This class contains methods used to format results of GOF test statistics, or to apply a series of tests simultaneously and format the results. More...

Plotting distribution functions
static final int	GNUPLOT = 0
	Data file format used for plotting functions with Gnuplot.

static final int	MATHEMATICA = 1
	Data file format used for creating graphics with Mathematica.

static int	graphSoft = GNUPLOT
	Environment variable that selects the type of software to be used for plotting the graphs of functions. More...

static String	drawCdf (ContinuousDistribution dist, double a, double b, int m, String desc)
	Formats data to plot the graph of the distribution function $F$ over the interval $[a,b]$, and returns the result as a String. More...

static String	drawDensity (ContinuousDistribution dist, double a, double b, int m, String desc)
	Formats data to plot the graph of the density $f(x)$ over the interval $[a,b]$, and returns the result as a String. More...

static String	graphDistUnif (DoubleArrayList data, String desc)
	Formats data to plot the empirical distribution of $U_{(1)},…,U_{(N)}$, which are assumed to be in `data[0...N-1]`, and to compare it with the uniform distribution. More...

Computing and printing $p$-values for EDF test statistics
static double	EPSILONP = 1.0E-15
	Environment variable used in formatp0 to determine which $p$-values are too close to 0 or 1 to be printed explicitly. More...

static double	SUSPECTP = 0.01
	Environment variable used in formatp1 to determine which $p$-values should be marked as suspect when printing test results. More...

static String	formatp0 (double p)
	Returns the $p$-value $p$ of a test, in the format "@f$1-p@f$" if $p$ is close to 1, and $p$ otherwise. More...

static String	formatp1 (double p)
	Returns the string "<tt>p-value of test : </tt>", then calls formatp0 to print $p$, and adds the marker "<tt>****</tt>" if $p$ is considered suspect (uses the environment variable `SUSPECTP` for this). More...

static String	formatp2 (double x, double p)
	Returns `x` on a single line, then go to the next line and calls formatp1. More...

static String	formatp3 (String testName, double x, double p)
	Formats the test statistic `x` for a test named `testName` with $p$-value `p`. More...

static String	formatChi2 (int k, int d, double chi2)
	Computes the $p$-value of the chi-square statistic `chi2` for a test with `k` intervals. More...

static String	formatKS (int n, double dp, double dm, double d)
	Computes the $p$-values of the three Kolmogorov-Smirnov statistics $D_N^+$, $D_N^-$, and $D_N$, whose values are in `dp, dm, d`, respectively, assuming a sample of size `n`. More...

static String	formatKS (DoubleArrayList data, ContinuousDistribution dist)
	Computes the KS test statistics to compare the empirical distribution of the observations in `data` with the theoretical distribution `dist` and formats the results. More...

static String	formatKSJumpOne (int n, double a, double dp)
	Similar to formatKS(int,double,double,double), but for the KS statistic $D_N^+(a)$ defined in ( KSPlusJumpOne ). More...

static String	formatKSJumpOne (DoubleArrayList data, ContinuousDistribution dist, double a)
	Similar to formatKS(DoubleArrayList,ContinuousDistribution), but for $D_N^+(a)$ defined in ( KSPlusJumpOne ). More...

Applying several tests at once and printing results
Higher-level tools for applying several EDF goodness-of-fit tests simultaneously are offered here. The environment variable `activeTests` specifies which tests in this list are to be performed when asking for several simultaneous tests via the functions `activeTests`, `formatActiveTests`, etc.
static final int	KSP = 0
	Kolmogorov-Smirnov+ test.

static final int	KSM = 1
	Kolmogorov-Smirnov $-$ test.

static final int	KS = 2
	Kolmogorov-Smirnov test.

static final int	AD = 3
	Anderson-Darling test.

static final int	CM = 4
	Cramér-von Mises test.

static final int	WG = 5
	Watson G test.

static final int	WU = 6
	Watson U test.

static final int	MEAN = 7
	Mean.

static final int	COR = 8
	Correlation.

