```{-# LANGUAGE BangPatterns #-}
-- |
-- Module    : Statistics.Math
-- Copyright : (c) 2009 Bryan O'Sullivan
--
-- Maintainer  : bos@serpentine.com
-- Stability   : experimental
-- Portability : portable
--
-- Mathematical functions for statistics.

module Statistics.Math
(
-- * Functions
chebyshev
, choose
-- ** Factorial functions
, factorial
, logFactorial
-- ** Gamma functions
, incompleteGamma
, logGamma
, logGammaL
-- * References
-- \$references
) where

import Data.Array.Vector
import Data.Word (Word64)
import Statistics.Constants (m_sqrt_2_pi)
import Statistics.Distribution (cumulative)
import Statistics.Distribution.Normal (standard)

data C = C {-# UNPACK #-} !Double {-# UNPACK #-} !Double {-# UNPACK #-} !Double

-- | Evaluate a series of Chebyshev polynomials. Uses Clenshaw's
-- algorithm.
chebyshev :: Double             -- ^ Parameter of each function.
-> UArr Double        -- ^ Coefficients of each polynomial
-- term, in increasing order.
-> Double
chebyshev x a = fini . foldlU step (C 0 0 0) .
enumFromThenToU (lengthU a - 1) (-1) \$ 0
where step (C u v w) k = C (x2 * v - w + indexU a k) u v
fini (C u _ w)   = (u - w) / 2
x2               = x * 2

-- | The binomial coefficient.
--
-- > 7 `choose` 3 == 35
choose :: Int -> Int -> Int
n `choose` k
| k > n = 0
| otherwise = ceiling . foldlU go 1 . enumFromToU 1 \$ k'
where go a i = a * (nk + j) / j
where j = fromIntegral i :: Double
k' | k > n `div` 2 = n - k
| otherwise     = k
nk = fromIntegral (n - k')
{-# INLINE choose #-}

data F = F {-# UNPACK #-} !Word64 {-# UNPACK #-} !Word64

-- | Compute the factorial function /n/!.  Returns &#8734; if the
-- input is above 170 (above which the result cannot be represented by
-- a 64-bit 'Double').
factorial :: Int -> Double
factorial n
| n < 0     = error "Statistics.Math.factorial: negative input"
| n <= 1    = 0
| n <= 14   = fini . foldlU goLong (F 1 1) \$ ns
| otherwise = foldlU goDouble 1 \$ ns
where goDouble t k = t * fromIntegral k
goLong (F z x) _ = F (z * x') x'
where x' = x + 1
fini (F z _) = fromIntegral z
ns = enumFromToU 2 n
{-# INLINE factorial #-}

-- | Compute the natural logarithm of the factorial function.  Gives
-- 16 decimal digits of precision.
logFactorial :: Int -> Double
logFactorial n
| n <= 14   = log (factorial n)
| otherwise = (x - 0.5) * log x - x + 9.1893853320467e-1 + z / x
where x = fromIntegral (n + 1)
y = 1 / (x * x)
z = ((-(5.95238095238e-4 * y) + 7.936500793651e-4) * y -
2.7777777777778e-3) * y + 8.3333333333333e-2
{-# INLINE logFactorial #-}

-- | Compute the incomplete gamma integral function &#947;(/s/,/x/).
-- Uses Algorithm AS 239 by Shea.
incompleteGamma :: Double       -- ^ /s/
-> Double       -- ^ /x/
-> Double
incompleteGamma x p
| x < 0 || p <= 0 = 1/0
| x == 0          = 0
| p >= 1000       = norm (3 * sqrt p * ((x/p) ** (1/3) + 1/(9*p) - 1))
| x >= 1e8        = 0
| x <= 1 || x < p = let a = p * log x - x - logGamma (p + 1)
g = a + log (pearson p 1 1)
in if g > limit then exp g else 0
| otherwise       = let g = p * log x - x - logGamma p + log cf
in if g > limit then 1 - exp g else 1
where
norm = cumulative standard
pearson !a !c !g
| c' <= tolerance = g'
| otherwise       = pearson a' c' g'
where a' = a + 1
c' = c * x / a'
g' = g + c'
cf = let a = 1 - p
b = a + x + 1
p3 = x + 1
p4 = x * b
in contFrac a b 0 1 x p3 p4 (p3/p4)
contFrac !a !b !c !p1 !p2 !p3 !p4 !g
| abs (g - rn) <= min tolerance (tolerance * rn) = g
| otherwise = contFrac a' b' c' (f p3) (f p4) (f p5) (f p6) rn
where a' = a + 1
b' = b + 2
c' = c + 1
an = a' * c'
p5 = b' * p3 - an * p1
p6 = b' * p4 - an * p2
rn = p5 / p6
f n | abs p5 > overflow = n / overflow
| otherwise         = n
limit     = -88
tolerance = 1e-14
overflow  = 1e37

-- | Compute the logarithm of the gamma function &#915;(/x/).  Uses
-- Algorithm AS 245 by Macleod.
