-- | See McHale, Vanessa [\"Hypergeometric Functions for Statistical Computing\"](http://vmchale.com/static/serve/hypergeometric.pdf) and especially Shaw, Ernest [\"Hypergeometric Functions and CDFs in J\"](https://www.jsoftware.com/papers/jhyper.pdf)
module Math.Hypergeometric ( hypergeometric
                           , erf
                           , ncdf
                           ) where

import           Data.Functor ((<$>))

risingFactorial :: Num a => a -> Int -> a
risingFactorial :: forall a. Num a => a -> Int -> a
risingFactorial a
_ Int
0 = a
1
risingFactorial a
a Int
n = (a
a a -> a -> a
forall a. Num a => a -> a -> a
+ Int -> a
forall a b. (Integral a, Num b) => a -> b
fromIntegral Int
n a -> a -> a
forall a. Num a => a -> a -> a
- a
1) a -> a -> a
forall a. Num a => a -> a -> a
* a -> Int -> a
forall a. Num a => a -> Int -> a
risingFactorial a
a (Int
nInt -> Int -> Int
forall a. Num a => a -> a -> a
-Int
1)

factorial :: Num a => Int -> a
factorial :: forall a. Num a => Int -> a
factorial Int
n = [a] -> a
forall (t :: * -> *) a. (Foldable t, Num a) => t a -> a
product (Int -> a
forall a b. (Integral a, Num b) => a -> b
fromIntegral (Int -> a) -> [Int] -> [a]
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> [Int
1..Int
n])

-- prop_cdf :: (Double -> Double) -> Double -> Bool
-- prop_cdf f x = f x <= 1

{-# SPECIALIZE ncdf :: Double -> Double #-}
-- | CDF of the standard normal \( N(0,1) \)
ncdf :: (Eq a, Floating a) => a -> a
ncdf :: forall a. (Eq a, Floating a) => a -> a
ncdf a
z = (a
1a -> a -> a
forall a. Fractional a => a -> a -> a
/a
2) a -> a -> a
forall a. Num a => a -> a -> a
* (a
1 a -> a -> a
forall a. Num a => a -> a -> a
+ a -> a
forall a. (Eq a, Floating a) => a -> a
erf (a
z a -> a -> a
forall a. Fractional a => a -> a -> a
/ a -> a
forall a. Floating a => a -> a
sqrt a
2))

{-# SPECIALIZE erf :: Double -> Double #-}
-- | [erf](https://mathworld.wolfram.com/Erf.html)
erf :: (Eq a, Floating a) => a -> a
erf :: forall a. (Eq a, Floating a) => a -> a
erf a
z = (a
2 a -> a -> a
forall a. Num a => a -> a -> a
* a
z a -> a -> a
forall a. Num a => a -> a -> a
* a -> a
forall a. Floating a => a -> a
exp (-a
za -> Int -> a
forall a b. (Num a, Integral b) => a -> b -> a
^(Int
2::Int)) a -> a -> a
forall a. Fractional a => a -> a -> a
/ a -> a
forall a. Floating a => a -> a
sqrt a
forall a. Floating a => a
pi) a -> a -> a
forall a. Num a => a -> a -> a
* [a] -> [a] -> a -> a
forall a. (Eq a, Fractional a) => [a] -> [a] -> a -> a
hypergeometric [a
1] [a
3a -> a -> a
forall a. Fractional a => a -> a -> a
/a
2] (a
za -> Int -> a
forall a b. (Num a, Integral b) => a -> b -> a
^(Int
2::Int))

{-# SPECIALIZE hypergeometric :: [Double] -> [Double] -> Double -> Double #-}
-- | \( _pF_q(a_1,\ldots,a_p;b_1,\ldots,b_q;z) = \displaystyle\sum_{n=0}^\infty\frac{(a_1)_n\cdots(a_p)_n}{(b_1)_b\cdots(b_q)_n}\frac{z^n}{n!} \)
--
-- This iterates until the result stabilizes.
hypergeometric :: (Eq a, Fractional a)
               => [a] -- ^ \( a_1,\ldots,a_p \)
               -> [a] -- ^ \( b_1,\ldots,b_q \)
               -> a -- ^ \( z \)
               -> a
hypergeometric :: forall a. (Eq a, Fractional a) => [a] -> [a] -> a -> a
hypergeometric [a]
as [a]
bs a
z = [a] -> a
forall a. (Eq a, Num a) => [a] -> a
sumUntilEq
    [ ([a] -> a
forall (t :: * -> *) a. (Foldable t, Num a) => t a -> a
product ((a -> a) -> [a] -> [a]
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap (a -> Int -> a
forall a. Num a => a -> Int -> a
`risingFactorial` Int
n) [a]
as) a -> a -> a
forall a. Fractional a => a -> a -> a
/ [a] -> a
forall (t :: * -> *) a. (Foldable t, Num a) => t a -> a
product ((a -> a) -> [a] -> [a]
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap (a -> Int -> a
forall a. Num a => a -> Int -> a
`risingFactorial` Int
n) [a]
bs)) a -> a -> a
forall a. Num a => a -> a -> a
* (a
z a -> Int -> a
forall a b. (Num a, Integral b) => a -> b -> a
^ Int
n) a -> a -> a
forall a. Fractional a => a -> a -> a
/ Int -> a
forall a. Num a => Int -> a
factorial Int
n | Int
n <- [Int
0..] ]

sumUntilEq :: (Eq a, Num a) => [a] -> a
sumUntilEq :: forall a. (Eq a, Num a) => [a] -> a
sumUntilEq = a -> [a] -> a
forall a. (Eq a, Num a) => a -> [a] -> a
sumUntilEqLoop a
0

sumUntilEqLoop :: (Eq a, Num a) => a -> [a] -> a
sumUntilEqLoop :: forall a. (Eq a, Num a) => a -> [a] -> a
sumUntilEqLoop a
acc (a
x:a
y:[a]
xs) =
    if a
step0 a -> a -> Bool
forall a. Eq a => a -> a -> Bool
== a
step1
        then a
step0
        else a -> [a] -> a
forall a. (Eq a, Num a) => a -> [a] -> a
sumUntilEqLoop a
step1 [a]
xs
    where step0 :: a
step0 = a
acc a -> a -> a
forall a. Num a => a -> a -> a
+ a
x
          step1 :: a
step1 = a
acc a -> a -> a
forall a. Num a => a -> a -> a
+ a
x a -> a -> a
forall a. Num a => a -> a -> a
+ a
y