Package org.apache.commons.math4.distribution

Class EmpiricalDistribution

  • All Implemented Interfaces:
    java.io.Serializable, RealDistribution, ContinuousDistribution
    public class EmpiricalDistribution
extends AbstractRealDistribution
implements ContinuousDistribution

    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. Must be strictly positive.
        Throws:
        NotStrictlyPositiveException - if binCount <= 0.

    • 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

      • load

        public void load​(java.net.URL url)
          throws java.io.IOException,
                 NullArgumentException,
                 ZeroException
        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:
        java.io.IOException - if an IO error occurs
        NullArgumentException - if url is null
        ZeroException - if URL contains no data

      • load

        public void load​(java.io.File file)
          throws java.io.IOException,
                 NullArgumentException
        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:
        java.io.IOException - if an IO error occurs
        NullArgumentException - if file is null

      • 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:
        java.lang.IllegalStateException - if the distribution has not been loaded

      • getBinCount

        public int getBinCount()
        Returns the number of bins.
        Returns:
        the number of bins.

      • getBinStats

        public java.util.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.

        Preconditions:
        • the distribution must be loaded before invoking this method

        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
        Throws:
        java.lang.NullPointerException - unless a load method has been called beforehand.
        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

      • 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 ContinuousDistribution
        Overrides:
        probability in class AbstractRealDistribution
        Parameters:
        x - 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.
        Specified by:
        density in interface ContinuousDistribution
        Parameters:
        x - Point at which the PDF is evaluated.
        Returns:
        the value of the probability density function at 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.
        If K is a constant distribution, we return P(B-) + P(B) (counting the full mass of B).
        Specified by:
        cumulativeProbability in interface ContinuousDistribution
        Parameters:
        x - 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)
                                    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 ContinuousDistribution
        Overrides:
        inverseCumulativeProbability in class AbstractRealDistribution
        Parameters:
        p - Cumulative probability.
        Returns:
        the smallest p-quantile of this distribution (largest 0-quantile for p = 0).
        Throws:
        OutOfRangeException
        Since:
        3.1

      • getMean

        public double getMean()
        Gets the mean of this distribution.
        Specified by:
        getMean in interface ContinuousDistribution
        Returns:
        the mean, or Double.NaN if it is not defined.
        Since:
        3.1

      • getVariance

        public double getVariance()
        Gets the variance of this distribution.
        Specified by:
        getVariance in interface ContinuousDistribution
        Returns:
        the variance, or Double.NaN if it is not defined.
        Since:
        3.1

      • getSupportLowerBound

        public double getSupportLowerBound()
        Gets the lower bound of the support. It must return the same value as inverseCumulativeProbability(0), i.e. inf {x in R | P(X <= x) > 0}.
        Specified by:
        getSupportLowerBound in interface ContinuousDistribution
        Returns:
        the lower bound of the support.
        Since:
        3.1

      • getSupportUpperBound

        public double getSupportUpperBound()
        Gets the upper bound of the support. It must return the same value as inverseCumulativeProbability(1), i.e. inf {x in R | P(X <= x) = 1}.
        Specified by:
        getSupportUpperBound in interface ContinuousDistribution
        Returns:
        the upper bound of the support.
        Since:
        3.1

      • isSupportConnected

        public boolean isSupportConnected()
        Indicates whether the support is connected, i.e. whether all values between the lower and upper bound of the support are included in the support.
        Specified by:
        isSupportConnected in interface ContinuousDistribution
        Returns:
        whether the support is connected.
        Since:
        3.1

      • getKernel

        protected ContinuousDistribution getKernel​(SummaryStatistics bStats)
        The within-bin smoothing kernel. Returns a Gaussian distribution parameterized by bStats, unless the bin contains only one observation, in which case a constant distribution is returned.
        Parameters:
        bStats - summary statistics for the bin
        Returns:
        within-bin kernel parameterized by bStats