Copyright | (c) 2008 Don Stewart 2009 Bryan O'Sullivan |
---|---|

License | BSD3 |

Maintainer | bos@serpentine.com |

Stability | experimental |

Portability | portable |

Safe Haskell | None |

Language | Haskell98 |

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
- welfordMean :: 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
- stdErrMean :: Vector v Double => v Double -> Double
- fastVariance :: Vector v Double => v Double -> Double
- fastVarianceUnbiased :: Vector v Double => v Double -> Double
- fastStdDev :: Vector v Double => v Double -> Double
- covariance :: (Vector v (Double, Double), Vector v Double) => v (Double, Double) -> Double
- correlation :: (Vector v (Double, Double), Vector v Double) => v (Double, Double) -> Double
- pair :: (Vector v a, Vector v b, Vector v (a, b)) => v a -> v b -> v (a, b)

# Types

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 Kahan-Babuška-Neumaier
summation, so is more accurate than `welfordMean`

unless the input
values are very large.

welfordMean :: 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.

Compared to `mean`

, this loses a surprising amount of precision
unless the inputs are very large.

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 `welfordMean`

.

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 *k*th 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 *k*th and *j*th 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.

## 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.

stdErrMean :: Vector v Double => v Double -> Double #

Standard error of the mean. This is the standard deviation divided by the square root of the sample size.

## 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.

# Joint distirbutions

covariance :: (Vector v (Double, Double), Vector v Double) => v (Double, Double) -> Double #

Covariance of sample of pairs. For empty sample it's set to zero

correlation :: (Vector v (Double, Double), Vector v Double) => v (Double, Double) -> Double #

Correlation coefficient for sample of pairs. Also known as Pearson's correlation. For empty sample it's set to zero.

pair :: (Vector v a, Vector v b, Vector v (a, b)) => v a -> v b -> v (a, b) #

Pair two samples. It's like `zip`

but requires that both
samples have equal size.

# 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