module Learning.HMM.Internal
( HMM (..)
, LogLikelihood
, init
, withEmission
, viterbi
, baumWelch
, baumWelch'
) where
import Control.Applicative ( (<$>) )
import Control.DeepSeq ( NFData, force, rnf )
import Control.Monad ( forM_, replicateM )
import Control.Monad.ST ( runST )
import qualified Data.Map.Strict as M ( findWithDefault )
import Data.Random.Distribution.Simplex ( stdSimplex )
import Data.Random.RVar ( RVar )
import qualified Data.Vector as V ( Vector, filter, foldl', foldl1', map, unsafeFreeze, unsafeIndex, unsafeTail, zip, zipWith3 )
import qualified Data.Vector.Generic as G ( convert )
import qualified Data.Vector.Generic.Extra as G ( frequencies )
import qualified Data.Vector.Mutable as MV ( unsafeNew, unsafeRead, unsafeWrite )
import qualified Data.Vector.Unboxed as U ( Vector, fromList, length, map, sum, unsafeFreeze, unsafeIndex, unsafeTail, zip )
import qualified Data.Vector.Unboxed.Mutable as MU ( unsafeNew, unsafeRead, unsafeWrite )
import qualified Numeric.LinearAlgebra.Data as H ( (!), Matrix, Vector, diag, fromColumns, fromList, fromLists, fromRows, konst, maxElement, maxIndex, toColumns, tr )
import qualified Numeric.LinearAlgebra.HMatrix as H ( (<>), (#>), sumElements )
import Prelude hiding ( init )
type LogLikelihood = Double
data HMM = HMM { nStates :: Int
, nOutputs :: Int
, initialStateDist :: H.Vector Double
, transitionDist :: H.Matrix Double
, emissionDistT :: H.Matrix Double
}
instance NFData HMM where
rnf HMM {..} = rnf nStates `seq`
rnf nOutputs `seq`
rnf initialStateDist `seq`
rnf transitionDist `seq`
rnf emissionDistT
init :: Int -> Int -> RVar HMM
init k l = do
pi0 <- H.fromList <$> stdSimplex (k1)
w <- H.fromLists <$> replicateM k (stdSimplex (k1))
phi <- H.fromLists <$> replicateM k (stdSimplex (l1))
return HMM { nStates = k
, nOutputs = l
, initialStateDist = pi0
, transitionDist = w
, emissionDistT = H.tr phi
}
withEmission :: HMM -> U.Vector Int -> HMM
withEmission (model @ HMM {..}) xs = model'
where
n = U.length xs
ss = [0..(nStates 1)]
os = [0..(nOutputs 1)]
step m = fst $ baumWelch1 (m { emissionDistT = H.tr phi }) n xs
where
phi :: H.Matrix Double
phi = let zs = fst $ viterbi m xs
fs = G.frequencies $ U.zip zs xs
hs = H.fromLists $ map (\s -> map (\o ->
M.findWithDefault 0 (s, o) fs) os) ss
hs' = hs + H.konst 1e-9 (nStates, nOutputs)
ns = hs' H.#> H.konst 1 nStates
in hs' / H.fromColumns (replicate nOutputs ns)
ms = iterate step model
ms' = tail ms
ds = zipWith euclideanDistance ms ms'
model' = fst $ head $ dropWhile ((> 1e-9) . snd) $ zip ms' ds
euclideanDistance :: HMM -> HMM -> Double
euclideanDistance model model' =
sqrt $ H.sumElements ((w w') ** 2) + H.sumElements ((phi phi') ** 2)
where
w = transitionDist model
w' = transitionDist model'
phi = emissionDistT model
phi' = emissionDistT model'
viterbi :: HMM -> U.Vector Int -> (U.Vector Int, LogLikelihood)
viterbi HMM {..} xs = (path, logL)
where
n = U.length xs
deltas :: V.Vector (H.Vector Double)
psis :: V.Vector (U.Vector Int)
(deltas, psis) = runST $ do
ds <- MV.unsafeNew n
ps <- MV.unsafeNew n
let x0 = U.unsafeIndex xs 0
MV.unsafeWrite ds 0 $ log (emissionDistT H.! x0) + log initialStateDist
forM_ [1..(n1)] $ \t -> do
d <- MV.unsafeRead ds (t1)
let x = U.unsafeIndex xs t
dws = map (\wj -> d + log wj) w'
