network models. As the next
step, the research will more focus on developing automatic higher order neural network models.
Acknowledgment
The authors would like to thank Prof. A. Martin Buoncristiani for his great support. The authors
would also like to express their thanks to Prof. David Hibler and Dr. Antonio Siochi for their
invaluable help.
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