matrix:regress

matrix:regress data-matrix

All three of the forecast primitives above are just special cases of performing an OLS (ordinary-least-squares) linear regression – the matrix:regress primitive provides a flexible/general-purpose approach. The input is a matrix data-matrix, with the first column being the observations on the dependent variable and each subsequent column being the observations on the (1 or more) independent variables. Thus each row consists of an observation of the dependent variable followed by the corresponding observations for each independent variable.

The output is a Logo nested list composed of two elements. The first element is a list containing the regression constant followed by the coefficients on each of the independent variables. The second element is a 3-element list containing the R² statistic, the total sum of squares, and the residual sum of squares. The following code example shows how the matrix:regress primitive can be used to perform the same function as the code examples shown in the matrix:forecast-*-growth primitives above. (However, keep in mind that the matrix:regress primitive is more powerful than this, and can have many more independent variables in the regression, as indicated in the fourth example below.)

;; this is equivalent to what the matrix:forecast-linear-growth does
let data-list [20 25 28 32 35 39]
let indep-var (n-values length data-list [ x -> x ]) ; 0,1,2...,5
let lin-output matrix:regress matrix:from-column-list (list data-list indep-var)
let lincnst item 0 (item 0 lin-output)
let linslpe item 1 (item 0 lin-output)
let linR2   item 0 (item 1 lin-output)
;;Note the "6" here is because we want to forecast the value at time t=6.
print (list (lincnst + linslpe * 6) (lincnst) (linslpe) (linR2))

;; this is equivalent to what the matrix:forecast-compound-growth does
let com-log-data-list  (map ln [20 25 28 32 35 39])
let com-indep-var2 (n-values length com-log-data-list [ x -> x ]) ; 0,1,2...,5
let com-output matrix:regress matrix:from-column-list (list com-log-data-list com-indep-var2)
let comcnst exp item 0 (item 0 com-output)
let comprop exp item 1 (item 0 com-output)
let comR2       item 0 (item 1 com-output)
;;Note the "6" here is because we want to forecast the value at time t=6.
print (list (comcnst * comprop ^ 6) (comcnst) (comprop) (comR2))

;; this is equivalent to what the matrix:forecast-continuous-growth does
let con-log-data-list  (map ln [20 25 28 32 35 39])
let con-indep-var2 (n-values length con-log-data-list [ x -> x ]) ; 0,1,2...,5
let con-output matrix:regress matrix:from-column-list (list con-log-data-list con-indep-var2)
let concnst exp item 0 (item 0 con-output)
let conrate     item 1 (item 0 con-output)
let conR2       item 0 (item 1 con-output)
print (list (concnst * exp (conrate * 6)) (concnst) (conrate) (conR2))

;; example of a regression with two independent variables:
;; Pretend we have a dataset, and we want to know how well happiness
;; is correlated to snack-food consumption and accomplishing goals.
let happiness [2 4 5 8 10]
let snack-food-consumed [3 4 3 7 8]
let goals-accomplished [2 3 5 8 9]
print matrix:regress matrix:from-column-list (list happiness snack-food-consumed goals-accomplished)
=> [[-0.14606741573033788 0.3033707865168543 0.8202247191011234] [0.9801718440185063 40.8 0.8089887640449439]]
;; linear regression: happiness = -0.146 + 0.303*snack-food-consumed + 0.820*goals-accomplished
;; (Since the 0.820 coefficient is higher than the 0.303 coefficient, it appears that each goal
;; accomplished yields more happiness than does each snack consumed, although both are positively
;; correlated with happiness.)
;; Also, we see that R^2 = 0.98, so the two factors together provide a good fit.