static final int	NTESTTYPES = 9
	Total number of test types.

static final String []	TESTNAMES
	Name of each `testType` test. More...

static boolean []	activeTests = null
	The set of EDF tests that are to be performed when calling the methods activeTests, formatActiveTests, etc. More...

static void	tests (DoubleArrayList sortedData, double[] sVal)
	Computes all EDF test statistics enumerated above (except `COR`) to compare the empirical distribution of $U_{(0)},…,U_{(N-1)}$ with the uniform distribution, assuming that these sorted observations are in `sortedData`. More...

static void	tests (DoubleArrayList data, ContinuousDistribution dist, double[] sVal)
	The observations $V$ are in `data`, not necessarily sorted, and their empirical distribution is compared with the continuous distribution `dist`. More...

static void	activeTests (DoubleArrayList sortedData, double[] sVal, double[] pVal)
	Computes the EDF test statistics by calling #tests(DoubleArrayList,double[]), then computes the $p$-values of those that currently belong to `activeTests`, and return these quantities in `sVal` and `pVal`, respectively. More...

static void	activeTests (DoubleArrayList data, ContinuousDistribution dist, double[] sVal, double[] pVal)
	The observations are in `data`, not necessarily sorted, and we want to compare their empirical distribution with the distribution `dist`. More...

static String	formatActiveTests (int n, double[] sVal, double[] pVal)
	Gets the $p$-values of the active EDF test statistics, which are in `activeTests`. More...

static String	iterSpacingsTests (DoubleArrayList sortedData, int k, boolean printval, boolean graph, PrintWriter f)
	Repeats the following `k` times: Applies the GofStat.iterateSpacings transformation to the $U_{(0)},…,U_{(N-1)}$, assuming that these observations are in `sortedData`, then computes the EDF test statistics and calls #activeTests(DoubleArrayList,double[],double[]) after each transformation. More...

static String	iterPowRatioTests (DoubleArrayList sortedData, int k, boolean printval, boolean graph, PrintWriter f)
	Similar to iterSpacingsTests, but with the GofStat.powerRatios transformation. More...

	[static initializer]

Detailed Description

This class contains methods used to format results of GOF test statistics, or to apply a series of tests simultaneously and format the results.

It is in fact a translation from C to Java of a set of functions that were specially written for the implementation of TestU01, a software package for testing uniform random number generators [133] .

Strictly speaking, applying several tests simultaneously makes the $p$-values "invalid" in the sense that the probability of having at least one $p$-value less than 0.01, say, is larger than 0.01. One must therefore be careful with the interpretation of these $p$-values (one could use, e.g., the Bonferroni inequality [118] ). Applying simultaneous tests is convenient in some situations, such as in screening experiments for detecting statistical deficiencies in random number generators. In that context, rejection of the null hypothesis typically occurs with extremely small $p$-values (e.g., less than $10^{-15}$), and the interpretation is quite obvious in this case.

The class also provides tools to plot an empirical or theoretical distribution function, by creating a data file that contains a graphic plot in a format compatible with the software specified by the environment variable graphSoft. NOTE: see also the more recent package umontreal.ssj.charts.

Note: This class uses the Colt library.

Member Function Documentation

◆ activeTests() [1/2]

static void activeTests	(	DoubleArrayList	sortedData,
		double []	sVal,
		double []	pVal
	)

static

Computes the EDF test statistics by calling #tests(DoubleArrayList,double[]), then computes the $p$-values of those that currently belong to activeTests, and return these quantities in sVal and pVal, respectively.

Assumes that $U_{(0)},…,U_{(N-1)}$ are in sortedData and that we want to compare their empirical distribution with the uniform distribution. If $N = 1$, only puts $1 - {}$sortedData.get (0) in sVal[KSP], pVal[KSP], and pVal[MEAN].

