Linear regression (regress) discrepancy with polynomial fit (polyfit) - matlab

Question

I have some data which comes from a linear function (y=mx+c) where m=4, c=1 (so: y=4x+1).
When I try to get back the coefficients using regress, I'm getting an R2<1 and a seemingly random m value:
x = [1 2 3 4]
y = [5 9 13 17]
[m,bint,r,rint,stats] = regress(y',x');
%{
>> R = stats(1) % Coefficient of determination
R =
1
>> m % Linear function coefficients
m =
4.333333333333333
%}
Whereas polyfit does this correctly:
P = polyfit(x,y,1);
%{
>> P(1)
ans =
4.000000000000000
>> P(2)
ans =
1.000000000000000
%}
Why is this happening?

Answer 1

The source of your problem is not following the documentation or regress which states that:
b = regress(y,X) returns a vector b of coefficient estimates for a multiple linear regression of the responses in vector y on the predictors in matrix X. The matrix X must include a column of ones.
If we include a column of ones in the 2nd input, we get the desired result:
x = [1 2 3 4].';
y = [5 9 13 17].';
[m,bint,r,rint,stats] = regress(y,[ones(size(x)) x]);
%{
Results:
m =
1.0000
4.0000
bint =
1.0000 1.0000
4.0000 4.0000
r =
1.0e-14 *
0.1776
0.1776
0.1776
0
rint =
1.0e-13 *
0.0178 0.0178
-0.2190 0.2545
-0.2190 0.2545
-0.2141 0.2141
stats =
1.0e+31 *
0.0000 1.6902 0.0000 0.0000
%}