Changes in AIDA Fitting Scheme

 

 

This page describes proposed changes to the AIDA Fitting. "Main Idea" gives an overview of proposed changes, "Use Cases" contains examples of usage and "Detailed Description" has full specification of interfaces.

 

Proposed Change in Structure:

 

IFitFunction - no longer needed

IFitter - NEW, does the job of fitting

IFitterFactory - NEW, creates IFitter of specified type

IFitResult - NEW, holds fit results, is created by IFitter

IVariable - NEW, represents variable or parameter, has "Value", "Error",

set of internal states, and can be connected to ITuple

 

 

Main Idea:

 

In the most general terms fitting process requires a set of Data (DataSource) and a function in order to produce a result. Fit result does not have to be part of the fit itself imagine making several fits and comparing results later. Schematically fitting can be represented by:

 

Result = Fitter.fit(DataSource, Function) ,

 

where different Fitters can do different kinds of fits. This approach provides a lot of flexibility and at the same time has enough structure to be easily used and implemented.

 

1. DataSource

With a few extensions ITuple interface can be used to represent a DataSource. New methods in ITupleFactory:

 

public ITuple create(IHistogram histogram)

public ITuple create(ICloud cloud)

 

provide ITuple wrapper around IHistogram and ICloud, allowing uniform access to the data. Points in ICloud and bins in IHistogram are mapped to ITuple rows.

 

ITuple based on IHistogram should have several standard columns:

int entries, double height, error, x (y, z)

where "x" is bin center of the IHistogram axis. ITuple based on ICloud should have columns:

double value, x (y, z)

Also convenience fit methods should be provided for simple cases, like

fit(IHistogram1D h, IFunction f);

 

 

2. Function

In AIDA context IFunction is used for three main purposes:

a)     to plot or calculate value

b)     to do chi-Square fits

c)      to do unbinned Maximum Likelihood (ML) fits

 

Item c) needs some clarification. In order to be used in a ML fit, function must be a PDF: always positive and normalized over the range of the dependent variables. The last requirement introduces normalization factor N that depends on analysis cuts.

 

Example: f(x) = ax+b, x variable, a, b - fit parameters

For a chi-Square fit we just have to find set of parameters (a, b) that

minimizes sum of squares of weighted deviations. For a ML fit f(x) has

to be normalized to some value N = integral[f(x)] from xmin to xmax. In this

case N = a(x2max x2min) + b(xmax xmin). Now N depends on parameters

a, b and the range of x, so in order to get correct result from the ML fit,

function f(x)/N should be used in the fit, not f(x).

 

In order for ML fit to be efficient, function should be able to calculate its own normalization factor for any point and any set of parameters that are likely to be used in a fit. To address this issue two new methods are included in IFunction interface:

 

public boolean supportsNormalization()

public void setNormalization(double N)

 

The first method returns "true" if function can calculate its own normalization and "false" otherwise. Second method sets the normalization constant N (usually N=1) and turns function into a PDF (in some cases amplitude parameter of the function has to be set fixed, ). These methods should be used by IFitter rather than by user.

 

3. Variable

IVariable is a new interface designed to help keep track of values, errors, states and ranges of various parameters and variables. IVariable can represent variable or parameter, it has "Value", "Error", set of internal states, and it also has ability to take its values from a DataSource in some orderly fashion. It is important that IVariable can support a set of ranges, not just one (x_min, x_max) pair.

 

IVariable does not have a factory and can not be created directly, instead IVariable can be returned by an IFunction or can be "derived" from ITuple:

ITuple: public IVariable variable(String name, String label, IEvaluator ev)

IFunction: public IVariable variable(String name)

It is also possible to connect IVariable to ITuple "by hand" using IVariable "connect" method:

public void connect(IEvaluator ev)

public void connect(ITuple data)

If IVariable is connected, its value is based upon the current row of ITuple.