Parameters

sortedData	array of sorted observations
sVal	array that will be filled with the results of the tests
pVal	array that will be filled with the $p$-values

◆ activeTests() [2/2]

static void activeTests	(	DoubleArrayList	data,
		ContinuousDistribution	dist,
		double []	sVal,
		double []	pVal
	)

static

The observations are in data, not necessarily sorted, and we want to compare their empirical distribution with the distribution dist.

If $N = 1$, only puts data.get(0) in sVal[MEAN], and $1 - {}$dist.cdf (data.get (0)) in sVal[KSP], pVal[KSP], and pVal[MEAN].

Parameters

data	array of observations to test
dist	assumed distribution of the observations
sVal	array that will be filled with the results of the tests
pVal	array that will be filled with the $p$-values

◆ drawCdf()

static String drawCdf	(	ContinuousDistribution	dist,
		double	a,
		double	b,
		int	m,
		String	desc
	)

static

Formats data to plot the graph of the distribution function $F$ over the interval $[a,b]$, and returns the result as a String.

The method dist.cdf(x) returns the value of $F$ at $x$. The String desc gives a short caption for the graphic plot. The method computes the $m+1$ points $(x_i, F (x_i))$, where $x_i = a + i (b-a)/m$ for $i=0,1,…,m$, and formats these points into a String in a format suitable for the software specified by graphSoft. NOTE: see also the more recent class umontreal.ssj.charts.ContinuousDistChart.

Parameters

dist	continuous distribution function to plot
a	lower bound of the interval to plot
b	upper bound of the interval to plot
m	number of points in the plot minus one
desc	short caption describing the plot

Returns: a string representation of the plot data

◆ drawDensity()

static String drawDensity	(	ContinuousDistribution	dist,
		double	a,
		double	b,
		int	m,
		String	desc
	)

static

Formats data to plot the graph of the density $f(x)$ over the interval $[a,b]$, and returns the result as a String.

The method dist.density(x) returns the value of $f(x)$ at $x$. The String desc gives a short caption for the graphic plot. The method computes the $m+1$ points $(x_i, f(x_i))$, where $x_i = a + i (b-a)/m$ for $i=0,1,…,m$, and formats these points into a String in a format suitable for the software specified by graphSoft. NOTE: see also the more recent class umontreal.ssj.charts.ContinuousDistChart.

Parameters

dist	continuous density function to plot
a	lower bound of the interval to plot
b	upper bound of the interval to plot
m	number of points in the plot minus one
desc	short caption describing the plot

Returns: a string representation of the plot data

◆ formatActiveTests()

static String formatActiveTests	(	int	n,
		double []	sVal,
		double []	pVal
	)

static

Gets the $p$-values of the active EDF test statistics, which are in activeTests.

It is assumed that the values of these statistics and their $p$-values are already computed, in sVal and pVal, and that the sample size is n. These statistics and $p$-values are formated using formatp2 for each one. If n=1, prints only pVal[KSP] using formatp1.

Parameters

n	sample size
sVal	array containing the results of the tests
pVal	array containing the $p$-values

Returns: the results formated as a string

◆ formatChi2()

static String formatChi2	(	int	k,
		int	d,
		double	chi2
	)

static

Computes the $p$-value of the chi-square statistic chi2 for a test with k intervals.

Uses $d$ decimal digits of precision in the calculations. The result of the test is returned as a string. The $p$-value is computed using GofStat.pDisc.

Parameters

k	number of subintervals for the chi-square test
chi2	chi-square statistic

Returns: the string representation of the test result and $p$-value

◆ formatKS() [1/2]

static String formatKS	(	int	n,
		double	dp,
		double	dm,
		double	d
	)

static

Computes the $p$-values of the three Kolmogorov-Smirnov statistics $D_N^+$, $D_N^-$, and $D_N$, whose values are in dp, dm, d, respectively, assuming a sample of size n.

Then formats these statistics and their $p$-values using formatp2 for each one.

