-- |
-- Module    : Statistics.Distribution.Poisson.Internal
-- Copyright : (c) 2011 Bryan O'Sullivan
-- License   : BSD3
--
-- Maintainer  : bos@serpentine.com
-- Stability   : experimental
-- Portability : portable
--
-- Internal code for the Poisson distribution.

module Statistics.Distribution.Poisson.Internal
    (
      probability, poissonEntropy
    ) where

import Data.List (unfoldr)
import Numeric.MathFunctions.Constants (m_sqrt_2_pi, m_tiny, m_epsilon)
import Numeric.SpecFunctions (logGamma, stirlingError {-, choose, logFactorial -})
import Numeric.SpecFunctions.Extra (bd0)

-- | An unchecked, non-integer-valued version of Loader's saddle point
-- algorithm.
probability :: Double -> Double -> Double
probability :: Double -> Double -> Double
probability 0      0     = 1
probability 0      1     = 0
probability lambda :: Double
lambda x :: Double
x
  | Double -> Bool
forall a. RealFloat a => a -> Bool
isInfinite Double
lambda    = 0
  | Double
x Double -> Double -> Bool
forall a. Ord a => a -> a -> Bool
< 0                = 0
  | Double
x Double -> Double -> Bool
forall a. Ord a => a -> a -> Bool
<= Double
lambda Double -> Double -> Double
forall a. Num a => a -> a -> a
* Double
m_tiny = Double -> Double
forall a. Floating a => a -> a
exp (-Double
lambda)
  | Double
lambda Double -> Double -> Bool
forall a. Ord a => a -> a -> Bool
< Double
x Double -> Double -> Double
forall a. Num a => a -> a -> a
* Double
m_tiny  = Double -> Double
forall a. Floating a => a -> a
exp (-Double
lambda Double -> Double -> Double
forall a. Num a => a -> a -> a
+ Double
x Double -> Double -> Double
forall a. Num a => a -> a -> a
* Double -> Double
forall a. Floating a => a -> a
log Double
lambda Double -> Double -> Double
forall a. Num a => a -> a -> a
- Double -> Double
logGamma (Double
xDouble -> Double -> Double
forall a. Num a => a -> a -> a
+1))
  | Bool
otherwise            = Double -> Double
forall a. Floating a => a -> a
exp (-(Double -> Double
stirlingError Double
x) Double -> Double -> Double
forall a. Num a => a -> a -> a
- Double -> Double -> Double
bd0 Double
x Double
lambda) Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/
                           (Double
m_sqrt_2_pi Double -> Double -> Double
forall a. Num a => a -> a -> a
* Double -> Double
forall a. Floating a => a -> a
sqrt Double
x)

-- -- | Compute entropy using Theorem 1 from "Sharp Bounds on the Entropy
-- -- of the Poisson Law".  This function is unused because 'directEntorpy'
-- -- is just as accurate and is faster by about a factor of 4.
-- alyThm1 :: Double -> Double
-- alyThm1 lambda =
--   sum (takeWhile (\x -> abs x >= m_epsilon * lll) alySeries) + lll
--   where lll = lambda * (1 - log lambda)
--         alySeries =
--           [ alyc k * exp (fromIntegral k * log lambda - logFactorial k)
--           | k <- [2..] ]

-- alyc :: Int -> Double
-- alyc k =
--   sum [ parity j * choose (k-1) j * log (fromIntegral j+1) | j <- [0..k-1] ]
--   where parity j
--           | even (k-j) = -1
--           | otherwise  = 1

