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International Journal of Advanced Research in Engineering RESEARCH IN ENGINEERING
INTERNATIONAL JOURNAL OF ADVANCED and Technology (IJARET), ISSN 0976 –
6480(Print), ISSN 0976 – 6499(Online)TECHNOLOGY (IJARET)
AND Volume 4, Issue 7, November – December (2013), © IAEME

ISSN 0976 - 6480 (Print)
ISSN 0976 - 6499 (Online)
Volume 4, Issue 7, November - December 2013, pp. 183-191
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Journal Impact Factor (2013): 5.8376 (Calculated by GISI)
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IJARET
©IAEME

ADAPTIVE AND REGRESSIVE MODEL FOR RAINFALL PREDICTION
Nizar Ali Charaniya1,

Dr. Sanjay V. Dudul2

1

2

(Electronics Engg Department, B.N.College of Engg, Pusad, India)
(Applied Electronics Department, Sant Gadge Baba Amravati University, Amravati, India)

ABSTRACT
Forecasting Indian summer monsoon rainfall (ISMR) has been a challenging task for the
research community. The variability of rainfall in time and space, however makes it extremely
difficult to have quantitative forecasting of rainfall. The depth of rainfall and its distribution in the
temporal and spatial dimensions depends on many ecological parameters. Due to the complexity of
the atmospheric processes by which rainfall is generated the accuracy of prediction is very low.
There are two possible methods for rainfall prediction. The first one is based on the study of the
rainfall processes and its dependence on other meteorological parameters such as pressure, humidity,
vapor pressure, temperature etc. However, this method is complex and non- feasible because rainfall
is the result of a number of complex atmospheric parameters which vary both in space and time and
these parameters are limited in both the spatial and temporal dimensions. Another method to forecast
rainfall is based on the pattern recognition methodology. In these method relevant spatial and
temporal features of rainfall series in past are extracted. These features are then utilized to predict the
rainfall in future. In this paper different models have been designed for the prediction of ISMR for
the next year based on the rainfall pattern for past four years. An adaptive neuro fuzzy inference
system (ANFIS), linear and nonlinear regressive model have been designed for prediction of rainfall.
Keywords: Indian Summer Monsoon Rainfall, Neuro-Fuzzy, Time Series Analysis, Pattern
Recognition.
I. INTRODUCTION
In India rainfall information is vital for planning crop production, water management and all
activity plans in the nature. The incident of extended dry period or heavy rain at the critical stages of
the crop growth and development may lead to noteworthy reduction in the crop yield and hence this
may affect the economy of the country. India is an agricultural country and its economy is largely
based upon agricultural product. Thus, rainfall prediction becomes a significant factor in agricultural
183

International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –
6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 7, November – December (2013), © IAEME

countries like India [1]. A wide range of rainfall forecast methods are employed in weather
forecasting at regional and national levels. Rainfall is a random process and therefore prediction of
rainfall is very difficult and cumbersome. Accurate and timely prediction of rainfall is a foremost
challenge for the research community [2]. Rainfall prediction modeling involves a combination of
computer models, inspection and information of trend and pattern. Using these methods, reasonably
accurate forecasts can be made up. Several recent research studies have reported rainfall prediction
using different weather and climate forecasting methods [3], [4], [5].
Because of strong non-linear, high degree of uncertainty, and time-varying characteristics of
the rainfall, it is very difficult to have a single superior model [6] for accurate prediction of rainfall.
Artificial neural networks (ANNs) have been accepted as a potentially useful tool for modeling
complex non-linear systems and widely used for prediction [7]. In the forecasting context, ANNs
have also proven to be an efficient alternative to traditional methods for rainfall forecasting [8], [9].
Alternate method for rainfall time series prediction is using pattern recognition. In these
method relevant spatial and temporal features of rainfall series in past are extracted. These features
are then utilized to estimate the rainfall pattern. If rainfall pattern can be estimated accurately then it
can help in predicting the rainfall for the next year [10] [11]. The adaptive neuro fuzzy inference
system (ANFIS) is a network which uses neural network learning algorithms and fuzzy reasoning in
order to map an input space to an output space. It has the ability to combine the verbal power of a
fuzzy system with the mathematical power of a neural system adaptive network. ANFIS has
tremendous capability of learning [12]. Therefore in this paper an attempt has been made to estimate
rainfall pattern using ANFIS model. In addition an attempt has also been made to develop linear and
nonlinear regressive model for prediction of rainfall using different nonlinear estimating functions.
II. ADAPTIVE NEURO-FUZZY INFERENCE SYSTEM (ANFIS)
2.1

