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  1. 1. 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 © IAEME: www.iaeme.com/ijaret.asp Journal Impact Factor (2013): 5.8376 (Calculated by GISI) www.jifactor.com 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
  2. 2. 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
  3. 3. 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 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
  4. 4. 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 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
  5. 5. 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 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
  6. 6. 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 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
  7. 7. 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 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)
  8. 8. 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 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
  9. 9. 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 VI. REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] Sulochana Gadgil and Siddhartha Gadgil, The Indian Monsoon, GDP and agriculture, Economic and Political Weekly, XLI, 2006,pp. 4887–4895. Rajeevan M., “Prediction of Indian summer monsoon: Status, problems and prospects”. Curr. Sci., 2001, 81, pp.1451–1457. Thapliyal, V. and Rajeevan, M., “Monsoon prediction”. Encyclopedia of Atmospheric Sciences (ed. Holton, J.), Academic Press, New York, 2003, pp. 1391–1400. Sikka, D.R., Some aspects of the large scale fluctuations of summer monsoon rainfall over India in relation to fluctuations in the planetary and regional scale circulation parameters, Proc. Ind.Acad.Sciences (Earth and Planet Sci), 89, 1980,179-195. F. W. Zwiers, J. V. Storch,”On the role of statistics in climate research,” Int. J. Climatology, 2004 Vol.24, pp. 665-680. Shamseldin AY. Application of a neural network technique to rainfall-runoff modeling. J Hydrol 1997;199:272–94. Ham FM, Kostanic I. Principles of neurocomputing for science and engineering New York: McGraw Hill; 2001. French MN, Krajewski WF, Cuykendall RR. Rainfall forecasting in space and time using a neural network. J Hydrol 1992;137:1–31. Grimes DIF, Coppola E, Verdecchia M, Visconti G. A neural network approach to real-time rainfall estimation for Africa using satellite data. J Hydrometeorol 2003;4:1119–33 Kin C. Luk, J. E. Ball AND A. Sharma, An Application of Artificial Neural Networks for Rainfall Forecasting, Mathematical and Computer Modelling 33, 2001,pp. 883-699. Poggio,T., and F. Girosi, Networks for approximation and learning, Proc. IEEE, vol. 78, 1990, pp. 1481–1497. Jang JSR, ANFIS: adaptive-network-based fuzzy inference system, IEEE Trans Syst, Man, Cybernet;23(3):1993,665–85. Takagi T, Sugeno M., “Fuzzy identification of systems and its applications to modeling and control,” IEEE Trans Syst, Man, Cybernet;15:1985,116–32. K. Hornik, M. Stinchcombe and H. White, Multilayer feedforward networks are universal approximators, Neural Networks 1989 2, pp.359-366. Sven F. Crone: A Business Forecasting Competition Approach to Modeling Artificial Neural Networksfor Time Series Prediction. IC-AI 2004, pp. 207-213. M.French, W. Krajewski and R.R. Cuykendall, Rainfall forecasting in Space and time using a neural network, Journal of Hydrology 1992 137, 1-31. K.L. Hsu, V. Gupta and S. Soroshian, Artificial neural network modeling of the rainfall-runoff process, Water Resources Research l995 31 (10), pp.2517-2530. Werbos, P. J., Backpropagation through time:What it does and how to do it,Proc. IEEE, vol. 78, 1990,pp. 1550–1560. Rumelhart, D. E., Hinton G. E., and Williams R. J., Learning internal representations by error propagation,” in D. E. Rumelhart and J. L. McCleland, eds. (Cambridge, MA: MIT Press), vol. 1, Chapter 8. Jiansheng. Wu, Long. Jin, Forecast Research and Applying of BP Neural Network Based on Genetic Algorithms, Mathematics in Practice and Theory, 2005, vol. 35(1), pp.83-88. V.Anandhi and Dr.R.Manicka Chezian, “Comparison of the Forecasting Techniques – Arima, Ann and SVM - A Review”, International Journal of Computer Engineering & Technology (IJCET), Volume 4, Issue 3, 2013, pp. 370 - 376, ISSN Print: 0976 – 6367, ISSN Online: 0976 – 6375. M. Nirmala and S. M. Sundaram, “Modeling and Predicting the Monthly Rainfall in Tamilnadu as a Seasonal Multivariate Arima Process”, International Journal of Computer Engineering & Technology (IJCET), Volume 1, Issue 1, 2010, pp. 103 - 111, ISSN Print: 0976 – 6367, ISSN Online: 0976 – 6375. 191