Parameters

n	sample size
dp	value of the $D_N^+$ statistic
dm	value of the $D_N^-$ statistic
d	value of the $D_N$ statistic

Returns: the string representation of the Kolmogorov-Smirnov statistics and their p-values

◆ formatKS() [2/2]

static String formatKS	(	DoubleArrayList	data,
		ContinuousDistribution	dist
	)

static

Computes the KS test statistics to compare the empirical distribution of the observations in data with the theoretical distribution dist and formats the results.

Parameters

data	array of observations to be tested
dist	assumed distribution of the observations

Returns: the string representation of the Kolmogorov-Smirnov statistics and their p-values

◆ formatKSJumpOne() [1/2]

static String formatKSJumpOne	(	int	n,
		double	a,
		double	dp
	)

static

Similar to formatKS(int,double,double,double), but for the KS statistic $D_N^+(a)$ defined in ( KSPlusJumpOne ).

Writes a header, computes the $p$-value and calls formatp2.

Parameters

n	sample size
a	size of the jump
dp	value of $D_N^+(a)$

Returns: the string representation of the Kolmogorov-Smirnov statistic and its p-value

◆ formatKSJumpOne() [2/2]

static String formatKSJumpOne	(	DoubleArrayList	data,
		ContinuousDistribution	dist,
		double	a
	)

static

Similar to formatKS(DoubleArrayList,ContinuousDistribution), but for $D_N^+(a)$ defined in ( KSPlusJumpOne ).

Parameters

data	array of observations to be tested
dist	assumed distribution of the data
a	size of the jump

Returns: string representation of the Kolmogorov-Smirnov statistic and its p-value

◆ formatp0()

static String formatp0 ( double p )

static

Returns the $p$-value $p$ of a test, in the format "@f$1-p@f$" if $p$ is close to 1, and $p$ otherwise.

Uses the environment variable EPSILONP and replaces $p$ by $\epsilon$ when it is too small.

Parameters

p	the $p$-value to be formated

Returns: the string representation of the $p$-value

◆ formatp1()

static String formatp1 ( double p )

static

Returns the string "<tt>p-value of test : </tt>", then calls formatp0 to print $p$, and adds the marker "<tt>****</tt>" if $p$ is considered suspect (uses the environment variable SUSPECTP for this).

Parameters

p	the $p$-value to be formated

Returns: the string representation of the p-value of test

◆ formatp2()

static String formatp2	(	double	x,
		double	p
	)

static

Returns x on a single line, then go to the next line and calls formatp1.

Parameters

x	value of the statistic for which the p-value is formated
p	the $p$-value to be formated

Returns: the string representation of the p-value of test

◆ formatp3()

static String formatp3	(	String	testName,
		double	x,
		double	p
	)

static

Formats the test statistic x for a test named testName with $p$-value p.

The first line of the returned string contains the name of the test and the statistic whereas the second line contains its p-value. The formated values of x and p are aligned.

Parameters

testName	name of the test that was performed
x	value of the test statistic
p	$p$-value of the test

Returns: the string representation of the test result

◆ graphDistUnif()

static String graphDistUnif	(	DoubleArrayList	data,
		String	desc
	)

static

Formats data to plot the empirical distribution of $U_{(1)},…,U_{(N)}$, which are assumed to be in data[0...N-1], and to compare it with the uniform distribution.

The $U_{(i)}$ must be sorted. The two endpoints $(0, 0)$ and $(1, 1)$ are always included in the plot. The string desc gives a short caption for the graphic plot. The data is printed in a format suitable for the software specified by graphSoft. NOTE: see also the more recent class umontreal.ssj.charts.EmpiricalChart.

Parameters

data	array of observations to plot
desc	short caption describing the plot

Returns: a string representation of the plot data

◆ iterPowRatioTests()

static String iterPowRatioTests	(	DoubleArrayList	sortedData,
		int	k,
		boolean	printval,
		boolean	graph,
		PrintWriter	f
	)

static

Similar to iterSpacingsTests, but with the GofStat.powerRatios transformation.