Architecture and Algorithm
ANFIS has the benefit of allowing the drawing out of fuzzy rules from numerical data or
expert knowledge and adaptively to constructs a rule base. Furthermore, it can tune the complicated
conversion of human intelligence to fuzzy systems.
The main drawback of the ANFIS predicting model is the time required for training structure
and determining parameters, is very long. For simplicity, we have assumed a fuzzy inference system
with two inputs, x and y, and one output, z. For a first-order Sugeno fuzzy model [13], a typical rule
set with two fuzzy if–then rules can be expressed as
Rule 1 : If x is A1 and y is B1 then f1=p1x+q1y+r1
Rule 2 : If x is A2 and y is B2 then f2=p2x+q2y+r2
where p, q and r denotes linear parameters in the then-part (consequent part) of the first-order
Sugeno fuzzy model. The architecture of ANFIS consists of five layers (Fig. 1), and a brief
introduction of the model is as follows.
Layer 1.input nodes. Each node of this layer generates membership grades to which they belong to
each of the appropriate fuzzy sets using membership functions.
1
O = µ ( x)
i
Ai

(1)

where x denotes the crisp inputs to node i, and Ai, (small, large, etc.) represents the linguistic
labels characterized by appropriate membership functions ߤ஺೔ . Due to softness and concise notation,
184


the Gaussian and bell-shaped membership functions are very popular for specifying fuzzy sets. If we
choose ߤ஺೔ ሺ‫ݔ‬ሻ to be bell shaped with maximum equal to 1 and minimum equal to 0 then
µ A ( x) =
i

1
 x −c 2bi 
i
1+ 
 
 ai  



(2)

where {ai,bi, ci} represent the parameter set. As the values of these parameter change, the bell shaped
function vary accordingly, thus exhibiting various forms of membership function on the linguistic
label Ai .
Layer 2: Every node in this layer gets multiplied with the incoming signal and gives the product
output:
O
= W = µ ( x ) × µ ( y ),
i = 1, 2..
(3)
2,i
i
A
B
i

i

Each node represents the firing strength of a rule.
Layer 3: The ith node calculates the ratio of the ith rule’s firing strength to the sum of all rule’s firing
strength:
wi
O
= wi =
,
3,i
w1 + w2

i = 1, 2

(4)

For convenience, the outputs of this layer are called normalized firing strengths.
Layer 4: The node function of the fourth layer computes the contribution of each ith rule toward the
total output and the function is defined as:
O
= wi f = w ( pi x + qi y + ri )
4,i
i
i

(5)

where ‫ݓ‬௜ denotes the ith node’s output from the previous layer. As for {pi,qi, ri}, they
ഥ
represent the coefficients of this linear combination and are also the parameter set in the consequent
part of the Sugeno fuzzy model.

Fig1. ANFIS architecture for two-input Sugeno fuzzy model with four rules

185


Layer 5: output nodes. The single node computes the overall output by summing all the incoming
signals. Accordingly, the defuzzification process transforms each fuzzy rule resulting into a crisp
output in this layer:
∑ w f
O
= ∑ wi fi = i i i
5,i
∑i wi
i

(6)

This network is trained based on supervised learning. So our objective is to train adaptive
networks to be able to estimate unknown functions given by training data and then to find the
specific values of the above parameters.
The distinguishing feature of the approach is that ANFIS applies a hybrid-learning algorithm,
the gradient descent method and the least-squares method, to update parameters. The gradient
descent method is used to tune premise non-linear parameters ({ai,bi, ci}), while the least-squares
method is employed to identify subsequent linear parameters ({pi,qi, ri}). As seen in Fig. 1, the
circular nodes are fixed (i.e., not adaptive) nodes without parameter variables, and the square nodes
have parameter variables (the parameters are changed during training). The task of the learning
procedure has two steps: In the first step, the least square method is used to recognize the consequent
parameters, while the antecedent parameters (membership functions) are assumed to be fixed for the
current cycle through the training set. Then, the error signals are propagated backward. Gradient
descent method is employed to update the premise parameters, through minimizing the overall
quadratic cost function, while the consequent parameters remain fixed. The detailed algorithm and
mathematical background of the hybrid-learning algorithm can be found in [12].
2.2

Non linear model
Dynamic models have complex functional dependence between the system's inputs u(t) and
outputs y(t). We can use these relationships to compute the current output from previous inputs and
outputs. The general form of a model in discrete time is:
y(t) = f(u(t - 1), y(t - 1), u(t - 2), y(t - 2), . . .)

