fr.cnes.sirius.patrius.math.random

Class EmpiricalDistribution

java.lang.Object

fr.cnes.sirius.patrius.math.distribution.AbstractRealDistribution

fr.cnes.sirius.patrius.math.random.EmpiricalDistribution

All Implemented Interfaces:

RealDistribution, Serializable

public class EmpiricalDistribution
extends AbstractRealDistribution

Represents an empirical probability distribution -- a probability distribution derived from observed data without making any assumptions about the functional form of the population distribution that the data come from.

An EmpiricalDistribution maintains data structures, called distribution digests, that describe empirical distributions and support the following operations:

• loading the distribution from a file of observed data values
• dividing the input data into "bin ranges" and reporting bin frequency counts (data for histogram)
• reporting univariate statistics describing the full set of data values as well as the observations within each bin
• generating random values from the distribution

Applications can use EmpiricalDistribution to build grouped frequency histograms representing the input data or to generate random values "like" those in the input file -- i.e., the values generated will follow the distribution of the values in the file.

The implementation uses what amounts to the Variable Kernel Method with Gaussian smoothing:

Digesting the input file

1. Pass the file once to compute min and max.
2. Divide the range from min-max into binCount "bins."
3. Pass the data file again, computing bin counts and univariate statistics (mean, std dev.) for each of the bins
4. Divide the interval (0,1) into subintervals associated with the bins, with the length of a bin's subinterval proportional to its count.

Generating random values from the distribution

1. Generate a uniformly distributed value in (0,1)
2. Select the subinterval to which the value belongs.
3. Generate a random Gaussian value with mean = mean of the associated bin and std dev = std dev of associated bin.

EmpiricalDistribution implements the RealDistribution interface as follows. Given x within the range of values in the dataset, let B be the bin containing x and let K be the within-bin kernel for B. Let P(B-) be the sum of the probabilities of the bins below B and let K(B) be the mass of B under K (i.e., the integral of the kernel density over B). Then set P(X < x) = P(B-) + P(B) * K(x) / K(B) where K(x) is the kernel distribution evaluated at x. This results in a cdf that matches the grouped frequency distribution at the bin endpoints and interpolates within bins using within-bin kernels.

USAGE NOTES:

• The binCount is set by default to 1000. A good rule of thumb is to set the bin count to approximately the length of the input file divided by 10.
• The input file must be a plain text file containing one valid numeric entry per line.

Version:

$Id: EmpiricalDistribution.java 18108 2017-10-04 06:45:27Z bignon $

See Also:

Serialized Form

Field Summary

Fields

Modifier and Type	Field and Description
static int	DEFAULT_BIN_COUNT Default bin count

Fields inherited from class fr.cnes.sirius.patrius.math.distribution.AbstractRealDistribution

random, SOLVER_DEFAULT_ABSOLUTE_ACCURACY

Constructor Summary

Constructors

Constructor and Description
EmpiricalDistribution() Creates a new EmpiricalDistribution with the default bin count.
EmpiricalDistribution(int binCountIn) Creates a new EmpiricalDistribution with the specified bin count.
EmpiricalDistribution(int binCountIn, RandomGenerator generator) Creates a new EmpiricalDistribution with the specified bin count using the provided RandomGenerator as the source of random data.
EmpiricalDistribution(RandomGenerator generator) Creates a new EmpiricalDistribution with default bin count using the provided RandomGenerator as the source of random data.

Method Summary

Methods

Modifier and Type	Method and Description
double	cumulativeProbability(double x) For a random variable X whose values are distributed according to this distribution, this method returns P(X <= x).
double	density(double x) Returns the probability density function (PDF) of this distribution evaluated at the specified point x.
int	getBinCount() Returns the number of bins.
List<SummaryStatistics>	getBinStats() Returns a List of SummaryStatistics instances containing statistics describing the values in each of the bins.
double[]	getGeneratorUpperBounds() Returns a fresh copy of the array of upper bounds of the subintervals of [0,1] used in generating data from the empirical distribution.
double	getNextValue() Generates a random value from this distribution.
double	getNumericalMean() Use this method to get the numerical value of the mean of this distribution.
double	getNumericalVariance() Use this method to get the numerical value of the variance of this distribution.
StatisticalSummary	getSampleStats() Returns a StatisticalSummary describing this distribution.
double	getSupportLowerBound() Access the lower bound of the support.
double	getSupportUpperBound() Access the upper bound of the support.
double[]	getUpperBounds() Returns a fresh copy of the array of upper bounds for the bins.
double	inverseCumulativeProbability(double p) Computes the quantile function of this distribution.
boolean	isLoaded() Property indicating whether or not the distribution has been loaded.
boolean	isSupportConnected() Use this method to get information about whether the support is connected, i.e.
boolean	isSupportLowerBoundInclusive() Returns true if support contains lower bound.
boolean	isSupportUpperBoundInclusive() Returns true if support contains upper bound.
void	load(double[] in) Computes the empirical distribution from the provided array of numbers.
void	load(File file) Computes the empirical distribution from the input file.
void	load(URL url) Computes the empirical distribution using data read from a URL.
double	probability(double x) For a random variable X whose values are distributed according to this distribution, this method returns P(X = x).
void	reSeed(long seed) Reseeds the random number generator used by getNextValue().
void	reseedRandomGenerator(long seed) Reseed the random generator used to generate samples.
double	sample() Generate a random value sampled from this distribution.

