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/*
* File: UnivariateStatisticsUtil.java
* Authors: Kevin R. Dixon
* Company: Sandia National Laboratories
* Project: Cognitive Foundry
*
* Copyright September 5, 2007, Sandia Corporation. Under the terms of Contract
* DE-AC04-94AL85000, there is a non-exclusive license for use of this work by
* or on behalf of the U.S. Government. Export of this program may require a
* license from the United States Government. See CopyrightHistory.txt for
* complete details.
*
*/
package gov.sandia.cognition.math;
import gov.sandia.cognition.annotation.PublicationReference;
import gov.sandia.cognition.annotation.PublicationType;
import gov.sandia.cognition.collection.NumberComparator;
import gov.sandia.cognition.util.DefaultPair;
import gov.sandia.cognition.util.Pair;
import gov.sandia.cognition.util.WeightedValue;
import java.util.ArrayList;
import java.util.Collection;
import java.util.Collections;
import java.util.Iterator;
/**
* Some static methods for computing generally useful univariate statistics.
*
* @author Kevin R. Dixon
* @since 2.0
*
*/
public class UnivariateStatisticsUtil
{
/**
* Computes the arithmetic mean (average, expectation, first central moment)
* of a dataset
* @param data
* Collection of Doubles to consider
* @return
* Arithmetic mean of the given dataset
*/
@PublicationReference(
title="Algorithms for calculating variance",
type=PublicationType.WebPage,
year=2010,
author="Wikipedia",
url="http://en.wikipedia.org/wiki/Algorithms_for_calculating_variance")
public static double computeMean(
Iterable data)
{
// Note: This is more compilcated than a straight-forward algorithm
// that just computes the sum and sum-of-squares to get around
// numerical precision issues.
int n = 0;
double mean = 0.0;
for (Number v : data)
{
final double x = v.doubleValue();
final double delta = x - mean;
n += 1;
mean += delta / n;
}
return mean;
}
/**
* Computes the arithmetic mean (average, expectation, first central moment)
* of a dataset. The absolute value of the weight is used to handle negative
* weights.
*
* @param data
* Collection of Doubles to consider.
* @return
* Arithmetic mean of the given dataset.
*/
@PublicationReference(
title="Algorithms for calculating variance",
type=PublicationType.WebPage,
year=2010,
author="Wikipedia",
url="http://en.wikipedia.org/wiki/Algorithms_for_calculating_variance")
public static double computeWeightedMean(
Iterable> data )
{
// Note: This is more compilcated than a straight-forward algorithm
// that just computes the sum and sum-of-squares to get around
// numerical precision issues.
double mean = 0.0;
double weightSum = 0.0;
for ( WeightedValue v : data)
{
final double x = v.getValue().doubleValue();
double weight = v.getWeight();
if (weight != 0.0)
{
if (weight < 0.0)
{
// Use the absolute value of weights.
weight = -weight;
}
final double newWeightSum = weightSum + weight;
final double delta = x - mean;
final double update = delta * weight / newWeightSum;
mean += update;
weightSum = newWeightSum;
}
}
return mean;
}
/**
* Computes the unbiased variance (second central moment,
* squared standard deviation) of a dataset. Computes the mean first,
* then computes the variance. If you already have the mean, then use the
* two-argument computeVariance(data,mean) method to save duplication of
* effort.
* @param data
* Data to consider
* @return
* Unbiased variance of the given dataset
*/
static public double computeVariance(
Collection data )
{
return computeMeanAndVariance(data).getSecond();
}
/**
* Computes the unbiased variance (second central moment,
* squared standard deviation) of a dataset
* @param data
* Data to consider
* @param mean
* Pre-computed mean (or central value) of the dataset
* @return
* Unbiased variance of the given dataset
*/
static public double computeVariance(
Collection data,
double mean )
{
int num = data.size();
if( num < 2 )
{
return 0.0;
}
double biasedVariance = computeCentralMoment(data, mean, 2);
double unbiasedVariance = (num/(num-1.0))*biasedVariance;
return unbiasedVariance;
}
/**
* Computes the standard deviation of a dataset, which is the square root
* of the unbiased variance. It computes the mean first and then computes
* the standard deviation. If you already have the mean, then use the
* two-argument computeStandardDeviation(data, mean) to save duplication of
* effort.
*
* @param data
* The data to consider.
* @return
* The standard deviation of the given data. Will not be negative.
*/
public static double computeStandardDeviation(
final Collection data)
{
return Math.sqrt(computeVariance(data));
}
/**
* Computes the standard deviation of a dataset, which is the square root
* of the unbiased variance.
