CHAPTER 3: NUMERICAL ALGORITHMS WITH MATLAB

Lecture 3.2: Least square curve fitting

Polynomial approximation:

Problem: Given a set of (n+1) data points:

(x1,y1); (x2,y2); …; (xn,yn); (xn+1,yn+1)

Find a polynomial of degree m < n, y = m(x),  that best fits to the given set of (n+1) data points.

Example: If the data set is corrupted by measurement noise, i.e. the data values are rounded off, the numerical interpolation would produce a very stiff curve with sharp corners and large interpolation error. Another technique is used instead: it is based on numerical approximation of given data values. The figure below shows numerical approximation of more than hundred of data values by a cubic polynomial:

Polynomial approximation is effective when the data values fit a simple curve, e.g. a polynomial of lower order. In many cases, the simple (polynomial) curves are predicted by solutions of theoretical models (e.g. differential equations). Methods of numerical approximation allow researchers to compare theoretical models and real data samples.

Methods:

·         Minimization of total square error

·         Over-determined linear system

·         MATLAB "polyfit" functions

Minimization of total square error:

Let y = m(x) be a function that best fits to the given data points in the sense that it minimizes the total square error between the data points and the values of the function:

(min) E = (  yj - m(xj) )2

Linear regression: y = 1(x) = c1 x + c2

Total error:   E = ( yj – c1 xj – c2 )2

Minimization:

= -2 xj ( yj – c1 xj – c2 ) = 0

= -2 ( yj – c1 xj – c2 ) = 0

x = [ 0.1,0.4,0.5,0.7,0.7,0.9 ]

y = [ 0.61,0.92,0.99,1.52,1.47,2.03]

a11 = sum(x.^2); a12 = sum(x); a21 = sum(x); a22 = sum(ones(1,length(x)));

A = [ a11,a12; a21,a22]; % the coefficient matrix of the minimization problem

b1 = sum(x.*y); b2 = sum(y);

b = [ b1; b2 ]; % right-hand-side of the minimization problem

c = A \ b  % solution of the minimization problem

xApr = 0 : 0.001 : 1; yApr = c(1)*xApr + c(2);

plot(x,y,'*g',xApr,yApr,'b');

x =    0.1000    0.4000    0.5000    0.7000    0.7000    0.9000

y =    0.6100    0.9200    0.9900    1.5200    1.4700    2.0300

c =    1.7646    0.2862

Over-determined linear system:

Suppose the approximating polynomial y = m(x) passes through all given points:m(xj) = yj. These conditions set an over-determined linear system A c = b, when the number of equations (n+1) exceeds the number of unknowns (m +1). Since the coefficient matrix A is not singular, no solution of such over-determined system exists. However, we can still solve the system in the approximation sense: (A'*A) c = (A'*b). The coefficient matrix (A'*A) is now 2-by-2 non-singular matrix, such that a unique solution exists for c. This solution is the same solution that gives minimum of the total square error E.

Linear regression:  c1 xj + c2 = yj,     j = 1,2,…,(n+1)

x = [ 0.1,0.4,0.5,0.7,0.7,0.9 ];

y = [ 0.61,0.92,0.99,1.52,1.47,2.03];

A = [ x', ones(length(x),1) ];

% the coefficient matrix of the over-determined problem

b = y'; % right-hand-side of the over-determined problem

c = (A'*A) \ (A'*b)  % solution of the over-determined problem

xApr = 0 : 0.001 : 1; yApr = c(1)*xApr + c(2);

c =    1.7646    0.2862

A, b, c = A\b

% MATLAB solves the over-determined system in the least square sense

E = sum((y-c(1)*x-c(2)).^2)

 A = 0.1000    1.0000     0.4000    1.0000     0.5000    1.0000     0.7000    1.0000     0.7000    1.0000     0.9000    1.0000 b = 0.6100     0.9200     0.9900     1.5200     1.4700     2.0300 c = 1.7646     0.2862   E = 0.0856

The total square error E becomes smaller, if the approximating polynomial y = m(x) has larger order m < n. The total square error E is zero, if m = n, i.e. the approximating polynomial y = m(x) is the interpolating polynomial y = Pn(x) that passes through all (n+1) data points.

x = [ 0.1,0.4,0.5,0.7,0.7,0.9 ];

y = [ 0.61,0.92,0.99,1.52,1.47,2.03];

n = length(x)-1;

for m = 1 : n

A = vander(x); A = A(:,n-m+1:n+1);

b = y'; c = A\b; yy = polyval(c,x);

E = sum((y-yy).^2) ;

fprintf('m = %d, E = %6.5f\n',m,E);

end

m = 2; A = vander(x); A = A(:,n-m+1:n+1); b = y';

c = A\b; xApr = 0 : 0.001 : 1; yApr = polyval(c,xApr);

plot(x,y,'*g',xApr,yApr,'b');

m = 1, E = 0.08564

m = 2, E = 0.00665

m = 3, E = 0.00646

m = 4, E = 0.00125

Warning: Matrix is singular to working precision.

m = 5, E =    Inf

The system of linear equations for least square approximation is ill-conditioned when n is large and also when data points include both very small and very large values of x. Systems of orthogonal polynomials are used regularly in those cases.

·         polyfit: computes coefficients of approximating polynomial when m < n and coefficients of interpolating polynomial when m = n

Nonlinear approximation:

Theoretical dependences fitting data samples can be expressed in terms of nonlinear functions rather than in terms of polynomial functions. For example, theoretical curves can be given by power functions or exponential functions. In both the cases, the axis (x,y) can be redefined such that the linear polynomial regression can still be used for nonlinear least square approximation.

Power function approximation: y = x

Take the logarithm of the power function:

log y = log x + log

In logarithmic axis (logx,logy), the data sample can be approximated by a linear regression.

x = [0.15,0.4,0.6,1.01,1.5,2.2,2.4,2.7,2.9,3.5,3.8,4.4,4.6,5.1,6.6,7.6];

y = [4.5,5.1,5.7,6.3,7.1,7.6,7.5,8.1,7.9,8.2,8.5,8.7,9.0,9.2,9.5,9.9];

c = polyfit(log(x),log(y),1) % linear regression in logarithmic axis

xInt = linspace(x(1),x(length(x)),100);

yInt = exp(c(2))*xInt.^(c(1));

plot(x,y,'g*',xInt,yInt,'b');  % nonlinear regression in (x,y)

loglog(x,y,'g*',xInt,yInt,'b'); % linear regression in (logx,logy)

c =    0.2094    1.8588

Exponential function approximation: y = e

Take the logarithm of the exponential function:

log y = x + log

In semi-logarithmic axis (x,logy), the data sample can be approximated by a linear regression

Approximation by a linear combination of nonlinear functions:

y = c1 f1(x) + c2 f2(x) + … + cn fn(x)

Trully nonlinear approximation: y = f(x;c1,c2,…,cn)

It represents a rather complicated problem of nonlinear optimization.