Fit¶

Performs a least-squares fit of data to a linear combination of basis functions.

$ wo 'Fit[{2, 4, 6}, {x}, x]'
2\.(0+\d*)?\*x (regex)

When the design matrix does not have full column rank there is no unique least-squares solution, and the minimum-norm one is returned (the norm taken after scaling each basis function's column of values to unit length). Fitting a line through a single point gives the shortest coefficient vector among the infinitely many lines through it:

$ wo 'Fit[{{1, 1}}, {1, x}, x]'
0\.(5(0+\d*)?|4999999999\d*) \+ 0\.(5(0+\d*)?|4999999999\d*)\*x (regex)

Repeated abscissae are the same situation — the fit passes through their mean:

$ wo 'Fit[{{1, 1}, {1, 2}}, {1, x}, x]'
0\.75(0+\d*)? \+ 0\.7(5(0+\d*)?|4999999999\d*)\*x (regex)

$ wo 'Fit[{{1, 1}, {1, 2}, {1, 3}}, {1, x}, x]'
(1\.(0+\d*)?|0\.9999999999999\d*) \+ (1\.(0+\d*)?|0\.9999999999999\d*)\*x (regex)

The scaling shows when the basis columns differ in length: at x = 2 the column of x is twice that of 1, so the fit is 1 + x/2 rather than the unscaled shortest 2/5 + 4/5 x:

$ wo 'Fit[{{2, 1}, {2, 2}, {2, 3}}, {1, x}, x]'
(1\.(0+\d*)?|0\.9999999999999\d*) \+ (0\.5(0+\d*)?|0\.4999999999999\d*)\*x (regex)