(7)

Such a model is nonlinear if the function f is a nonlinear function, which might include
nonlinear components representing arbitrary nonlinearities. Nonlinear models might be necessary to
represent systems that operate over a range of operating points. In some cases, We might fit several
linear models, where each model is accurate at specific operating conditions.
If We know the nonlinear equations describing a system, We can represent this system as a
nonlinear grey-box model and estimate the coefficients from experimental data. In this case, the
coefficients are the parameters of the model.
This block diagram shows the structure of a nonlinear ARX model:

Fig 2. Structure of Nonlinear Auto Regressive (ARX) Models
186


The nonlinear ARX model computes the output y in two stages:
1. It computes regressors from the current and past input values and past output data.
In the simplest case, regressors are delayed inputs and outputs, such as u(t-1) and y(t-3)—
called standard regressors. Custom regressors can also be specified, which are nonlinear
functions of delayed inputs and outputs. For example, tan(u(t-1)) or u(t-1)*y(t-3). By default,
all regressors are inputs to both the linear and the nonlinear function blocks of the
nonlinearity estimator. We can select a subset of regressors as inputs to the nonlinear function
block.
2. The nonlinearity estimator block maps the regressors to the model output using a
combination of nonlinear and linear functions. One can select from available nonlinearity
estimators, such as tree-partition networks, wavelet networks, and multi-layer neural
networks. One can also exclude either the linear or the nonlinear function block from the
nonlinearity estimator.
The nonlinearity estimator block can include linear and nonlinear blocks in parallel. For instance:
F (x) = LT ( x − r ) + d + g ( x − r )

(8)

x denotes a vector of the regressors. LT ( x) + d is the output of the linear function block and is affine
when d≠0. d is a scalar offset. g (Q(x − r)) represents the output of the nonlinear function block. r
denotes the mean of the regressors x. Q is a projection matrix that makes the calculations well
conditioned. The exact form of F(x) depends on our choice of the nonlinearity estimator. Estimating
a nonlinear ARX model computes the model parameter values, such as L, r, d, Q, and other
parameters specifying g.
Most nonlinearity estimators represent the nonlinear function as a sum of series of nonlinear
units, such as wavelet networks or sigmoid functions. One can configure the number of nonlinear
units n for estimation.
A sigmoidnet define a nonlinear function y=F(x) where y is scalar and x is an m-dimensional
row vector. The sigmoid network function is based on the following expansion:

F(x) =(x−r)PL+aa f ((x−r)Q 1+c1))+...an f ((x−r)Q n +cn))+d
b
b

(9)

where f is the sigmoid function, given by the following equation:
f (z) =

1
e +1
−z

P and Q are m-by-p and m-by-q projection matrices. The projection matrices P and Q are determined
by principal component analysis of estimation data. If the components of x in the estimation data are
linearly dependent, then p<m. The number of columns of Q, q, corresponds to the number of
components of x used in the sigmoid function.

187


A wavenet defines a nonlinear function y=f(x), where y is scalar and x is an m-dimensional
row vector. The wavelet network function is based on the following function expansion:
F (x) = (x − r) PL + a s −1 f (b s _ 1 ((x − r) Q − c s _ 1 )) + .. + a w _ 1 g (b w _ 1 ((x − rQ − c w _ 1 )))

(10)

where:
f is a scaling function.
• g is the wavelet function.
• P and Q are m-by-p and m-by-q projection matrices, respectively.
The projection matrices P and Q are determined by principal component analysis of
estimation data. If the components of x in the estimation data are linearly dependent, then
p<m. The number of columns of Q, q, corresponds to the number of components of x used in
the scaling and wavelet function.
r is a 1-by-m vector and represents the mean value of the regressor vector computed from
estimation data.
• as, bs, aw, and bw are scalars. Parameters with the s subscript are scaling parameters, and
parameters with the w subscript are wavelet parameters.
• L is a p-by-1 vector.
• cs and cw are 1-by-q vectors.
III. DATA
142 years of rainfall data set (1871-2012) was obtained from the Indian Institute of Tropical
Meteorology website (ftp://www.tropmet.res.in/pub/data/rain/iitm-regionrf.txt), with the original
source as referred by the department being the Indian Meteorological Department (IMD), is used for
the analysis. Linear transformation is used to normalize rainfall series data. The outputs of the
normalization function are real numbers between 0 and 1. The equation can be described as:
D

Norm

=

Do − Dmin
Dmax − Dmin

(11)

where D0 is for the observed data, Dmax and Dmin denote maximum and minimum of the
observed data, respectively.
IV. WORK PERFORMED
Different prediction models have been designed such as
•
•
•

Adaptive neuro fuzzy model,
Non linear auto regressive model
Linear regressive model.

