-----------------------------------------------------------------------------
-- |
-- Module      :  Data.Random.Distribution.Static.MultivariateNormal
-- Copyright   :  (c) 2016 FP Complete Corporation
-- Maintainer  :  dominic@steinitz.org
--
-- Sample from the multivariate normal distribution with a given
-- vector-valued $$\mu$$ and covariance matrix $$\Sigma$$. For
-- example, the chart below shows samples from the bivariate normal
-- distribution. The dimension of the mean $$n$$ is statically checked
-- to be compatible with the dimension of the covariance matrix $$n \times n$$.
--
-- <<diagrams/src_Data_Random_Distribution_Static_MultivariateNormal_diagMS.svg#diagram=diagMS&height=600&width=500>>
--
-- Example code to generate the chart:
--
-- > {-# LANGUAGE DataKinds #-}
-- >
-- > import qualified Graphics.Rendering.Chart as C
-- > import Graphics.Rendering.Chart.Backend.Diagrams
-- >
-- > import Data.Random.Distribution.Static.MultivariateNormal
-- >
-- > import qualified Data.Random as R
-- > import Data.Random.Source.PureMT
-- > import Numeric.LinearAlgebra.Static
-- >
-- > nSamples :: Int
-- > nSamples = 10000
-- >
-- > sigma1, sigma2, rho :: Double
-- > sigma1 = 3.0
-- > sigma2 = 1.0
-- > rho = 0.5
-- >
-- > singleSample :: R.RVarT (State PureMT) (R 2)
-- > singleSample = R.sample $Normal (vector [0.0, 0.0]) -- > (sym$ matrix [ sigma1, rho * sigma1 * sigma2
-- >                              , rho * sigma1 * sigma2, sigma2])
-- >
-- > multiSamples :: [R 2]
-- > multiSamples = evalState (replicateM nSamples $R.sample singleSample) (pureMT 3) -- > -- > pts = map f multiSamples -- > where -- > f z = (x, y) -- > where -- > (x, t) = headTail z -- > (y, _) = headTail t -- > -- > chartPoint pointVals n = C.toRenderable layout -- > where -- > -- > fitted = C.plot_points_values .~ pointVals -- >$ C.plot_points_style  . C.point_color .~ opaque red
-- >               $C.plot_points_title .~ "Sample" -- >$ def
-- >
-- >     layout = C.layout_title .~ "Sampling Bivariate Normal (" ++ (show n) ++ " samples)"
-- >            $C.layout_y_axis . C.laxis_generate .~ C.scaledAxis def (-3,3) -- >$ C.layout_x_axis . C.laxis_generate .~ C.scaledAxis def (-3,3)
-- >
-- >            $C.layout_plots .~ [C.toPlot fitted] -- >$ def
-- >
-- > diagMS = do
-- >   denv <- defaultEnv C.vectorAlignmentFns 600 500
-- >   return $fst$ runBackend denv (C.render (chartPoint pts nSamples) (500, 500))
--
-----------------------------------------------------------------------------

{-# OPTIONS_GHC -Wall                     #-}
{-# OPTIONS_GHC -fno-warn-type-defaults   #-}
{-# OPTIONS_GHC -fno-warn-unused-do-bind  #-}
{-# OPTIONS_GHC -fno-warn-missing-methods #-}
{-# OPTIONS_GHC -fno-warn-orphans         #-}

{-# LANGUAGE MultiParamTypeClasses        #-}
{-# LANGUAGE TypeFamilies                 #-}
{-# LANGUAGE ScopedTypeVariables          #-}
{-# LANGUAGE DataKinds                    #-}

module Data.Random.Distribution.Static.MultivariateNormal
( Normal(..)
) where

import           Data.Random hiding ( StdNormal, Normal )
import qualified Data.Random as R
import qualified Numeric.LinearAlgebra.HMatrix as H
import           Numeric.LinearAlgebra.Static as S
( R, vector, extract, Sq, Sym, col,
tr, linSolve, uncol, chol, (<.>),
ℝ, (<>), diag, (#>), eigensystem
)
import          GHC.TypeLits ( KnownNat, natVal )
import          Data.Maybe ( fromJust )

normalMultivariate :: KnownNat n =>
R n -> Sym n -> RVarT m (R n)
normalMultivariate mu bigSigma = do
z <- replicateM (fromIntegral $natVal mu) (rvarT R.StdNormal) return$ mu + bigA #> (vector z)
where
(vals, bigU) = eigensystem bigSigma
lSqrt = diag $mapVector sqrt vals bigA = bigU S.<> lSqrt mapVector :: KnownNat n => (ℝ -> ℝ) -> R n -> R n mapVector f = vector . H.toList . H.cmap f . extract sumVector :: KnownNat n => R n -> ℝ sumVector x = x <.> 1 data family Normal k :: * data instance Normal (R n) = Normal (R n) (Sym n) instance KnownNat n => Distribution Normal (R n) where rvar (Normal m s) = normalMultivariate m s normalLogPdf :: KnownNat n => R n -> Sym n -> R n -> Double normalLogPdf mu bigSigma x = - sumVector (mapVector log (diagonals dec)) - 0.5 * (fromIntegral$ natVal mu) * log (2 * pi)
- 0.5 * s
where
dec = chol bigSigma
t = uncol $fromJust$ linSolve (tr dec) (col $x - mu) u = mapVector (\x -> x * x) t s = sumVector u normalPdf :: KnownNat n => R n -> Sym n -> R n -> Double normalPdf mu sigma x = exp$ normalLogPdf mu sigma x

diagonals :: KnownNat n => Sq n -> R n
diagonals = vector . H.toList . H.takeDiag . extract

instance KnownNat n => PDF Normal (R n) where
pdf (Normal m s) = normalPdf m s
logPdf (Normal m s) = normalLogPdf m s