IFunction should also use IVariables for parameters and variables. Instead of calling IFunction's current "parameterNames/parameterValues/setParameterValue" methods, user should get IVariable and use IVariable methods to set its value and limits directly.

 

 

4. Fitting

Here we introduce three new interfaces: IFitter, IFitterFactory and IFitResult.

In a standard AIDA fashion, IFitter is created by IFitterFactory with "create" method:

 

public IFitter create(String type),

 

where "type" describes what kind of fitter is created. Among other methods, IFitter has "fit" method that returns IFitResult:

 

public IFitResult fit(ITuple data, IFunction function).

 

IFitResult contains all results relevant to this particular fit. IFitter should have several "setup" methods, like what results to save in IFitResult, how to do the fit, etc.

 

Note that there is a problem with displaying fit results IFunction can retain parameter values obtained in the fit, and we can even put a switch in IPlotter (or in IFunction) to display values and errors of IFunction's parameters, but IFunction can not have any fit-related information. Such information (like chi-2, fit quality, parameter scans, etc.) belongs in IFitResult and it looks like this information has to be extracted and displayed "by hand". There is a possibility of making IFitResult a hash-bag, so that IFitter can just drop there any result it is configured to save and mark each individual result "displayable" or "non-displayable". It would be convenient for IFitResult to retain reference to histogram and function used in the fit. Then we can have IPlotter.plot(IFitResult result) method that automatically displays histogram, function plus all "displayable" information in the current region.


Use Cases:

 

1. Below is a simple example of fitting histogram with a Gaussian function, as it stands in the current version of AIDA: creating function factory, then function and then asking function to do fit.

 

import hep.aida.*;

import java.util.Random;

 

public class FitExample

{

public static void main(String[] args)

{

// Create factories

IAnalysisFactory analysisFactory = IAnalysisFactory.create();

ITreeFactory treeFactory = analysisFactory.createTreeFactory();

ITree tree = treeFactory.create();

IPlotter plotter = analysisFactory.createPlotterFactory().create("Plot");

IHistogramFactory histogramFactory = analysisFactory.createHistogramFactory(tree);

IFunctionFactory functionFactory = analysisFactory.createFunctionFactory(tree);

 

// Create 1D histogram

IHistogram1D h1d = histogramFactory.create1D("Histogram 1D",50,-3,3);

 

// Fill 1D histogram with Gaussian

Random r = new Random();

for (int i=0; i<5000; i++)

{

h1d.fill(r.nextGaussian());

}

 

// Create Gaussian fitting function

IFitFunction f = functionFactory.createFit("Gaussian Fit", "Gaussian Fit", "G","amplitude=1., mean=0., sigma=1.");

 

///////////////////////////////////////////////////////////////////////////////////////////////////////////////

// Other supported functions:

// f = functionFactory.createFit("Exp Fit", "Exponential Fit", "E", "amplitude= ; origin= ; exponent= ");

// f = functionFactory.createFit("BW Fit", "Breit-Wigner Fit", "BW","amplitude= ; origin= ; width= ");

// f = functionFactory.createFit("Poly Fit", "Polynomial Fit", "P", "a4= , a3= , a2= , a1= , a0= ");

//////////////////////////////////////////////////////////////////////////////////////////////////////////////

 

// Do Fit

f.fit(h1d);

 

// Show results

plotter.createRegions(1,1,0);

plotter.plot(h1d);

plotter.plot(f);

plotter.show();

}

}

 

 

2. With proposed AIDA modifications the fit-related part of example will change.

There are several extra steps - create fitter factory and fitter, configure fitter, ask fitter to do fitting. "IFitter.resetFunction(boolean state)" is set to "false" here so that after fitting is done, function is not reset to its original state. We can make it a default, I just wanted to show where fitter configuration goes. Only fit-related part here:

 

 

IFunctionFactory functionFactory = analysisFactory.createFunctionFactory(tree);

IFitterFactory fitterFactory = analysisFactory.createFitterFactory(tree);

 

IFunction f = functionFactory.create("Gauss", "1D Gaussian", "G");

