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Numerical Computing with MATLAB: least squares

Numerical

Computing,

MATLAB

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Numerical Computing with MATLAB

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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(t

), i = 1, . . . , m

The idea is to model y(t) by a linear combination of n basis functions,

y(t) ≈ β

(t) + . . . + β

(t)

The design matrix X is a rectangular matrix of order m-by-n with elements

i,j

= φ

)

The design matrix usually has more rows than columns. In matrix-vector notation,

the model is

y ≈ Xβ

2 Chapter 5. Least Squares

The symbol ≈ stands for “is approximately equal to.” We are more precise about

this in the next section, but our emphasis is on least squares approximation.

The basis functions φ

(t) can be nonlinear functions of t, but the unknown

parameters, β

, appear in the model linearly. The system of linear equations

Xβ ≈ y

is overdetermined if there are more equations than unknowns. The Matlab back-

slash operator computes a least squares solution to such a system.

beta = X\y

The basis functions might also involve some nonlinear parameters, α

, . . . , α

The problem is separable if it involves both linear and nonlinear parameters.

y(t) ≈ β

(t, α) + . . . + β

(t, α)

The elements of the design matrix depend upon both t and α.

i,j

= φ

, α)

Separable problems can be solved by combining backslash with the Matlab func-

tion fminsearch or one of the nonlinear minimizers available in the Optimization

Toolbox. The new Curve Fitting Toolbox provides a graphical interface for solving

nonlinear ﬁtting problems.

Some common models include:

• Straight line: If the model is also linear in t, it is a straight line.

y(t) ≈ β

t + β

• Polynomials: The coeﬃcients β

appear linearly. Matlab orders polynomials

with the highest power ﬁrst.

(t) = t

n−j

, j = 1, . . . , n

y(t) ≈ β

n−1

+ . . . + β

n−1

t + β

The Matlab function polyfit computes least squares polynomial ﬁts by

setting up the design matrix and using backslash to ﬁnd the coeﬃcients.

• Rational functions: The coeﬃcients in the numerator appear linearly; the

coeﬃcients in the denominator appear nonlinearly.

(t) =

n−j

n−1

+ . . . + α

n−1

t + α

y(t) ≈

n−1

+ . . . + β

n−1

t + β

n−1

+ . . . + α

n−1

t + α

5.2. Norms 3

• Exponentials: The decay rates, λ

, appear nonlinearly.

(t) = e

−λ

y(t) ≈ β

−λ

+ . . . + β

−λ

• Log-linear: If there is only one exponential, taking logs makes the model

linear, but changes the ﬁt criterion.

y(t) ≈ Ke

λt

log y ≈ β

t + β

, with β

= λ, β

= log K

• Gaussians: The means and variances appear nonlinearly.

(t) = e

−

t−µ

y(t) ≈ β

−

t−µ

+ . . . β

−

(

t−µ

)

5.2 Norms

The residuals are the diﬀerences between the observations and the model,

= y

−

, α), i = 1, . . . , m

Or, in matrix-vector notation,

r = y − X(α)β

We want to ﬁnd the α’s and β’s that make the residuals as small as possible.

What do we mean by “small”? In other words, what do we mean when we use the

“≈” symbol? There are several possibilities.

• Interpolation: If the number of parameters is equal to the number of obser-

vations, we might be able to make the residuals zero. For linear problems,

this will mean that m = n and that the design matrix X is square. If X is

nonsingular, the β’s are the solution to a square system of linear equations.

β = X\y

• Least squares: Minimize the sum of the squares of the residuals.

krk

4 Chapter 5. Least Squares

• Weighted least squares: If some observations are more important or more

accurate than others, then we might associate diﬀerent weights, w

, with

diﬀerent observations and minimize

krk

For example, if the error in the ith observation is approximately e

, then

choose w

= 1/e

Any algorithm for solving an unweighted least squares problem can be used

to solve a weighted problem by scaling the observations and design matrix.

We simply multiply both y

and the ith row of X by w

. In Matlab, this

can be accomplished with

X = diag(w)*X

y = diag(w)*y

• One-norm: Minimize the sum of the absolute values of the residuals.

krk

This problem can be reformulated as a linear programming problem, but it is

computationally more diﬃcult than least squares. The resulting parameters

are less sensitive to the presence of spurious data points or outliers. .

• Inﬁnity-norm: Minimize the largest residual.

krk

∞

= max

This is also known as a Chebyshev ﬁt and can be reformulated as a linear

programming problem. Chebyshev ﬁts are frequently used in the design of

digital ﬁlters and in the development of approximations for use in mathemat-

ical function libraries.

The Matlab Optimization and Curve Fitting toolboxes include functions for

one-norm and inﬁnity-norm problems. We will limit ourselves to least squares in

this book.

5.3 censusgui

The NCM program censusgui involves several diﬀerent linear models. The data is

the total population of the United States, as determined by the U. S. Census, for

the years 1900 to 2000. The units are millions of people.

5.3. censusgui 5

t y

1900 75.995

1910 91.972

1900 105.711

1930 123.203

1940 131.669

1950 150.697

1960 179.323

1970 203.212

1980 226.505

1990 249.633

2000 281.422

The task is to mo del the population growth and predict the population when t =

2010. The default model in censusgui is a cubic polynomial in t.

y ≈ β

+ β

t + β

There are four unknown coeﬃcients, appearing linearly.

1900 1920 1940 1960 1980 2000

100

150

200

250

300

350

400

Predict U.S. Population in 2010

Millions

312.691

Figure 5.1. censusgui

Numerically, it’s a bad idea to use powers of t as basis functions when t is

around 1900 or 2000. The design matrix is badly scaled and its columns are nearly

linearly dependent. A much better basis is provided by powers of a translated and

scaled t,

s = (t − 1950)/50

剩余26页未读，继续阅读

$message/rfc822\011$

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