## org.apache.commons.math3.random Class EmpiricalDistribution

```java.lang.Object
org.apache.commons.math3.distribution.AbstractRealDistribution
org.apache.commons.math3.random.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:

• 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 1422350 2012-12-15 20:47:47Z psteitz \$
Serialized Form

Field Summary
`static int` `DEFAULT_BIN_COUNT`
Default bin count

Fields inherited from class org.apache.commons.math3.distribution.AbstractRealDistribution
`random, SOLVER_DEFAULT_ABSOLUTE_ACCURACY`

Constructor Summary
`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, RandomDataImpl randomData)```
Deprecated. As of 3.1. Please use `EmpiricalDistribution(int,RandomGenerator)` instead.
```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(RandomDataImpl randomData)`
Deprecated. As of 3.1. Please use `EmpiricalDistribution(RandomGenerator)` instead.
`EmpiricalDistribution(RandomGenerator generator)`
Creates a new EmpiricalDistribution with default bin count using the provided `RandomGenerator` as the source of random data.

Method Summary
` 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()`
Whether or not the lower bound of support is in the domain of the density function.
` boolean` `isSupportUpperBoundInclusive()`
Whether or not the upper bound of support is in the domain of the density function.
` 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 org.apache.commons.math3.distribution.AbstractRealDistribution
`cumulativeProbability, 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

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)
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

### EmpiricalDistribution

```@Deprecated
public 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)
Since:
3.0

### EmpiricalDistribution

```@Deprecated
public 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)
Since:
3.0
Method Detail

```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

```public void load(URL url)
throws 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:
`IOException` - if an IO error occurs
`NullArgumentException` - if url is null
`ZeroException` - if URL contains no data

```public void load(File file)
throws 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:
`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 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

`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)
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`
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()`
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
Since:
3.1

### 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
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