Numerical Computing with MATLAB: least squares
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2009-08-08
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Chapter 5
Least Squares
The term least squares describes a frequently used approach to solving overdeter-
mined or inexactly specified systems of equations in an approximate sense. Instead
of solving the equations exactly, we seek only to minimize the sum of the squares
of the residuals.
The least squares criterion has important statistical interpretations. If ap-
propriate probabilistic assumptions about underlying error distributions are made,
least squares produces what is known as the maximum-likelihood estimate of the pa-
rameters. Even if the probabilistic assumptions are not satisfied, years of experience
have shown that least squares produces useful results.
The computational techniques for linear least squares problems make use of
orthogonal matrix factorizations.
5.1 Models and Curve Fitting
A very common source of least squares problems is curve fitting. Let t be the
independent variable and let y(t) denote an unknown function of t that we want
to approximate. Assume there are m observations, i.e. values of y measured at
specified values of t.
y
i
= y(t
i
), i = 1, . . . , m
The idea is to model y(t) by a linear combination of n basis functions,
y(t) ≈ β
1
φ
1
(t) + . . . + β
n
φ
n
(t)
The design matrix X is a rectangular matrix of order m-by-n with elements
x
i,j
= φ
j
(t
i
)
The design matrix usually has more rows than columns. In matrix-vector notation,
the model is
y ≈ Xβ
1
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