EmpiricalDistribution
org.apache.commons.math3.random

Class EmpiricalDistribution

  • All Implemented Interfaces:
    Serializable, RealDistribution


    public class EmpiricalDistributionextends 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.

    See Also:
    Serialized Form
    • 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 binCount)
        Creates a new EmpiricalDistribution with the specified bin count.
        Parameters:
        binCount - number of bins
      • 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
        generator - random data generator (may be null, resulting in default JDK generator)
      • 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)
      • EmpiricalDistribution

        @Deprecatedpublic EmpiricalDistribution(int binCount,                                RandomDataImpl randomData)
        Deprecated. As of 3.1. Please use EmpiricalDistribution(int,RandomGenerator) instead.
        Creates a new EmpiricalDistribution with the specified bin count using the provided RandomDataImpl instance as the source of random data.
        Parameters:
        binCount - number of bins
        randomData - random data generator (may be null, resulting in default JDK generator)
      • EmpiricalDistribution

        @Deprecatedpublic EmpiricalDistribution(RandomDataImpl randomData)
        Deprecated. As of 3.1. Please use EmpiricalDistribution(RandomGenerator) instead.
        Creates a new EmpiricalDistribution with default bin count using the provided RandomDataImpl as the source of random data.
        Parameters:
        randomData - random data generator (may be null, resulting in default JDK generator)
    • Method Detail

      • load

        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
      • 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 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
      • 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
      • 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
      • 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.
      • 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.

        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 OutOfRangeException
        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:
        OutOfRangeException - 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)
      • isSupportLowerBoundInclusive

        public boolean isSupportLowerBoundInclusive()
        Whether or not the lower bound of support is in the domain of the density function. Returns true iff getSupporLowerBound() is finite and density(getSupportLowerBound()) returns a non-NaN, non-infinite value.
        Returns:
        true if the lower bound of support is finite and the density function returns a non-NaN, non-infinite value there
      • isSupportUpperBoundInclusive

        public boolean isSupportUpperBoundInclusive()
        Whether or not the upper bound of support is in the domain of the density function. Returns true iff getSupportUpperBound() is finite and density(getSupportUpperBound()) returns a non-NaN, non-infinite value.
        Returns:
        true if the upper bound of support is finite and the density function returns a non-NaN, non-infinite value there
      • 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

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