An Application of the Kronecker Product of Matrices in Multiple Regression

In a recent climatological investigation, the author had occasion to compute the regression of monthly rainfall on the altitude, latitude and longitude of certain observing stations in South Australia. The original data comprised the monthly rainfall at each of fifty stations for the years 1888-1949 inclusive, and regression formulae were calculated for both mean monthly rainfall and monthly totals in individual years. For any particular month, the analysis of variance of the 3100 (50 X 62) rainfall totals then assumed the form shown in Table 1. Designating the determining variates altitude, latitude and longitude by xi , x2 and X3 respectively, with corresponding means l x and x3 and regression coefficients b, , b2 and b3 , and writing