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

-- |
--   Module      :  ELynx.Tree.Distribution.BirthDeathNearlyCritical
--   Description :  Birth and death distribution
--   Copyright   :  2021 Dominik Schrempf
--   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 values of the point process such that it corresponds to
-- reconstructed trees under the birth and death process; nearly critical birth and
-- death process with lambda~mu.
--
-- Basically, this is a Taylor expansion of Eq. (2) and Eq. (3).
module ELynx.Tree.Distribution.BirthDeathNearlyCritical
  ( BirthDeathNearlyCriticalDistribution (..),
    cumulative,
    density,
    quantile,
  )
where

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

-- | Distribution of the values of the point process such that it corresponds to
-- a reconstructed tree of the birth and death process.
data BirthDeathNearlyCriticalDistribution = BDNCD
  { -- | Time to origin of the tree.
    BirthDeathNearlyCriticalDistribution -> Double
bdncdTOr :: Time,
    -- | Birth and death rate.
    BirthDeathNearlyCriticalDistribution -> Double
bdncdLa :: Rate,
    -- | Birth and death rate.
    BirthDeathNearlyCriticalDistribution -> Double
bdncdMu :: Rate
  }
  deriving (BirthDeathNearlyCriticalDistribution
-> BirthDeathNearlyCriticalDistribution -> Bool
forall a. (a -> a -> Bool) -> (a -> a -> Bool) -> Eq a
/= :: BirthDeathNearlyCriticalDistribution
-> BirthDeathNearlyCriticalDistribution -> Bool
$c/= :: BirthDeathNearlyCriticalDistribution
-> BirthDeathNearlyCriticalDistribution -> Bool
== :: BirthDeathNearlyCriticalDistribution
-> BirthDeathNearlyCriticalDistribution -> Bool
$c== :: BirthDeathNearlyCriticalDistribution
-> BirthDeathNearlyCriticalDistribution -> Bool
Eq, Typeable, Typeable BirthDeathNearlyCriticalDistribution
BirthDeathNearlyCriticalDistribution -> DataType
BirthDeathNearlyCriticalDistribution -> Constr
(forall b. Data b => b -> b)
-> BirthDeathNearlyCriticalDistribution
-> BirthDeathNearlyCriticalDistribution
forall a.
Typeable a
-> (forall (c :: * -> *).
    (forall d b. Data d => c (d -> b) -> d -> c b)
    -> (forall g. g -> c g) -> a -> c a)
-> (forall (c :: * -> *).
    (forall b r. Data b => c (b -> r) -> c r)
    -> (forall r. r -> c r) -> Constr -> c a)
-> (a -> Constr)
-> (a -> DataType)
-> (forall (t :: * -> *) (c :: * -> *).
    Typeable t =>
    (forall d. Data d => c (t d)) -> Maybe (c a))
-> (forall (t :: * -> * -> *) (c :: * -> *).
    Typeable t =>
    (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c a))
-> ((forall b. Data b => b -> b) -> a -> a)
-> (forall r r'.
    (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> a -> r)
-> (forall r r'.
    (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> a -> r)
-> (forall u. (forall d. Data d => d -> u) -> a -> [u])
-> (forall u. Int -> (forall d. Data d => d -> u) -> a -> u)
