statistics-0.10.2.0: A library of statistical types, data, and functions

Portability | portable
Stability | experimental
Maintainer | bos@serpentine.com
Safe Haskell | None

Statistics.Sample

Contents

- Types
- Descriptive functions
- Statistics of location
- Statistics of dispersion
  - Functions over central moments
  - Two-pass functions (numerically robust)
  - Single-pass functions (faster, less safe)
- References

Description

Commonly used sample statistics, also known as descriptive statistics.

Synopsis

- type Sample = Vector Double
- type WeightedSample = Vector (Double, Double)
- range :: Vector v Double => v Double -> Double
- mean :: Vector v Double => v Double -> Double
- meanWeighted :: Vector v (Double, Double) => v (Double, Double) -> Double
- harmonicMean :: Vector v Double => v Double -> Double
- geometricMean :: Vector v Double => v Double -> Double
- centralMoment :: Vector v Double => Int -> v Double -> Double
- centralMoments :: Vector v Double => Int -> Int -> v Double -> (Double, Double)
- skewness :: Vector v Double => v Double -> Double
- kurtosis :: Vector v Double => v Double -> Double
- variance :: Vector v Double => v Double -> Double
- varianceUnbiased :: Vector v Double => v Double -> Double
- meanVariance :: Vector v Double => v Double -> (Double, Double)
- meanVarianceUnb :: Vector v Double => v Double -> (Double, Double)
- stdDev :: Vector v Double => v Double -> Double
- varianceWeighted :: Vector v (Double, Double) => v (Double, Double) -> Double
- fastVariance :: Vector v Double => v Double -> Double
- fastVarianceUnbiased :: Vector v Double => v Double -> Double
- fastStdDev :: Vector v Double => v Double -> Double

# Types

type Sample = Vector Double

Sample data.

type WeightedSample = Vector (Double, Double)

Sample with weights. First element of sample is data, second is weight

# Descriptive functions

range :: Vector v Double => v Double -> Double

O(n) Range. The difference between the largest and smallest elements of a sample.

# Statistics of location

mean :: Vector v Double => v Double -> Double

O(n) Arithmetic mean. This uses Welford's algorithm to provide numerical stability, using a single pass over the sample data.

meanWeighted :: Vector v (Double, Double) => v (Double, Double) -> Double

O(n) Arithmetic mean for weighted sample. It uses a single-pass algorithm analogous to the one used by `mean`.

harmonicMean :: Vector v Double => v Double -> Double

O(n) Harmonic mean. This algorithm performs a single pass over the sample.

geometricMean :: Vector v Double => v Double -> Double

O(n) Geometric mean of a sample containing no negative values.

# Statistics of dispersion

The variance—and hence the standard deviation—of a sample of fewer than two elements are both defined to be zero.

## Functions over central moments

centralMoment :: Vector v Double => Int -> v Double -> Double

Compute the kth central moment of a sample. The central moment is also known as the moment about the mean.

This function performs two passes over the sample, so is not subject to stream fusion.

For samples containing many values very close to the mean, this function is subject to inaccuracy due to catastrophic cancellation.

centralMoments :: Vector v Double => Int -> Int -> v Double -> (Double, Double)

Compute the kth and jth central moments of a sample.

This function performs two passes over the sample, so is not subject to stream fusion.

For samples containing many values very close to the mean, this function is subject to inaccuracy due to catastrophic cancellation.

skewness :: Vector v Double => v Double -> Double

Compute the skewness of a sample. This is a measure of the asymmetry of its distribution.

A sample with negative skew is said to be left-skewed. Most of its mass is on the right of the distribution, with the tail on the left.

     skewness $ U.to [1,100,101,102,103]
     ==> -1.497681449918257

A sample with positive skew is said to be right-skewed.

     skewness $ U.to [1,2,3,4,100]
     ==> 1.4975367033335198

A sample's skewness is not defined if its `variance` is zero.

This function performs two passes over the sample, so is not subject to stream fusion.

For samples containing many values very close to the mean, this function is subject to inaccuracy due to catastrophic cancellation.

kurtosis :: Vector v Double => v Double -> Double

Compute the excess kurtosis of a sample. This is a measure of the "peakedness" of its distribution. A high kurtosis indicates that more of the sample's variance is due to infrequent severe deviations, rather than more frequent modest deviations.

A sample's excess kurtosis is not defined if its `variance` is zero.

This function performs two passes over the sample, so is not subject to stream fusion.

For samples containing many values very close to the mean, this function is subject to inaccuracy due to catastrophic cancellation.

## Two-pass functions (numerically robust)

These functions use the compensated summation algorithm of Chan et al. for numerical robustness, but require two passes over the sample data as a result.

Because of the need for two passes, these functions are not subject to stream fusion.

variance :: Vector v Double => v Double -> Double

Maximum likelihood estimate of a sample's variance. Also known as the population variance, where the denominator is n.

varianceUnbiased :: Vector v Double => v Double -> Double

Unbiased estimate of a sample's variance. Also known as the sample variance, where the denominator is n-1.

meanVariance :: Vector v Double => v Double -> (Double, Double)

Calculate mean and maximum likelihood estimate of variance. This function should be used if both mean and variance are required since it will calculate mean only once.

meanVarianceUnb :: Vector v Double => v Double -> (Double, Double)

Calculate mean and unbiased estimate of variance. This function should be used if both mean and variance are required since it will calculate mean only once.

stdDev :: Vector v Double => v Double -> Double

Standard deviation. This is simply the square root of the unbiased estimate of the variance.

varianceWeighted :: Vector v (Double, Double) => v (Double, Double) -> Double

Weighted variance. This is biased estimation.

## Single-pass functions (faster, less safe)

The functions prefixed with the name `fast` below perform a single pass over the sample data using Knuth's algorithm. They usually work well, but see below for caveats. These functions are subject to array fusion.

Note: in cases where most sample data is close to the sample's mean, Knuth's algorithm gives inaccurate results due to catastrophic cancellation.

fastVariance :: Vector v Double => v Double -> Double

Maximum likelihood estimate of a sample's variance.

fastVarianceUnbiased :: Vector v Double => v Double -> Double

Unbiased estimate of a sample's variance.

fastStdDev :: Vector v Double => v Double -> Double

Standard deviation. This is simply the square root of the maximum likelihood estimate of the variance.

# References

- Chan, T. F.; Golub, G.H.; LeVeque, R.J. (1979) Updating formulae and a pairwise algorithm for computing sample variances. Technical Report STAN-CS-79-773, Department of Computer Science, Stanford University. ftp://reports.stanford.edu/pub/cstr/reports/cs/tr/79/773/CS-TR-79-773.pdf
- Knuth, D.E. (1998) The art of computer programming, volume 2: seminumerical algorithms, 3rd ed., p. 232.
- Welford, B.P. (1962) Note on a method for calculating corrected sums of squares and products. Technometrics 4(3):419–420. http://www.jstor.org/stable/1266577
- West, D.H.D. (1979) Updating mean and variance estimates: an improved method. Communications of the ACM 22(9):532–535. http://doi.acm.org/10.1145/359146.359153