View source for DMelt:DataAnalysis/5 Fitting Data


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= Fitting data with functions =

DataMelt offers many reach classes for linear and non-linear regressions. Read [[Curve fitting]]. In particular, DataMelt support:

* Linear-regression fit (fit with a straight lines)
* Non-linear regression (fitting with more complex functions)
* Fitting data with shapes (circles, ellipses).

For fits with analytic functions, fit minimisation  can include statistical errors on input data points.
In addition, several approaches to minimization procedure can be used. 

== Linear regression ==

DataMelt offers many reach classes for linear and non-linear regressions.
To perform a linear regression, use the class  <javadoc>jhplot.stat.LinReg</javadoc>. The example below shows 
how to use this class to perform the linear regression:

<jcode lang="python">
dmelt 5395777.py
</jcode>

The result of this fit is shown below:

<jput>
image 5395777
</jput>

<jnote>
See  [https://datamelt.org/code/index.php?keyword=linreg LinReg code examples]
</jnote>

== Linear regression with  HFitter ==
Now we can do a bit more flexible fit defining a linear function analytically.
We will use <javadoc>jhplot.HFitter</javadoc> which allows to define any fit function.
We simulate a linear  dependence of the variable Y on X using random numbers:

<jcode lang="python">
dmelt 30415157.py
</jcode>

<jput>
image 30415157
</jput>

The fit returns exactly the same a and b values which were used to generate data:

<jcode lang="python">
a = 9.99526486454, b = 1.99636040719
</jcode>

== Non-linear fitting ==

Data fits can be done by using the Java class <javadoc>jhplot.HFitter</javadoc>.
But before, one needs to construct a function and then set initial values
for free parameters. Fitting can be done either in the interactive mode or using
a script.

By default, HFitter fits data using chi2 methods. It is important to specify errors on Y-values of input data. You can print the 
available methods as: 

<jcode lang="python">
from jhplot import *
f=HFitter()
print f.getFitMethod()
</jcode>

which prints "leastsquares", "cleverchi2", "chi2", "bml" fitting methods. You can set the fit method when initialize the fitter:

<jcode lang="python">
from jhplot import *
f=HFitter("leastsquares")
</jcode>
In this case, instead "chi2", "leastsquares" fit method is used. In this case, you  can fit "X-Y" data arrays without specifying errors on Y.

Let's make a simple chi2 fit of experimental data using an analytic function <math>Tu + (Ta - Tu) * exp(-kk * x)</math>, 
where parameters Tu, Ta and kk need to be determined by fitting data stored as array. The data has experimental (statistical or 
systematic) uncertainties. This is mandatory for chi2 minimization. 
The resulting fit is shown below (example is provided by Klaus Rohe).

<jput>
image 373333
</jput>

This image is generated by the code given below where we use <javadoc>jhplot.P1D</javadoc> container to store input data.
You can access fit errors and the fit quality (chi2/ndf) as described by the <javadoc>jhplot.HFitter</javadoc> class.

<ifauth !@member>
<jcode lang="python">
dmelt 373333.py
</jcode>
</ifauth>


Below we will illustrate how to perform a rather complicated fit using the chi2 method. 
The fit will be done in several steps.
In this example we fit data which can be described by multiple Gaussians, in which next Gaussians fit takes the values from the previous fit:

<jcode lang="python">
dmelt 682494.py
</jcode>


The output is shown here:

<jput>
image 682494
</jput>

<jput>
See [https://datamelt.org/code/index.php?keyword=hfitter HFitter code examples]
</jput>

== Numerical interpolation ==
 

One can use numerical way to describe data using [[Interpolation]] and <wp>Smoothing</wp>
One example is shown on this figure:

<jput>
image 44604062
</jput>

where we attempted to smooth data using a non-analytical approach. Such approach is often considered for various predictions when to find an appropriate analytical function is
difficult or impossible. DataMelt provides a several flexible methods to smooth data and perform interpolation. The code of this example is given below:

<ifauth>
<jcode lang="python">
dmelt 44604062.py
</jcode>
</ifauth>

== Interactive  fit ==
 
One simple way to fit data using interactive fitter is to use the class <javadoc sc>jhplot.HFit</javadoc>. This fitter creates a dialog that is shown in

<jput>
image 47104544
</jput>



The code that creates this dialog window is shown here:

<ifauth !@member>
<jcode lang="python">
from java.awt import Color
from jhplot  import *

c1 = HPlot()
c1.visible(); c1.setAutoRange()
c1.setGTitle("&chi;^{2} curve fitting",Color.blue)
c1.setGrid(0,0)
c1.setGrid(1,0)

p1 = P1D("Linear fit")
p1.add(1, 2, 1) # x, y and dy  
p1.add(2, 2, 1)
p1.add(3, 5, 1)
p1.add(4, 5, 1)
p1.add(5, 7, 2)
p1.setErr(1)
p1.setErrFillColor(Color.yellow,0.3)
c1.draw(p1)

a=HFit(c1,p1) # run interactive curve fitter 
a.addFunc("User1","My fit function", "a*x[0]+b","a,b")
</jcode>             
</ifauth>
                                           
Data can be fitted using many predefined functions in an more interactive way.
Let's create a few data containers (1D array, 1D histograms) and start <javadoc>jhplot.HPlotJas</javadoc> plotter  based on JAS2 program from Slac:

<jcode lang="python">
from java.util import Random
from jhplot  import * 

h1 = H1D("1D  histogram",100, -2, 2.0)
p1=P1D("data with errors")

rand = Random()
for i in range(600):
      h1.fill(rand.nextGaussian())     
p0=P0D("Normal distribution")
p0.randomNormal(1000,0.0,10.0)
p1.add(1,10,3)
p1.add(2,5,1)
p1.add(3,12,2)
from  java.util import ArrayList
a=ArrayList([h1,p1,p0])
c=HPlotJas("JAS",ArrayList([h1,p1,p0]))
</jcode>

This will bring up a "JAS" program with all objects shown on the left side.

<jput>
image 36993847
</jput>

You can expand the tree (left) and click on each object to plot on the canvas. Then try to fit the data: using the mouse pop-up dialog, press "Add function" (for example, a Gaussian function) and click on "Fit". The program will perform chi2 minimisation and you will see a Gaussian line on to of the data. You can adjust the initial values of the function by dragging 3 points of this function.

<jnote>
Look at the example [http://www.youtube.com/watch?v=Ovhd_-vnwgw Fit video].
</jnote>

In the above example we used pre-built functions to perform fits. You can add your own custom fit function to the menu and use  it
for fitting as well. You can do this directly in the Jython script and pass this function to HplotJas canvas.
Below we show an example in which we create 2 custom functions (a new Gaussian and a parabola) and passed them to the HPlotJas for interactive fitting. 

<ifauth !@member>
<jcode lang="python">
dmelt 41988602.py
</jcode>
</ifauth>

<jput>
image 41988602
</jput>

== Advanced fitting of data  ==

Section [[DMelt:DataAnalysis/6_Advanced_Data_Fitting|Advanced data fitting]] discusses how to
perform non-linear fit in Java using data in many dimensions and using 
any complex function that can be defined
not as a string, but totally programically.  

[[Category:Linear regression]]
[[Category:Curve fitting]]
[[Category:Mathematical optimization]]
[[Category:Regression analysis]]
[[Category:Numerical analysis]]
[[Category:Interpolation]]