Class to construct histogram of a pair of random variables. More...

#include <ocsHistogram2D.hh>

Public Member Functions
	Histogram2D (string InFileName, float x_min2, float x_max2, float y_min2, float y_max2)
	Constructor.
void	InitializeParameters (void)
	Internal method called by constructors.
	Histogram2D (string InFileName)
	Constructor.
	Histogram2D (string InFileName, OptSignal *oOptSignal)
	Constructor to be used when making eye diagram.
	Histogram2D (Histogram2D &OrigHist)
	Copy constructor.
Histogram2D &	operator= (Histogram2D &RHSHist)
	Assignment operator.
	~Histogram2D ()
	Destructor.
void	Clear ()
	Sets all dynamic variables to their initial state. For use when want to output multiple independent histograms during course of a simulation.
void	SetMaxX (double x_max2)
void	SetMaxY (double y_max2)
void	SetMinMaxXY (double x_min2, double x_max2, double y_min2, double y_max2)
float	GetMaxX (void)
float	GetMaxY (void)
float	GetMinX (void)
float	GetMinY (void)
float	GetDeltaX (void)
float	GetDeltaY (void)
int	GetNumBinsX (void)
int	GetNumBinsY (void)
float	GetSamplesMaxX (void)
float	GetSamplesMaxY (void)
void	SetBin (int x_index, int y_index, double value)
	Internal method.
double	GetBin (int x_index, int y_index)
double	GetBinSquare (int x_index, int y_index)
int	GetIndex (int x_index, int y_index)
double	Get_pdfConfidence (int x_index, int y_index)
	Internal method.
void	ObtainConfidence (void)
int	GetQtSamplesBin (int x_index, int y_index)
void	IncrementBin (int x_index, int y_index, double LikelihoodRatio)
int	GetQtSamples (void)
	Returns total number of samples.
void	AddSample (double x, double y)
	Method to add a sample in a standard Monte Carlo simulation.
void	AddSample (double x, double y, double LikelihoodRatio)
	Method to add sample in a Monte Carlo simulation with importance sampling.
void	WriteFile_pdf2D (string OutFileName)
	Writes out pdf computed by the histogram to a file.
double	GetMeanX (void)
	Returns mean of X calculated from samples that are in range.
double	GetMeanY (void)
	Returns mean of Y calculated from samples that are in range.
void	UpdateEyeDiagram (OptSignal oOptSignal, Photodetector oPhotodetector)
	For applications to eye diagrams in Monte Carlo simulations.
void	WriteEyeDiagramHistogram (string OutFileName)
void	MultipleSelectMaxRelNumHitsImport (string InFileName, int Convert2dB_Scale)
void	GetLevelCurvesMultipleSelectMaxRelNumHitsImport (string InFileName, int Convert2dB_Scale, int CosnvertParam2dB_Scale)
void	MultipleHeuristicBalanceImport (string InFileName, int Convert2dB_Scale)
	Use this function for merging 2D histograms named in InFileName.
void	MergeMultipleConditionalHeuristicBalance (string InFileName, double z_min, double z_max, int Convert2dB_Scale)
void	MergeMultipleConditionalHeuristicBalance (string InFileName, double z_min, double z_max, double w_min, double w_max, int Convert2dB_Scale)
void	GetLevelCurvesMultipleHeuristicBalanceImport (string InFileName, int Convert2dB_Scale, int ConvertParam2dB_Scale)
void	GetLevelCurvesMultipleHeuristicBalanceImportVarianceWeighted (string InFileName, int Convert2dB_Scale, int ConvertParam2dB_Scale)
void	ComputePDFfromHistogram (void)
void	NormalizePDF (void)
void	SetPDF (int x_index, int y_index, double value)
double	GetPDF (int x_index, int y_index)
void	ComputeXMarginalPDF (void)
void	ComputeYMarginalPDF (void)
double	ComputeIntegralPDF (void)
double	ComputeMeanX_PDF (void)
double	ComputeMeanY_PDF (void)
double	GetMaxPDF (void)
double	SetPDFToZeroBelowThreshold (double Threshold)
void	GaussianPDF (double mean_x, double mean_y, double std)
	Analytic formula for Gaussian PDF. For debugging purposes.
void	WritePDF (string OutFileName)
	Writes out data in array PDF.
double	BilinearInterpolationPDF (double x, double y)
void	RotatePDF (double CenterX, double CenterY, double Angle, double *OutData)
Public Attributes
double *	PDF
	The 2D PDF array.
Private Attributes
float	x_min
	Minimum x value for 2D Histogram of x and y.
float	x_max
	Maximum x value for 2D Histogram of x and y.
float	y_min
	Minimum y value for 2D Histogram of x and y.
float	y_max
	Maximum y value for 2D Histogram of x and y.
float	x_window_size
float	y_window_size
int	NumBinsX
	Number of bins in X variable.
int	NumBinsY
	Number of bins in Y variable.
float	delta_x
float	delta_y
double *	Bin
	The 2D Histogram array.
int *	qtSamplesBin
double *	BinSquare
double *	pdfConfidence
bool	PreviouslyComputed_pdfConfidenceFlag
double *	XData
double *	YData
double *	XMarginal
double *	YMarginal
int	NumSamplesTotal
int	NumSamplesInRange
int	NumSamplesWithOutOfRangeX
int	NumSamplesWithOutOfRangeY
float	SamplesMinX
	Minimum X value of all samples (even if lie outside Histogram).
float	SamplesMaxX
	Maximum X value of all samples (even if lie outside Histogram).
float	SamplesMinY
	Minimum Y value of all samples (even if lie outside Histogram).
float	SamplesMaxY
	Maximum Y value of all samples (even if lie outside Histogram).
string	HistInFileName
	Used by merge algorithms in importance sampling applications.
double	MeanX
double	MeanY

