Statistics.Sample.Powers

Very fast statistics over simple powers of a sample. These can all be computed efficiently in just a single pass over a sample, with that pass subject to stream fusion.

The tradeoff is that some of these functions are less numerically robust than their counterparts in the Statistics.Sample module. Where this is the case, the alternatives are noted.

Types

data Powers

Instances

Eq Powers

Show Powers

Constructor

powers

Arguments

:: Vector v Double

=> Int

n, the number of powers, where n >= 2.

-> v Double

-> Powers

O(n) Collect the n simple powers of a sample.

Functions computed over a sample's simple powers require at least a certain number (or order) of powers to be collected.

• To compute the kth centralMoment, at least k simple powers must be collected.
• For the variance, at least 2 simple powers are needed.
• For skewness, we need at least 3 simple powers.
• For kurtosis, at least 4 simple powers are required.

This function is subject to stream fusion.

Descriptive functions

order :: Powers -> Int

The order (number) of simple powers collected from a sample.

count :: Powers -> Int

The number of elements in the original Sample. This is the sample's zeroth simple power.

sum :: Powers -> Double

The sum of elements in the original Sample. This is the sample's first simple power.

Statistics of location

mean :: Powers -> Double

The arithmetic mean of elements in the original Sample.

This is less numerically robust than the mean function in the Statistics.Sample module, but the number is essentially free to compute if you have already collected a sample's simple powers.

Statistics of dispersion

variance :: Powers -> Double

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

This is less numerically robust than the variance function in the Statistics.Sample module, but the number is essentially free to compute if you have already collected a sample's simple powers.

Requires Powers with order at least 2.

stdDev :: Powers -> Double

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

varianceUnbiased :: Powers -> Double

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

Requires Powers with order at least 2.

Functions over central moments

centralMoment :: Int -> Powers -> Double

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

skewness :: Powers -> 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 . powers 3 $ U.to [1,100,101,102,103]
 ==> -1.497681449918257

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

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

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

Requires Powers with order at least 3.

kurtosis :: Powers -> Double

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

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

Requires Powers with order at least 4.

References

• Besset, D.H. (2000) Elements of statistics. Object-oriented implementation of numerical methods ch. 9, pp. 311–331. http://www.elsevier.com/wps/product/cws_home/677916
• Anderson, G. (2009) Compute kth central moments in one pass. quantblog. http://quantblog.wordpress.com/2009/02/07/compute-kth-central-moments-in-one-pass/