-> (forall (m :: * -> *).
    Monad m =>
    (forall d. Data d => d -> m d) -> a -> m a)
-> (forall (m :: * -> *).
    MonadPlus m =>
    (forall d. Data d => d -> m d) -> a -> m a)
-> (forall (m :: * -> *).
    MonadPlus m =>
    (forall d. Data d => d -> m d) -> a -> m a)
-> Data a
forall u.
Int
-> (forall d. Data d => d -> u)
-> BirthDeathNearlyCriticalDistribution
-> u
forall u.
(forall d. Data d => d -> u)
-> BirthDeathNearlyCriticalDistribution -> [u]
forall r r'.
(r -> r' -> r)
-> r
-> (forall d. Data d => d -> r')
-> BirthDeathNearlyCriticalDistribution
-> r
forall r r'.
(r' -> r -> r)
-> r
-> (forall d. Data d => d -> r')
-> BirthDeathNearlyCriticalDistribution
-> r
forall (m :: * -> *).
Monad m =>
(forall d. Data d => d -> m d)
-> BirthDeathNearlyCriticalDistribution
-> m BirthDeathNearlyCriticalDistribution
forall (m :: * -> *).
MonadPlus m =>
(forall d. Data d => d -> m d)
-> BirthDeathNearlyCriticalDistribution
-> m BirthDeathNearlyCriticalDistribution
forall (c :: * -> *).
(forall b r. Data b => c (b -> r) -> c r)
-> (forall r. r -> c r)
-> Constr
-> c BirthDeathNearlyCriticalDistribution
forall (c :: * -> *).
(forall d b. Data d => c (d -> b) -> d -> c b)
-> (forall g. g -> c g)
-> BirthDeathNearlyCriticalDistribution
-> c BirthDeathNearlyCriticalDistribution
forall (t :: * -> *) (c :: * -> *).
Typeable t =>
(forall d. Data d => c (t d))
-> Maybe (c BirthDeathNearlyCriticalDistribution)
forall (t :: * -> * -> *) (c :: * -> *).
Typeable t =>
(forall d e. (Data d, Data e) => c (t d e))
-> Maybe (c BirthDeathNearlyCriticalDistribution)
gmapMo :: forall (m :: * -> *).
MonadPlus m =>
(forall d. Data d => d -> m d)
-> BirthDeathNearlyCriticalDistribution
-> m BirthDeathNearlyCriticalDistribution
$cgmapMo :: forall (m :: * -> *).
MonadPlus m =>
(forall d. Data d => d -> m d)
-> BirthDeathNearlyCriticalDistribution
-> m BirthDeathNearlyCriticalDistribution
gmapMp :: forall (m :: * -> *).
MonadPlus m =>
(forall d. Data d => d -> m d)
-> BirthDeathNearlyCriticalDistribution
-> m BirthDeathNearlyCriticalDistribution
$cgmapMp :: forall (m :: * -> *).
MonadPlus m =>
(forall d. Data d => d -> m d)
-> BirthDeathNearlyCriticalDistribution
-> m BirthDeathNearlyCriticalDistribution
gmapM :: forall (m :: * -> *).
Monad m =>
(forall d. Data d => d -> m d)
-> BirthDeathNearlyCriticalDistribution
-> m BirthDeathNearlyCriticalDistribution
$cgmapM :: forall (m :: * -> *).
Monad m =>
(forall d. Data d => d -> m d)
-> BirthDeathNearlyCriticalDistribution
-> m BirthDeathNearlyCriticalDistribution
gmapQi :: forall u.
Int
-> (forall d. Data d => d -> u)
-> BirthDeathNearlyCriticalDistribution
-> u
$cgmapQi :: forall u.
Int
-> (forall d. Data d => d -> u)
-> BirthDeathNearlyCriticalDistribution
-> u
gmapQ :: forall u.
(forall d. Data d => d -> u)
-> BirthDeathNearlyCriticalDistribution -> [u]
