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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 Adri�n 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 Adri�n Vargas Aguilera. nubeobscura@hotmail.com
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