--
-- Gives an accuracy of 10&#8211;12 significant decimal digits, except
-- for small regions around /x/ = 1 and /x/ = 2, where the function
-- goes to zero.  For greater accuracy, use 'logGammaL'.
--
-- Returns &#8734; if the input is outside of the range (0 < /x/
-- &#8804; 1e305).
logGamma :: Double -> Double
logGamma x
| x <= 0    = 1/0
| x < 1.5   = a + c *
((((r1_4 * b + r1_3) * b + r1_2) * b + r1_1) * b + r1_0) /
((((b + r1_8) * b + r1_7) * b + r1_6) * b + r1_5)
| x < 4     = (x - 2) *
((((r2_4 * x + r2_3) * x + r2_2) * x + r2_1) * x + r2_0) /
((((x + r2_8) * x + r2_7) * x + r2_6) * x + r2_5)
| x < 12    = ((((r3_4 * x + r3_3) * x + r3_2) * x + r3_1) * x + r3_0) /
((((x + r3_8) * x + r3_7) * x + r3_6) * x + r3_5)
| x > 5.1e5 = k
| otherwise = k + x1 *
((r4_2 * x2 + r4_1) * x2 + r4_0) /
((x2 + r4_4) * x2 + r4_3)
where
a :*: b :*: c
| x < 0.5   = -y :*: x + 1 :*: x
| otherwise = 0  :*: x     :*: x - 1

y      = log x
k      = x * (y-1) - 0.5 * y + alr2pi
alr2pi = 0.918938533204673

x1 = 1 / x
x2 = x1 * x1

r1_0 = -2.66685511495; r1_1 = -24.4387534237; r1_2 = -21.9698958928
r1_3 = 11.1667541262; r1_4 = 3.13060547623; r1_5 = 0.607771387771
r1_6 = 11.9400905721; r1_7 = 31.4690115749; r1_8 = 15.2346874070

r2_0 = -78.3359299449; r2_1 = -142.046296688; r2_2 = 137.519416416
r2_3 = 78.6994924154; r2_4 = 4.16438922228; r2_5 = 47.0668766060
r2_6 = 313.399215894; r2_7 = 263.505074721; r2_8 = 43.3400022514

r3_0 = -2.12159572323; r3_1 = 2.30661510616; r3_2 = 2.74647644705
r3_3 = -4.02621119975; r3_4 = -2.29660729780; r3_5 = -1.16328495004
r3_6 = -1.46025937511; r3_7 = -2.42357409629; r3_8 = -5.70691009324

r4_0 = 0.279195317918525; r4_1 = 0.4917317610505968;
r4_2 = 0.0692910599291889; r4_3 = 3.350343815022304
r4_4 = 6.012459259764103

data L = L {-# UNPACK #-} !Double {-# UNPACK #-} !Double

-- | Compute the logarithm of the gamma function, &#915;(/x/).  Uses a
-- Lanczos approximation.
--
-- This function is slower than 'logGamma', but gives 14 or more
-- significant decimal digits of accuracy, except around /x/ = 1 and
-- /x/ = 2, where the function goes to zero.
--
-- Returns &#8734; if the input is outside of the range (0 < /x/
-- &#8804; 1e305).
logGammaL :: Double -> Double
logGammaL x
| x <= 0    = 1/0
| otherwise = fini . foldlU go (L 0 (x+7)) \$ a
where fini (L l _) = log (l+a0) + log m_sqrt_2_pi - x65 + (x-0.5) * log x65
go (L l t) k = L (l + k / t) (t-1)
x65 = x + 6.5
a0  = 0.9999999999995183
a   = toU [ 0.1659470187408462e-06
, 0.9934937113930748e-05
, -0.1385710331296526
, 12.50734324009056
, -176.6150291498386
, 771.3234287757674
, -1259.139216722289
, 676.5203681218835
]

-- \$references
--
-- * Clenshaw, C.W. (1962) Chebyshev series for mathematical
--   functions. /National Physical Laboratory Mathematical Tables 5/,
--   Her Majesty's Stationery Office, London.
--
-- * Lanczos, C. (1964) A precision approximation of the gamma
--   function.  /SIAM Journal on Numerical Analysis B/
--   1:86&#8211;96. <http://www.jstor.org/stable/2949767>
--
-- * Macleod, A.J. (1989) Algorithm AS 245: A robust and reliable
--   algorithm for the logarithm of the gamma function.
--   /Journal of the Royal Statistical Society, Series C (Applied Statistics)/
--   38(2):397&#8211;402. <http://www.jstor.org/stable/2348078>
--
-- * Shea, B. (1988) Algorithm AS 239: Chi-squared and incomplete
--   gamma integral. /Applied Statistics/
--   37(3):466&#8211;473. <http://www.jstor.org/stable/2347328>
```