-- AlexanderFedotov - 2015-03-03

Linear Least Squares

Wikipedia links

Uncorrelated measurements

Let with variances be measurements of functions with the known matrix and unknown parameters .

In linear least square method, one estimates the parameter vector by minimizing over the expression

where the weight matrix of dimension is diagonal and defined as the inverse of the diagonal covariance matrix for : i.e. .
In matrix notation (considering and as columns and respectively), one has

The estimate is the solution of the system of equations

or
In a general case of linear transformation , the covariance matrice for is transformed into that for via . Hence,
With by the definition of , that simplifies to

Note, that

and

Correlated measurements

Let be uncorrelated measurements as those in the previous section, and ( is an invertible matrix). Then are generally correlated and have the covariance matrix .

With and , one gets

where .

Similarly,

and

Thus, all the formulae for the correlated measurements are similar to those for the uncorrelated , with the only complication being the replacement of a diagonal weight matrix with a non-diagonal one:

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Topic revision: r3 - 2015-03-05 - AlexanderFedotov
 
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