Markov

Three type classes for deterministically analyzing Markov chains with known parameters. Markov0 is intended to list possible outcomes, Markov should allow for more sophisticated analysis. A more general definition can be found in Markov.Generic that allows for containers other than lists. See Markov.Example for examples. See README for a detailed description.

Markov0

class Eq s => Markov0 s where

A basic implementation of Markov chains.

Minimal complete definition

transition0 | step0

Methods

transition0 :: s -> [s -> s]

step0 :: s -> [s]

Instances

Markov0 Simple

Defined in Markov.Example

Methods

transition0 :: Simple -> [Simple -> Simple]

step0 :: Simple -> [Simple]

chain0 :: Markov0 s => [s] -> [[s]]

Iterated steps, with equal states combined.

Markov

class (Applicative t, Comonad t) => Markov t s where

An implementation of Markov chains.

Minimal complete definition

transition | step | sequential

Methods

transition :: s -> [t (s -> s)]

step :: t s -> [t s]

sequential :: [s -> [t (s -> s)]]

Instances

Markov ((,) (Product Rational, Merge String)) Room

Defined in Markov.Example

Methods

transition :: Room -> [((Product Rational, Merge String), Room -> Room)]

step :: ((Product Rational, Merge String), Room) -> [((Product Rational, Merge String), Room)]

sequential :: [Room -> [((Product Rational, Merge String), Room -> Room)]]

Markov ((,) (Sum Int, Product Rational)) Extinction

Defined in Markov.Example

Methods

transition :: Extinction -> [((Sum Int, Product Rational), Extinction -> Extinction)]

step :: ((Sum Int, Product Rational), Extinction) -> [((Sum Int, Product Rational), Extinction)]

sequential :: [Extinction -> [((Sum Int, Product Rational), Extinction -> Extinction)]]

Markov ((,) (Product Double)) FillBin

Defined in Markov.Example

Methods

transition :: FillBin -> [(Product Double, FillBin -> FillBin)]

step :: (Product Double, FillBin) -> [(Product Double, FillBin)]

sequential :: [FillBin -> [(Product Double, FillBin -> FillBin)]]

Markov ((,) (Product Double)) Tidal

Defined in Markov.Example

Methods

transition :: Tidal -> [(Product Double, Tidal -> Tidal)]

step :: (Product Double, Tidal) -> [(Product Double, Tidal)]

sequential :: [Tidal -> [(Product Double, Tidal -> Tidal)]]

Markov ((,) (Product Double)) Urn

Defined in Markov.Example

Methods

transition :: Urn -> [(Product Double, Urn -> Urn)]

step :: (Product Double, Urn) -> [(Product Double, Urn)]

sequential :: [Urn -> [(Product Double, Urn -> Urn)]]

Markov ((,) (Product Double)) Simple

Defined in Markov.Example

Methods

transition :: Simple -> [(Product Double, Simple -> Simple)]

step :: (Product Double, Simple) -> [(Product Double, Simple)]

sequential :: [Simple -> [(Product Double, Simple -> Simple)]]

Markov ((,) (Product Double)) FromLists

Defined in Markov.Example

Methods

transition :: FromLists -> [(Product Double, FromLists -> FromLists)]

step :: (Product Double, FromLists) -> [(Product Double, FromLists)]

sequential :: [FromLists -> [(Product Double, FromLists -> FromLists)]]

Markov ((,) (Product Int)) Simple

Defined in Markov.Example

Methods

transition :: Simple -> [(Product Int, Simple -> Simple)]

step :: (Product Int, Simple) -> [(Product Int, Simple)]

sequential :: [Simple -> [(Product Int, Simple -> Simple)]]

Markov ((,) (Sum Int)) Simple

Defined in Markov.Example

Methods

transition :: Simple -> [(Sum Int, Simple -> Simple)]

step :: (Sum Int, Simple) -> [(Sum Int, Simple)]

sequential :: [Simple -> [(Sum Int, Simple -> Simple)]]

chain :: (Combine (t s), Ord (t s), Markov t s) => [t s] -> [[t s]]

Iterated steps, with equal states combined using summarize operation.

