FORECASTING YIELD PER HECTARE OF RICE IN ANDHRA PRADESH

Transcription

1 International Journal of Mathematics and Computer Applications Research (IJMCAR) ISSN Vol. 3, Issue 1, Mar 013, 9-14 TJPRC Pvt. Ltd. FORECASTING YIELD PER HECTARE OF RICE IN ANDHRA PRADESH R. RAMAKRISHNA 1 & NAVEEN KUMAR BOIROJU 1 Vidya Jyothi Institute of Technology, CB Post, Aziznagar, Hyderabad, Andhra Pradesh, India Department of Statistics, Osmania University, Hyderabad, Andhra Pradesh, India ABSTRACT In this paper, forecasting of yield per hectare of Rice (in Kg) in Andhra Pradesh using Box-Jenkins methodology and feedforward neural networks are presented. The forecasting performance of the models evaluated using mean absolute error, mean absolute percentage error and root mean squared errors. Neural networks model outperforms than that of the Box-Jenkins model. KEYWORDS: Box-Jenkins Methodology, ARIMA, Rice, Neural Networks INTRODUCTION Andhra Pradesh is the fifth largest state in India accounting for 9 and 8 per cent of the country s area and population, respectively. Rice is the Principal food crop cultivated throughout the state providing food for its growing population, fodder to the cattle and employment to the rural masses. Any decline in its hectarage and production will have a perceivable impact on the state s economy and food security. In A.P rice is mostly cultivated under irrigated eco-system under canals (5%), tube wells (19.31) tanks (16.%), other wells (8.8%) and other sources (3.7%). (Cheralu, 011). Cheralu (011) presented the status paper on rice in Andhra Pradesh. Sarma et.al. (008) developed an agroclimatic model for the estimation of rice yield in Andhra Pradesh. A multiple regression model is proposed for the estimation of rice yield based on the atmospheric and oceanic indices. Raghavender (009) presented an autoregressive integrated moving average (ARIMA) model for the forecasting of rice yield per hectare in Andhra Pradesh based on the data set during 1955 to 007. The data used in this paper is collected from the Directorate of Economics and Statistics, Andhra Pradesh and which consists of yearly average yield per hectare of rice in Kilograms during the years to In this paper, forecasting of the yield per hectare of Rice in Andhra Pradesh using Box-Jenkins methodology and feedforward neural networks is discussed. A comparative study is carried out to investigate the forecasting ability of neural networks and the results of the neural networks model is compared with that of the ARIMA model. Section, presents the ARIMA model building using Box-Jenkins methodology. The development of feedforward neural networks is discussed in Section 3. Section 4 presents the final results and conclusion. BOX-JENKINS METHODOLOGY In this Section, the modeling of yield per hectare of Rice in Andhra Pradesh using Box-Jenkins methodology is discussed. The Box-Jenkins procedure is concerned with the fitting of an ARIMA model of the following form to a given set of data{ Z t : t = 1,,..., N} and the general form of ARIMA (p,d,q) model is given by φ d ( B) Z t θ ( B) a t, (1) φ = φ φ φ pb p q where ( B) = 1 B B L, ( B) = 1 B B L, d = ( B) d 1 θ θ 1 θ θ qb 1,

2 10 R. Ramakrishna & Naveen Kumar Boiroju B k Z t = Z and at is a white noise process with zero mean and variance t k σ a. The Box-Jenkins procedure consists of the following four stages: (1) model identification, where the orders d, p, q are determined by observing the behaviour of the corresponding autocorrelation function (ACF) and partial autocorrelation function (PACF); () estimation, where the parameters of the model are estimated by the maximum likelihood method; (3) diagnostic checking by the Portmanteau test, where the adequacy of the fitted model is checked by the Ljung-Box statistic applied to the residuals of the model; (4) forecasts are obtained from an adequate model using minimum mean squared error method. If the model is judged to be inadequate, stages 1-3 are repeated with different values of d, p and q until an adequate model is obtained (Box et al. 1994). to The following Figure -1, presents the time plot of the yield per hectare of rice in Andhra Pradesh during Figure 1: Yield per Hectare of Rice (in Kg) in Andhra Pradesh The sample autocorrelation function for the given data is displayed in the following figure. Figure : Sample Autocorrelation Function From the above sample ACF it is evident that the autocorrelations not dies out quickly for higher lags and also the time plot of the given series shows an increasing trend, it indicates that the given time series is a non-stationary series. The non-stationarity in mean corrected through the successive differencing of order one (d=1) is enough to achieve stationary series. The newly constructed variable 1 W t Z = forms a stationary series. The next step is to identify the values of p t and q. Autocorrelations and partial autocorrelations for 5 lags of W t are computed for the identification of the parameters of ARIMA model.

