```-- |
-- Module    : Statistics.Sample
-- Copyright : (c) 2008 Don Stewart, 2009 Bryan O'Sullivan
--
-- Maintainer  : bos@serpentine.com
-- Stability   : experimental
-- Portability : portable
--
-- Commonly used sample statistics, also known as descriptive
-- statistics.

module Statistics.Sample
(
-- * Types
Sample
-- * Statistics of location
, mean
, harmonicMean
, geometricMean

-- * Statistics of dispersion
-- \$variance

-- ** Two-pass functions (numerically robust)
-- \$robust
, variance
, varianceUnbiased
, stdDev

-- ** Single-pass functions (faster, less safe)
-- \$cancellation
, fastVariance
, fastVarianceUnbiased
, fastStdDev

-- * References
-- \$references
) where

import Data.Array.Vector (foldlU)
import Statistics.Types (Sample)

-- | Arithmetic mean.  This uses Welford's algorithm to provide
-- numerical stability, using a single pass over the sample data.
mean :: Sample -> Double
mean = fini . foldlU 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 #-}

-- | Harmonic mean.  This algorithm performs a single pass over the
-- sample.
harmonicMean :: Sample -> Double
harmonicMean = fini . foldlU 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 #-}

-- | Geometric mean of a sample containing no negative values.
geometricMean :: Sample -> Double
geometricMean = fini . foldlU go (T 1 0)
where
fini (T p n) = p ** (1 / fromIntegral n)
go (T p n) a = T (p * a) (n + 1)
{-# INLINE geometricMean #-}

-- \$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.

robustVar :: Sample -> T
robustVar samp = fini . foldlU go (T1 0 0 0) \$ samp
where
go (T1 n s c) x = T1 n' s' c'
where n' = n + 1
s' = s + d * d
c' = c + d
d  = x - m
fini (T1 n s c) = T (s - c ** (2 / fromIntegral n)) n
m = mean samp

-- | Maximum likelihood estimate of a sample's variance.
variance :: Sample -> Double
variance = fini . robustVar
where fini (T v n)
| n > 1     = v / fromIntegral n
| otherwise = 0
{-# INLINE variance #-}

-- | Unbiased estimate of a sample's variance.
varianceUnbiased :: Sample -> Double
varianceUnbiased = fini . robustVar
where fini (T v n)
| n > 1     = v / fromIntegral (n-1)
| otherwise = 0
{-# INLINE varianceUnbiased #-}

-- | Standard deviation.  This is simply the square root of the
-- maximum likelihood estimate of the variance.
stdDev :: Sample -> Double
stdDev = sqrt . varianceUnbiased

-- \$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 :: Sample -> T1
fastVar = foldlU 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 :: Sample -> 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 :: Sample -> 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 :: Sample -> Double
fastStdDev = sqrt . fastVariance
{-# INLINE fastStdDev #-}

------------------------------------------------------------------------
-- Helper code. Monomorphic unpacked accumulators.

-- 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>
```