Detailed Description

Class to construct histogram of a pair of random variables.

For use in computing two-dimensional histograms, particularly in Monte Carlo sampling. The main applications the code was written for include construction of eye diagrams in optical communications systems and quantification of statistics related to random polarization effects using importance sampling. However the class is generic enough to be used in other situations.

For simple applications call as follows:

Histogram2D MyHist("Histogram2D.in");
for i=1 to N, Generate samples x_i, y_i and call MyHist.AddSample(x_i,y_i)
MyHist.WriteFile_pdf2D("TwoDimPDF.dat");

For applications to eyediagrams call as

Histogram2D MyHist("Histogram2D.in",Signal);
MyHist.UpdateEyeDiagram(Signal,oPhotodetector);
MyHist.WriteEyeDiagramHistogram("EyeDiagram.dat");

The two random variables are refered to as x and y throughout the code.

If we think of x and y as being on a 2D coordinate system as usual then the bins are indexed by a pair of integers that describes the position of the lower left corner of each bin, with the indices starting at (0,0).

So x-index goes from 0 to NumBinsX - 1.

Samples out of range are discarded and their number is tracked.

For importance sampling applications the pdfs for each bias choice are first written to files using WriteFile_pdf2D in one application. Then in a second aplication they are read back in and merged together usually using the balanaced heuristic method by calling a method like MultipleHeuristicBalanceImport. Finally the merged pdf is written out once more using WriteFile_pdf2D

Constructor & Destructor Documentation

Histogram2D::Histogram2D	(	string	InFileName,
		float	x_min2,
		float	x_max2,
		float	y_min2,
		float	y_max2
	)

Constructor.

See also:: x_min, x_max, y_min, y_max If you update this constructor update other two as well as copy constructor and assignement operator

References HistInFileName, InitializeParameters(), LogFile, LogFileSeparator(), LOWER_ONLY, NumBinsX, NumBinsY, ReadInt(), x_max, x_min, y_max, and y_min.

Referenced by GetLevelCurvesMultipleHeuristicBalanceImport(), GetLevelCurvesMultipleHeuristicBalanceImportVarianceWeighted(), GetLevelCurvesMultipleSelectMaxRelNumHitsImport(), MergeMultipleConditionalHeuristicBalance(), MultipleHeuristicBalanceImport(), and MultipleSelectMaxRelNumHitsImport().

Histogram2D::Histogram2D ( string InFileName )

Constructor.

References HistInFileName, InitializeParameters(), LogFile, LogFileSeparator(), LOWER_ONLY, NO_BOUNDS, NumBinsX, NumBinsY, ReadDouble(), ReadInt(), x_max, x_min, y_max, and y_min.

Histogram2D::Histogram2D	(	string	InFileName,
		OptSignal *	oOptSignal
	)

Constructor to be used when making eye diagram.