3 Forecasting Yield Per Hectare of Rice in Andhra Pradesh 11 Figure 3: Sample ACF and PACF of W t From the above ACF and PACF, it is observed that the order of autoregressive parameters is at most (p=) and the order of moving average parameters at most one (q=1). The following tentative models are entertained and chosen a suitable model which has minimum normalized Baysian information criterion (BIC) value. Table 1: Tentative ARIMA Models ARIMA(p, d, q) Model Normalized BIC ARIMA(,1,1) 10.1 ARIMA(,1,0) ARIMA(0,1,1) The suitable model is ARIMA (0,1,1) for forecasting the yield per hectare of rice in Andhra Pradesh. The model parameters are estimated using SPSS software and the results are presented in the following table. Table : ARIMA Model Parameters Variable Transformation Parameter Estimate SE t Sig. Constant Yield per Hectare of No Difference 1 Rice (in Kg) Transformation MA Lag From the above table it is observed that all the parameters are significant at 5% level. So the fitted model for the yield per hectare of Rice in Andhra Pradesh is Z ˆ. t = Zt 1 + at at 1 model. The adequacy of the model is checked using the ACF and PACF of the residuals of various orders of the selected

4 1 R. Ramakrishna & Naveen Kumar Boiroju Figure 4: ACF and PACF of Residuals From the above figure, it is observed that none of these autocorrelations is significantly different from zero at 5% level. This proves that the model is an appropriate model. The adequacy of the model is tested using Ljung-Box Q- test statistic (Ljung and Box, 1978). Ljung-Box statistic value is for 17 degrees of freedom and the significant probability corresponding to Box-Ljung Q-statistic is which is greater than 0.05, therefore, H o is accepted and we may conclude that the selected ARIMA (0,1,1) model is an adequate model for the given time series. Forecasts from the ARIMA for the years from 011 to 015 are presented in the Section 4. NEURAL NETWORKS MODEL In this Section, we develop a feedforward neural networks (FFNN) model for forecasting of the yield per hectare of Rice in Andhra Pradesh. Artificial Neural Networks (ANN) are a biologically inspired information processing systems. Inspired by the Brain, they have similar individual information processing elements called artificial neurons which are interconnected to form a complex network. The processing of information is based on mathematical modeling and depending upon the connectivity and different ways they process the information, ANN s are classified into a number of Networks (Haykin, 1999). Use of neural network-based models is an alternative option available to researchers for capturing the underlying non-linearity in the time series. There are several features of the artificial neural network based models that make them attractive as a forecasting tool. First, as opposed to the traditional model-based methods, ANN-based models are datadriven and self-adaptive. Second, ANNs are universal function approximators. It has been shown that a network can approximate any continuous function to any desired accuracy. Finally, ANNs are non-linear models. The fact that realworld systems are often non-linear has led to the development of several non-linear time series models during the last decade (Hornik, 1993; Ramakrishna et.al., 011). The given data is partitioned into two samples namely training and testing samples. The training sample comprises the data records used to train the neural networks; the testing sample is an independent set of data records used to track errors during training in order to prevent over training. The model is a three layer feed forward neural network and it consists of an input layer, a hidden layer and an output layer. Total number of input neurons needed in this model is one, and it representing the values of lag 1 (previous year yield of rice). In this model only one output unit is needed and it indicates the forecasts of rice yield. The following table displays information about the neural networks model, including the dependent variable, number of input and output units, rescaling method, number of hidden layers and units, and activation functions.