$cgmapQ :: forall u.
(forall d. Data d => d -> u)
-> BirthDeathNearlyCriticalDistribution -> [u]
gmapQr :: forall r r'.
(r' -> r -> r)
-> r
-> (forall d. Data d => d -> r')
-> BirthDeathNearlyCriticalDistribution
-> r
$cgmapQr :: forall r r'.
(r' -> r -> r)
-> r
-> (forall d. Data d => d -> r')
-> BirthDeathNearlyCriticalDistribution
-> r
gmapQl :: forall r r'.
(r -> r' -> r)
-> r
-> (forall d. Data d => d -> r')
-> BirthDeathNearlyCriticalDistribution
-> r
$cgmapQl :: forall r r'.
(r -> r' -> r)
-> r
-> (forall d. Data d => d -> r')
-> BirthDeathNearlyCriticalDistribution
-> r
gmapT :: (forall b. Data b => b -> b)
-> BirthDeathNearlyCriticalDistribution
-> BirthDeathNearlyCriticalDistribution
$cgmapT :: (forall b. Data b => b -> b)
-> BirthDeathNearlyCriticalDistribution
-> BirthDeathNearlyCriticalDistribution
dataCast2 :: forall (t :: * -> * -> *) (c :: * -> *).
Typeable t =>
(forall d e. (Data d, Data e) => c (t d e))
-> Maybe (c BirthDeathNearlyCriticalDistribution)
$cdataCast2 :: forall (t :: * -> * -> *) (c :: * -> *).
Typeable t =>
(forall d e. (Data d, Data e) => c (t d e))
-> Maybe (c BirthDeathNearlyCriticalDistribution)
dataCast1 :: forall (t :: * -> *) (c :: * -> *).
Typeable t =>
(forall d. Data d => c (t d))
-> Maybe (c BirthDeathNearlyCriticalDistribution)
$cdataCast1 :: forall (t :: * -> *) (c :: * -> *).
Typeable t =>
(forall d. Data d => c (t d))
-> Maybe (c BirthDeathNearlyCriticalDistribution)
dataTypeOf :: BirthDeathNearlyCriticalDistribution -> DataType
$cdataTypeOf :: BirthDeathNearlyCriticalDistribution -> DataType
toConstr :: BirthDeathNearlyCriticalDistribution -> Constr
$ctoConstr :: BirthDeathNearlyCriticalDistribution -> Constr
gunfold :: forall (c :: * -> *).
(forall b r. Data b => c (b -> r) -> c r)
-> (forall r. r -> c r)
-> Constr
-> c BirthDeathNearlyCriticalDistribution
$cgunfold :: forall (c :: * -> *).
(forall b r. Data b => c (b -> r) -> c r)
-> (forall r. r -> c r)
-> Constr
-> c BirthDeathNearlyCriticalDistribution
gfoldl :: forall (c :: * -> *).
(forall d b. Data d => c (d -> b) -> d -> c b)
-> (forall g. g -> c g)
-> BirthDeathNearlyCriticalDistribution
-> c BirthDeathNearlyCriticalDistribution
$cgfoldl :: forall (c :: * -> *).
(forall d b. Data d => c (d -> b) -> d -> c b)
-> (forall g. g -> c g)
-> BirthDeathNearlyCriticalDistribution
-> c BirthDeathNearlyCriticalDistribution
Data, forall x.
Rep BirthDeathNearlyCriticalDistribution x
-> BirthDeathNearlyCriticalDistribution
forall x.
BirthDeathNearlyCriticalDistribution
-> Rep BirthDeathNearlyCriticalDistribution x
forall a.
(forall x. a -> Rep a x) -> (forall x. Rep a x -> a) -> Generic a
$cto :: forall x.
Rep BirthDeathNearlyCriticalDistribution x
-> BirthDeathNearlyCriticalDistribution
$cfrom :: forall x.
BirthDeathNearlyCriticalDistribution
-> Rep BirthDeathNearlyCriticalDistribution x
Generic)