Parameters

sortedData	array containing the sorted observations
k	number of times the tests are applied
printval	if `true`, stores all the values of the observations at each iteration
graph	if `true`, the distribution of the $U_i$ will be plotted after each iteration
f	stream where the plots are written to

Returns: a string representation of the test results

◆ iterSpacingsTests()

static String iterSpacingsTests	(	DoubleArrayList	sortedData,
		int	k,
		boolean	printval,
		boolean	graph,
		PrintWriter	f
	)

static

Repeats the following k times: Applies the GofStat.iterateSpacings transformation to the $U_{(0)},…,U_{(N-1)}$, assuming that these observations are in sortedData, then computes the EDF test statistics and calls #activeTests(DoubleArrayList,double[],double[]) after each transformation.

The function returns the original array sortedData (the transformations are applied on a copy of sortedData). If printval = true, stores all the values into the returned String after each iteration. If graph = true, calls graphDistUnif after each iteration to print to stream f the data for plotting the distribution function of the $U_i$.

Parameters

sortedData	array containing the sorted observations
k	number of times the tests are applied
printval	if `true`, stores all the values of the observations at each iteration
graph	if `true`, the distribution of the $U_i$ will be plotted after each iteration
f	stream where the plots are written to

Returns: a string representation of the test results

◆ tests() [1/2]

static void tests	(	DoubleArrayList	sortedData,
		double []	sVal
	)

static

Computes all EDF test statistics enumerated above (except COR) to compare the empirical distribution of $U_{(0)},…,U_{(N-1)}$ with the uniform distribution, assuming that these sorted observations are in sortedData.

If $N > 1$, returns sVal with the values of the KS statistics $D_N^+$, $D_N^-$ and $D_N$, of the Cramér-von Mises statistic $W_N^2$, Watson’s $G_N$ and $U_N^2$, Anderson-Darling’s $A_N^2$, and the average of the $U_i$’s, respectively. If $N = 1$, only puts $1 - {}$sortedData.get (0) in sVal[KSP]. Calling this method is more efficient than computing these statistics separately by calling the corresponding methods in GofStat.

Parameters

sortedData	array of sorted observations
sVal	array that will be filled with the results of the tests

◆ tests() [2/2]

static void tests	(	DoubleArrayList	data,
		ContinuousDistribution	dist,
		double []	sVal
	)

static

The observations $V$ are in data, not necessarily sorted, and their empirical distribution is compared with the continuous distribution dist.

If $N = 1$, only puts data.get (0) in sVal[MEAN], and $1 - {}$dist.cdf (data.get (0)) in sVal[KSP].

Parameters

data	array of observations to test
dist	assumed distribution of the observations
sVal	array that will be filled with the results of the tests

Member Data Documentation

◆ activeTests

boolean [] activeTests = null

static

The set of EDF tests that are to be performed when calling the methods activeTests, formatActiveTests, etc.

By default, this set contains KSP, KSM, and AD. Note: MEAN and COR are always excluded from this set of active tests. The valid indices for this array are KSP, KSM, KS, AD, CM, WG, WU, MEAN, and COR.

◆ EPSILONP

double EPSILONP = 1.0E-15

static

Environment variable used in formatp0 to determine which $p$-values are too close to 0 or 1 to be printed explicitly.

If EPSILONP $= \epsilon$, then any $p$-value less than $\epsilon$ or larger than $1-\epsilon$ is not written explicitly; the program simply writes "<tt>eps</tt>" or "<tt>1-eps</tt>". The default value is $10^{-15}$.

◆ graphSoft

int graphSoft = GNUPLOT

static

Environment variable that selects the type of software to be used for plotting the graphs of functions.

The data files produced by #graphFunc and graphDistUnif will be in a format suitable for this selected software. The default value is GNUPLOT. To display a graphic in file f using gnuplot, for example, one can use the command "<tt>plot f with steps, x with lines</tt>" in gnuplot. graphSoft can take the values GNUPLOT or MATHEMATICA.

◆ SUSPECTP

double SUSPECTP = 0.01

static

Environment variable used in formatp1 to determine which $p$-values should be marked as suspect when printing test results.