References OptSignal::GetBitLength(), HistInFileName, InitializeParameters(), LogFile, LogFileSeparator(), LOWER_ONLY, NumBinsX, NumBinsY, ReadDouble(), ReadInt(), x_max, x_min, y_max, and y_min.

Histogram2D::Histogram2D ( Histogram2D & OrigHist )

Copy constructor.

References BinSquare, GetBin(), GetPDF(), HistInFileName, InitializeParameters(), MeanX, MeanY, NumBinsX, NumBinsY, NumSamplesInRange, NumSamplesTotal, NumSamplesWithOutOfRangeX, NumSamplesWithOutOfRangeY, pdfConfidence, PreviouslyComputed_pdfConfidenceFlag, qtSamplesBin, SamplesMaxX, SamplesMaxY, SamplesMinX, SamplesMinY, SetBin(), SetPDF(), x_max, x_min, y_max, and y_min.

Histogram2D::~Histogram2D ( )

Destructor.

References Bin, BinSquare, pdfConfidence, and qtSamplesBin.

Referenced by ReceiverSubsystem::~ReceiverSubsystem().

Member Function Documentation

void Histogram2D::AddSample	(	double	x,
		double	y
	)

Method to add a sample in a standard Monte Carlo simulation.

Referenced by GetLevelCurvesMultipleHeuristicBalanceImport(), GetLevelCurvesMultipleHeuristicBalanceImportVarianceWeighted(), GetLevelCurvesMultipleSelectMaxRelNumHitsImport(), MergeMultipleConditionalHeuristicBalance(), MultipleHeuristicBalanceImport(), MultipleSelectMaxRelNumHitsImport(), and UpdateEyeDiagram().

void Histogram2D::AddSample	(	double	x,
		double	y,
		double	LikelihoodRatio
	)

Method to add sample in a Monte Carlo simulation with importance sampling.

References delta_x, delta_y, IncrementBin(), MeanX, MeanY, NumBinsX, NumBinsY, NumSamplesInRange, NumSamplesTotal, NumSamplesWithOutOfRangeX, NumSamplesWithOutOfRangeY, SamplesMaxX, SamplesMaxY, SamplesMinX, SamplesMinY, x_min, and y_min.

double Histogram2D::BilinearInterpolationPDF	(	double	x,
		double	y
	)

If the data of one of the grid points is off the grid then the value at that grid point is set to zero.

References delta_x, delta_y, GetPDF(), x_min, and y_min.

Referenced by RotatePDF().

void Histogram2D::Clear ( )

Sets all dynamic variables to their initial state. For use when want to output multiple independent histograms during course of a simulation.

References BinSquare, LogFile, LogFileSeparator(), MeanX, MeanY, NumBinsX, NumBinsY, NumSamplesInRange, NumSamplesTotal, NumSamplesWithOutOfRangeX, NumSamplesWithOutOfRangeY, pdfConfidence, qtSamplesBin, SetBin(), SetPDF(), XData, XMarginal, YData, and YMarginal.

Referenced by GetLevelCurvesMultipleHeuristicBalanceImport(), GetLevelCurvesMultipleHeuristicBalanceImportVarianceWeighted(), GetLevelCurvesMultipleSelectMaxRelNumHitsImport(), InitializeParameters(), MergeMultipleConditionalHeuristicBalance(), MultipleHeuristicBalanceImport(), MultipleSelectMaxRelNumHitsImport(), ReceiverSubsystem::ReceiveSignalBackToBack(), ReceiverSemiAnalytical::ReceiveSignalBackToBackMonteCarlo(), SetMaxY(), and SetMinMaxXY().

double Histogram2D::ComputeIntegralPDF ( void )

References ComputeXMarginalPDF(), delta_x, NumBinsX, Quadrature(), x_max, x_min, and XMarginal.

Referenced by NormalizePDF().

double Histogram2D::ComputeMeanX_PDF ( void )

References ComputeXMarginalPDF(), delta_x, NumBinsX, Quadrature(), x_max, x_min, XData, and XMarginal.

double Histogram2D::ComputeMeanY_PDF ( void )

References ComputeYMarginalPDF(), delta_y, NumBinsY, Quadrature(), y_max, y_min, YData, and YMarginal.

void Histogram2D::ComputePDFfromHistogram ( void )

For applications where want to manipulate PDF rather than histogram first call this method to compute PDF from histogram.

