```{-# LANGUAGE DerivingStrategies         #-}
{-# LANGUAGE FlexibleContexts           #-}
{-# LANGUAGE FlexibleInstances          #-}
{-# LANGUAGE GeneralizedNewtypeDeriving #-}
{-# LANGUAGE LambdaCase                 #-}
{-# LANGUAGE MultiParamTypeClasses      #-}
{-# LANGUAGE TypeOperators              #-}

{-|
Module      : Markov.Example
Description : Examples of Markov chains implemented using "Markov".
Maintainer  : atloomis@math.arizona.edu
Stability   : Experimental

Several examples of Markov chains.
-}

module Markov.Example
( FromLists (..)
, Simple (..)
, Urn (..)
, Extinction (..)
, Tidal (..)
, Room (..)
, FillBin
, initial
, expectedLoss
) where

import           Markov
import           Markov.Extra
import qualified Markov.Generic as MG

---------------------------------------------------------------
-- From a matrix
---------------------------------------------------------------

-- |An example defined from a matrix.
--
-- >>> chain [pure (FromMatrix 't') :: (Product Double, FromMatrix)] !! 100
-- [ (0.5060975609756099,'a')
-- , (0.201219512195122,'t')
-- , (0.29268292682926833,'l') ]
newtype FromLists = FromLists Char
deriving newtype (Eq, Ord, Show)

instance Combine FromLists where combine = const

instance Markov ((,) (Product Double)) FromLists where
transition = let mat = [ [0.4, 0.3, 0.3]
, [0.2, 0.1, 0.7]
, [0.9, 0.1, 0.0] ]
chars = map FromLists ['a','t','l']
in fromLists mat chars

---------------------------------------------------------------
-- Simple random walk
---------------------------------------------------------------

-- |A simple random walk.
-- Possible outcomes of the first three steps:
--
-- >>> take 3 \$ chain0 [Simple 0]
-- [ 
-- , [-1,1]
-- , [-2,0,2] ]
--
-- Probability of each outcome:
--
-- >>> take 3 \$ chain [pure 0 :: (Product Double, Simple)]
-- [ [(1.0,0)]
-- , [(0.5,-1),(0.5,1)]
-- , [(0.25,-2),(0.5,0),(0.25,2)] ]
--
-- Number of ways to achieve each outcome:
--
-- >>> take 3 \$ chain [pure 0 :: (Product Int, Simple)]
-- [ [(1,0)]
-- , [(1,-1),(1,1)]
-- , [(1,-2),(2,0),(1,2)] ]
--
-- Number of times @pred@ was applied,
-- allowing steps in place (@id@)
-- for more interesting output:
--
-- >>> chain [pure 0 :: (Sum Int, Simple)] !! 2
-- [ (2,-2), (1,-1), (1,0), (0,0), (0,1), (0,2) ]

newtype Simple = Simple Int
deriving newtype (Num, Enum, Eq, Ord, Show)

instance Combine Simple where combine = const

instance Markov0 Simple where
transition0 _ = [pred, succ]

instance Markov ((,) (Product Double)) Simple where
transition _ = [ 0.5 >*< pred
, 0.5 >*< succ ]

instance Markov ((,) (Product Int)) Simple where
transition _ = [ 1 >*< pred
, 1 >*< succ ]

instance Markov ((,) (Sum Int)) Simple where
transition _ = [ 1 >*< pred
, 0 >*< id
, 0 >*< succ ]

---------------------------------------------------------------
-- Urn model
---------------------------------------------------------------

-- |An urn contains balls of two colors.
-- At each step, a ball is chosen uniformly at random from the urn
-- and a ball of the same color is added.
--
-- >>> randomPath (mkStdGen 70) (Urn (2,5)) !! 8 :: (Product Double, Urn)
-- (0.1648351648351649, Urn (2,13))
newtype Urn = Urn (Int,Int)
deriving newtype (Eq, Ord, Show)

instance Combine Urn where combine = const

instance Markov ((,) (Product Double)) Urn where
transition x = [ probLeft x >*< addLeft
, 1 - probLeft x >*< addRight ]

instance MG.Markov [] ((,) (Product Double)) Urn where
transition x = [ probLeft x >*< addLeft
, 1 - probLeft x >*< addRight ]

addLeft  (Urn (a,b)) = Urn (a+1,b)

addRight (Urn (a,b)) = Urn (a,b+1)

probLeft :: Fractional a => Urn -> a
probLeft (Urn (a,b)) =  fromIntegral a / fromIntegral (a + b)

---------------------------------------------------------------
-- Tutorial
---------------------------------------------------------------

-- |This is the chain from the README.
