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<h1 class="title"><a name="f3-12899" id="f3-12899"></a>Non-linear least squares fitting of time series</h1>
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<h1 class="title"><a name="f3-12899" id="f3-12899"></a>Non-linear least squares fitting of time series</h1>
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<p>
<p>
Non-linear least square fitting of time-series exploits the function <tt>ao/xfit</tt>.
</p>
<h2> Non-linear least square fitting of time series </h2>
<p>
During this exercise we will:
<ol>
<li> Load time series data
<li> Fit data with <tt>ao/xfit</tt>
<li> Check results
<li> Refine the fit with a Monte Carlo search
</ol>
</p>
<p>
Let us open a new editor window and load test data.
</p>
<div class="fragment"><pre>
a = ao(plist(<span class="string">'filename'</span>, <span class="string">'topic5/T5_Ex05_TestNoise.xml'</span>));
a.setName(<span class="string">'data'</span>);
iplot(a)
</pre></div>
<p>
As can be seen this is a chirped sine wave with some noise.
<div align="center">
<IMG src="images/ltpda_training_1/topic5/ltpda_training_5_4_1.png" align="center" border="0">
</div>
We could now try the fit. The first parameter to pass to <tt>xfit</tt>
is a fit model. In this case we assume that we are dealing with a linearly
chirped sine wave according to the equation: <br/>
<br/>
<div align="center">
<IMG src="images/ltpda_training_1/topic5/ex4_chirpsine.gif" align="center" border="0">
</div>
<br/>
The previous function can be stored within a <tt>smodel</tt> analysis object to pass to the fitting machinery:
<div class="fragment"><pre>
mdl = smodel(plist(<span class="string">'Name'</span>, <span class="string">'chirp'</span>, ...
<span class="string">'expression'</span>, <span class="string">'A.*sin(2*pi*(f + f0.*t).*t + p) + c'</span>, ...
<span class="string">'params'</span>, <span class="string">{'A','f','f0','p','c'}</span>, ...
<span class="string">'xvar'</span>, <span class="string">'t'</span>, ...
<span class="string">'xunits'</span>, <span class="string">'s'</span>, <span class="string">'yunits'</span>, <span class="string">'m'</span>));
</pre></div>
We need to specify a starting guess for the model parameters.
The output of <tt>ao/xfit</tt> is a <tt>pest</tt> analysis objects containing fit parameters.
</p>
<div class="fragment"><pre>
plfit1 = plist(<span class="string">'Function'</span>, <span class="string">mdl</span>, ...
<span class="string">'P0'</span>, [5,9e-5,9e-6,0,5]);
params1 = xfit(a, plfit1);
</pre></div>
<p>
Once the fit is done. We can evaluate our model to check fit results.
</p>
<div class="fragment"><pre>
b = eval(params1, plist(<span class="string">'xdata'</span>, a, <span class="string">'xfield'</span>, <span class="string">'x'</span>));
b.setName;
iplot(a,b)
</pre></div>
<p>
As you can see, the fit is not accurate. One of the great problems of
non-linear least square methods is that they easily find a local minimum of
the chi square function and stop there without finding the global minimum.
There are two possibile solutions to such kind of problems: the first one is
to refine step by step the fit by looking at the data; the second one is to
perform a Monte Carlo search in the parameter space. This way, the fitting machinery
extracts the number of points you define in the <tt>'Npoints'</tt>
key, evaluates the chi square at those points, reoders by ascending chi square, selects
the first guesses and fit starting from them.
</p>
<div class="fragment"><pre>
plfit2 = plist(<span class="string">'Function'</span>, <span class="string">mdl</span>, ...
<span class="string">'MonteCarlo'</span>, <span class="string">'yes'</span>, ...
<span class="string">'Npoints'</span>, 1000, ...
<span class="string">'LB'</span>, [1,5e-5,5e-6,0,2], ...
<span class="string">'UB'</span>, [10,5e-4,5e-5,2*pi,7]);
params2 = xfit(a, plfit2);
c = eval(params2, plist(<span class="string">'xdata'</span>, a, <span class="string">'xfield'</span>, <span class="string">'x'</span>));
c.setName;
iplot(a,c)
</pre></div>
<p>
The fit now looks like better...
<div align="center">
<IMG src="images/ltpda_training_1/topic5/ltpda_training_5_4_3.png" align="center" border="0">
</div>
Let us compare fit results with nominal parameters. <br/>
Data were generated with the following set of parameters:
</p>
<div class="fragment"><pre>
A = 3
f = 1e-4
f0 = 1e-5
p = 0.3
c = 5
</pre></div>
<p>
Fitted parameters are instead:
</p>
<div class="fragment"><pre>
A = 3.02 +/- 0.05
f = (7 +/- 3)e-5
f0 = (1.003 +/- 0.003)e-5
p = 0.33 +/- 0.04
c = 4.97 +/- 0.03
</pre></div>
<p>
The correlation matrix of the parameters, the chi square, the degree of freedom, the covariance matrix are
store in the output <tt>pest</tt>. Other useful information are stored in the <tt>procinfo</tt> (processing information)
field. This field is a <tt>plist</tt> and is used to additional information that can be
returned from algorithms. For example, to extract the chi square, we write:
</p>
<div class="fragment"><pre>
params2.chi2
1.0253740840052
</pre></div>
<p>
And to know the correlation matrix:
</p>
<div class="fragment"><pre>
params2.corr
Columns 1 through 3
1 0.120986348157139 -0.0970894969803509
0.120986348157139 1 -0.966114904879414
-0.0970894969803509 -0.966114904879414 1
-0.156801230958825 -0.848296014553159 0.717376893765734
-0.0994358284166703 0.187645552903433 -0.169496082635319
Columns 4 through 5
-0.156801230958825 -0.0994358284166703
-0.848296014553159 0.187645552903433
0.717376893765734 -0.169496082635319
1 -0.199286767157984
-0.199286767157984 1
</pre></div>
<p>
Not so bad!
</p>
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