PDF data is stored in separate 2D array double * PDF All integrals involving PDF data done using (iterated) Simpson's Rule

References GetBin(), NormalizePDF(), NumBinsX, NumBinsY, and SetPDF().

void Histogram2D::ComputeXMarginalPDF ( void )

References delta_y, GetPDF(), NumBinsX, NumBinsY, Quadrature(), XMarginal, y_max, y_min, and YData.

Referenced by ComputeIntegralPDF(), and ComputeMeanX_PDF().

void Histogram2D::ComputeYMarginalPDF ( void )

References delta_x, GetPDF(), NumBinsX, NumBinsY, Quadrature(), x_max, x_min, XData, and YMarginal.

Referenced by ComputeMeanY_PDF().

void Histogram2D::GaussianPDF	(	double	mean_x,
		double	mean_y,
		double	std
	)

Analytic formula for Gaussian PDF. For debugging purposes.

References delta_x, delta_y, NumBinsX, NumBinsY, pi, SetPDF(), x_min, and y_min.

double Histogram2D::Get_pdfConfidence	(	int	x_index,
		int	y_index
	)

Internal method.

References GetBin(), GetBinSquare(), NumSamplesTotal, sq(), and sqrt().

Referenced by GetLevelCurvesMultipleHeuristicBalanceImportVarianceWeighted(), GetLevelCurvesMultipleSelectMaxRelNumHitsImport(), MergeMultipleConditionalHeuristicBalance(), MultipleHeuristicBalanceImport(), and ObtainConfidence().

double Histogram2D::GetBin	(	int	x_index,
		int	y_index
	)

References Bin, and NumBinsY.

Referenced by ComputePDFfromHistogram(), Get_pdfConfidence(), GetLevelCurvesMultipleHeuristicBalanceImport(), GetLevelCurvesMultipleHeuristicBalanceImportVarianceWeighted(), GetLevelCurvesMultipleSelectMaxRelNumHitsImport(), Histogram2D(), MultipleSelectMaxRelNumHitsImport(), operator=(), WriteEyeDiagramHistogram(), and WriteFile_pdf2D().

double Histogram2D::GetBinSquare	(	int	x_index,
		int	y_index
	)

References BinSquare, and NumBinsY.

Referenced by Get_pdfConfidence(), and GetLevelCurvesMultipleHeuristicBalanceImport().

float Histogram2D::GetDeltaX ( void ) [inline]

References delta_x.

float Histogram2D::GetDeltaY ( void ) [inline]

References delta_y.

int Histogram2D::GetIndex	(	int	x_index,
		int	y_index
	)			`[inline]`

References NumBinsY.

Referenced by GetLevelCurvesMultipleHeuristicBalanceImport(), GetLevelCurvesMultipleHeuristicBalanceImportVarianceWeighted(), GetLevelCurvesMultipleSelectMaxRelNumHitsImport(), MergeMultipleConditionalHeuristicBalance(), MultipleHeuristicBalanceImport(), ObtainConfidence(), and WriteFile_pdf2D().

void Histogram2D::GetLevelCurvesMultipleHeuristicBalanceImport	(	string	InFileName,
		int	Convert2dB_Scale,
		int	ConvertParam2dB_Scale
	)

void Histogram2D::GetLevelCurvesMultipleHeuristicBalanceImportVarianceWeighted	(	string	InFileName,
		int	Convert2dB_Scale,
		int	ConvertParam2dB_Scale
	)

void Histogram2D::GetLevelCurvesMultipleSelectMaxRelNumHitsImport	(	string	InFileName,
		int	Convert2dB_Scale,
		int	CosnvertParam2dB_Scale
	)

double Histogram2D::GetMaxPDF ( void )

Returns the maximum value of the PDF obtained using

See also:: Histogram2D::ComputePDFfromHistogram()

References GetPDF(), NumBinsX, and NumBinsY.

float Histogram2D::GetMaxX ( void ) [inline]

References x_max.

float Histogram2D::GetMaxY ( void ) [inline]

References y_max.

double Histogram2D::GetMeanX ( void )

Returns mean of X calculated from samples that are in range.

Warning:: Not correct for importance sampling!

References MeanX, and NumSamplesInRange.

double Histogram2D::GetMeanY ( void )

Returns mean of Y calculated from samples that are in range.