newtype Extinction = Extinction Int
deriving newtype (Eq, Num, Show)

instance Combine Extinction where combine = const

instance Markov ((,) (Sum Int, Product Rational)) Extinction where
transition = \case
0 -> [ 0 >*< (q+r) >*< id
, 0 >*< s >*< (+) 1 ]
_ -> [ 1 >*< q >*< const 0
, 0 >*< r >*< id
, 0 >*< s >*< (+) 1 ]
where q = 0.1; r = 0.3; s = 0.6

---------------------------------------------------------------
-- More complex random walk
---------------------------------------------------------------

-- |A time inhomogenous random walk that vaguely models tides
-- by periodically switching directions
-- and falling back from a shore at the origin.
data Tidal = Tidal { time     :: Double
, position :: Int }
deriving (Eq, Ord, Show)

instance Combine Tidal where combine = const

instance Markov ((,) (Product Double)) Tidal where
transition tw = [ probRight tw >*< stepPos (+1)
, 1 - probRight tw >*< stepPos (flip (-) 1) ]

stepPos :: (Int -> Int) -> Tidal -> Tidal
stepPos f tw = Tidal (time tw + 1) (f \$ position tw)

probRight :: Tidal -> Product Double
probRight tw = Product \$ timeBias * positionBias
where timeBias = (1 + sin (2 * pi * time tw / stepsPerCycle))/2
positionBias
| position tw >= 0 = 1 / steepness
| otherwise       = 1
stepsPerCycle = 10
steepness     = 1.3 -- Double from 1 (flat) to +infty

---------------------------------------------------------------
-- Hidden Markov Model
---------------------------------------------------------------

-- |A hidden Markov model.
--
-- >>> :{ filter (\((_,Merge xs),_) -> xs == "aaa") \$ chain
--        [1 >*< Merge "" >*< 1 :: Product Rational :* Merge String :* Room] !! 3
--     :}
-- [ ((3243 % 200000,"aaa"),Room 1)
-- , ((9729 % 500000,"aaa"),Room 2)
-- , ((4501 % 250000,"aaa"),Room 3) ]
--
-- Given that all three tokens recieved were @"a"@,
-- there is a probability of approximately @0.34@
-- that the current room is @Room 3@.
newtype Room = Room Int
deriving Show
deriving newtype (Eq, Num, Ord)

instance Combine Room where combine = const

-- Note that changeState is applied before giveToken.
-- In spirit, we have @transition = giveToken . changeState@
instance Markov ((,) (Product Rational, Merge String)) Room where
sequential = [giveToken, changeState]
where changeState = \case
1 -> [ 0.3 >*< mempty >*< const 1
, 0.6 >*< mempty >*< const 2
, 0.1 >*< mempty >*< const 3 ]
2 -> [ 1.0 >*< mempty >*< const 3 ]
3 -> [ 0.3 >*< mempty >*< const 1
, 0.6 >*< mempty >*< const 2
, 0.1 >*< mempty >*< const 3 ]
_ -> error "State out of bounds in transition"
giveToken = \case
1 -> [ 0.5 >*< Merge "a" >*< const 1
, 0.5 >*< Merge "b" >*< const 1 ]
2 -> [ 0.3 >*< Merge "a" >*< const 2
, 0.7 >*< Merge "b" >*< const 2 ]
3 -> [ 0.4 >*< Merge "a" >*< const 3
, 0.4 >*< Merge "b" >*< const 3
, 0.2 >*< Merge "c" >*< const 3 ]
_ -> error "State out of bounds in transition"

---------------------------------------------------------------
-- Yet more complex example
---------------------------------------------------------------

-- |Represents bins with free slots and items.
type Bin   = (Open,Full)
type Index = Int
-- |Represents space between bins where they can expand.
type Gap   = Int
type Full  = Int
type Open  = Int
type Trans = FillBin -> FillBin

-- |A collection of bins with gaps between them.
-- At each step an empty space is chosen
-- form a bin or from a gap.
-- If it is in a bin, the space is filled.
-- If it is in a gap, it is assigned to an adjacent bin,
-- which expands to contain it and any intervening spaces,
-- and then the space filled.
data FillBin = End Gap | Ext Gap Bin FillBin deriving (Eq, Ord)

instance Show FillBin where
show (Ext g b s) = show g ++ " " ++ show b ++ " " ++ show s
show (End g)     = show g

instance Combine FillBin where combine = const

instance Markov ((,) (Product Double)) FillBin where
transition x = case probId x of
0 -> filter (\(Product y,_) -> y /= 0) -- Careful, Product _ == Product _ = True
++ [probGrowL i x >*< addItem i . growLeft  j i
| i <- indices, j <- [1..gapN (i-1) x]]
++ [probGrowR i x >*< addItem i . growRight j i
| i <- indices, j <- [1..gapN i x]]
1 -> [pure id]
_ -> error "Pattern not matched in transition"
where indices = [1..size x]

-- |>>> fBFromLists [1,3,5,10] [(3,5),(9,9),(8,3)]
-- 1 (3,5) 3 (9,9) 5 (8,3) 10
fBFromLists :: [Gap] -> [Bin] -> FillBin
fBFromLists gaps bins = case (gaps,bins) of
(g:_  , []  ) -> End g
([g]  , _   ) -> End g
(g:gs , b:bs) -> Ext g b \$ fBFromLists gs bs
([]   , _   ) -> End 0

-- |Create state where all bins start as (0,0).
