An Inverted Matrix Approach for Determining Crop-Weather Regression Equations

Abstract

We would like to know whether year-to-year changes, or month-tomonth changes in crop yields or prospects are consistent with observed weather data. Generally, historical weather records extend back farther than records of crop yields. We wish to make use of weather data for the entire period of record even though yield data may be available for a much shorter period. This paper reports on an exploratory inverted matrix approach used in one phase of a crop-weather study. The application of multiple regression methods in the study of relationships between crop yields and weather factors is, of course, not new, but the large amount of computational labor involved has discouraged many workers and our people from attempting correlations studies on a very extensive scale. As pointed out by R. A. Fisher, the use of the inverse matrix of a set of normal equations greatly reduces the amount of computations when the same set of independent variables is used repeatedly; in addition, it serves to simplify the calculation of sampling errors of the regression coefficients. However, a large amount of computational work is still required when the various dependent variables are available for only relatively few years, and these periods vary from crop to crop because of the fact that the data or series were started at different points in time. We would like some way of utilizing all the weather and crop yield data available. Therefore, we would like to devise what might be called inverse matrix solution for a given State or area which could be used whenever the given set of weather factors were appropriate. However, the sampling errors of the regression coefficients cannot be computed using the elements of this generalized where the dependent variables are used for only a subperiod. The inverse matrix is obtained for a given set of independent variables (i.e., weather factors) for the entire period of the weather