Warning:: Not correct for importance sampling!

References MeanY, and NumSamplesInRange.

float Histogram2D::GetMinX ( void ) [inline]

References x_min.

float Histogram2D::GetMinY ( void ) [inline]

References y_min.

int Histogram2D::GetNumBinsX ( void ) [inline]

References NumBinsX.

int Histogram2D::GetNumBinsY ( void ) [inline]

References NumBinsX.

double Histogram2D::GetPDF	(	int	x_index,
		int	y_index
	)

See also:: Histogram2D::ComputePDFfromHistogram()

References NumBinsX, NumBinsY, and PDF.

Referenced by BilinearInterpolationPDF(), ComputeXMarginalPDF(), ComputeYMarginalPDF(), GetMaxPDF(), Histogram2D(), NormalizePDF(), operator=(), SetPDFToZeroBelowThreshold(), and WritePDF().

int Histogram2D::GetQtSamples ( void ) [inline]

Returns total number of samples.

See also:: NumSamplesTotal

References NumSamplesTotal.

Referenced by GetLevelCurvesMultipleHeuristicBalanceImportVarianceWeighted(), GetLevelCurvesMultipleSelectMaxRelNumHitsImport(), and MultipleSelectMaxRelNumHitsImport().

int Histogram2D::GetQtSamplesBin	(	int	x_index,
		int	y_index
	)

References NumBinsY, and qtSamplesBin.

Referenced by GetLevelCurvesMultipleHeuristicBalanceImport(), GetLevelCurvesMultipleSelectMaxRelNumHitsImport(), and MultipleSelectMaxRelNumHitsImport().

float Histogram2D::GetSamplesMaxX ( void ) [inline]

See also:: SamplesMaxX

References SamplesMaxX.

float Histogram2D::GetSamplesMaxY ( void ) [inline]

See also:: SamplesMaxY

References SamplesMaxY.

void Histogram2D::IncrementBin	(	int	x_index,
		int	y_index,
		double	LikelihoodRatio
	)

References Bin, BinSquare, NumBinsY, qtSamplesBin, and sq().

Referenced by AddSample().

void Histogram2D::InitializeParameters ( void )

Internal method called by constructors.

References Bin, BinSquare, Clear(), delta_x, delta_y, LogFile, NumBinsX, NumBinsY, PDF, pdfConfidence, PreviouslyComputed_pdfConfidenceFlag, qtSamplesBin, x_max, x_min, x_window_size, XData, XMarginal, y_max, y_min, y_window_size, YData, and YMarginal.

Referenced by Histogram2D().

void Histogram2D::MergeMultipleConditionalHeuristicBalance	(	string	InFileName,
		double	z_min,
		double	z_max,
		double	w_min,
		double	w_max,
		int	Convert2dB_Scale
	)

Use this function for merging 2D histograms named in InFileName Computes the 2D probability density function f(X=x,Y=y| z_min < Z < z_max, w_min < W < w_max), Samples of the random variables X,Y,Z,W have been obtained from an importance sampling simulation, and that the names of the files containing the samples are stored in the file InFileName.

Computes the 2D probability density function f(X=x,Y=y| z_min < Z < z_max, w_min < W < w_max), Samples of the random variables X,Y,Z,W have been obtained from an importance sampling simulation, and that the names of the files containing the samples are stored in the file InFileName.

The input parameter Convert2dB_Scale converts the pdfs of X and Y to a plus/minus dB scale before adding them to this 2D histogram.

We only add a (X,Y) sample to this 2D histogram if z_min < Z < z_max and w_min < W < w_max

Note: I'm not sure whether or not the Confidence Interval is correctly computed.

By JZ, 23/2/2005 and adapted from code by Ivan T. Lima Jr. and Aurenice O. Lima (7/28/02)

References AddSample(), Clear(), Get_pdfConfidence(), GetIndex(), HistInFileName, Histogram2D(), Linear2dB(), LogFile, NumBinsX, NumBinsY, ObtainConfidence(), pdfConfidence, PreviouslyComputed_pdfConfidenceFlag, sgn(), sq(), and sqrt().

void Histogram2D::MergeMultipleConditionalHeuristicBalance	(	string	InFileName,
		double	z_min,
		double	z_max,
		int	Convert2dB_Scale
	)

Use this function for merging 2D histograms named in InFileName Computes the 2D probability density function f(X=x,Y=y| z_min < Z < z_max), the joint probability density for X and Y given that z_min < Z < z_max. It is assumed that samples of the random variables X,Y,Z have been obtained from an importance sampling simulation, and that the names of the files containing the samples are stored in the file InFileName.

