lscov

Least-squares solution in presence of known covariance

Syntax

x = lscov(A,B)
x = lscov(A,B,w)
x = lscov(A,B,V)
x = lscov(A,B,V,alg)
[x,stdx] = lscov(...)
[x,stdx,mse] = lscov(...)
[x,stdx,mse,S] = lscov(...)

Description

x = lscov(A,B) returns the ordinary least squares solution to the linear system of equations A*x = B, i.e., x is the n-by-1 vector that minimizes the sum of squared errors (B - A*x)'*(B - A*x), where A is m-by-n, and B is m-by-1. B can also be an m-by-k matrix, and lscov returns one solution for each column of B. When rank(A) < n, lscov sets the maximum possible number of elements of x to zero to obtain a "basic solution".

x = lscov(A,B,w), where w is a vector length m of real positive weights, returns the weighted least squares solution to the linear system A*x = B, that is, x minimizes (B - A*x)'*diag(w)*(B - A*x). w typically contains either counts or inverse variances.

x = lscov(A,B,V), where V is an m-by-m real symmetric positive definite matrix, returns the generalized least squares solution to the linear system A*x = B with covariance matrix proportional to V, that is, x minimizes (B - A*x)'*inv(V)*(B - A*x).

More generally, V can be positive semidefinite, and lscov returns x that minimizes e'*e, subject to A*x + T*e = B, where the minimization is over x and e, and T*T' = V. When V is semidefinite, this problem has a solution only if B is consistent with A and V (that is, B is in the column space of [A T]), otherwise lscov returns an error.

By default, lscov computes the Cholesky decomposition of V and, in effect, inverts that factor to transform the problem into ordinary least squares. However, if lscov determines that V is semidefinite, it uses an orthogonal decomposition algorithm that avoids inverting V.

x = lscov(A,B,V,alg) specifies the algorithm used to compute x when V is a matrix. alg can have the following values:

  • 'chol' uses the Cholesky decomposition of V.

  • 'orth' uses orthogonal decompositions, and is more appropriate when V is ill-conditioned or singular, but is computationally more expensive.

[x,stdx] = lscov(...) returns the estimated standard errors of x. When A is rank deficient, stdx contains zeros in the elements corresponding to the necessarily zero elements of x.

[x,stdx,mse] = lscov(...) returns the mean squared error. If B is assumed to have covariance matrix σ2V (or (σ2diag(1./W)), then mse is an estimate of σ2.

[x,stdx,mse,S] = lscov(...) returns the estimated covariance matrix of x. When A is rank deficient, S contains zeros in the rows and columns corresponding to the necessarily zero elements of x. lscov cannot return S if it is called with multiple right-hand sides, that is, if size(B,2) > 1.

The standard formulas for these quantities, when A and V are full rank, are

  • x = inv(A'*inv(V)*A)*A'*inv(V)*B

  • mse = B'*(inv(V) - inv(V)*A*inv(A'*inv(V)*A)*A'*inv(V))*B./(m-n)

  • S = inv(A'*inv(V)*A)*mse

  • stdx = sqrt(diag(S))

However, lscov uses methods that are faster and more stable, and are applicable to rank deficient cases.

lscov assumes that the covariance matrix of B is known only up to a scale factor. mse is an estimate of that unknown scale factor, and lscov scales the outputs S and stdx appropriately. However, if V is known to be exactly the covariance matrix of B, then that scaling is unnecessary. To get the appropriate estimates in this case, you should rescale S and stdx by 1/mse and sqrt(1/mse), respectively.

Examples

Example 1 — Computing Ordinary Least Squares

The MATLAB® backslash operator (\) enables you to perform linear regression by computing ordinary least-squares (OLS) estimates of the regression coefficients. You can also use lscov to compute the same OLS estimates. By using lscov, you can also compute estimates of the standard errors for those coefficients, and an estimate of the standard deviation of the regression error term:

x1 = [.2 .5 .6 .8 1.0 1.1]'; 
x2 = [.1 .3 .4 .9 1.1 1.4]'; 
X = [ones(size(x1)) x1 x2]; 
y = [.17 .26 .28 .23 .27 .34]';

a = X\y
a =
    0.1203
    0.3284
   -0.1312

[b,se_b,mse] = lscov(X,y) 
b =
    0.1203
    0.3284
   -0.1312
se_b =
    0.0643
    0.2267
    0.1488
mse =
    0.0015

Example 2 — Computing Weighted Least Squares

Use lscov to compute a weighted least-squares (WLS) fit by providing a vector of relative observation weights. For example, you might want to downweight the influence of an unreliable observation on the fit:

w = [1 1 1 1 1 .1]'; 

[bw,sew_b,msew] = lscov(X,y,w)
bw =
    0.1046
    0.4614
   -0.2621
sew_b =
    0.0309
    0.1152
    0.0814
msew =
  3.4741e-004

Example 3 — Computing General Least Squares

Use lscov to compute a general least-squares (GLS) fit by providing an observation covariance matrix. For example, your data may not be independent:

V = .2*ones(length(x1)) + .8*diag(ones(size(x1))); 

[bg,sew_b,mseg] = lscov(X,y,V)
bg =
    0.1203
    0.3284
   -0.1312
sew_b =
    0.0672
    0.2267
    0.1488
mseg =
    0.0019

Example 4 — Estimating the Coefficient Covariance Matrix

Compute an estimate of the coefficient covariance matrix for either OLS, WLS, or GLS fits. The coefficient standard errors are equal to the square roots of the values on the diagonal of this covariance matrix:

[b,se_b,mse,S] = lscov(X,y); 

S
S =
    0.0041   -0.0130    0.0075
   -0.0130    0.0514   -0.0328
    0.0075   -0.0328    0.0221

[se_b sqrt(diag(S))] 
ans =
    0.0643    0.0643
    0.2267    0.2267
    0.1488    0.1488

More About

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Algorithms

The vector x minimizes the quantity (A*x-B)'*inv(V)*(A*x-B). The classical linear algebra solution to this problem is

 x = inv(A'*inv(V)*A)*A'*inv(V)*B

but the lscov function instead computes the QR decomposition of A and then modifies Q by V.

References

[1] Strang, G., Introduction to Applied Mathematics, Wellesley-Cambridge, 1986, p. 398.

See Also

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