extrema.m

function [xmax,imax,xmin,imin] = extrema(x)
%EXTREMA   Gets the global extrema points from a time series.
%   [XMAX,IMAX,XMIN,IMIN] = EXTREMA(X) returns the global minima and maxima 
%   points of the vector X ignoring NaN's, where
%    XMAX - maxima points in descending order
%    IMAX - indexes of the XMAX
%    XMIN - minima points in descending order
%    IMIN - indexes of the XMIN
%
%   DEFINITION (from http://en.wikipedia.org/wiki/Maxima_and_minima):
%   In mathematics, maxima and minima, also known as extrema, are points in
%   the domain of a function at which the function takes a largest value
%   (maximum) or smallest value (minimum), either within a given
%   neighbourhood (local extrema) or on the function domain in its entirety
%   (global extrema).
%
%   Example:
%      x = 2*pi*linspace(-1,1);
%      y = cos(x) - 0.5 + 0.5*rand(size(x)); y(40:45) = 1.85; y(50:53)=NaN;
%      [ymax,imax,ymin,imin] = extrema(y);
%      plot(x,y,x(imax),ymax,'g.',x(imin),ymin,'r.')
%
%   See also EXTREMA2, MAX, MIN

%   Written by
%   Lic. on Physics Carlos Adrián Vargas Aguilera
%   Physical Oceanography MS candidate
%   UNIVERSIDAD DE GUADALAJARA 
%   Mexico, 2004
%
%   nubeobscura@hotmail.com

% From       : http://www.mathworks.com/matlabcentral/fileexchange
% File ID    : 12275
% Submited at: 2006-09-14
% 2006-11-11 : English translation from spanish. 
% 2006-11-17 : Accept NaN's.
% 2007-04-09 : Change name to MAXIMA, and definition added.


xmax = [];
imax = [];
xmin = [];
imin = [];

% Vector input?
Nt = numel(x);
if Nt ~= length(x)
 error('Entry must be a vector.')
end

% NaN's:
inan = find(isnan(x));
indx = 1:Nt;
if ~isempty(inan)
 indx(inan) = [];
 x(inan) = [];
 Nt = length(x);
end

% Difference between subsequent elements:
dx = diff(x);

% Is an horizontal line?
if ~any(dx)
 return
end

% Flat peaks? Put the middle element:
a = find(dx~=0);              % Indexes where x changes
lm = find(diff(a)~=1) + 1;    % Indexes where a do not changes
d = a(lm) - a(lm-1);          % Number of elements in the flat peak
a(lm) = a(lm) - floor(d/2);   % Save middle elements
a(end+1) = Nt;

% Peaks?
xa  = x(a);             % Serie without flat peaks
b = (diff(xa) > 0);     % 1  =>  positive slopes (minima begin)  
                        % 0  =>  negative slopes (maxima begin)
xb  = diff(b);          % -1 =>  maxima indexes (but one) 
                        % +1 =>  minima indexes (but one)
imax = find(xb == -1) + 1; % maxima indexes
imin = find(xb == +1) + 1; % minima indexes
imax = a(imax);
imin = a(imin);

nmaxi = length(imax);
nmini = length(imin);                

% Maximum or minumim on a flat peak at the ends?
if (nmaxi==0) && (nmini==0)
 if x(1) > x(Nt)
  xmax = x(1);
  imax = indx(1);
  xmin = x(Nt);
  imin = indx(Nt);
 elseif x(1) < x(Nt)
  xmax = x(Nt);
  imax = indx(Nt);
  xmin = x(1);
  imin = indx(1);
 end
 return
end

% Maximum or minumim at the ends?
if (nmaxi==0) 
 imax(1:2) = [1 Nt];
elseif (nmini==0)
 imin(1:2) = [1 Nt];
else
 if imax(1) < imin(1)
  imin(2:nmini+1) = imin;
  imin(1) = 1;
 else
  imax(2:nmaxi+1) = imax;
  imax(1) = 1;
 end
 if imax(end) > imin(end)
  imin(end+1) = Nt;
 else
  imax(end+1) = Nt;
 end
end
xmax = x(imax);
xmin = x(imin);

% NaN's:
if ~isempty(inan)
 imax = indx(imax);
 imin = indx(imin);
end

% Same size as x:
imax = reshape(imax,size(xmax));
imin = reshape(imin,size(xmin));

% Descending order:
[temp,inmax] = sort(-xmax); clear temp
xmax = xmax(inmax);
imax = imax(inmax);
[xmin,inmin] = sort(xmin);
imin = imin(inmin);


% Carlos Adrián Vargas Aguilera. nubeobscura@hotmail.com