instance D.Distribution BirthDeathNearlyCriticalDistribution where
  cumulative :: BirthDeathNearlyCriticalDistribution -> Double -> Double
cumulative = BirthDeathNearlyCriticalDistribution -> Double -> Double
cumulative

-- | Cumulative distribution function section 2.1.2, second formula.
cumulative :: BirthDeathNearlyCriticalDistribution -> Time -> Double
cumulative :: BirthDeathNearlyCriticalDistribution -> Double -> Double
cumulative (BDNCD Double
t Double
l Double
m) Double
s
  | Double
s forall a. Ord a => a -> a -> Bool
<= Double
0 = Double
0
  | Double
s forall a. Ord a => a -> a -> Bool
> Double
t = Double
1
  | Bool
otherwise = Double
o0 forall a. Num a => a -> a -> a
+ Double
o1
  where
    o0 :: Double
o0 = Double
s forall a. Num a => a -> a -> a
* (Double
1.0 forall a. Num a => a -> a -> a
+ Double
t forall a. Num a => a -> a -> a
* Double
l) forall a. Fractional a => a -> a -> a
/ Double
t forall a. Fractional a => a -> a -> a
/ (Double
1.0 forall a. Num a => a -> a -> a
+ Double
s forall a. Num a => a -> a -> a
* Double
l)
    o1 :: Double
o1 = (-Double
s forall a. Num a => a -> a -> a
* Double
s forall a. Num a => a -> a -> a
+ Double
s forall a. Num a => a -> a -> a
* Double
t) forall a. Num a => a -> a -> a
* (Double
m forall a. Num a => a -> a -> a
- Double
l) forall a. Fractional a => a -> a -> a
/ (Double
2.0 forall a. Num a => a -> a -> a
* Double
t forall a. Num a => a -> a -> a
* (Double
1.0 forall a. Num a => a -> a -> a
+ Double
s forall a. Num a => a -> a -> a
* Double
l) forall a. Floating a => a -> a -> a
** Double
2)

instance D.ContDistr BirthDeathNearlyCriticalDistribution where
  density :: BirthDeathNearlyCriticalDistribution -> Double -> Double
density = BirthDeathNearlyCriticalDistribution -> Double -> Double
density
  quantile :: BirthDeathNearlyCriticalDistribution -> Double -> Double
quantile = BirthDeathNearlyCriticalDistribution -> Double -> Double
quantile

-- | Density function section 2.1.2, first formula.
density :: BirthDeathNearlyCriticalDistribution -> Time -> Double
density :: BirthDeathNearlyCriticalDistribution -> Double -> Double
density (BDNCD Double
t Double
l Double
m) Double
s
  | Double
s forall a. Ord a => a -> a -> Bool
< Double
0 = Double
0
  | Double
s forall a. Ord a => a -> a -> Bool
> Double
t = Double
0
  | Bool
otherwise = Double
o0 forall a. Num a => a -> a -> a
+ Double
o1
  where
    o0 :: Double
o0 = (Double
1.0 forall a. Num a => a -> a -> a
+ Double
t forall a. Num a => a -> a -> a
* Double
l) forall a. Fractional a => a -> a -> a
/ (Double
t forall a. Num a => a -> a -> a
* (Double
1.0 forall a. Num a => a -> a -> a
+ Double
s forall a. Num a => a -> a -> a
* Double
l) forall a. Floating a => a -> a -> a
** Double
2)
    o1 :: Double
o1 = (-Double
2.0 forall a. Num a => a -> a -> a
* Double
s forall a. Num a => a -> a -> a
+ Double
t forall a. Num a => a -> a -> a
- Double
s forall a. Num a => a -> a -> a
* Double
t forall a. Num a => a -> a -> a
* Double
l) forall a. Num a => a -> a -> a
* (Double
m forall a. Num a => a -> a -> a
- Double
l) forall a. Fractional a => a -> a -> a
/ (Double
2.0 forall a. Num a => a -> a -> a
* Double
t forall a. Num a => a -> a -> a
* (Double
1.0 forall a. Num a => a -> a -> a
+ Double
s forall a. Num a => a -> a -> a
* Double
l) forall a. Floating a => a -> a -> a
** Double
3)

