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{-# LANGUAGE FlexibleContexts #-}
-- |
-- Module : Statistics.Sample
-- Copyright : (c) 2008 Don Stewart, 2009 Bryan O'Sullivan
-- License : BSD3
--
-- Maintainer : bos@serpentine.com
-- Stability : experimental
-- Portability : portable
--
-- Commonly used sample statistics, also known as descriptive
-- statistics.
module Statistics.Sample
(
-- * Types
Sample
, WeightedSample
-- * Descriptive functions
, range
-- * Statistics of location
, mean
, meanWeighted
, harmonicMean
, geometricMean
-- * Statistics of dispersion
-- $variance
-- ** Functions over central moments
, centralMoment
, centralMoments
, skewness
, kurtosis
-- ** Two-pass functions (numerically robust)
-- $robust
, variance
, varianceUnbiased
, meanVariance
, meanVarianceUnb
, stdDev
, varianceWeighted
-- ** Single-pass functions (faster, less safe)
-- $cancellation
, fastVariance
, fastVarianceUnbiased
, fastStdDev
-- * References
-- $references
) where
import Statistics.Function (minMax)
import Statistics.Types (Sample,WeightedSample)
import qualified Data.Vector.Generic as G
-- Operator ^ will be overriden
import Prelude hiding ((^))
-- | /O(n)/ Range. The difference between the largest and smallest
-- elements of a sample.
range :: (G.Vector v Double) => v Double -> Double
range s = hi - lo
where (lo , hi) = minMax s
{-# INLINE range #-}
-- | /O(n)/ Arithmetic mean. This uses Welford's algorithm to provide
-- numerical stability, using a single pass over the sample data.
mean :: (G.Vector v Double) => v Double -> Double
mean = fini . G.foldl' go (T 0 0)
where
fini (T a _) = a
go (T m n) x = T m' n'
where m' = m + (x - m) / fromIntegral n'
n' = n + 1
{-# INLINE mean #-}
-- | /O(n)/ Arithmetic mean for weighted sample. It uses a single-pass
-- algorithm analogous to the one used by 'mean'.
meanWeighted :: (G.Vector v (Double,Double)) => v (Double,Double) -> Double
meanWeighted = fini . G.foldl' go (V 0 0)
where
fini (V a _) = a
go (V m w) (x,xw) = V m' w'
where m' | w' == 0 = 0
| otherwise = m + xw * (x - m) / w'
w' = w + xw
{-# INLINE meanWeighted #-}
-- | /O(n)/ Harmonic mean. This algorithm performs a single pass over
-- the sample.
harmonicMean :: (G.Vector v Double) => v Double -> Double
harmonicMean = fini . G.foldl' go (T 0 0)
where
fini (T b a) = fromIntegral a / b
go (T x y) n = T (x + (1/n)) (y+1)
{-# INLINE harmonicMean #-}
-- | /O(n)/ Geometric mean of a sample containing no negative values.
geometricMean :: (G.Vector v Double) => v Double -> Double
geometricMean = exp . mean . G.map log
{-# INLINE geometricMean #-}
-- | 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.
centralMoment :: (G.Vector v Double) => Int -> v Double -> Double
centralMoment a xs
| a < 0 = error "Statistics.Sample.centralMoment: negative input"
| a == 0 = 1
| a == 1 = 0
| otherwise = G.sum (G.map go xs) / fromIntegral (G.length xs)
where
go x = (x-m) ^ a
m = mean xs
{-# INLINE centralMoment #-}
-- | 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.
centralMoments :: (G.Vector v Double) => Int -> Int -> v Double -> (Double, Double)
centralMoments a b xs
| a < 2 || b < 2 = (centralMoment a xs , centralMoment b xs)
| otherwise = fini . G.foldl' go (V 0 0) $ xs
where go (V i j) x = V (i + d^a) (j + d^b)
where d = x - m
fini (V i j) = (i / n , j / n)
m = mean xs
n = fromIntegral (G.length xs)
{-# INLINE centralMoments #-}
-- | 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.
skewness :: (G.Vector v Double) => v Double -> Double
skewness xs = c3 * c2 ** (-1.5)
where (c3 , c2) = centralMoments 3 2 xs
{-# INLINE skewness #-}
-- | 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.
kurtosis :: (G.Vector v Double) => v Double -> Double
kurtosis xs = c4 / (c2 * c2) - 3
where (c4 , c2) = centralMoments 4 2 xs
{-# INLINE kurtosis #-}
-- $variance
--
-- The variance&#8212;and hence the standard deviation&#8212;of a
-- sample of fewer than two elements are both defined to be zero.
