Class EmpiricalDistribution
- All Implemented Interfaces:
Serializable
,RealDistribution
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
- Pass the file once to compute min and max.
- Divide the range from min-max into
binCount
"bins." - Pass the data file again, computing bin counts and univariate statistics (mean, std dev.) for each of the bins
- Divide the interval (0,1) into subintervals associated with the bins, with the length of a bin's subinterval proportional to its count.
- Generate a uniformly distributed value in (0,1)
- Select the subinterval to which the value belongs.
- 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:
-
Field Summary
Modifier and TypeFieldDescriptionstatic final int
Default bin countprotected final RandomDataGenerator
RandomDataGenerator instance to use in repeated calls to getNext()Fields inherited from class org.hipparchus.distribution.continuous.AbstractRealDistribution
DEFAULT_SOLVER_ABSOLUTE_ACCURACY
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Constructor Summary
ConstructorDescriptionCreates 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 providedRandomGenerator
as the source of random data.EmpiricalDistribution
(RandomGenerator generator) Creates a new EmpiricalDistribution with default bin count using the providedRandomGenerator
as the source of random data. -
Method Summary
Modifier and TypeMethodDescriptiondouble
cumulativeProbability
(double x) For a random variableX
whose values are distributed according to this distribution, this method returnsP(X <= x)
.double
density
(double x) Returns the probability density function (PDF) of this distribution evaluated at the specified pointx
.int
Returns the number of bins.Returns a List ofStreamingStatistics
instances containing statistics describing the values in each of the bins.double[]
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
Generates a random value from this distribution.double
Use this method to get the numerical value of the mean of this distribution.double
Use this method to get the numerical value of the variance of this distribution.Returns aStatisticalSummary
describing this distribution.double
Access the lower bound of the support.double
Access the upper bound of the support.double[]
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
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
Computes the empirical distribution from the input file.void
Computes the empirical distribution using data read from a URL.void
reSeed
(long seed) Reseeds the random number generator used bygetNextValue()
.void
reseedRandomGenerator
(long seed) Reseed the underlying PRNG.Methods inherited from class org.hipparchus.distribution.continuous.AbstractRealDistribution
getSolverAbsoluteAccuracy, logDensity, probability
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Field Details
-
DEFAULT_BIN_COUNT
public static final int DEFAULT_BIN_COUNTDefault bin count- See Also:
-
randomData
RandomDataGenerator instance to use in repeated calls to getNext()
-
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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
- ifbinCount <= 0
.
-
EmpiricalDistribution
Creates a new EmpiricalDistribution with the specified bin count using the providedRandomGenerator
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
- ifbinCount <= 0
.
-
EmpiricalDistribution
Creates a new EmpiricalDistribution with default bin count using the providedRandomGenerator
as the source of random data.- Parameters:
generator
- random data generator (may be null, resulting in default JDK generator)
-
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Method Details
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load
Computes the empirical distribution from the provided array of numbers.- Parameters:
in
- the input data array- Throws:
NullArgumentException
- if in is null
-
load
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 occursNullArgumentException
- if url is nullMathIllegalArgumentException
- if URL contains no data
-
load
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 occursNullArgumentException
- if file is null
-
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
Returns aStatisticalSummary
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.
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getBinStats
Returns a List ofStreamingStatistics
instances containing statistics describing the values in each of the bins. The list is indexed on the bin number.- Returns:
- List of bin statistics.
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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
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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 aload
method has been called beforehand.
-
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 bygetNextValue()
.- Parameters:
seed
- random generator seed
-
density
public double density(double x) Returns the probability density function (PDF) of this distribution evaluated at the specified pointx
. In general, the PDF is the derivative of theCDF
. If the derivative does not exist atx
, 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:
- Find the bin B that x belongs to.
- Compute K(B) = the mass of B with respect to the within-bin kernel (i.e., the integral of the kernel density over B).
- 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
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cumulativeProbability
public double cumulativeProbability(double x) For a random variableX
whose values are distributed according to this distribution, this method returnsP(X <= x)
. In other words, this method represents the (cumulative) distribution function (CDF) for this distribution.Algorithm description:
- Find the bin B that x belongs to.
- Compute P(B) = the mass of B and P(B-) = the combined mass of the bins below B.
- 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
- 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
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inverseCumulativeProbability
Computes the quantile function of this distribution. For a random variableX
distributed according to this distribution, the returned value isinf{x in R | P(X<=x) >= p}
for0 < p <= 1
,inf{x in R | P(X<=x) > 0}
forp = 0
.
RealDistribution.getSupportLowerBound()
forp = 0
,RealDistribution.getSupportUpperBound()
forp = 1
.
Algorithm description:
- Find the smallest i such that the sum of the masses of the bins through i is at least p.
-
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-)
- Return the inverse of K evaluated at
K(B-) + pCrit * K(B) / P(B)
- Specified by:
inverseCumulativeProbability
in interfaceRealDistribution
- Overrides:
inverseCumulativeProbability
in classAbstractRealDistribution
- Parameters:
p
- the cumulative probability- Returns:
- the smallest
p
-quantile of this distribution (largest 0-quantile forp = 0
) - Throws:
MathIllegalArgumentException
- ifp < 0
orp > 1
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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
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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 inTDistribution
) orDouble.NaN
if it is not defined
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getSupportLowerBound
public double getSupportLowerBound()Access the lower bound of the support. This method must return the same value asinverseCumulativeProbability(0)
. In other words, this method must returninf {x in R | P(X <= x) > 0}
.- Returns:
- lower bound of the support (might be
Double.NEGATIVE_INFINITY
)
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getSupportUpperBound
public double getSupportUpperBound()Access the upper bound of the support. This method must return the same value asinverseCumulativeProbability(1)
. In other words, this method must returninf {x in R | P(X <= x) = 1}
.- Returns:
- upper bound of the support (might be
Double.POSITIVE_INFINITY
)
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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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reseedRandomGenerator
public void reseedRandomGenerator(long seed) Reseed the underlying PRNG.- Parameters:
seed
- new seed value
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getKernel
The within-bin smoothing kernel. Returns a Gaussian distribution parameterized bybStats
, 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
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