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-3.0-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.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