TimeOfOriginNearCritical.hs

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

{- |
   Module      :  ELynx.Distribution.TimeOfOriginNearCritical
   Description :  Distribution of time of origin for birth and death trees
   Copyright   :  (c) Dominik Schrempf 2018
   License     :  GPL-3

   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.TimeOfOriginNearCritical
  ( TimeOfOriginNearCriticalDistribution(..)
  , 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 TimeOfOriginNearCriticalDistribution = TONCD
  { todTN :: Int           -- ^ Number of leaves of the tree.
  , todLa :: Rate          -- ^ Birth rate.
  , todMu :: Rate          -- ^ Death rate.
  } deriving (Eq, Typeable, Data, Generic)

instance D.Distribution TimeOfOriginNearCriticalDistribution where
    cumulative = cumulative

-- | Cumulative distribution function; see Mathematica notebook.
cumulative :: TimeOfOriginNearCriticalDistribution -> Time -> Double
cumulative (TONCD n' l m) t
  | t <= 0    = 0
  | otherwise = t1 + t2
  where d  = l - m
        n  = fromIntegral n'
        t1 = (t*l/(1.0 + t*l)) ** n
        t2 = (n * t * t1) * d / (2.0 * (1.0 + t*l))

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

-- | The density function Eq. (5).
density :: TimeOfOriginNearCriticalDistribution -> Time -> Double
density (TONCD n' l m) t
  | t < 0     = 0
  | otherwise = nom/den
  where n  = fromIntegral n'
        nom = n * (t*l/(1+t*l))**n * (2+(3+n)*t*l - (1+n)*t*m)
        den = 2*t*(1+t*l)**2

-- | The inverted cumulative probability distribution 'cumulative'. See also
-- 'D.ContDistr'.
quantile :: TimeOfOriginNearCriticalDistribution -> Double -> Time
quantile (TONCD n' l m) p
  | p >= 0 && p <= 1 = t1 + t2nom/t2den
  | otherwise        =
    error $ "PointProcess.quantile: p must be in [0,1] range. Got: " ++ show p ++ "."
  where n = fromIntegral n'
        t1 = - p**(1/n)/((-1+p**(1/n))*l)
        t2nom = p**(2/n)*(m-l)
        t2den = 2*(-1+p**(1/n))**2 * l**2

instance D.ContGen TimeOfOriginNearCriticalDistribution where
  genContVar = D.genContinuous