IFitter fitter = fitterFactory.create("LSF");

 

// Configure fitter

fitter.resetFunction(false);

 

// Do fitting

IFitResult result = fitter.fit(h1d, f);

 

// Plot results just the same as before

plotter.createRegions(1,1,0);

plotter.plot(h1d);

plotter.plot(f);

plotter.show();

 

 


3. Here is more complicated example of chi-Square fit using data from ITuple.

Suppose ITuple contains data from beam position scan with following columns:

x wire position in ADC counts

signal absolute beam intensity

sigma error with which "signal" was measured

Each row of ITuple corresponds to the next step in "x" direction. We want to see how well normalized beam intensity can be described by a double Gaussian in real detector coordinates between 1.5 and 1.5 cm. One way to do it would be to create an IHistogram and fill it with {x="0.05*x+18.6", Weight=signal/maxSignal, Error=sigma/maxSignal}, but currently IHistogram does not have "setError" method; so we'll just fit ITuple. The IXYData interface would be very useful in this particular case. IEvaluator factory is needed here:

 

IEvaluatorFactory evaluatorFactory = analysisFactory.createEvaluatorFactory(tree);

IFunctionFactory functionFactory = analysisFactory.createFunctionFactory(tree);

IFitterFactory fitterFactory = analysisFactory.createFitterFactory(tree);

 

// Derive x coordinate, measurement, and error variables from tuple

IVariable x_cm = tuple.variable("x", "x in cm", evaluatorFactory.create(tuple, "0.05*x + 18.6"));

x_cm.setMinRange(-1.5);

x_cm.setMaxRange(1.5);

x_cm.setUnits("cm");

 

double maxSignal = tuple.columnMax(2);

IVariable meas = tuple.variable("meas", "scaled signal", evaluatorFactory.create(tuple, "signal/maxSignal");

IVariable error = tuple.variable("error", "scaled sigma", evaluatorFactory.create(tuple, "sigma/maxSignal");

 

// Create functions with variable already connected to tuple

IFunction f1 = functionFactory.create("Gauss1", "First Gaussian", "G", IVariable[] {x_cm} );

IFunction f2 = functionFactory.create("Gauss2", "Second Gaussian", "G", IVariable[] {x_cm} );

IFunction f_sum = functionFactory.add(f1,f2, 0.9);

 

// Create and configure fitter

IFitter fitter = fitterFactory.create("LSF");

 

fitter.resetFunction(false);

 

// Do fitting. This fit method takes function, measurement and measurement error variables

// The only variable, x_cm, is already connected to tuple.

IFitResult result = fitter.fit(f_sum, meas, error);

 

// Can plot only resulting function and histogram with wrong errors (or without errors)

// Also there is no corresponding "ITuple.project()" method to fill histogram with weight

plotter.createRegions(1,1,0);

plotter.plot(h1d);

plotter.plot(f_sum);

plotter.show();

 

 


4. Example of unbinned Maximum Likelihood analysis.

 

IEvaluatorFactory evaluatorFactory = analysisFactory.createEvaluatorFactory(tree);

IFunctionFactory functionFactory = analysisFactory.createFunctionFactory(tree);

IFitterFactory fitterFactory = analysisFactory.createFitterFactory(tree);

 

// Derive variables from tuple

IVariable cosTheta = tuple.variable("cosTheta", "cos of Theta");

cosTheta.setMinRange(-0.7);

cosTheta.setMaxRange(0.7);

 

IVariable prob_b = tuple.variable("prob_b", "probability to be b event", evaluatorFactory.create(tuple, "pB"));

 

IVariable prob_c = tuple.variable("prob_c", "probability to be c event", evaluatorFactory.create(tuple, "pC"));

 

IVariable ap_b= tuple.variable("ap_b", "analyzing power, b event", evaluatorFactory.create(tuple, "1/(1+exp(-abs(Qdiff)*alphaB))");

 