Combine

class Combine a where

Within equivalence classes, combine should be associative, commutative, and idempotent (up to equivalence). I.e. if x == y == z,

(x `combine` y) `combine` z = x `combine` (y `combine` z)

x `combine` y = y `combine` x

x `combine` x == x

Minimal complete definition

combine | summarize

Methods

combine :: a -> a -> a

summarize :: NonEmpty a -> a

Instances

Combine FillBin

Defined in Markov.Example

Methods

combine :: FillBin -> FillBin -> FillBin

summarize :: NonEmpty FillBin -> FillBin

Combine Room

Defined in Markov.Example

Methods

combine :: Room -> Room -> Room

summarize :: NonEmpty Room -> Room

Combine Tidal

Defined in Markov.Example

Methods

combine :: Tidal -> Tidal -> Tidal

summarize :: NonEmpty Tidal -> Tidal

Combine Extinction

Defined in Markov.Example

Methods

combine :: Extinction -> Extinction -> Extinction

summarize :: NonEmpty Extinction -> Extinction

Combine Urn

Defined in Markov.Example

Methods

combine :: Urn -> Urn -> Urn

summarize :: NonEmpty Urn -> Urn

Combine Simple

Defined in Markov.Example

Methods

combine :: Simple -> Simple -> Simple

summarize :: NonEmpty Simple -> Simple

Combine FromLists

Defined in Markov.Example

Methods

combine :: FromLists -> FromLists -> FromLists

summarize :: NonEmpty FromLists -> FromLists

Num a => Combine (Product a)

Defined in Markov

Methods

combine :: Product a -> Product a -> Product a

summarize :: NonEmpty (Product a) -> Product a

Combine (Sum a)

Defined in Markov

Methods

combine :: Sum a -> Sum a -> Sum a

summarize :: NonEmpty (Sum a) -> Sum a

Combine (Merge a)

Defined in Markov

Methods

combine :: Merge a -> Merge a -> Merge a

summarize :: NonEmpty (Merge a) -> Merge a

(Combine a, Combine b) => Combine (a, b)

Defined in Markov

Methods

combine :: (a, b) -> (a, b) -> (a, b)

summarize :: NonEmpty (a, b) -> (a, b)

(Combine a, Combine b, Combine c) => Combine (a, b, c)

Defined in Markov

Methods

combine :: (a, b, c) -> (a, b, c) -> (a, b, c)

summarize :: NonEmpty (a, b, c) -> (a, b, c)

newtype Merge a

Values from a Monoid which have the respective binary operation applied each step, where different values mean states should not be combined. E.g., strings with concatenation.

Constructors

Merge a

Instances

Enum a => Enum (Merge a)

Defined in Markov

Methods

succ :: Merge a -> Merge a

pred :: Merge a -> Merge a

toEnum :: Int -> Merge a

fromEnum :: Merge a -> Int

enumFrom :: Merge a -> [Merge a]

enumFromThen :: Merge a -> Merge a -> [Merge a]

enumFromTo :: Merge a -> Merge a -> [Merge a]

enumFromThenTo :: Merge a -> Merge a -> Merge a -> [Merge a]

Eq a => Eq (Merge a)

Defined in Markov

Methods

(==) :: Merge a -> Merge a -> Bool

(/=) :: Merge a -> Merge a -> Bool

Fractional a => Fractional (Merge a)

Defined in Markov

Methods

(/) :: Merge a -> Merge a -> Merge a

recip :: Merge a -> Merge a

fromRational :: Rational -> Merge a

Num a => Num (Merge a)

Defined in Markov

Methods

(+) :: Merge a -> Merge a -> Merge a

(-) :: Merge a -> Merge a -> Merge a

(*) :: Merge a -> Merge a -> Merge a

negate :: Merge a -> Merge a

abs :: Merge a -> Merge a

signum :: Merge a -> Merge a

fromInteger :: Integer -> Merge a

Ord a => Ord (Merge a)

Defined in Markov

Methods

compare :: Merge a -> Merge a -> Ordering

(<) :: Merge a -> Merge a -> Bool

(<=) :: Merge a -> Merge a -> Bool

(>) :: Merge a -> Merge a -> Bool

(>=) :: Merge a -> Merge a -> Bool

max :: Merge a -> Merge a -> Merge a

min :: Merge a -> Merge a -> Merge a

Show a => Show (Merge a)

Defined in Markov

Methods

showsPrec :: Int -> Merge a -> ShowS

show :: Merge a -> String

showList :: [Merge a] -> ShowS

Semigroup a => Semigroup (Merge a)

Defined in Markov

Methods

(<>) :: Merge a -> Merge a -> Merge a

sconcat :: NonEmpty (Merge a) -> Merge a

stimes :: Integral b => b -> Merge a -> Merge a

Monoid a => Monoid (Merge a)

Defined in Markov

Methods

mempty :: Merge a

mappend :: Merge a -> Merge a -> Merge a

mconcat :: [Merge a] -> Merge a

Combine (Merge a)

Defined in Markov

Methods

combine :: Merge a -> Merge a -> Merge a

summarize :: NonEmpty (Merge a) -> Merge a

Markov ((,) (Product Rational, Merge String)) Room

Defined in Markov.Example

Methods

transition :: Room -> [((Product Rational, Merge String), Room -> Room)]

step :: ((Product Rational, Merge String), Room) -> [((Product Rational, Merge String), Room)]

sequential :: [Room -> [((Product Rational, Merge String), Room -> Room)]]

newtype Sum a

Values which are added each step, where different values mean states should not be combined. E.g., number of times a red ball is picked from an urn.