-- | Returns [x, x^2, x^3, x^4, ...]
powers :: Double -> [Double]
powers :: Double -> [Double]
powers x :: Double
x = (Double -> Maybe (Double, Double)) -> Double -> [Double]
forall b a. (b -> Maybe (a, b)) -> b -> [a]
unfoldr (\y :: Double
y -> (Double, Double) -> Maybe (Double, Double)
forall a. a -> Maybe a
Just (Double
yDouble -> Double -> Double
forall a. Num a => a -> a -> a
*Double
x,Double
yDouble -> Double -> Double
forall a. Num a => a -> a -> a
*Double
x)) 1

-- | Returns an upper bound according to theorem 2 of "Sharp Bounds on
-- the Entropy of the Poisson Law"
alyThm2Upper :: Double -> [Double] -> Double
alyThm2Upper :: Double -> [Double] -> Double
alyThm2Upper lambda :: Double
lambda coefficients :: [Double]
coefficients =
  1.4189385332046727 Double -> Double -> Double
forall a. Num a => a -> a -> a
+ 0.5 Double -> Double -> Double
forall a. Num a => a -> a -> a
* Double -> Double
forall a. Floating a => a -> a
log Double
lambda Double -> Double -> Double
forall a. Num a => a -> a -> a
+
  Double -> [Double] -> Double
zipCoefficients Double
lambda [Double]
coefficients

-- | Returns the average of the upper and lower bounds accounding to
-- theorem 2.
alyThm2 :: Double -> [Double] -> [Double] -> Double
alyThm2 :: Double -> [Double] -> [Double] -> Double
alyThm2 lambda :: Double
lambda upper :: [Double]
upper lower :: [Double]
lower =
  Double -> [Double] -> Double
alyThm2Upper Double
lambda [Double]
upper Double -> Double -> Double
forall a. Num a => a -> a -> a
+ 0.5 Double -> Double -> Double
forall a. Num a => a -> a -> a
* (Double -> [Double] -> Double
zipCoefficients Double
lambda [Double]
lower)

zipCoefficients :: Double -> [Double] -> Double
zipCoefficients :: Double -> [Double] -> Double
zipCoefficients lambda :: Double
lambda coefficients :: [Double]
coefficients =
  ([Double] -> Double
forall (t :: * -> *) a. (Foldable t, Num a) => t a -> a
sum ([Double] -> Double) -> [Double] -> Double
forall a b. (a -> b) -> a -> b
$ ((Double, Double) -> Double) -> [(Double, Double)] -> [Double]
forall a b. (a -> b) -> [a] -> [b]
map ((Double -> Double -> Double) -> (Double, Double) -> Double
forall a b c. (a -> b -> c) -> (a, b) -> c
uncurry Double -> Double -> Double
forall a. Num a => a -> a -> a
(*)) ([Double] -> [Double] -> [(Double, Double)]
forall a b. [a] -> [b] -> [(a, b)]
zip (Double -> [Double]
powers (Double -> [Double]) -> Double -> [Double]
forall a b. (a -> b) -> a -> b
$ Double -> Double
forall a. Fractional a => a -> a
recip Double
lambda) [Double]
coefficients))

-- Mathematica code deriving the coefficients below:
--
-- poissonMoment[0, s_] := 1
-- poissonMoment[1, s_] := 0
-- poissonMoment[k_, s_] :=
--   Sum[s * Binomial[k - 1, j] * poissonMoment[j, s], {j, 0, k - 2}]
--
-- upperSeries[m_]  :=
--  Distribute[Integrate[
--    Sum[(-1)^(j - 1) *
--      poissonMoment[j, \[Lambda]] / (j * (j - 1)* \[Lambda]^j),
--     {j, 3, 2 m - 1}],
--    \[Lambda]]]
--
-- lowerSeries[m_] :=
--  Distribute[Integrate[
--    poissonMoment[
--      2 m + 2, \[Lambda]] / ((2 m +
--         1)*\[Lambda]^(2 m + 2)), \[Lambda]]]
--
-- upperBound[m_] := upperSeries[m] + (Log[2*Pi*\[Lambda]] + 1)/2
--
-- lowerBound[m_] := upperBound[m] + lowerSeries[m]