If SUSPECTP $= \alpha$, then any $p$-value less than $\alpha$ or larger than $1-\alpha$ is considered suspect and is "singled out" by formatp1. The default value is 0.01.

◆ TESTNAMES

final String [] TESTNAMES

static

Initial value:

= {
    "KolmogorovSmirnovPlus", "KolmogorovSmirnovMinus",
    "KolmogorovSmirnov", "Anderson-Darling",
    "CramerVon-Mises", "Watson G", "Watson U",
    "Mean", "Correlation"
   }

Name of each testType test.

Could be used for printing the test results, for example.

The documentation for this class was generated from the following file:

GofFormat.java

Computing and printing \(p\)-values for EDF test statistics
static double	EPSILONP = 1.0E-15
	Environment variable used in formatp0 to determine which \(p\)-values are too close to 0 or 1 to be printed explicitly. More...

static double	SUSPECTP = 0.01
	Environment variable used in formatp1 to determine which \(p\)-values should be marked as suspect when printing test results. More...

static String	formatp0 (double p)
	Returns the \(p\)-value \(p\) of a test, in the format "@f$1-p@f$" if \(p\) is close to 1, and \(p\) otherwise. More...

static String	formatp1 (double p)
	Returns the string "<tt>p-value of test : </tt>", then calls formatp0 to print \(p\), and adds the marker "<tt>****</tt>" if \(p\) is considered suspect (uses the environment variable `SUSPECTP` for this). More...

static String	formatp2 (double x, double p)
	Returns `x` on a single line, then go to the next line and calls formatp1. More...

static String	formatp3 (String testName, double x, double p)
	Formats the test statistic `x` for a test named `testName` with \(p\)-value `p`. More...

static String	formatChi2 (int k, int d, double chi2)
	Computes the \(p\)-value of the chi-square statistic `chi2` for a test with `k` intervals. More...

static String	formatKS (int n, double dp, double dm, double d)
	Computes the \(p\)-values of the three Kolmogorov-Smirnov statistics \(D_N^+\), \(D_N^-\), and \(D_N\), whose values are in `dp, dm, d`, respectively, assuming a sample of size `n`. More...

static String	formatKS (DoubleArrayList data, ContinuousDistribution dist)
	Computes the KS test statistics to compare the empirical distribution of the observations in `data` with the theoretical distribution `dist` and formats the results. More...

static String	formatKSJumpOne (int n, double a, double dp)
	Similar to formatKS(int,double,double,double), but for the KS statistic \(D_N^+(a)\) defined in ( KSPlusJumpOne ). More...

static String	formatKSJumpOne (DoubleArrayList data, ContinuousDistribution dist, double a)
	Similar to formatKS(DoubleArrayList,ContinuousDistribution), but for \(D_N^+(a)\) defined in ( KSPlusJumpOne ). More...

n	sample size
dp	value of the \(D_N^+\) statistic
dm	value of the \(D_N^-\) statistic
d	value of the \(D_N\) statistic

x	value of the statistic for which the p-value is formated
p	the \(p\)-value to be formated

Plotting distribution functions

Computing and printing \(p\)-values for EDF test statistics

Applying several tests at once and printing results

Detailed Description

Member Function Documentation

◆ activeTests() [1/2]

◆ activeTests() [2/2]

◆ drawCdf()

◆ drawDensity()

◆ formatActiveTests()

◆ formatChi2()

◆ formatKS() [1/2]

◆ formatKS() [2/2]

◆ formatKSJumpOne() [1/2]

◆ formatKSJumpOne() [2/2]

◆ formatp0()

◆ formatp1()

◆ formatp2()

◆ formatp3()

◆ graphDistUnif()

◆ iterPowRatioTests()

◆ iterSpacingsTests()

◆ tests() [1/2]

◆ tests() [2/2]

Member Data Documentation

◆ activeTests

◆ EPSILONP

◆ graphSoft

◆ SUSPECTP

◆ TESTNAMES