finding peaks and troughs, Part II (with corresponding definition) - algorithm

Question

this is an update to the previous question that I had about locating peaks and troughs. The previous question was this:
peaks and troughs in MATLAB (but with corresponding definition of a peak and trough)
This time around, I did the suggested answer, but I think there is still something wrong with the final algorithm. Can you please tell me what I did wrong in my code? Thanks.
function [vectpeak, vecttrough]=peaktroughmodified(x,cutoff)
% This function is a modified version of the algorithm used to identify
% peaks and troughs in a series of prices. This will be used to identify
% the head and shoulders algorithm. The function gives you two vectors:
% PEAKS - an indicator vector that identifies the peaks in the function,
% and TROUGHS - an indicator vector that identifies the troughs of the
% function. The input is the vector of exchange rate series, and the cutoff
% used for refining possible peaks and troughs.
% Finding all possible peaks and troughs of our vector.
[posspeak,possploc]=findpeaks(x);
[posstrough,posstloc]=findpeaks(-x);
posspeak=posspeak';
posstrough=posstrough';
% Initialize vector of peaks and troughs.
numobs=length(x);
prelimpeaks=zeros(numobs,1);
prelimtroughs=zeros(numobs,1);
numpeaks=numel(possploc);
numtroughs=numel(posstloc);
% Indicator for possible peaks and troughs.
for i=1:numobs
for j=1:numpeaks
if i==possploc(j);
prelimpeaks(i)=1;
end
end
end
for i=1:numobs
for j=1:numtroughs
if i==posstloc(j);
prelimtroughs(i)=1;
end
end
end
% Vector that gives location.
location=1:1:numobs;
location=location';
% From the list of possible peaks and troughs, find the peaks and troughs
% that fit Chang and Osler [1999] definition.
% "A peak is a local minimum at least x percent higher than the preceding
% trough, and a trough is a local minimum at least x percent lower than the
% preceding peak." [Chang and Osler, p.640]
% cutoffs
peakcutoff=1.0+cutoff; % cutoff for peaks
troughcutoff=1.0-cutoff; % cutoff for troughs
% First peak and first trough are initialized as previous peaks/troughs.
prevpeakloc=possploc(1);
prevtroughloc=posstloc(1);
% Initialize vectors of final peaks and troughs.
vectpeak=zeros(numobs,1);
vecttrough=zeros(numobs,1);
% We first check whether we start looking for peaks and troughs.
for i=1:numobs
if prelimpeaks(i)==1;
if i>prevtroughloc;
ratio=x(i)/x(prevtroughloc);
if ratio>peakcutoff;
vectpeak(i)=1;
prevpeakloc=location(i);
else vectpeak(i)=0;
end
end
elseif prelimtroughs(i)==1;
if i>prevpeakloc;
ratio=x(i)/x(prevpeakloc);
if ratio<troughcutoff;
vecttrough(i)=1;
prevtroughloc=location(i);
else vecttrough(i)=0;
end
end
else
vectpeak(i)=0;
vecttrough(i)=0;
end
end
end

Answer 1

I just ran it, and it seems to work if you make this change:
peakcutoff= 1/cutoff; % cutoff for peaks
troughcutoff= cutoff; % cutoff for troughs
I tested it with the following code, with a cutoff of 0.1 (peaks must be 10 times larger than troughs), and it looks reasonable
x = randn(1,100).^2;
[vectpeak,vecttrough] = peaktroughmodified(x,0.1);
peaks = find(vectpeak);
troughs = find(vecttrough);
plot(1:100,x,peaks,x(peaks),'o',troughs,x(troughs),'o')
I strongly urge you to read up on vectorization in matlab. There are many wasted lines in your program, and it makes it difficult to read and will also make it very slow with big datasets. For instance, prelimpeaks and prelimtroughs can be completely defined without loops, in a single line for each:
prelimpeaks(possploc) = 1;
prelimtroughs(posstloc) = 1;

Answer 2

I think there are better techniques for finding peaks and troughs than the percentage threshold technique given above. Fit the least squares fit parabola to the data set, a technique for doing this is in the 1946 Frank Peters paper, "Parabolic Correlation, a New Descrptive Statistic." The fitted parabola will likely have an index of curvature, as Peters defines it. Find peaks and troughs by testing which points, when eliminated, minimize the absolute value of the index of curvature of the parabola. Once these points are discovered, test for which are peaks and which are troughs by considering how the index of curvature changes when the point is excluded, which will depend on whether the original parabola had a positive or negative index of curvature. If you become concerned about contiguous points the elimination of which achieves the minimum absolute value curvature, constrain by setting a minimum distance the identified points must be from each other. Another constraint would have to be the number of points identified. Without this constraint, this algorithm would remove all but two points, a straight line without curvature.
Sometimes there are steep changes between contiguous points and both should be included in extreme points. Perhaps a percentage threshold test for contiguous points that overrides the minimum distance constraint would be useful.
Another solution might be to compute the Fast Fourier Transform of the series and remove points that minimize the lower spectra. FFT functions are more readily available than code that finds least square fits parabola. There is a matrix manipulation technique for determining the least square fit parabola that is easier to manage than Peter's approach. I saw it documented on the web someplace, but lost the link. Advice from anybody able to arrive at a least square fit parabola using matrix vector notation would be appreciated.