upperCoefficients4 :: [Double]
upperCoefficients4 :: [Double]
upperCoefficients4 = [1Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/12, 1Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/24, -103Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/180, -13Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/40, -1Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/210]

lowerCoefficients4 :: [Double]
lowerCoefficients4 :: [Double]
lowerCoefficients4 = [0,0,0, -105Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/4, -210, -2275Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/18, -167Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/21, -1Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/72]

upperCoefficients6 :: [Double]
upperCoefficients6 :: [Double]
upperCoefficients6 = [1Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/12, 1Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/24, 19Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/360, 9Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/80, -38827Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/2520,
                      -74855Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/1008, -73061Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/2520, -827Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/720, -1Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/990]

lowerCoefficients6 :: [Double]
lowerCoefficients6 :: [Double]
lowerCoefficients6 = [0,0,0,0,0, -3465Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/2, -45045, -466235Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/4, -531916Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/9,
                      -56287Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/10, -629Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/11, -1Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/156]

upperCoefficients8 :: [Double]
upperCoefficients8 :: [Double]
upperCoefficients8 = [1Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/12, 1Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/24, 19Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/360, 9Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/80, 863Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/2520, 1375Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/1008,
                      -3023561Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/2520, -15174047Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/720, -231835511Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/5940,
                      -18927611Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/1320, -58315591Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/60060, -23641Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/3640,
                      -1Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/2730]

lowerCoefficients8 :: [Double]
lowerCoefficients8 :: [Double]
lowerCoefficients8 = [0,0,0,0,0,0,0, -2027025Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/8, -15315300, -105252147,
                      -178127950, -343908565Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/4, -10929270, -3721149Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/14,
                      -7709Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/15, -1Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/272]

upperCoefficients10 :: [Double]
upperCoefficients10 :: [Double]
upperCoefficients10 = [1Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/12, 1Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/24, 19Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/360, 9,80, 863Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/2520, 1375Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/1008,
                       33953Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/5040, 57281Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/1440, -2271071617Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/11880,
                       -1483674219Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/176, -31714406276557Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/720720,
                       -7531072742237Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/131040, -1405507544003Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/65520,
                       -21001919627Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/10080, -1365808297Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/36720,
                       -26059Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/544, -1Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/5814]

lowerCoefficients10 :: [Double]
lowerCoefficients10 :: [Double]
lowerCoefficients10 = [0,0,0,0,0,0,0,0,0,-130945815Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/2, -7638505875,
                       -438256243425Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/4, -435477637540, -3552526473925Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/6,
                       -857611717105Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/3, -545654955967Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/12, -5794690528Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/3,
                       -578334559Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/42, -699043Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/133, -1Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/420]

upperCoefficients12 :: [Double]
upperCoefficients12 :: [Double]
upperCoefficients12 = [1Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/12, 1Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/24, 19Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/360, 863Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/2520, 1375Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/1008,
                       33953Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/5040, 57281Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/1440, 3250433Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/11880,
                       378351Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/176, -37521922090657Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/720720,
                       -612415657466657Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/131040, -3476857538815223Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/65520,
                       -243882174660761Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/1440, -34160796727900637Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/183600,
                       -39453820646687Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/544, -750984629069237Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/81396,
                       -2934056300989Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/9576, -20394527513Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/12540,
                       -3829559Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/9240, -1Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/10626]

lowerCoefficients12 :: [Double]
lowerCoefficients12 :: [Double]
lowerCoefficients12 = [0,0,0,0,0,0,0,0,0,0,0,
                       -105411381075Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/4, -5270569053750, -272908057767345Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/2,
                       -1051953238104769, -24557168490009155Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/8,
                       -3683261873403112, -5461918738302026Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/3,
                       -347362037754732, -2205885452434521Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/100,
                       -12237195698286Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/35, -16926981721Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/22,
                       -6710881Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/155, -1Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/600]