IVariable ap_c= tuple.variable("ap_c", "analyzing power, c event", evaluatorFactory.create(tuple, "1/(1+exp(-abs(Qdiff)*alphaC))");

 

// Register my function factory and create my function. Registering does not make much sense here

functionFactory.register(new MyFunctionFactory());

IFunction func = functionFactory.create("myFunction", "My fit Function", "myFunc", new IVariable[] {cosTheta, prob_b, prob_c, ap_b, ap_c} );

 

// Create filters and add them to the tuple

IFilter filter1 = filterFactory.create("Evis>20. && nTracks>=7 && abs(cosTheta)<0.7");

IFilter filter2 = filterFactory.create("(Q1!=0 || Q2!=0) && abs(Qdiff)<20");

 

tuple.addFilter("event cuts", filter1);

tuple.addFilter("extra cuts", filter2);

 

// Create and configure fitter

IFitter fitter = fitterFactory.create("MLF");

 

fitter.resetFunction(false);

fitter.keepScan("asymmetry");

 

// Do fitting

IFitResult result = fitter.fit(func);

 

IHistogram1D h = result.scan("asymmetry");

 


Detailed Description:

 

1. IVariables

 

public interface IVariable

{

public String label();

public String name();

 

public void setValue(double value);

public double value();

public double error();

 

// Set variable ranges

public void setRange(double lower, double upper);

public void addRange(double lower, double upper);

public double[][2] range();

 

// If dependent is "true", this is a variable, if "false" - parameter

public void setDependent(boolean state);

public boolean isDependent();

 

// States related to fitting

public void setStep(double step);

public void setFixed(boolean state);

public boolean isFixed();

 

// Even if bounds are defined, they don't have to be used

public void setUseBounds(boolean state);

public boolean useBounds();

 

// IVariable can be connected to ITuple to derive its value

// from the current ITuple row

public boolean isConnected();

public ITuple connection();

public void connect(ITuple data);

public void connect(IEvaluator ev);

 

// Units can be used to annotate plot axis

public void setUnits(String units);

public String units();

}

 

There is no IVariableFactory class. IFunction can return its variables and parameters in a form of IVariable, or IVariable can be "derived" from ITuple.

 

 

2. IFunctions

 

IFunctions should now be based on IVariables. To make it easier for user we still keep "lazy" create(String name, String label, String type) method for IFunction creation. In that case IVariables will be created by IFunction internally. Alternatively user can supply a vector of pre-configured IVariables. I can imagine that this vector can be shorter than required, in which case missing IVariables will be created by IFunction internally.

 

 

public interface IFunctionFactory

{

public IFunction create(String name, String label, String type);

public IFunction create(String name, String label, String type,

String options);

public IFunction create(String name, String label, String type,

IVariable[] variables);

public IFunction create(String name, String label, String type,

IVariable[] variables, IVariable[] parameters);

public IFunction createScripted(String name, String label,

String script) ;

public IFunction createScripted(String name, String label,

String script, IVariable[] variables) ;

public IFunction createScripted(String name, String label,

String script, IVariable[] variables, IVariable[] parameters) ;

 

// Allows user to register new function factory

// IUserFunctionFactory should have method "String[] types()" that

// return all IFunction types that this user factory can make

public void register(IUserFunctionFactory userFunctionFactory);

 

// Do arithmetics with functions, Example: "add" f_new=f1*p+f2*(1-p)

// Note that "div" does not make much sence for PDFs

public IFunction add(IFunction f1, IFunction f2, double p);

public IFunction add(IFunction[] f, double[] p);

public IFunction mul(IFunction f1, IFunction f2, double p);

public IFunction mul(IFunction[] f, double p);

public IFunction div(IFunction f1, IFunction f2);

}

 

public interface IFunction

{

public int dimension();

public String label();

 

// Deal with variables and parameters

public IVariable variable(String name);

public IVariable[] variables();

public IVariable parameter(String name);

public IVariable[] parameters();

 

// Get value based on internal state of the function

public double value();

 

// Get value of the function without changing its internal state

public double value(double[] point);

public double value(double[] point, double[] parameters);

 

// Can be used by IFitter, maybe should be excluded from the user

// interface together with support for derivatives

public boolean supportsNormalization();

public void setNormalization(double N);

}

 

Note that in order to change variable or parameter settings (names, values, limits, fixed, connection, ...) user now have to talk to the corresponding IVariable directly.

