# Class EmpiricalDistribution

All Implemented Interfaces:
Serializable, RealDistribution

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:

• 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.
• ## Field Summary

Fields
Modifier and Type
Field
Description
static final int
DEFAULT_BIN_COUNT
Default bin count
protected final RandomDataGenerator
randomData
RandomDataGenerator instance to use in repeated calls to getNext()

### Fields inherited from class org.hipparchus.distribution.continuous.AbstractRealDistribution

DEFAULT_SOLVER_ABSOLUTE_ACCURACY
• ## Constructor Summary

Constructors
Constructor
Description
EmpiricalDistribution()
Creates a new EmpiricalDistribution with the default bin count.
EmpiricalDistribution(int binCount)
Creates a new EmpiricalDistribution with the specified bin count.
EmpiricalDistribution(int binCount, 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

Modifier and Type
Method
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<StreamingStatistics>
getBinStats()
Returns a List of StreamingStatistics 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.
protected RealDistribution
getKernel(StreamingStatistics bStats)
The within-bin smoothing kernel.
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.
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.
void
reSeed(long seed)
Reseeds the random number generator used by getNextValue().
void
reseedRandomGenerator(long seed)
Reseed the underlying PRNG.

### Methods inherited from class org.hipparchus.distribution.continuous.AbstractRealDistribution

getSolverAbsoluteAccuracy, logDensity, probability

### Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait
• ## Field Details

• ### DEFAULT_BIN_COUNT

public static final int DEFAULT_BIN_COUNT
Default bin count
• ### randomData

protected final RandomDataGenerator randomData
RandomDataGenerator instance to use in repeated calls to getNext()
• ## Constructor Details

• ### EmpiricalDistribution

public EmpiricalDistribution()
Creates a new EmpiricalDistribution with the default bin count.
• ### EmpiricalDistribution

public EmpiricalDistribution(int binCount)
Creates a new EmpiricalDistribution with the specified bin count.
Parameters:
binCount - number of bins. Must be strictly positive.
Throws:
MathIllegalArgumentException - if binCount <= 0.
• ### EmpiricalDistribution

public EmpiricalDistribution(int binCount, RandomGenerator generator)
Creates a new EmpiricalDistribution with the specified bin count using the provided RandomGenerator as the source of random data.
Parameters:
binCount - number of bins. Must be strictly positive.
generator - random data generator (may be null, resulting in default JDK generator)
Throws:
MathIllegalArgumentException - if binCount <= 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)
• ## Method Details

public void load(double[] in) throws NullArgumentException
Computes the empirical distribution from the provided array of numbers.
Parameters:
in - the input data array
Throws:
NullArgumentException - if in is null

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
MathIllegalArgumentException - if URL contains no data

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() throws MathIllegalStateException
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  getBinStats()
Returns a List of StreamingStatistics 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].

Returns:
array of bin upper bounds
• ### 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.

Preconditions:
• the distribution must be loaded before invoking this method
Returns:
array of upper bounds of subintervals used in data generation
Throws:
NullPointerException - unless a load method has been called beforehand.

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
• ### 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
• ### 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.

If K is a constant distribution, we return P(B-) + P(B) (counting the full mass of B).

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
• ### inverseCumulativeProbability

public double inverseCumulativeProbability(double p) throws MathIllegalArgumentException
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

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)
Throws:
MathIllegalArgumentException - if p < 0 or p > 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
• ### 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
• ### 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)
• ### 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)
• ### 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
• ### reseedRandomGenerator

public void reseedRandomGenerator(long seed)
Reseed the underlying PRNG.
Parameters:
seed - new seed value
• ### getKernel

protected RealDistribution getKernel(StreamingStatistics bStats)
The within-bin smoothing kernel. Returns a Gaussian distribution parameterized by bStats, unless the bin contains less than 2 observations, in which case a constant distribution is returned.
Parameters:
bStats - summary statistics for the bin
Returns:
within-bin kernel parameterized by bStats