Methods inherited from class fr.cnes.sirius.patrius.math.distribution.AbstractRealDistribution

getSolverAbsoluteAccuracy, probability, sample

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Field Detail

DEFAULT_BIN_COUNT

public static final int DEFAULT_BIN_COUNT

Default bin count

See Also:

Constant Field Values

Constructor Detail

EmpiricalDistribution

public EmpiricalDistribution()

Creates a new EmpiricalDistribution with the default bin count.

EmpiricalDistribution

public EmpiricalDistribution(int binCountIn)

Creates a new EmpiricalDistribution with the specified bin count.

Parameters:

binCountIn - number of bins

EmpiricalDistribution

public EmpiricalDistribution(int binCountIn,
                     RandomGenerator generator)

Creates a new EmpiricalDistribution with the specified bin count using the provided RandomGenerator as the source of random data.

Parameters:

binCountIn - number of bins

generator - random data generator (may be null, resulting in default JDK generator)

Since:

3.0

EmpiricalDistribution

public EmpiricalDistribution(RandomGenerator generator)

Creates a new EmpiricalDistribution with default bin count using the provided RandomGenerator as the source of random data.

Parameters:

generator - random data generator (may be null, resulting in default JDK generator)

Since:

3.0

Method Detail

load

public void load(double[] in)

Computes the empirical distribution from the provided array of numbers.

Parameters:

in - the input data array

Throws:

NullArgumentException - if in is null

load

public void load(URL url)
          throws IOException

Computes the empirical distribution using data read from a URL.

The input file must be an ASCII text file containing one valid numeric entry per line.

Parameters:

url - url of the input file

Throws:

IOException - if an IO error occurs

NullArgumentException - if url is null

ZeroException - if URL contains no data

load

public void load(File file)
          throws IOException

Computes the empirical distribution from the input file.

The input file must be an ASCII text file containing one valid numeric entry per line.

Parameters:

file - the input file

Throws:

IOException - if an IO error occurs

NullArgumentException - if file is null

getNextValue

public double getNextValue()

Generates a random value from this distribution. Preconditions:

• the distribution must be loaded before invoking this method

Returns:

the random value.

Throws:

MathIllegalStateException - if the distribution has not been loaded

getSampleStats

public StatisticalSummary getSampleStats()

Returns a StatisticalSummary describing this distribution. Preconditions:

• the distribution must be loaded before invoking this method

Returns:

the sample statistics

Throws:

IllegalStateException - if the distribution has not been loaded

getBinCount

public int getBinCount()

Returns the number of bins.

Returns:

the number of bins.

getBinStats

public List<SummaryStatistics> getBinStats()

Returns a List of SummaryStatistics instances containing statistics describing the values in each of the bins. The list is indexed on the bin number.

Returns:

List of bin statistics.

getUpperBounds

public double[] getUpperBounds()

Returns a fresh copy of the array of upper bounds for the bins. Bins are:
[min,upperBounds[0]],(upperBounds[0],upperBounds[1]],..., (upperBounds[binCount-2], upperBounds[binCount-1] = max].

Note: In versions 1.0-2.0 of commons-math, this method incorrectly returned the array of probability generator upper bounds now returned by getGeneratorUpperBounds().

Returns:

array of bin upper bounds

Since:

2.1

getGeneratorUpperBounds

public double[] getGeneratorUpperBounds()

Returns a fresh copy of the array of upper bounds of the subintervals of [0,1] used in generating data from the empirical distribution. Subintervals correspond to bins with lengths proportional to bin counts.

In versions 1.0-2.0 of commons-math, this array was (incorrectly) returned by getUpperBounds().

Returns:

array of upper bounds of subintervals used in data generation

Since:

2.1

isLoaded

public boolean isLoaded()

Property indicating whether or not the distribution has been loaded.

Returns:

true if the distribution has been loaded

reSeed

public void reSeed(long seed)

Reseeds the random number generator used by getNextValue().

Parameters:

seed - random generator seed

Since:

3.0

probability

public double probability(double x)

For a random variable X whose values are distributed according to this distribution, this method returns P(X = x). In other words, this method represents the probability mass function (PMF) for the distribution.

Specified by:

probability in interface RealDistribution

Overrides:

probability in class AbstractRealDistribution

Parameters:

x - the point at which the PMF is evaluated

Returns:

zero.