 


3. ITuple

 

public interface ITupleFactory

{

public ITuple create(String name, String label, String[] ColumnNames,

Class[] columnType, string options);

public ITuple create(String name, String label, String columns,

String options);

 

// New methods

public ITuple create(String name, String label, IHistogram hist);

public ITuple create(String name, String label, ICloud cloud);

public ITuple create(String name, String label, ITuple tuple,

IFilter filter);

public ITuple create(String name, String label, ITuple tuple,

IFilter[] filters);

public ITuple chain(ITuple[] set);

public ITuple merge(ITuple[] set);

}

 

Method "chain()" makes a chain of ITuples with the same internal structure while "merge()" creates a union of columns from all ITuples in the set. Exact rules how to do it must be defined. Can create ITuple from another ITuple by applying filters.

 

Also we need to include following methods in ITuple:

 

public interface ITuple

{

...

 

// New ITuple methods here:

// Need ability to "derive" variable from an ITuple

public IVariable variable(String name, String label);

public IVariable variable(String name, String label, IEvaluator ev);

 

// Need more "project" methods. Same with 2D and 3D histograms

public boolean project(IHistogram1D hist, IEvaluator value);

public boolean project(IHistogram1D hist, IEvaluator value,

IEvaluator weight);

public boolean project(IHistogram1D hist, IEvaluator value,

IEvaluator weight, IFilter filter);

public boolean project(IHistogram1D hist, IVariable value);

public boolean project(IHistogram1D hist, IVariable value,

IVariable weight);

public boolean project(IHistogram1D hist, IVariable value,

IVariable weight, IFilter filter);

}

 


4. Fitting

 

public interface IFitterFactory

{

// "type" can be something like LSF, UMLF ...

public IFitter create(String type);

 

// Allows user to register new fitter types and fitter factory

// that makes them

public IFitter register(String[] userTypes, IFitterFactory

userFitterFactory);

}

 

public interface IFitter

{

// Various fit control methods here

public void keepScan(String parameterName);

public void keepScan(String parameterName1, String parameterName2);

public void resetFunction(boolean state);

...

 

// Does fitting and returns results in a form of IFitResult object.

// Fitter first connects all function's not-yet-connected variables to

// ITuple

public IFitResult fit(ITuple data, IFunction function);

public IFitResult fit(IHistogram data, IFunction function);

public IFitResult fit(ICloud data, IFunction function);

 

// If all function's variables are already connected, can use this

// method

public IFitResult fit(IFunction function);

 

// Specific for chi-Square fit

public IFitResult fit(IHistogram data, IFunction f, IVariable meas, IVariable error);

public IFitResult fit(IHistogram data, IFunction f, IVariable[] meas, IVariable[] error);

 

// Returns IFitter type

public String getType();

}

 

// Need more thinking here

public interface IFitResult

{

public double chiSquared();

public double degreeOfFreedom();

 

public IVariable[] parameters();

 

public IVariable parameter(String name);

public IHistogram1D scan(String parameterName);

public IHistogram2D scan(String parameterName1, String parameterName2);

}

 

 


Questions:

 

1.     Should there be single interface for chi-Square and ML fitters? Configuration and fitting procedure can be quite different.

2.     How should IEvaluator handle external parameters and do we need both IEvaluator and IVariable? Maybe IEvaluator should be hidden from user?

3.     How do we do simultaneous fits of several functions and fits with extra constrains?

4.     It is possible to split IVariable into two different interfaces, one describes variable, another one describes parameter (see IVariable alternative).

 

...