*
* @param data
* The data to consider.
* @param mean
* The pre-computed mean of the given data.
* @return
* The standard deviation of the given data. Will not be negative.
*/
public static double computeStandardDeviation(
final Collection data,
final double mean)
{
return Math.sqrt(computeVariance(data, mean));
}
/**
* Computes the Root mean-squared (RMS) error between the data and its mean.
* Computes the mean first, then computes the RMS error. If
* you already have the mean, then use the two-argument
* computeRootMeanSquaredError(data,mean) method to save computation
*
* @return RMS error of the dataset about the mean
* @param data Dataset to consider
*/
static public double computeRootMeanSquaredError(
Collection data )
{
double mean = UnivariateStatisticsUtil.computeMean( data );
return UnivariateStatisticsUtil.computeRootMeanSquaredError( data, mean );
}
/**
* Computes the Root mean-squared (RMS) error between the data and its mean
* @param data
* Dataset to consider
* @param mean
* Mean value about which to compute the sum-squared error
* @return
* RMS error of the dataset about the mean
*/
static public double computeRootMeanSquaredError(
Collection data,
double mean )
{
double sse = UnivariateStatisticsUtil.computeSumSquaredDifference( data, mean );
double rms;
if (data.size() > 0)
{
rms = Math.sqrt( sse / data.size() );
}
else
{
rms = 0.0;
}
return rms;
}
/**
* Computes the arithmetic sum of the dataset
* @param data
* Dataset to consider
* @return
* Arithmetic sum of the given dataset
*/
static public double computeSum(
Iterable data )
{
double sum = 0.0;
for (Number value : data)
{
sum += value.doubleValue();
}
return sum;
}
/**
* Computes the sum-squared difference between the data and a target
* @param data
* Dataset to consider
* @param target
* Target about which to compute the difference
* @return
* Sum-squared difference between the dataset and the target
*/
static public double computeSumSquaredDifference(
Iterable data,
double target )
{
double sum = 0.0;
double delta;
for (Number value : data)
{
delta = value.doubleValue() - target;
sum += delta * delta;
}
return sum;
}
/**
* Computes the correlation coefficient in a single pass. However, this
* algorithm can become numerically unstable but is about twice as fast
* as the non-single-pass method.
* @param data1
* First dataset to consider, must have same size as data2
* @param data2
* Second dataset to consider
* @return
* Normalized correlation coefficient, [-1,+1]
*/
@PublicationReference(
author="Wikipedia",
title="Pearson product-moment correlation coefficient",
type=PublicationType.WebPage,
year=2011,
url="http://en.wikipedia.org/wiki/Pearson_product-moment_correlation_coefficient"
)
static public double computeCorrelation(
Collection data1,
Collection data2 )
{
if (data1.size() != data2.size())
{
throw new IllegalArgumentException(
"Datasets must be the same size!" );
}
int num = data1.size();
if( num <= 0 )
{
return 1.0;
}
Iterator i1 = data1.iterator();
Iterator i2 = data2.iterator();
double sum1 = 0.0;
double sum2 = 0.0;
double sum11 = 0.0;
double sum22 = 0.0;
double sum12 = 0.0;
for( int n = 0; n < num; n++ )
{
final double value1 = i1.next().doubleValue();
final double value2 = i2.next().doubleValue();
sum1 += value1;
sum2 += value2;
sum11 += value1*value1;
sum22 += value2*value2;
sum12 += value1*value2;
}
double top = num * (sum12 - ((sum1/num)*sum2));
double b1 = Math.sqrt( num * (sum11 - (sum1/num)*sum1) );
double b2 = Math.sqrt( num * (sum22 - (sum2/num)*sum2) );
double bottom = b1*b2;
double retval;
if( bottom <= 0.0 )
{
retval = 1.0;
}
else
{
retval = top/bottom;
}
return retval;
}
/**
* Computes the median of the given data.
* @param data
* Data from which to compute the median.
* @return
* Median of the sample.
*/
public static double computeMedian(
Collection data )
{
return computePercentile(data, 0.5);
}
/**
* Computes the percentile value of the given data. For example, if
* data has 101 values and "percentile" is 0.27, then the return value
* would be the ascending-sorted value in the 26th zero-based index.
* @param data
* Data from which to compute the percentile.
* @param percentile
* Percentile to choose, must be on the closed interval 0.0 to 1.0.
* @return
* Requested percentile from the data.
