The multiple linear Ordinary Least Squares (OLS) regression models the relationship or linear dependence between a dependent (predictand) variable and and one or more independent (predictor) variables. The procedure estimates the linear regression coefficients by minimising the sum of squared residuals.
The estimated regression model is of the form:
where is the predictand variable, are the predictor variables (x)
with nd observations, is the y-intercept (b0), (b)
are the regression coefficients, and (nv) is the number of predictors (excluding the intercept).
The subroutine constructs a full matrix internally by prepending a column of ones to account for the intercept. The regression coefficients are estimated as:
where is the extended design matrix including the intercept term.
The coefficient of determination (r2) which represents the proportion of the total
variance of (y) explained by the predictors. The predicted values (y_hat),
standard errors (se) of the coefficients, and the covariance matrix of the predictors (cov_b)
can optionally be returned by the procedure, too.
Note: This subroutine uses eigh from the stdlib_linalg module.
Note: The intercept and predictor coefficients are computed separately and returned explicitly.
Multiple Linear Ordinary Least Squares (OLS) Regression with intercept. NOTE: OLS could be wrapper for ridge (with lambda = 0 or presence checks if mande an optional argument). However, it would increase computation slightly and make code less readable. OLS is often used in teaching and therefore, an easily readable standalone is kept.
| Type | Intent | Optional | Attributes | Name | ||
|---|---|---|---|---|---|---|
| real(kind=wp), | intent(in) | :: | x(nd,nv) |
predictor data matrix (no intercept column) |
||
| real(kind=wp), | intent(in) | :: | y(nd) |
response vector |
||
| integer(kind=i4), | intent(in) | :: | nd |
number of datapoints |
||
| integer(kind=i4), | intent(in) | :: | nv |
number of predictors (excluding intercept) |
||
| real(kind=wp), | intent(out) | :: | b0 |
intercept coefficient |
||
| real(kind=wp), | intent(out) | :: | b(nv) |
predictor coefficients |
||
| real(kind=wp), | intent(out) | :: | r2 |
coefficient of determination R² |
||
| real(kind=wp), | intent(out), | optional | :: | y_hat(nd) |
predicted y values |
|
| real(kind=wp), | intent(out), | optional | :: | se(nv) |
standard errors of predictor coefficients |
|
| real(kind=wp), | intent(out), | optional | :: | cov_b(nv,nv) |
covariance matrix of predictor coefficients |