regress (Statistics Toolbox)

Multiple linear regression.

Syntax

b = regress(y,X)
[b,bint,r,rint,stats] = regress(y,X)
[b,bint,r,rint,stats] = regress(y,X,alpha)

Description

b = regress(y,X) returns the least squares fit of y on X by solving the linear model

for , where:

y is an n-by-1 vector of observations
X is an n-by-p matrix of regressors
is a p-by-1 vector of parameters
is an n-by-1 vector of random disturbances

[b,bint,r,rint,stats] = regress(y,X) returns an estimate of in b, a 95% confidence interval for in the p-by-2 vector bint. The residuals are returned in r and a 95% confidence interval for each residual is returned in the n-by-2 vector rint. The vector stats contains the R² statistic along with the F and p values for the regression.

[b,bint,r,rint,stats] = regress(y,X,alpha) gives 100(1-alpha)% confidence intervals for bint and rint. For example, alpha = 0.2 gives 80% confidence intervals.

Examples

Suppose the true model is

where I is the identity matrix.

X = [ones(10,1) (1:10)']
X =
     1     1
     1     2
     1     3
     1     4
     1     5
     1     6
     1     7
     1     8
     1     9
     1    10
y = X * [10;1] + normrnd(0,0.1,10,1)
y =
   11.1165
   12.0627
   13.0075
   14.0352
   14.9303
   16.1696
   17.0059
   18.1797
   19.0264
   20.0872
[b,bint] = regress(y,X,0.05)
b =
   10.0456
    1.0030
bint =
    9.9165   10.1747
    0.9822    1.0238

Compare b to [10 1]'. Note that bint includes the true model values.

Reference

Chatterjee, S. and A. S. Hadi. Influential Observations, High Leverage Points, and Outliers in Linear Regression. Statistical Science, 1986. pp. 379-416.

refline regstats