*/
public static double computePercentile(
Collection data,
double percentile )
{
ProbabilityUtil.assertIsProbability(percentile);
ArrayList sortedData = new ArrayList( data );
Collections.sort( sortedData, new NumberComparator() );
int num = data.size();
double numPct = (num-1) * percentile;
double remainder = Math.abs(Math.IEEEremainder(numPct, 1.0));
int lowerIndex = (int) Math.floor(numPct);
double lower = sortedData.get(lowerIndex).doubleValue();
if( remainder == 0.0 )
{
return lower;
}
int upperIndex = lowerIndex + 1;
double upper = sortedData.get(upperIndex).doubleValue();
return lower * (1.0-remainder) + upper*remainder;
}
/**
* Finds the minimum value of a data set.
* @param data
* Data set to consider
* @return
* Minimum value of the data set.
*/
public static double computeMinimum(
Iterable data )
{
double minimum = Double.POSITIVE_INFINITY;
for( Number value : data )
{
double x = value.doubleValue();
if( minimum > x )
{
minimum = x;
}
}
return minimum;
}
/**
* Finds the maximum value of a data set.
* @param data
* Data set to consider
* @return
* Maximum value of the data set.
*/
public static double computeMaximum(
Iterable data )
{
double maximum = Double.NEGATIVE_INFINITY;
for( Number value : data )
{
double x = value.doubleValue();
if( maximum < x )
{
maximum = x;
}
}
return maximum;
}
/**
* Computes the minimum and maximum of a set of data in a single pass.
* @param data
* Data to consider
* @return
* Minimum and Maximum
*/
public static Pair computeMinAndMax(
Iterable data )
{
double min = Double.POSITIVE_INFINITY;
double max = Double.NEGATIVE_INFINITY;
for( Number value : data )
{
double x = value.doubleValue();
if( min > x )
{
min = x;
}
if( max < x )
{
max = x;
}
}
return DefaultPair.create( min, max );
}
/**
* Computes the unbiased skewness of the dataset.
* @param data
* Data from which to compute the unbiased skewness.
* @return
* Unbiased skewness.
*/
@PublicationReference(
author="Wikipedia",
title="Skewness",
type=PublicationType.WebPage,
year=2009,
url="http://en.wikipedia.org/wiki/Skewness"
)
public static double computeSkewness(
Collection data )
{
int num = data.size();
if( num < 3 )
{
return 0.0;
}
Pair pair = computeMeanAndVariance(data);
double mean = pair.getFirst();
double stddev = Math.sqrt(pair.getSecond());
double moment3 = computeCentralMoment(data, mean, 3);
double biasedSkewness = moment3 / Math.pow(stddev, 3.0);
return biasedSkewness;
}
/**
* Computes the desired biased estimate central moment of the given dataset.
* @param data
* Data to compute the moment of.
* @param mean
* Mean of the data (to prevent redundant computation).
* @param moment
* Desired moment of the data, must be greater than or equal to 1.
* @return
* Biased estimate of the desired central moment.
*/
public static double computeCentralMoment(
Iterable data,
double mean,
int moment )
{
if( moment < 1 )
{
throw new IllegalArgumentException( "Moment must be >= 1" );
}
int num = 0;
double sum = 0.0;
for( Number value : data )
{
double delta = value.doubleValue() - mean;
sum += Math.pow(delta, moment);
num++;
}
if( num < 1 )
{
return 0.0;
}
else
{
return sum / num;
}
}
/**
* Computes the desired biased estimate central moment of the given dataset.
* The absolute value of the weight is used to handle negative weights.
*
* @param data
* Data to compute the moment of.
* @param mean
* Mean of the data (to prevent redundant computation).
* @param moment
* Desired moment of the data, must be greater than or equal to 1.
* @return
* Biased estimate of the desired central moment.
*/
public static double computeWeightedCentralMoment(
Iterable> data,
double mean,
int moment )
{
if( moment < 1 )
{
throw new IllegalArgumentException( "Moment must be >= 1" );
}
int num = 0;
double sum = 0.0;
double weightSum = 0.0;
for( WeightedValue value : data )
{
final double weight = Math.abs(value.getWeight());
if( weight != 0.0 )
{
double delta = value.getValue().doubleValue() - mean;
sum += weight*Math.pow(delta, moment);
weightSum += weight;
}
num++;
}
if( num < 1 )
{
return 0.0;
}
else
{
return sum / weightSum;
}
}
/**
* Computes the biased excess kurtosis of the given dataset. Intuitively,
* kurtosis quantifies the pointiness of the data by normalizing the fourth
* central moment.
* @param data
* Dataset to compute its kurtosis.
* @return
* Biased excess kurtosis of the given dataset.
