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The command SEGMENT segments data that are generated from systems that may undergo abrupt changes. Typical applications for data segmentation are segmentation of speech signals (each segment corresponds to a phonem), failure detection (the segments correspond to operation with and without failures) and estimating different working modes of a system. We shall study a system whose time delay changes from two to one.
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echo on
echo off
% Lennart Ljung
% Copyright (c) 1986-98 by The MathWorks, Inc.
% $Revision: 3.6 $ $Date: 1997/12/02 03:43:02 $
echo on
load iddemo6m.mat
% First, take a look at the data:
newplot;
pause, idplot(z), pause % Press any key for plot.
% The change takes place at sample number 20, but this is not
% so easy to see.
%
% We would like to estimate the system as an ARX-structure model
% with one a-parameter, two b-parameters and one delay:
% y(t) + a*y(t-1) = b1*u(t-1) + b2*u(t-2)
% The information to be given is the data, the model orders
% and a guess of the variance (r2) of the noise that affects
% the system. If this is entirely unknown, it can be estimated
% automatically. Here we set it to 0.1:
nn = [1 2 1];
[seg,v,tvmod] = segment(z,nn,0.1);
pause % Press any key to continue.
% Let's take a look at the segmented model. Blue(yellow) line
% is for the a-parameter. Green(magneta) is for b1 and Red(cyan)
% for the parameter b2.
pause, plot(seg), pause % Press any key for plot.
hold on
% We see clearly the jump around sample number 19. b1 goes from
% 0 to 1 and b2 vice versa, which shows the change of the
% delay. The true values can also be shown:
pause, plot(pars), pause % Press any key for plot.
hold off
pause
% The method for segmentation is based on AFMM (adaptive
% forgetting through multiple models), Andersson, Int. J.
% Control Nov 1985. A multi-model approach is used in a first
% step to track the time varying system. The resulting tracking
% model could be of interest in its own right, and are given by
% the third output argument of SEGMENT (tvmod in our case). They
% look as follows:
pause, plot(tvmod), pause % Press any key for plot.
% The SEGMENT M-file is thus an alternative to the recursive
% algorithms RPEM, RARX etc for tracking time varying systems.
% It is particularly suited for systems that may change rapidly.
% From the tracking model, SEGMENT estimates the time points when
% jumps have occurred, and constructs the segmented model by a
% smoothing procedure over the tracking model.
% The two most important "knobs" for the algorithm are r2, as
% mentioned before, and the guessed probability of jumps, q, the
% fourth input argument to SEGMENT. The smaller r2 and the larger
% q, the more willing SEGMENT will be to indicate segmentation
% points. In an off line situation, the user will have to try a
% couple of choices (r2 is usually more sensitive than q). The
% second output argument to SEGMENT, v, is the loss function for
% the segmented model (i.e. the estimated prediction error variance
% for the segmented model). A goal will be to minimize this value.
pause % Press any key to continue.
% OBJECT DETECTION IN LASER RANGE DATA
%
% The reflected signal from a laser (or radar) beam contains
% information about the distance to the reflecting object. The
% signals can be quite noisy. The presence of objects affects
% both the distance information and the correlation between
% neighbouring points. (A smooth object increases the
% correlation between nearby points.)
%
% In the following we study some quite noisy laser range data.
% They are obtained by one horizontal sweep, like one line on
% a TV-screen. The value is the distance to the reflecting object.
% We happen to know that an object of interest hides between
% sample numbers 17 and 48.
pause, plot(hline), pause % Press any key for plot.
% The eye is not good at detecting the object. We shall use
% "segment". First we detrend and normalize the data to a
% variance about one. (This is not necessary, but it means that
% the default choices in the algorithm are better tuned.)
hline = dtrend(hline)/200;
% We shall now build a model of the kind:
%
% y(t) + a y(t-1) = b
%
% The coefficient 'a' will pick up correlation information. The
% value 'b' takes up the possible changes in level. We thus
% introduce a fake input of all ones:
[m,n]=size(hline);
zline = [hline ones(m,n)];
s = segment(zline,[1 1 1],0.2);
pause % Press any key to check segmentation.
subplot(211),plot(hline),title('LASER RANGE DATA')
subplot(212),plot(s)
title('SEGMENTED MODELS, blue/solid: correlation, green/dashed: distance'),pause
% The segmentation has thus been quite successful.
% SEGMENT is capable of handling multi-input systems, and of using
% ARMAX models for the added noise. We can try this on the test
% data iddata1.mat (which contains no jumps):
load iddata1.mat
s = segment(z1(1:100,:),[2 2 2 1],1);
newplot;
pause, plot(s),hold on, pause % Press any key for plot.
% Compare this with the true values:
pause % Press any key for plot.
plot([ ones(100,1)*[-1.5 0.7],ones(100,1)*[1 0.5],ones(100,1)*[-1 0.2]]),pause
hold off
% SEGMENT thus correctly finds that no jumps have occurred, and
% also gives good estimates of the parameters.
pause % Press any key to continue.
echo off
set(gcf,'NextPlot','replace');