A grid based ANFIS model has been developed with a view to predict the rainfall for the next
year based on rainfall pattern for last four year data. Model has been designed and tested with
different membership functions and different number of members. It is found that Gaussian
membership function delivers better performance.
A linear regressive model has been designed for prediction with different number of delays
and order. To find the best value, performance parameters of the model were evaluated with different
delay and order. It is found that delay of 3 and order of 2 was found to be the best. The rainfall
188


pattern has a nonlinear relationship therefore a nonlinear auto regressive model with various
nonlinearity estimator such as sigmoidnet and wavenet was designed with different parameters and
was tested with different order and delay.
Nonlinearity

structure
n

g (x) = ∑ ak K ( β k ( x − γ k ) ) where K(s) is
k =1

Wavelet network
(default)

the wavelet function and k s the number of
unit.
n

g (x) = ∑ ak K ( β k ( x − γ k ) ) where
k =1

sigmoid network

K(s)=(es+1) -1 is the sigmoid function

V. PERFORMANCE PARAMETERS
To assess the models’ performance, following criteria are used as given below.
Correlation coefficient (CC): It indicates the strength of relationships between observed and
estimated rainfall. The correlation coefficient is a number between 0 and 1, and the higher the
correlation coefficient the better is the accuracy of the model.

N
∑ xact (i ) − x act x pre (i ) − x pre
i =1
N
2N
2
∑ xact (i ) − x act ∑ x pre (i ) − x pre
i =1
i =1

(

CC =

)(

(

)

)

(

)

(12)

Where xact (i) denotes actual rainfall and x pre (i ) denotes predicted rainfall value at point i.

xact and x pre denotes the mean value of actual and predicted rainfall series. N denotes the
sample size.
Root mean square error (RMSE): It evaluates the residual between observed and forecasted
rainfall. This index assumes that larger forecast errors are of greater importance than smaller ones;
hence they are given a more than proportionate penalty

2
1 N
∑ xact ( i )− x pre (i )
N i =1

(

RMSE =

(

Max x pre

)
( Max( x pre )−Min( x pre ))

) indicates maximum value of predicted rainfall ,

(

Min x pre

) denotes minimum value of

predicted rainfall It is noticed that RMSE equal to zero represents a perfect fit.
189

(13)


V. RESULT AND DISCUSSION
During the development of the prediction model, various configurations were explored in
order to achieve enhanced performance parameter value. ANFIS model was tested for different
membership function such as Gaussian, trapezoidal, etc and different number of members. Gaussian
membership function with three members is found to be optimum. Linear model with different
number of delay and order were designed and tested. Nonlinear model was designed using different
type of non-linear estimating function. Table 1.1 shows the performance parameter for the model
designed .It is observed that ANFIS model has better prediction capability due to combined power of
fuzzy logic and neural network. But the execution time taken is more. As rainfall process is nonlinear
in nature, It is found that nonlinear model has good predicting capabilities. Nonlinear model is found
to be better as compare to linear model. As the order of the model is increased it become more
complex and hence execution time increases. In nonlinear model sigmoidnet estimator is superior in
making prediction as compared to wavelet estimator. The graph of observed rainfall and predicted
rainfall by different model is shown in the fig 3.

Table 1.1 The performance of parameters of proposed models during different phases
Training period
Model type

parameter
NRMSE

Testing period

Correlation
Correlation
NRMSE
Coefficient
Coefficient

ANFIS model

Member function
‘Gaussian’, three
member

0.0011

0.998

0.0046

0.95

Linear regressive

Delay=2,order=3

0.0080

0.94

0.0088

0.84

Nonlinear
regressive model

Wavenet estimator
Delay=2,order=3

0.0054

0.96

0.0075

0.86

Nonlinear
regressive model

Sigmoidnet estimator
Delay =2,order=3

0.0033

0.97

0.0049

0.93

Obserevd RF
ANFIS Model
NL-Sigmoid
NL-wavelet
Linear Model

0.95

A ra e R in ll
ve g a fa

0.9

0.85

0.8

0.75
1997

1999

2001

2003
2005
Years

190

2007

2009

2011


VI. REFERENCES
[1]
[2]
[3]
[4]
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[8]
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[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
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