-- $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.
data V = V {-# UNPACK #-} !Double {-# UNPACK #-} !Double
robustSumVar :: (G.Vector v Double) => Double -> v Double -> Double
robustSumVar m samp = G.sum . G.map (square . subtract m) $ samp
where square x = x * x
{-# INLINE robustSumVar #-}
-- | Maximum likelihood estimate of a sample's variance. Also known
-- as the population variance, where the denominator is /n/.
variance :: (G.Vector v Double) => v Double -> Double
variance samp
| n > 1 = robustSumVar (mean samp) samp / fromIntegral n
| otherwise = 0
where
n = G.length samp
{-# INLINE variance #-}
-- | Unbiased estimate of a sample's variance. Also known as the
-- sample variance, where the denominator is /n/-1.
varianceUnbiased :: (G.Vector v Double) => v Double -> Double
varianceUnbiased samp
| n > 1 = robustSumVar (mean samp) samp / fromIntegral (n-1)
| otherwise = 0
where
n = G.length samp
{-# INLINE varianceUnbiased #-}
-- | 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.
meanVariance :: (G.Vector v Double) => v Double -> (Double,Double)
meanVariance samp
| n > 1 = (m, robustSumVar m samp / fromIntegral n)
| otherwise = (m, 0)
where
n = G.length samp
m = mean samp
{-# INLINE meanVariance #-}
-- | 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.
meanVarianceUnb :: (G.Vector v Double) => v Double -> (Double,Double)
meanVarianceUnb samp
| n > 1 = (m, robustSumVar m samp / fromIntegral (n-1))
| otherwise = (m, 0)
where
n = G.length samp
m = mean samp
{-# INLINE meanVarianceUnb #-}
-- | Standard deviation. This is simply the square root of the
-- unbiased estimate of the variance.
stdDev :: (G.Vector v Double) => v Double -> Double
stdDev = sqrt . varianceUnbiased
{-# INLINE stdDev #-}
robustSumVarWeighted :: (G.Vector v (Double,Double)) => v (Double,Double) -> V
robustSumVarWeighted samp = G.foldl' go (V 0 0) samp
where
go (V s w) (x,xw) = V (s + xw*d*d) (w + xw)
where d = x - m
m = meanWeighted samp
{-# INLINE robustSumVarWeighted #-}
-- | Weighted variance. This is biased estimation.
varianceWeighted :: (G.Vector v (Double,Double)) => v (Double,Double) -> Double
varianceWeighted samp
| G.length samp > 1 = fini $ robustSumVarWeighted samp
| otherwise = 0
where
fini (V s w) = s / w
{-# INLINE varianceWeighted #-}
-- $cancellation
--
-- 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.
fastVar :: (G.Vector v Double) => v Double -> T1
fastVar = G.foldl' go (T1 0 0 0)
where
go (T1 n m s) x = T1 n' m' s'
where n' = n + 1
m' = m + d / fromIntegral n'
s' = s + d * (x - m')
d = x - m
-- | Maximum likelihood estimate of a sample's variance.
fastVariance :: (G.Vector v Double) => v Double -> Double
fastVariance = fini . fastVar
where fini (T1 n _m s)
| n > 1 = s / fromIntegral n
| otherwise = 0
{-# INLINE fastVariance #-}
-- | Unbiased estimate of a sample's variance.
fastVarianceUnbiased :: (G.Vector v Double) => v Double -> Double
fastVarianceUnbiased = fini . fastVar
where fini (T1 n _m s)
| n > 1 = s / fromIntegral (n - 1)
| otherwise = 0
{-# INLINE fastVarianceUnbiased #-}
-- | Standard deviation. This is simply the square root of the
-- maximum likelihood estimate of the variance.
fastStdDev :: (G.Vector v Double) => v Double -> Double
fastStdDev = sqrt . fastVariance
{-# INLINE fastStdDev #-}
------------------------------------------------------------------------
-- Helper code. Monomorphic unpacked accumulators.
-- (^) operator from Prelude is just slow.
(^) :: Double -> Int -> Double
x ^ 1 = x
x ^ n = x * (x ^ (n-1))
{-# INLINE (^) #-}
-- don't support polymorphism, as we can't get unboxed returns if we use it.
data T = T {-# UNPACK #-}!Double {-# UNPACK #-}!Int
data T1 = T1 {-# UNPACK #-}!Int {-# UNPACK #-}!Double {-# UNPACK #-}!Double
{-
Consider this core:
with data T a = T !a !Int
$wfold :: Double#
-> Int#
-> Int#
-> (# Double, Int# #)
and without,
$wfold :: Double#
-> Int#
-> Int#
-> (# Double#, Int# #)
yielding to boxed returns and heap checks.
-}
-- $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&#8211;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&#8211;535. <http://doi.acm.org/10.1145/359146.359153>
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