Class reg_linear_least_squares

This solves min! |a*x - y|^2 + |regstr*(x-regtgt)|^2, and returns (x, the_fit, rank, s). Normally, a.shape==(m,n) and y.shape==(m,q), where m is the number of data to be fit, n is the number of parameters to use in a fit (equivalently, the number of basis functions), and q is the number of separate sets of equations that you are fitting. Then, x has shape (n,q) and the_fit has shape (m,q).

The regularization target, regtgt is the same shape as x, that is (n,q). (It must be a vector if and only if y is a vector.) Regstr, the strength of the regularization is normally an (n,n) matrix, though (*,n) will work, as will a scalar.

Y may be a 1-D matrix (a vector), in which case the fit is a vector. This is the normal case where you are fitting one equation. If y is a 2-D matrix, each column (second index) in y is a separate fit, and each column in the solution is a separate result.

Overrides: object.__init__

Returns something like the rank of the solution, but rather than counting how many dimensions can be solved at all, it reports how many dimensions can be solved with a relatively good accuracy.

Returns: float

Overrides: lls_base.eff_rank

(inherited documentation)

Hat Matrix Diagonal Data points that are far from the centroid of the X-space are potentially influential. A measure of the distance between a data point, xi, and the centroid of the X-space is the data point's associated diagonal element hi in the hat matrix. Belsley, Kuh, and Welsch (1980) propose a cutoff of 2 p/n for the diagonal elements of the hat matrix, where n is the number of observations used to fit the model, and p is the number of parameters in the model. Observations with hi values above this cutoff should be investigated. For linear models, the hat matrix

H = X (X'X)-1 X'

can be used as a projection matrix. The hat matrix diagonal variable contains the diagonal elements of the hat matrix

hi = xi (X'X)-1 xi'

Overrides: lls_base.hat