Computes the 2D probability density function f(X=x,Y=y| z_min < Z < z_max), the joint probability density for X and Y given that z_min < Z < z_max. It is assumed that samples of the random variables X,Y,Z have been obtained from an importance sampling simulation, and that the names of the files containing the samples are stored in the file InFileName.

The input parameter Convert2dB_Scale converts the pdfs of X and Y to a plus/minus dB scale before adding them to this 2D histogram.

We only add a (X,Y) sample to this 2D histogram if z_min < Z < z_max.

Note: I'm not sure whether or not the Confidence Interval is correctly computed for the case that Prob(z_min < Z < z_max) != 1

By JZ, 15/11/2004, modified 23/2/2005 and adapted from code by Ivan T. Lima Jr. and Aurenice O. Lima (7/28/02)

References AddSample(), Clear(), Get_pdfConfidence(), GetIndex(), HistInFileName, Histogram2D(), Linear2dB(), LogFile, NumBinsX, NumBinsY, ObtainConfidence(), pdfConfidence, PreviouslyComputed_pdfConfidenceFlag, sgn(), sq(), and sqrt().

void Histogram2D::MultipleHeuristicBalanceImport	(	string	InFileName,
		int	Convert2dB_Scale
	)

Use this function for merging 2D histograms named in InFileName.

References AddSample(), Clear(), Get_pdfConfidence(), GetIndex(), HistInFileName, Histogram2D(), Linear2dB(), LogFile, NumBinsX, NumBinsY, ObtainConfidence(), pdfConfidence, PreviouslyComputed_pdfConfidenceFlag, sgn(), sq(), and sqrt().

void Histogram2D::MultipleSelectMaxRelNumHitsImport	(	string	InFileName,
		int	Convert2dB_Scale
	)

References AddSample(), Clear(), GetBin(), GetQtSamples(), GetQtSamplesBin(), HistInFileName, Histogram2D(), Linear2dB(), LogFile, NumBinsX, NumBinsY, NumSamplesInRange, NumSamplesTotal, SetBin(), and sgn().

void Histogram2D::NormalizePDF ( void )

Normalize so integrates to 1. Useful after numerical operations are performed on PDF

References ComputeIntegralPDF(), GetPDF(), NumBinsX, NumBinsY, and SetPDF().

Referenced by ComputePDFfromHistogram(), and SetPDFToZeroBelowThreshold().

void Histogram2D::ObtainConfidence ( void )

Computes width of confidence interval for each bin and stores result in 2D array pdfConfidence

References Get_pdfConfidence(), GetIndex(), NumBinsX, NumBinsY, pdfConfidence, and PreviouslyComputed_pdfConfidenceFlag.

Referenced by MergeMultipleConditionalHeuristicBalance(), and MultipleHeuristicBalanceImport().

Histogram2D & Histogram2D::operator= ( Histogram2D & RHSHist )

Assignment operator.

References BinSquare, delta_x, delta_y, GetBin(), GetPDF(), HistInFileName, MeanX, MeanY, NumBinsX, NumBinsY, NumSamplesInRange, NumSamplesTotal, NumSamplesWithOutOfRangeX, NumSamplesWithOutOfRangeY, pdfConfidence, PreviouslyComputed_pdfConfidenceFlag, qtSamplesBin, SamplesMaxX, SamplesMaxY, SamplesMinX, SamplesMinY, SetBin(), SetPDF(), x_max, x_min, x_window_size, y_max, y_min, and y_window_size.

void Histogram2D::RotatePDF	(	double	CenterX,
		double	CenterY,
		double	Angle,
		double *	OutData
	)

Rotates PDF about (CenterX,CenterY) throuhg angle Angle and stores result in OutData.

OutData should be the 1D data array double * PDF from another Histogram2D object. BilinearInterpolationPDF() is used to get a value at ecah rotated point

References BilinearInterpolationPDF(), delta_x, delta_y, NumBinsX, NumBinsY, x_min, and y_min.