-- | Compute entropy directly from its definition. This is just as accurate
-- as 'alyThm1' for lambda <= 1 and is faster, but is slow for large lambda,
-- and produces some underestimation due to accumulation of floating point
-- error.
directEntropy :: Double -> Double
directEntropy :: Double -> Double
directEntropy lambda :: Double
lambda =
  Double -> Double
forall a. Num a => a -> a
negate (Double -> Double) -> ([Double] -> Double) -> [Double] -> Double
forall b c a. (b -> c) -> (a -> b) -> a -> c
. [Double] -> Double
forall (t :: * -> *) a. (Foldable t, Num a) => t a -> a
sum ([Double] -> Double) -> [Double] -> Double
forall a b. (a -> b) -> a -> b
$
  (Double -> Bool) -> [Double] -> [Double]
forall a. (a -> Bool) -> [a] -> [a]
takeWhile (Double -> Double -> Bool
forall a. Ord a => a -> a -> Bool
< Double -> Double
forall a. Num a => a -> a
negate Double
m_epsilon Double -> Double -> Double
forall a. Num a => a -> a -> a
* Double
lambda) ([Double] -> [Double]) -> [Double] -> [Double]
forall a b. (a -> b) -> a -> b
$
  (Double -> Bool) -> [Double] -> [Double]
forall a. (a -> Bool) -> [a] -> [a]
dropWhile (Bool -> Bool
not (Bool -> Bool) -> (Double -> Bool) -> Double -> Bool
forall b c a. (b -> c) -> (a -> b) -> a -> c
. (Double -> Double -> Bool
forall a. Ord a => a -> a -> Bool
< Double -> Double
forall a. Num a => a -> a
negate Double
m_epsilon Double -> Double -> Double
forall a. Num a => a -> a -> a
* Double
lambda)) ([Double] -> [Double]) -> [Double] -> [Double]
forall a b. (a -> b) -> a -> b
$
  [ let x :: Double
x = Double -> Double -> Double
probability Double
lambda Double
k in Double
x Double -> Double -> Double
forall a. Num a => a -> a -> a
* Double -> Double
forall a. Floating a => a -> a
log Double
x | Double
k <- [0..]]

-- | Compute the entropy of a poisson distribution using the best available
-- method.
poissonEntropy :: Double -> Double
poissonEntropy :: Double -> Double
poissonEntropy lambda :: Double
lambda
  | Double
lambda Double -> Double -> Bool
forall a. Eq a => a -> a -> Bool
== 0 = 0
  | Double
lambda Double -> Double -> Bool
forall a. Ord a => a -> a -> Bool
<= 10 = Double -> Double
directEntropy Double
lambda
  | Double
lambda Double -> Double -> Bool
forall a. Ord a => a -> a -> Bool
<= 12 = Double -> [Double] -> [Double] -> Double
alyThm2 Double
lambda [Double]
upperCoefficients4 [Double]
lowerCoefficients4
  | Double
lambda Double -> Double -> Bool
forall a. Ord a => a -> a -> Bool
<= 18 = Double -> [Double] -> [Double] -> Double
alyThm2 Double
lambda [Double]
upperCoefficients6 [Double]
lowerCoefficients6
  | Double
lambda Double -> Double -> Bool
forall a. Ord a => a -> a -> Bool
<= 24 = Double -> [Double] -> [Double] -> Double
alyThm2 Double
lambda [Double]
upperCoefficients8 [Double]
lowerCoefficients8
  | Double
lambda Double -> Double -> Bool
forall a. Ord a => a -> a -> Bool
<= 30 = Double -> [Double] -> [Double] -> Double
alyThm2 Double
lambda [Double]
upperCoefficients10 [Double]
lowerCoefficients10
  | Bool
otherwise = Double -> [Double] -> [Double] -> Double
alyThm2 Double
lambda [Double]
upperCoefficients12 [Double]
lowerCoefficients12