Forecasting By Discriminant Function Weather Based Analysis

Weather is one of the most important factors influencing crop growth. It may influence production directly through affecting the growth structural characteristics of crop such as plant population, number of tillers leaf area etc., and indirectly through its effect on incidence of pests and diseases. The effect of weather parameters at different stages of growth of crop may help in understanding their response in term of final yield and also provide a forecast of crop yields in advance before the harvest. Changes in the timing of phonological events are among the most important indicators of global warming Parmesan and Yohe [1]. The extent of weather influence on crop yields depends not only on magnitude of weather parameters but also on their frequency distribution. Menzel and Fabian [2] reported on phonological change due to increasing of temperature.


Introduction
Weather is one of the most important factors influencing crop growth. It may influence production directly through affecting the growth structural characteristics of crop such as plant population, number of tillers leaf area etc., and indirectly through its effect on incidence of pests and diseases. The effect of weather parameters at different stages of growth of crop may help in understanding their response in term of final yield and also provide a forecast of crop yields in advance before the harvest. Changes in the timing of phonological events are among the most important indicators of global warming Parmesan and Yohe [1]. The extent of weather influence on crop yields depends not only on magnitude of weather parameters but also on their frequency distribution. Menzel and Fabian [2] reported on phonological change due to increasing of temperature.
The alternation in global warming has dramatically affected agriculture and its productivity. The increase in temperature has significantly led to change in the agricultural zones and shift in the growing season. Fisher [3] has been used by biologists to solve the classificatory problems involving multiple measures in different contexts. Models based on weather parameters can provide reliable forecast of crop yield in advance of harvest Agrawal and Mehta [4]. The forecasting equations have also been developed for wheat yield in Kanpur district U.P. Agarwal [5], Rai, Chandrahas [6] made use Discriminant function of weather variables to develop statistical models for pre-harvest forecasting of rice yield in Raipur district of Chhattisgarh. The model on the basis of weather variable have been done by Agrawal [7,8] A lot of works have been done for the development of the model with the weather variables but no work has been done in this direction for the eastern Uttar Pradesh for rice crop. In the present paper, an attempt has been made to develop suitable statistical model for forecasting of pre-harvest rice yield in faizabad district using Discriminant scores from Discriminant functions obtained from the weekly data on weather variables with a few modifications.

Materials and Methods
The study has been conducted for Faizabad district of Eastern Uttar Pradesh, which is situated between 260 47' N latitude and We will first describe briefly the technique of Discriminant function analysis. The Discriminant function analysis has been discussed in many books, to mention a few, Anderson [9], Hair [10], Sharma [11], Johnson, Wichern [12] etc.

Consider a linear function of the form
Where Z is discriminant Function, l'= (l 1 ,l 2 ,……l p ) X'= (X 1 ,X 2 ,……X p ) X i is the i th weather variable used to discriminate the groups and l i is the corresponding discriminant coefficient, p is the number of variables.
Let nj be the size of jth group ( j = l ,….., k) and xijm be the mth observation of ith variable for jth group. Then mean of j th group for i th variable is x ij = x ijm and overall average for i th variable is given by be between group matrix of sum of squares and cross products, and let W = be the pooled matrix of Sum of squares and Sum of products, where S j is the matrix of sum of squares and product in j th group.
The Fisher's sample linear Discriminant functions can therefore, be obtained as follows.
Let denote the S ≤ min (g-1, p) non-zero Eigen Values of and be the corresponding eigenvectors (scaled so that ).
Then the vector of coefficients lˆ that maximize the ratio

Development of forecast models
The crop years have been developed into three groups namely, congenial normal and adverse on the basis of crop yield, which is adjusted for trend effect. Here, only the first 19 year data from 1990 to 2008 have been utilized for the model fitting and remaining two years were left for the validation of the model. Weekly data on weather variables corresponding to three pre defined groups have been used for the development of scores for each year through function analysis technique. In the present study the number of groups is three and number of weather variable is seven. Therefore only two scores will be obtained. Discriminant analysis approach predicts the future observations qualitatively in different groups. For quantitative forecasting, regression models are fitted by taking the scores and the trend variable as and crop yield as the entire 19 weeks data from 23 rd to 36 th (Standard meteorological week) have been utilized for development of the model.

Development of the Model
In this procedure, function analysis have been carried out using the data on the first weather variables spread over 14 weeks using 23 rd to 36 th SMW. Using two scores obtained function of the data on the first weather variable and 14 week data on second variable, function analysis has been again performed and two sets of scores are obtained (here the discriminating variables will now become 16). Using these two sets of scores and 14 week data of third variable have been again used to analysis and subsequently two sets of scores have been obtained. up to seventh weather variables, and ultimately we get two set of scores. These two sets of scores and the trend as the variable and crop yield as were utilized to develop forecast model by fitting the following model: Where y is detrended crop yield β i 's ( i =1,2,3) are model parameters, ds 1 and ds 2 are two sets of discriminant scores, T is the trend variable and e is error term assumed to follow N (0, σ 2 ). This model utilized the complete data over 14 weeks and also considers relative importance of weather variables in different weeks. It is also a measure for comparing two models. The formula of RMSE is given bellow

Comparison and validation of forecast models
Oi and the Ei are the observed and forecasted value of the crop yield respectively and n is the number of years for which forecasting has been done.

d) Percent Standard error of forecast:
Let ŷf be forecast value of crop yield and X 0 be the column vector of P independent variable at which y is forecasted then variance ŷf is given by (Draper and Smith, 1998) is obtained as

Results and Discussion
The forecast models for the rice crop yield have been developed under this procedure along with R adj 2 and RMSE are given in (Table 1). First Discriminant score ds1 has been found to be significant at one percent probability level of significance (p < 0.01) in the model and the second discriminant score has been found to be significant at one percent probability level of significance (p < 0.01) in model. Adjusted coefficient of determination (R adj 2 ) has been found to be 87.5% in the model. Thus, it can be concluded that the proposed model is most suitable model to forecast rice yield in Faizabad district of Eastern Uttar Pradesh. Hence, a reliable forecast of rice yield about two months before the harvest can be obtained from the proposed model.
Minhajuddin [13] proposed a method to simulate the joint distribution which have equal to positive pair-wise correlations and the method was illustrated for the p-dimensional families of beta and gamma distributions. Sever [14] compaired fisher's discriminant analysis under normal and skewed curved normal distribution based on the apparent error rates, which were used as a measure of classification performance and found that fisher's linear discriminant analysis to be highly robust under skewed curved normal distribution. Rausch, Kelley [15] compared different methods for discriminant analysis with respect to classification accuracy under non normality through Monte Carlo simulation. Pandey [16] compared different distribution as normal, lognormal, and pearson's type on the basis of weather variable on wheat yield for Faizabad district of Eastern U.P. Raman [17] compared non-normal rice and maize yield with linear discriminant function analysis under multivariate analysis for New Delhi. Ito and Schull [18] discuss the robustness of T02 Statistics, when the conditions of equality of covariance matrices are not satisfied.
The Dirichlet distribution is a multivariate generalization of beta distribution Kotz [19]. Almost similar results, as observed in this study, have also been reported by Kandiannan [20] for Coimbatore in Tamil Nadu, where temperature, rainfall and radiation entered significantly in a stepwise prediction equation of rice yield. In Andhra Pradesh also, rainfall and temperature have been reported to affect rice yield significantly Barnwal and Kotani [21], Lal [22] also observed that maximum temperature, minimum temperature and moisture stress were crucial weather variables affecting soybean yield. Temperature, rainfall and relative humidity were found significantly correlated with sugarcane yield, Srivastava [23,24].