robustfit.m

function varargout = robustfit(X,y,wfun,tune,const)
%ROBUSTFIT Robust linear regression
%   B = ROBUSTFIT(X,Y) returns the vector B of regression coefficients,
%   obtained by performing robust regression to estimate the linear model
%   Y = Xb.  X is an n-by-p matrix of predictor variables, and Y is an
%   n-by-1 vector of observations.  The algorithm uses iteratively
%   reweighted least squares with the bisquare weighting function.  By
%   default, ROBUSTFIT adds a column of ones to X, corresponding to a
%   constant term in the first element of B.  Do not enter a column of ones
%   directly into the X matrix.
%
%   B = ROBUSTFIT(X,Y,'WFUN',TUNE) uses the weighting function 'WFUN' and
%   tuning constant TUNE.  'WFUN' can be any of 'andrews', 'bisquare',
%   'cauchy', 'fair', 'huber', 'logistic', 'talwar', or 'welsch'.
%   Alternatively 'WFUN' can be a function that takes a residual vector as
%   input and produces a weight vector as output.  The residuals are scaled
%   by the tuning constant and by an estimate of the error standard
%   deviation before the weight function is called.  'WFUN' can be
%   specified using @ (as in @myfun).  TUNE is a tuning constant that is
%   divided into the residual vector before computing the weights, and it
%   is required if 'WFUN' is specified as a function.
% 
%   B = ROBUSTFIT(X,Y,'WFUN',TUNE,'CONST') controls whether or not the
%   model will include a constant term.  'CONST' is 'on' (the default) to
%   include the constant term, or 'off' to omit it.
%
%   [B,STATS] = ROBUSTFIT(...) also returns a STATS structure
%   containing the following fields:
%       'ols_s'     sigma estimate (rmse) from least squares fit
%       'robust_s'  robust estimate of sigma
%       'mad_s'     MAD estimate of sigma; used for scaling
%                   residuals during the iterative fitting
%       's'         final estimate of sigma, the larger of robust_s
%                   and a weighted average of ols_s and robust_s
%       'se'        standard error of coefficient estimates
%       't'         ratio of b to stats.se
%       'p'         p-values for stats.t
%       'covb'      estimated covariance matrix for coefficient estimates
%       'coeffcorr' estimated correlation of coefficient estimates
%       'w'         vector of weights for robust fit
%       'h'         vector of leverage values for least squares fit
%       'dfe'       degrees of freedom for error
%       'R'         R factor in QR decomposition of X matrix
%
%   The ROBUSTFIT function estimates the variance-covariance matrix of the
%   coefficient estimates as V=inv(X'*X)*STATS.S^2.  The standard errors
%   and correlations are derived from V.
%
%   ROBUSTFIT treats NaNs in X or Y as missing values, and removes them.
%
%   Example:
%      x = (1:10)';
%      y = 10 - 2*x + randn(10,1); y(10) = 0;
%      bls = regress(y,[ones(10,1) x])
%      brob = robustfit(x,y)
%      scatter(x,y)
%      hold on
%      plot(x,brob(1)+brob(2)*x,'r-', x,bls(1)+bls(2)*x,'m:')
%
%   See also REGRESS, ROBUSTDEMO.

% References:
%   DuMouchel, W.H., and F.L. O'Brien (1989), "Integrating a robust
%     option into a multiple regression computing environment,"
%     Computer Science and Statistics:  Proceedings of the 21st
%     Symposium on the Interface, American Statistical Association.
%   Holland, P.W., and R.E. Welsch (1977), "Robust regression using
%     iteratively reweighted least-squares," Communications in
%     Statistics - Theory and Methods, v. A6, pp. 813-827.
%   Huber, P.J. (1981), Robust Statistics, New York: Wiley.
%   Street, J.O., R.J. Carroll, and D. Ruppert (1988), "A note on
%     computing robust regression estimates via iteratively
%     reweighted least squares," The American Statistician, v. 42,
%     pp. 152-154.

%   Copyright 1993-2006 The MathWorks, Inc. 
%   $Revision: 1.1.6.2 $  $Date: 2010/10/08 17:26:28 $

if  nargin < 2      
    error(message('stats:robustfit:TooFewInputs'));      
end 

if (nargin<3 || isempty(wfun)), wfun = 'bisquare'; end
if nargin<4, tune = []; end
[eid,emsg,wfun,tune] = statrobustwfun(wfun,tune);
if ~isempty(eid)
    error(sprintf('stats:robustfit:%s',eid), emsg);
end
if (nargin<5), const='on'; end
switch(const)
 case {'on' 1},  doconst = 1;
 case {'off' 0}, doconst = 0;
 otherwise,  error(message('stats:robustfit:BadConst'));
end

% Remove missing values and check dimensions
[anybad wasnan y X] = statremovenan(y,X);
if (anybad==2)
   error(message('stats:robustfit:InputSizeMismatch'));
end

varargout=cell(1,max(1,nargout));
[varargout{:}] = statrobustfit(X,y,wfun,tune,wasnan,doconst);