statistics-0.13.2.1: A library of statistical types, data, and functions

Portability portable experimental bos@serpentine.com None

Statistics.Sample

Description

Commonly used sample statistics, also known as descriptive statistics.

Synopsis

# Types

Sample data.

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

# Descriptive functions

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

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

# Statistics of location

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

O(n) Arithmetic mean. This uses Kahan-Babuška-Neumaier summation, so is more accurate than `welfordMean` unless the input values are very large.

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) -> DoubleSource

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

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

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

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)Source

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.

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.

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.

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

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)Source

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)Source

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 -> DoubleSource

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

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

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.

Maximum likelihood estimate of a sample's variance.

Unbiased estimate of a sample's variance.

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