Since:

3.1

density

public double density(double x)

Returns the probability density function (PDF) of this distribution evaluated at the specified point x. In general, the PDF is the derivative of the CDF. If the derivative does not exist at x, then an appropriate replacement should be returned, e.g. Double.POSITIVE_INFINITY, Double.NaN, or the limit inferior or limit superior of the difference quotient.

Returns the kernel density normalized so that its integral over each bin equals the bin mass.

Algorithm description:

1. Find the bin B that x belongs to.
2. Compute K(B) = the mass of B with respect to the within-bin kernel (i.e., the integral of the kernel density over B).
3. Return k(x) * P(B) / K(B), where k is the within-bin kernel density and P(B) is the mass of B.

Parameters:

x - the point at which the PDF is evaluated

Returns:

the value of the probability density function at point x

Since:

3.1

cumulativeProbability

public double cumulativeProbability(double x)

For a random variable X whose values are distributed according to this distribution, this method returns P(X <= x). In other words, this method represents the (cumulative) distribution function (CDF) for this distribution.

Algorithm description:

1. Find the bin B that x belongs to.
2. Compute P(B) = the mass of B and P(B-) = the combined mass of the bins below B.
3. Compute K(B) = the probability mass of B with respect to the within-bin kernel and K(B-) = the kernel distribution evaluated at the lower endpoint of B
4. Return P(B-) + P(B) * [K(x) - K(B-)] / K(B) where K(x) is the within-bin kernel distribution function evaluated at x.

Parameters:

x - the point at which the CDF is evaluated

Returns:

the probability that a random variable with this distribution takes a value less than or equal to x

Since:

3.1

inverseCumulativeProbability

public double inverseCumulativeProbability(double p)

Computes the quantile function of this distribution. For a random variable X distributed according to this distribution, the returned value is

• inf{x in R | P(X<=x) >= p} for 0 < p <= 1,
• inf{x in R | P(X<=x) > 0} for p = 0.

The default implementation returns

• RealDistribution.getSupportLowerBound() for p = 0,
• RealDistribution.getSupportUpperBound() for p = 1.

Algorithm description:

1. Find the smallest i such that the sum of the masses of the bins through i is at least p.
2. Let K be the within-bin kernel distribution for bin i.
   Let K(B) be the mass of B under K.
   Let K(B-) be K evaluated at the lower endpoint of B (the combined mass of the bins below B under K).
   Let P(B) be the probability of bin i.
   Let P(B-) be the sum of the bin masses below bin i.
   Let pCrit = p - P(B-)
   
3. Return the inverse of K evaluated at
   K(B-) + pCrit * K(B) / P(B)

Specified by:

inverseCumulativeProbability in interface RealDistribution

Overrides:

inverseCumulativeProbability in class AbstractRealDistribution

Parameters:

p - the cumulative probability

Returns:

the smallest p-quantile of this distribution (largest 0-quantile for p = 0)

Since:

3.1

getNumericalMean

public double getNumericalMean()

Use this method to get the numerical value of the mean of this distribution.

Returns:

the mean or Double.NaN if it is not defined

Since:

3.1

getNumericalVariance

public double getNumericalVariance()

Use this method to get the numerical value of the variance of this distribution.

Returns:

the variance (possibly Double.POSITIVE_INFINITY as for certain cases in TDistribution) or Double.NaN if it is not defined

Since:

3.1

getSupportLowerBound

public double getSupportLowerBound()

Access the lower bound of the support. This method must return the same value as inverseCumulativeProbability(0). In other words, this method must return

inf {x in R | P(X <= x) > 0}.

Returns:

lower bound of the support (might be Double.NEGATIVE_INFINITY)

Since:

3.1

getSupportUpperBound

public double getSupportUpperBound()

Access the upper bound of the support. This method must return the same value as inverseCumulativeProbability(1). In other words, this method must return

inf {x in R | P(X <= x) = 1}.

Returns:

upper bound of the support (might be Double.POSITIVE_INFINITY)

Since:

3.1

isSupportLowerBoundInclusive

public boolean isSupportLowerBoundInclusive()

Returns true if support contains lower bound.

Returns:

true

Since:

3.1

isSupportUpperBoundInclusive

public boolean isSupportUpperBoundInclusive()

Returns true if support contains upper bound.

Returns:

true

Since:

3.1

isSupportConnected

public boolean isSupportConnected()

Use this method to get information about whether the support is connected, i.e. whether all values between the lower and upper bound of the support are included in the support.

Returns:

whether the support is connected or not

Since:

3.1

sample

public double sample()

Generate a random value sampled from this distribution. The default implementation uses the inversion method.

Specified by:

sample in interface RealDistribution

Overrides:

sample in class AbstractRealDistribution

Returns:

a random value.

Since:

3.1

reseedRandomGenerator

public void reseedRandomGenerator(long seed)

Reseed the random generator used to generate samples.

Specified by:

reseedRandomGenerator in interface RealDistribution

Overrides:

reseedRandomGenerator in class AbstractRealDistribution

Parameters:

seed - the new seed

Since:

3.1