*/
@PublicationReference(
author="Wikipedia",
title="Kurtosis",
type=PublicationType.WebPage,
year=2009,
url="http://en.wikipedia.org/wiki/Kurtosis"
)
public static double computeKurtosis(
Collection data )
{
if( data.size() < 2 )
{
return 0.0;
}
Pair pair = computeMeanAndVariance(data);
double mean = pair.getFirst();
double variance = pair.getSecond();
double moment4 = computeCentralMoment(data, mean, 4);
double biasedKurtosis = moment4 / (variance*variance) - 3.0;
return biasedKurtosis;
}
/**
* Computes the biased excess kurtosis of the given dataset. Intuitively,
* kurtosis quantifies the pointiness of the data by normalizing the fourth
* central moment. The absolute value of the weight is used to handle
* negative weights.
*
* @param data
* Dataset to compute its kurtosis.
* @return
* Biased excess kurtosis of the given dataset.
*/
@PublicationReference(
author="Wikipedia",
title="Kurtosis",
type=PublicationType.WebPage,
year=2009,
url="http://en.wikipedia.org/wiki/Kurtosis"
)
public static double computeWeightedKurtosis(
Collection> data )
{
if( data.size() < 2 )
{
return 0.0;
}
Pair pair = computeWeightedMeanAndVariance(data);
double mean = pair.getFirst();
double variance = pair.getSecond();
double moment4 = computeWeightedCentralMoment(data, mean, 4);
double biasedKurtosis = moment4 / (variance*variance) - 3.0;
return biasedKurtosis;
}
/**
* Computes the information-theoretic entropy of the PMF in bits (base 2).
* @param data
* Data to compute the entropy.
* @return
* Entropy in bits of the given PMF.
*/
@PublicationReference(
author="Wikipedia",
title="Entropy (information theory)",
type=PublicationType.WebPage,
year=2009,
url="http://en.wikipedia.org/wiki/Entropy_(Information_theory)"
)
public static double computeEntropy(
Iterable data )
{
// Compute the entropy by looping over the values in the map.s
double entropy = 0.0;
for( Number value : data )
{
double p = value.doubleValue();
if( p != 0.0 )
{
entropy -= p * MathUtil.log2( p );
}
}
// Return the computed entropy.
return entropy;
}
/**
* Computes the mean and unbiased variance of a Collection of data using
* the one-pass approach.
* @param data
* Data to consider
* @return
* Mean and unbiased Variance Pair.
*/
@PublicationReference(
title="Algorithms for calculating variance",
type=PublicationType.WebPage,
year=2010,
author="Wikipedia",
url="http://en.wikipedia.org/wiki/Algorithms_for_calculating_variance")
public static Pair computeMeanAndVariance(
Iterable data)
{
// Note: This is more compilcated than a straight-forward algorithm
// that just computes the sum and sum-of-squares to get around
// numerical precision issues.
int n = 0;
double mean = 0.0;
double m2 = 0.0;
for (Number v : data)
{
final double x = v.doubleValue();
final double delta = x - mean;
n += 1;
mean += delta / n;
m2 += delta * (x - mean);
}
double variance;
if (n >= 2)
{
variance = m2 / (n - 1);
}
else
{
variance = 0.0;
}
return DefaultPair.create(mean, variance);
}
/**
* Computes the mean and unbiased variance of a Collection of data using
* the one-pass approach. The absolute value is used to handle negative
* weights.
*
* @param data
* Data to consider.
* @return
* Mean and unbiased Variance Pair.
*/
@PublicationReference(
title="Algorithms for calculating variance",
type=PublicationType.WebPage,
year=2010,
author="Wikipedia",
url="http://en.wikipedia.org/wiki/Algorithms_for_calculating_variance")
public static Pair computeWeightedMeanAndVariance(
Iterable> data )
{
// Note: This is more compilcated than a straight-forward algorithm
// that just computes the sum and sum-of-squares to get around
// numerical precision issues.
double mean = 0.0;
double weightSum = 0.0;
double m2 = 0.0;
for ( WeightedValue v : data)
{
final double x = v.getValue().doubleValue();
double weight = v.getWeight();
if (weight != 0.0)
{
if (weight < 0.0)
{
// Use the absolute value of weights.
weight = -weight;
}
final double newWeightSum = weightSum + weight;
final double delta = x - mean;
final double update = delta * weight / newWeightSum;
m2 += weightSum * delta * update;
mean += update;
weightSum = newWeightSum;
}
}
double variance;
if (weightSum > 0.0)
{
variance = m2 / weightSum;
}
else
{
variance = 0.0;
}
return DefaultPair.create(mean, variance);
}
}