5 Forecasting Yield Per Hectare of Rice in Andhra Pradesh 13 Input Layer Hidden Layer(s) Output Layer Table 3: Network Information Covariates 1 Lag1 Number of Units a 1 Rescaling Method for Covariates Adjusted normalized Number of Hidden Layers 1 Number of Units in Hidden Layer 1 a Activation Function Dependent Variables 1 Hyperbolic tangent Yield per Hectare of Rice (in Number of Units 1 Rescaling Method for Scale Dependents Activation Function Error Function a. Excluding the bias unit Kg) Standardized Identity Sum of Squares The network is trained using backpropagation algorithm until the sum of squares of error is small for the training set. The network parameters are presented in the following table. Input Layer Hidden Layer 1 Predictor Table 4: Parameter Estimates Predicted Hidden Layer 1 Output Layer H(1:1) H(1:) Rice_Yield (Bias) lag (Bias) H(1:1) 1.4 H(1:).8 The forecasting model is given by Zˆ t I ( H ( 1:1).8H ( 1: ) ) = + + where H(1:1)=TANH( Z % t 1 ), H(1:)=TANH( Z % t 1), I(.) identity function and Z % t 1is an adjusted normalized lag variable. Forecasts from FFNN model for the years from 011 to 015 are presented in the following section. CONCLUSIONS This section presents the error measures and the forecasts of the rice yield (in Kg) using the two models. We computed the forecasts for the given data using both the models and computed the mean absolute error (MAE), root mean squared error (RMSE), mean absolute percentage error (MAPE) and are presented in the following table. Table 5: Measures of Error Model MAE RMSE MAPE ARIMA FFNN From the above table it is clear that, FFNN model has very less error comparing to errors of the ARIMA model. Hence the FFNN model is suitable for the forecasting of rice yield. Forecasts for the future years from 011 to 015 by using the ARIMA and FFNN models presented in the following table.

6 14 R. Ramakrishna & Naveen Kumar Boiroju Table 6: Forecasts of Rice Yield Year ARIMA Forecasts FFNN Forecasts From the above table it is observed that the forecasts using ARIMA model shows an increasing trend but where as in FFNN model shows a decreasing trend in the yield of rice in Andhra Pradesh. But FFNN model performed well compare to the ARIMA model at fitting stage, so one can consider the forecasts from the FFNN model for the future course of action. Here the two models producing the two different forecasts, so the researcher has to plan the future by combining these forecasts or by considering the two situations. The validity of these forecasts can be checked when the actual data is available for the lead years. From the above forecasts for the lead periods shows that there is a small change in the forecasts of yield of rice in Andhra Pradesh. There is a need to adopt the high yielding varieties of rice and improved package of practices for increasing the yield of rice in Andhra Pradesh. REFERENCES 1. Box, G. E. P., Jenkins, G. M. And Reinsel, G. C., (1994), Time Series Analysis Forecasting and Control, 3rd ed., Englewood Cliffs, N.J. Prentice Hall.. Cheralu, C. (011), Status paper on rice in Andhra Pradesh, Rice Knowledge Management Portal, Directorate of Rice Research, Hyderabad. ( 3. Haykin, S. S., (1999), Neural Networks: A Comprehensive Foundation, Upper Saddle River, N.J., Prentice Hall. 4. Hornik, K, (1993), Some new results on neural network approximation, Neural Networks, 6, Ljung, G. M. and Box, G. E. P., (1978), On A Measure of Lack of Fit in Time Series Models, Biometrika, Raghavender, M. (009), Forecasting paddy yield in Andhra Pradesh using seasonal time series model, Bulletin of pure and applied sciences. 7. Ramakrishna, R., Naveen Kumar, B. and Krishna Reddy, M. (011), Forecasting daily electricity load using neural networks, International Journal of Mathematical Archive, Vol., Sarma, A.A.L.N, Lakshmi Kurmar, T.V and Koteswararao, K. (008), Development of an agroclimatic model for the estimation of rice yield, J. Ind. Geophys. Union, Vol.1,