TimeOfOrigin.hs

{-# LANGUAGE DeriveDataTypeable #-}
{-# LANGUAGE DeriveGeneric      #-}

{- |
   Module      :  ELynx.Distribution.TimeOfOrigin
   Description :  Distribution of time of origin for birth and death trees
   Copyright   :  (c) Dominik Schrempf 2018
   License     :  GPL-3.0-or-later

   Maintainer  :  dominik.schrempf@gmail.com
   Stability   :  unstable
   Portability :  portable

Creation date: Tue Feb 13 13:16:18 2018.

See Gernhard, T. (2008). The conditioned reconstructed process. Journal of
Theoretical Biology, 253(4), 769–778. http://doi.org/10.1016/j.jtbi.2008.04.005.

Distribution of the time of origin for birth and death trees. See corollary 3.3
in the paper cited above.

-}

module ELynx.Distribution.TimeOfOrigin
  ( TimeOfOriginDistribution(..)
  , cumulative
  , density
  , quantile
  )
where

import           Data.Data                      ( Data
                                                , Typeable
                                                )
import           GHC.Generics                   ( Generic )
import qualified Statistics.Distribution       as D

import           ELynx.Distribution.Types

-- | Distribution of the time of origin for a phylogenetic tree evolving under
-- the birth and death process and conditioned on observing n leaves today.
data TimeOfOriginDistribution = TOD
  { todTN :: Int           -- ^ Number of leaves of the tree.
  , todLa :: Rate          -- ^ Birth rate.
  , todMu :: Rate          -- ^ Death rate.
  } deriving (Eq, Typeable, Data, Generic)

instance D.Distribution TimeOfOriginDistribution where
  cumulative = cumulative

-- | Cumulative distribution function Corollary 3.3.
cumulative :: TimeOfOriginDistribution -> Time -> Double
cumulative (TOD n l m) x | x <= 0    = 0
                         | otherwise = te ** fromIntegral n
 where
  d  = l - m
  te = l * (1.0 - exp (-d * x)) / (l - m * exp (-d * x))

instance D.ContDistr TimeOfOriginDistribution where
  density  = density
  quantile = quantile

-- | The density function Eq. (5).
density :: TimeOfOriginDistribution -> Time -> Double
density (TOD nn l m) x
  | x < 0     = 0
  | otherwise = n * l ** n * d ** 2 * t1 ** (n - 1.0) * ex / (t2 ** (n + 1.0))
 where
  d  = l - m
  n  = fromIntegral nn
  ex = exp (-d * x)
  t1 = 1.0 - ex
  t2 = l - m * ex

-- | The inverted cumulative probability distribution 'cumulative'. See also
-- 'D.ContDistr'.
quantile :: TimeOfOriginDistribution -> Double -> Time
quantile (TOD n' l m) p
  | p >= 0 && p <= 1
  = -1.0 / d * log (t1 / t2)
  | otherwise
  = error
    $  "PointProcess.quantile: p must be in [0,1] range. Got: "
    ++ show p
    ++ "."
 where
  d  = l - m
  n  = fromIntegral n'
  t1 = l * (1.0 - p ** (1.0 / n))
  t2 = l - p ** (1.0 / n) * m

instance D.ContGen TimeOfOriginDistribution where
  genContVar = D.genContinuous