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 ELynx.Distribution.Types
import GHC.Generics (Generic)
import qualified Statistics.Distribution as D

-- | 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
  { -- | Number of leaves of the tree.
    todTN :: Int,
    -- | Birth rate.
    todLa :: Rate,
    -- | Death rate.
    todMu :: 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