Numerical methods for linear least squares entails the numerical analysis of linear least squares problems.
A general approach to the least squares problem can be described as follows. Suppose that we can find an n by m matrix S
such that XS is an
orthogonal projection onto the image of X. Then a solution to our minimization problem is given by
simply because
is exactly a sought for orthogonal projection of onto an image of X
(see the picture below and note that as explained in the
next section the image of X is just a subspace generated by column vectors of X).
A few popular ways to find such a matrix S are described below.
The equation
is known as the normal equation.
The algebraic solution of the normal equations with a full-rank matrix XTX can be written as
where X+ is the Moore–Penrose pseudoinverse of X. Although this equation is correct and can work in many applications, it is not computationally efficient to invert the normal-equations matrix (the Gramian matrix). An exception occurs in numerical smoothing and differentiation where an analytical expression is required.
If the matrix XTX is well-conditioned and positive definite, implying that it has full rank, the normal equations can be solved directly by using the Cholesky decomposition RTR, where R is an upper triangular matrix, giving:
The solution is obtained in two stages, a forward substitution step, solving for z:
followed by a backward substitution, solving for :
Both substitutions are facilitated by the triangular nature of R.
Orthogonal decomposition methods of solving the least squares problem are slower than the normal equations method but are more numerically stable because they avoid forming the product XTX.
The residuals are written in matrix notation as
The matrix X is subjected to an orthogonal decomposition, e.g., the QR decomposition as follows.
where Q is an m×m orthogonal matrix (QTQ=I) and R is an n×n upper triangular matrix with .
The residual vector is left-multiplied by QT.
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Numerical methods for linear least squares entails the numerical analysis of linear least squares problems. A general approach to the least squares problem can be described as follows. Suppose that we can find an n by m matrix S such that XS is an orthogonal projection onto the image of X. Then a solution to our minimization problem is given by simply because is exactly a sought for orthogonal projection of onto an image of X (see the picture below and note that as explained in the next section the image of X is just a subspace generated by column vectors of X).
In the mathematical subfield of numerical analysis, numerical stability is a generally desirable property of numerical algorithms. The precise definition of stability depends on the context. One is numerical linear algebra and the other is algorithms for solving ordinary and partial differential equations by discrete approximation. In numerical linear algebra, the principal concern is instabilities caused by proximity to singularities of various kinds, such as very small or nearly colliding eigenvalues.
Linear least squares (LLS) is the least squares approximation of linear functions to data. It is a set of formulations for solving statistical problems involved in linear regression, including variants for ordinary (unweighted), weighted, and generalized (correlated) residuals. Numerical methods for linear least squares include inverting the matrix of the normal equations and orthogonal decomposition methods. The three main linear least squares formulations are: Ordinary least squares (OLS) is the most common estimator.
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