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Statistics.Sample.Powers | Portability | portable | Stability | experimental | Maintainer | bos@serpentine.com |
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Description |
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.
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Synopsis |
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Types
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Instances | |
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Constructor
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:: 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.
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Descriptive functions
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The order (number) of simple powers collected from a sample.
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The number of elements in the original Sample. This is the
sample's zeroth simple power.
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The sum of elements in the original Sample. This is the
sample's first simple power.
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Statistics of location
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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.
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Statistics of dispersion
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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.
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Standard deviation. This is simply the square root of the
maximum likelihood estimate of the variance.
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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.
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Functions over central moments
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Compute the kth central moment of a sample. The central
moment is also known as the moment about the mean.
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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.
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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.
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References
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Produced by Haddock version 2.4.2 |