-- | Inverted cumulative probability distribution 'cumulative'. See also
-- 'D.ContDistr'.
quantile :: BirthDeathNearlyCriticalDistribution -> Double -> Time
quantile :: BirthDeathNearlyCriticalDistribution -> Double -> Double
quantile (BDNCD Double
t Double
l Double
m) Double
p
  | Double
p forall a. Ord a => a -> a -> Bool
>= Double
0 Bool -> Bool -> Bool
&& Double
p forall a. Ord a => a -> a -> Bool
<= Double
1 =
      Double
res
  | Bool
otherwise =
      forall a. HasCallStack => [Char] -> a
error forall a b. (a -> b) -> a -> b
$
        [Char]
"PointProcess.quantile: p must be in [0,1] range. Got: "
          forall a. [a] -> [a] -> [a]
++ forall a. Show a => a -> [Char]
show Double
p
          forall a. [a] -> [a] -> [a]
++ [Char]
"."
  where
    den :: Double
den = Double
l forall a. Num a => a -> a -> a
* (-Double
3.0 forall a. Num a => a -> a -> a
+ Double
2.0 forall a. Num a => a -> a -> a
* Double
t forall a. Num a => a -> a -> a
* (-Double
1.0 forall a. Num a => a -> a -> a
+ Double
p) forall a. Num a => a -> a -> a
* Double
l) forall a. Num a => a -> a -> a
+ Double
m
    t1 :: Double
t1 = (Double
2.0 forall a. Num a => a -> a -> a
+ Double
t forall a. Num a => a -> a -> a
* (Double
l forall a. Num a => a -> a -> a
- Double
4.0 forall a. Num a => a -> a -> a
* Double
p forall a. Num a => a -> a -> a
* Double
l forall a. Num a => a -> a -> a
+ Double
m)) forall a. Fractional a => a -> a -> a
/ Double
den
    t2Nom :: Double
t2Nom =
      Double
4.0
        forall a. Num a => a -> a -> a
+ Double
t
          forall a. Num a => a -> a -> a
* ( Double
l
                forall a. Num a => a -> a -> a
* (Double
4.0 forall a. Num a => a -> a -> a
+ Double
t forall a. Num a => a -> a -> a
* Double
l forall a. Num a => a -> a -> a
+ Double
8.0 forall a. Num a => a -> a -> a
* Double
p forall a. Num a => a -> a -> a
* (Double
1.0 forall a. Num a => a -> a -> a
+ Double
t forall a. Num a => a -> a -> a
* Double
l))
                forall a. Num a => a -> a -> a
+ Double
2.0
                  forall a. Num a => a -> a -> a
* (Double
2.0 forall a. Num a => a -> a -> a
+ Double
t forall a. Num a => a -> a -> a
* Double
l forall a. Num a => a -> a -> a
- Double
4.0 forall a. Num a => a -> a -> a
* Double
p forall a. Num a => a -> a -> a
* (Double
1.0 forall a. Num a => a -> a -> a
+ Double
t forall a. Num a => a -> a -> a
* Double
l))
                  forall a. Num a => a -> a -> a
* Double
m
                forall a. Num a => a -> a -> a
+ Double
t
                  forall a. Num a => a -> a -> a
* Double
m
                  forall a. Num a => a -> a -> a
* Double
m
            )
    t2 :: Double
t2 = Double
t2Nom forall a. Fractional a => a -> a -> a
/ (Double
den forall a. Floating a => a -> a -> a
** Double
2)
    res :: Double
res = Double
0.5 forall a. Num a => a -> a -> a
* (Double
t1 forall a. Num a => a -> a -> a
+ forall a. Floating a => a -> a
sqrt Double
t2)

instance D.ContGen BirthDeathNearlyCriticalDistribution where
  genContVar :: forall g (m :: * -> *).
StatefulGen g m =>
BirthDeathNearlyCriticalDistribution -> g -> m Double
genContVar = forall d g (m :: * -> *).
(ContDistr d, StatefulGen g m) =>
d -> g -> m Double
D.genContinuous