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verification. (July 2018) |
In constrained least squares one solves a linear least squares problem with an additional constraint on the solution. [1] [2] This means, the unconstrained equation must be fit as closely as possible (in the least squares sense) while ensuring that some other property of is maintained.
There are often special-purpose algorithms for solving such problems efficiently. Some examples of constraints are given below:
If the constraint only applies to some of the variables, the mixed problem may be solved using separable least squares by letting and represent the unconstrained (1) and constrained (2) components. Then substituting the least-squares solution for , i.e.
(where + indicates the Moore–Penrose pseudoinverse) back into the original expression gives (following some rearrangement) an equation that can be solved as a purely constrained problem in .
where is a projection matrix. Following the constrained estimation of the vector is obtained from the expression above.
This article needs additional citations for
verification. (July 2018) |
In constrained least squares one solves a linear least squares problem with an additional constraint on the solution. [1] [2] This means, the unconstrained equation must be fit as closely as possible (in the least squares sense) while ensuring that some other property of is maintained.
There are often special-purpose algorithms for solving such problems efficiently. Some examples of constraints are given below:
If the constraint only applies to some of the variables, the mixed problem may be solved using separable least squares by letting and represent the unconstrained (1) and constrained (2) components. Then substituting the least-squares solution for , i.e.
(where + indicates the Moore–Penrose pseudoinverse) back into the original expression gives (following some rearrangement) an equation that can be solved as a purely constrained problem in .
where is a projection matrix. Following the constrained estimation of the vector is obtained from the expression above.