--
-- >>> initial [5,7,0]
-- 5 (0,0) 7 (0,0) 0
initial :: [Int] -> FillBin
initial gs = fBFromLists gs \$ repeat (0,0)

-- |The number of bins.
size :: FillBin -> Int
size = \case
End _ -> 0
Ext _ _ s -> 1 + size s

-- |The bins of a state.
getBins :: FillBin -> [Bin]
getBins = \case
End _ -> []
Ext _ b s -> b:getBins s

-- |The open values of a state.
getOpen :: FillBin -> [Open]
getOpen x = map fst \$ getBins x

-- |The open value of the Nth bin.
openN :: Index -> FillBin -> Open
openN i x = getOpen x !!(i-1)

-- |The full values of a state.
getFull :: FillBin -> [Full]
getFull x = map snd \$ getBins x

-- |The full value of the Nth bin.
fullN :: Index -> FillBin -> Full
fullN i x = getFull x !!(i-1)

-- |The gap values of a state.
getGap :: FillBin -> [Gap]
getGap = \case
End g -> [g]
Ext g _ s -> g:getGap s

-- |Warning! Indexed from zero!
gapN :: Index -> FillBin -> Gap
gapN i x = getGap x !! i

-- |The command @iApply i f s@ is analagous to
-- @take i s ++ f (drop i s)@.
iApply :: Trans -> Index -> Trans
iApply f idx x = case (idx,x) of
(1, y)         -> f y
(i, Ext g b s) -> Ext g b \$ iApply f (i-1) s
_              -> error "Pattern not matched in iApply"

-- |Add an item to the ith bin.
where h (Ext g (o,f) s) = Ext g (o-1,f+1) s
h _               = error "pattern not matched in h in addItem"

-- |Expand the ith bin to the left by j.
-- The Markov chain will use @addItem i . growLeft j i@.
growLeft :: Int -> Index -> Trans
growLeft j = iApply h
where h (Ext g (o,f) s) = Ext (g-j) (o+j,f) s
h _               = error "pattern not matched in h in growLeft"

growRight :: Int -> Index -> Trans
growRight j = iApply h
where h (Ext g (o,f) s) = Ext g (o+j,f) (shrink s)
h _               = error "pattern not matched in h in growRight"
shrink = \case
End g -> End (g-j)
Ext g b t -> Ext (g-j) b t

-- |The sum of all open slots in bins and gaps.
slots :: FillBin -> Int
slots x = sum \$ getGap x ++ getOpen x

-- |The probability that a state returns to itself.
probId :: Num a => FillBin -> a
probId x
| slots x == 0 = 1
| otherwise    = 0

divInt :: Fractional a => Int -> Int -> a
divInt x y = fromIntegral x / fromIntegral y

-- |The probability that the ith bin gains an item.
probAdd :: Fractional a => Index -> FillBin -> a
probAdd i x = openN i x `divInt` slots x

-- |The probability that the ith bin expands to the left.
probGrowL :: Fractional a => Index -> FillBin -> a
probGrowL i x
| test      = 1 `divInt` slots x
| otherwise = 0
where test = i == 1 || fullN i x < fullN (i-1) x

-- |The probability that the ith bin expands to the right.
probGrowR :: Fractional a => Index -> FillBin -> a
probGrowR i x
| test      = 1 `divInt` slots x
| otherwise = 0
where test = i == size x || fullN i x <= fullN (i+1) x

---------------------------------------------------------------
-- Several functions to help study the previous process
---------------------------------------------------------------

-- |The \(l^2\) distance between a finished state
-- and a state with perfectly balanced bins.
individualLoss :: Fractional a => FillBin -> a
individualLoss x = sum . map f . getFull \$ x
where f y = (fromIntegral y - ideal)^2
ideal = sum (getFull x) `divInt` size x

probLoss :: Fractional a => (Product a, FillBin) -> a
probLoss (Product x, y) = x * individualLoss y

-- |Expected loss of a set of states of @['FillBin']@.
-- Loss is the \(l^2\) distance between a finished state
-- and a state with perfectly balanced bins.
--
-- >>> expectedLoss [pure \$ initial [1,0,3] :: (Product Double, FillBin)]
-- 2.0
expectedLoss :: (Fractional a, Markov ((,) (Product a)) FillBin)
=> [Product a :* FillBin] -> a
expectedLoss xs = sum . map probLoss \$ chain xs !! idx
where idx = slots . snd . head \$ xs
```