Multivariate Linear Models

For example, we may have (i) measurements of d = 5 air pollutants (CO, NO, etc.) on n = 42 widely-separated days, (ii) d test scores for n different students, (iii) best results for d Olympic events for teams from n different countries, or (iv) d different physical measurements for n individuals (human or animal) that we are trying to classify. In each case, the i row corresponds to the i multivariate observation, and the j column corresponds to the j variable measured. As in the univariate (d = 1) case, we can also assume that we have r covariates for each observation (day or student or country or individual). For air pollution, these might be wind strength and solar intensity (r = 2), age, sex, and income for students (r = 3), or species or country of origin for physical measurements. These are connected in the regression model