%% Least-squares data fitting and POLYFIT
% Nick Trefethen, 1st October 2011

%%
% (Chebfun example stats/LeastSquares.m)

%%
% In Matlab, a standard command for least-squares fitting by a polynomial
% to a set of discrete data points is POLYFIT.  The polynomial returned by
% POLYFIT is represented in Matlab's usual manner by a vector of
% coefficients in the monomial basis.

%%
% In Chebfun, there is an overloaded POLYFIT command in the domain class
% that does the same thing, except that the polynomial is returned as a
% chebfun rather than a coefficient vector. Here is an example:
npts = 100;
x = linspace(-1,1,npts);
y = 1./(1+25*x.^2) + 1e-1*randn(1,npts);
f = polyfit(x,y,10,domain(-1,1));
plot(x,y,'xk','markersize',12)
hold on, plot(f,'r','linewidth',2)
title('Discrete polynomial least-squares fit','fontsize',16)

%%
% Chebfun also has a POLYFIT command in the chebfun class, and this is for
% continuous rather than discrete polynomial least-squares fitting.  For
% example, here is a least-squares fit to a jagged function:
f = chebfun('abs(x+.2)-.5*sign(x-.5) ','splitting','on');
r = polyfit(f,10);
hold off, plot(f,'k',r,'r','linewidth',2)
title('Continuous polynomial least-squares fit','fontsize',16)