Constructors

Sum a

Instances

Enum a => Enum (Sum a)

Defined in Markov

Methods

succ :: Sum a -> Sum a

pred :: Sum a -> Sum a

toEnum :: Int -> Sum a

fromEnum :: Sum a -> Int

enumFrom :: Sum a -> [Sum a]

enumFromThen :: Sum a -> Sum a -> [Sum a]

enumFromTo :: Sum a -> Sum a -> [Sum a]

enumFromThenTo :: Sum a -> Sum a -> Sum a -> [Sum a]

Eq a => Eq (Sum a)

Defined in Markov

Methods

(==) :: Sum a -> Sum a -> Bool

(/=) :: Sum a -> Sum a -> Bool

Fractional a => Fractional (Sum a)

Defined in Markov

Methods

(/) :: Sum a -> Sum a -> Sum a

recip :: Sum a -> Sum a

fromRational :: Rational -> Sum a

Num a => Num (Sum a)

Defined in Markov

Methods

(+) :: Sum a -> Sum a -> Sum a

(-) :: Sum a -> Sum a -> Sum a

(*) :: Sum a -> Sum a -> Sum a

negate :: Sum a -> Sum a

abs :: Sum a -> Sum a

signum :: Sum a -> Sum a

fromInteger :: Integer -> Sum a

Ord a => Ord (Sum a)

Defined in Markov

Methods

compare :: Sum a -> Sum a -> Ordering

(<) :: Sum a -> Sum a -> Bool

(<=) :: Sum a -> Sum a -> Bool

(>) :: Sum a -> Sum a -> Bool

(>=) :: Sum a -> Sum a -> Bool

max :: Sum a -> Sum a -> Sum a

min :: Sum a -> Sum a -> Sum a

Show a => Show (Sum a)

Defined in Markov

Methods

showsPrec :: Int -> Sum a -> ShowS

show :: Sum a -> String

showList :: [Sum a] -> ShowS

Num a => Semigroup (Sum a)

Defined in Markov

Methods

(<>) :: Sum a -> Sum a -> Sum a

sconcat :: NonEmpty (Sum a) -> Sum a

stimes :: Integral b => b -> Sum a -> Sum a

Num a => Monoid (Sum a)

Defined in Markov

Methods

mempty :: Sum a

mappend :: Sum a -> Sum a -> Sum a

mconcat :: [Sum a] -> Sum a

Combine (Sum a)

Defined in Markov

Methods

combine :: Sum a -> Sum a -> Sum a

summarize :: NonEmpty (Sum a) -> Sum a

Markov ((,) (Sum Int, Product Rational)) Extinction

Defined in Markov.Example

Methods

transition :: Extinction -> [((Sum Int, Product Rational), Extinction -> Extinction)]

step :: ((Sum Int, Product Rational), Extinction) -> [((Sum Int, Product Rational), Extinction)]

sequential :: [Extinction -> [((Sum Int, Product Rational), Extinction -> Extinction)]]

Markov ((,) (Sum Int)) Simple

Defined in Markov.Example

Methods

transition :: Simple -> [(Sum Int, Simple -> Simple)]

step :: (Sum Int, Simple) -> [(Sum Int, Simple)]

sequential :: [Simple -> [(Sum Int, Simple -> Simple)]]

newtype Product a

Values which are multiplied each step, and combined additively for equal states. E.g., probabilities.

Constructors

Product a

Instances

Markov [] ((,) (Product Double)) Urn

Defined in Markov.Example

Methods

transition :: Urn -> [(Product Double, Urn -> Urn)]

step :: (Product Double, Urn) -> [(Product Double, Urn)]

sequential :: [Urn -> [(Product Double, Urn -> Urn)]]

Enum a => Enum (Product a)

Defined in Markov

Methods

succ :: Product a -> Product a

pred :: Product a -> Product a

toEnum :: Int -> Product a

fromEnum :: Product a -> Int

enumFrom :: Product a -> [Product a]

enumFromThen :: Product a -> Product a -> [Product a]

enumFromTo :: Product a -> Product a -> [Product a]

enumFromThenTo :: Product a -> Product a -> Product a -> [Product a]

Eq (Product a)

WARNING! Defined _ == _ = True!

Defined in Markov

Methods

(==) :: Product a -> Product a -> Bool

(/=) :: Product a -> Product a -> Bool

Fractional a => Fractional (Product a)

Defined in Markov

Methods

(/) :: Product a -> Product a -> Product a

recip :: Product a -> Product a

fromRational :: Rational -> Product a

Integral a => Integral (Product a)

Defined in Markov

Methods

quot :: Product a -> Product a -> Product a

rem :: Product a -> Product a -> Product a

div :: Product a -> Product a -> Product a

mod :: Product a -> Product a -> Product a

quotRem :: Product a -> Product a -> (Product a, Product a)

divMod :: Product a -> Product a -> (Product a, Product a)

toInteger :: Product a -> Integer

Num a => Num (Product a)

Defined in Markov

Methods

(+) :: Product a -> Product a -> Product a

(-) :: Product a -> Product a -> Product a

(*) :: Product a -> Product a -> Product a

negate :: Product a -> Product a

abs :: Product a -> Product a

signum :: Product a -> Product a

fromInteger :: Integer -> Product a

Ord (Product a)

WARNING! Defined compare _ _ = EQ!

Defined in Markov

Methods

compare :: Product a -> Product a -> Ordering

(<) :: Product a -> Product a -> Bool

(<=) :: Product a -> Product a -> Bool

(>) :: Product a -> Product a -> Bool

(>=) :: Product a -> Product a -> Bool

max :: Product a -> Product a -> Product a

min :: Product a -> Product a -> Product a

Real a => Real (Product a)

Defined in Markov

Methods

toRational :: Product a -> Rational

Show a => Show (Product a)

Defined in Markov

Methods

showsPrec :: Int -> Product a -> ShowS

show :: Product a -> String

showList :: [Product a] -> ShowS

Num a => Semigroup (Product a)

Defined in Markov

Methods

(<>) :: Product a -> Product a -> Product a

sconcat :: NonEmpty (Product a) -> Product a

stimes :: Integral b => b -> Product a -> Product a

Num a => Monoid (Product a)

Defined in Markov

Methods

mempty :: Product a

mappend :: Product a -> Product a -> Product a

mconcat :: [Product a] -> Product a

Num a => Combine (Product a)

Defined in Markov

Methods

combine :: Product a -> Product a -> Product a

summarize :: NonEmpty (Product a) -> Product a

Markov ((,) (Product Rational, Merge String)) Room

Defined in Markov.Example

Methods

transition :: Room -> [((Product Rational, Merge String), Room -> Room)]

step :: ((Product Rational, Merge String), Room) -> [((Product Rational, Merge String), Room)]

sequential :: [Room -> [((Product Rational, Merge String), Room -> Room)]]

Markov ((,) (Sum Int, Product Rational)) Extinction

Defined in Markov.Example

Methods

transition :: Extinction -> [((Sum Int, Product Rational), Extinction -> Extinction)]

step :: ((Sum Int, Product Rational), Extinction) -> [((Sum Int, Product Rational), Extinction)]

sequential :: [Extinction -> [((Sum Int, Product Rational), Extinction -> Extinction)]]

Markov ((,) (Product Double)) FillBin

Defined in Markov.Example

Methods

transition :: FillBin -> [(Product Double, FillBin -> FillBin)]

step :: (Product Double, FillBin) -> [(Product Double, FillBin)]

sequential :: [FillBin -> [(Product Double, FillBin -> FillBin)]]

Markov ((,) (Product Double)) Tidal

Defined in Markov.Example

Methods

transition :: Tidal -> [(Product Double, Tidal -> Tidal)]

step :: (Product Double, Tidal) -> [(Product Double, Tidal)]

sequential :: [Tidal -> [(Product Double, Tidal -> Tidal)]]

Markov ((,) (Product Double)) Urn

Defined in Markov.Example

Methods

transition :: Urn -> [(Product Double, Urn -> Urn)]

step :: (Product Double, Urn) -> [(Product Double, Urn)]

sequential :: [Urn -> [(Product Double, Urn -> Urn)]]

Markov ((,) (Product Double)) Simple

Defined in Markov.Example

Methods

transition :: Simple -> [(Product Double, Simple -> Simple)]

step :: (Product Double, Simple) -> [(Product Double, Simple)]

sequential :: [Simple -> [(Product Double, Simple -> Simple)]]

Markov ((,) (Product Double)) FromLists

Defined in Markov.Example

Methods

transition :: FromLists -> [(Product Double, FromLists -> FromLists)]

step :: (Product Double, FromLists) -> [(Product Double, FromLists)]

sequential :: [FromLists -> [(Product Double, FromLists -> FromLists)]]

Markov ((,) (Product Int)) Simple

Defined in Markov.Example

Methods

transition :: Simple -> [(Product Int, Simple -> Simple)]

step :: (Product Int, Simple) -> [(Product Int, Simple)]

sequential :: [Simple -> [(Product Int, Simple -> Simple)]]