MV.unsafeWrite ds t $ log (emissionDistT H.! x) + H.fromList (map H.maxElement dws)
MV.unsafeWrite ps t $ U.fromList (map H.maxIndex dws)
ds' <- V.unsafeFreeze ds
ps' <- V.unsafeFreeze ps
return (ds', ps')
where
w' = H.toColumns transitionDist
deltaE = V.unsafeIndex deltas (n1)
path = runST $ do
ix <- MU.unsafeNew n
MU.unsafeWrite ix (n1) $ H.maxIndex deltaE
forM_ [nl | l <- [1..(n1)]] $ \t -> do
i <- MU.unsafeRead ix t
let psi = V.unsafeIndex psis t
MU.unsafeWrite ix (t1) $ U.unsafeIndex psi i
U.unsafeFreeze ix
logL = H.maxElement deltaE
baumWelch :: HMM -> U.Vector Int -> [(HMM, LogLikelihood)]
baumWelch model xs = zip models (tail logLs)
where
n = U.length xs
step (m, _) = baumWelch1 m n xs
(models, logLs) = unzip $ iterate step (model, undefined)
baumWelch' :: HMM -> U.Vector Int -> (HMM, LogLikelihood)
baumWelch' model xs = go (10000 :: Int) (undefined, 1/0) (baumWelch1 model n xs)
where
n = U.length xs
go k (m, l) (m', l')
| k > 0 && l' l > 1.0e-9 = go (k 1) (m', l') (baumWelch1 m' n xs)
| otherwise = (m, l')
baumWelch1 :: HMM -> Int -> U.Vector Int -> (HMM, LogLikelihood)
baumWelch1 (model @ HMM {..}) n xs = force (model', logL)
where
(alphas, cs) = forward model n xs
betas = backward model n xs cs
(gammas, xis) = posterior model n xs alphas betas cs
pi0 = V.unsafeIndex gammas 0
w = let ds = V.foldl1' (+) xis
ns = ds H.#> H.konst 1 nStates
in H.diag (H.konst 1 nStates / ns) H.<> ds
phi' = let gs' o = V.map snd $ V.filter ((== o) . fst) $ V.zip (G.convert xs) gammas
ds = V.foldl' (+) 0 . gs'
ns = V.foldl1' (+) gammas
in H.fromRows $ map (\o -> ds o / ns) [0..(nOutputs 1)]
model' = model { initialStateDist = pi0
, transitionDist = w
, emissionDistT = phi'
}
logL = (U.sum $ U.map log cs)
forward :: HMM -> Int -> U.Vector Int -> (V.Vector (H.Vector Double), U.Vector Double)
forward HMM {..} n xs = runST $ do
as <- MV.unsafeNew n
cs <- MU.unsafeNew n
let x0 = U.unsafeIndex xs 0
a0 = (emissionDistT H.! x0) * initialStateDist
c0 = 1 / H.sumElements a0
MV.unsafeWrite as 0 (H.konst c0 nStates * a0)
MU.unsafeWrite cs 0 c0
forM_ [1..(n1)] $ \t -> do
a <- MV.unsafeRead as (t1)
let x = U.unsafeIndex xs t
a' = (emissionDistT H.! x) * (w' H.#> a)
c' = 1 / H.sumElements a'
MV.unsafeWrite as t (H.konst c' nStates * a')
MU.unsafeWrite cs t c'
as' <- V.unsafeFreeze as
cs' <- U.unsafeFreeze cs
return (as', cs')
where
w' = H.tr transitionDist
backward :: HMM -> Int -> U.Vector Int -> U.Vector Double -> V.Vector (H.Vector Double)
backward HMM {..} n xs cs = runST $ do
bs <- MV.unsafeNew n
let bE = H.konst 1 nStates
cE = U.unsafeIndex cs (n1)
MV.unsafeWrite bs (n1) (H.konst cE nStates * bE)
forM_ [nl | l <- [1..(n1)]] $ \t -> do
b <- MV.unsafeRead bs t
let x = U.unsafeIndex xs t
b' = transitionDist H.#> ((emissionDistT H.! x) * b)
c' = U.unsafeIndex cs (t1)
MV.unsafeWrite bs (t1) (H.konst c' nStates * b')
V.unsafeFreeze bs
posterior :: HMM -> Int -> U.Vector Int -> V.Vector (H.Vector Double) -> V.Vector (H.Vector Double) -> U.Vector Double -> (V.Vector (H.Vector Double), V.Vector (H.Matrix Double))
posterior HMM {..} _ xs alphas betas cs = (gammas, xis)
where
gammas = V.zipWith3 (\a b c -> a * b / H.konst c nStates)
alphas betas (G.convert cs)
xis = V.zipWith3 (\a b x -> H.diag a H.<> transitionDist H.<> H.diag (b * (emissionDistT H.! x)))
alphas (V.unsafeTail betas) (G.convert $ U.unsafeTail xs)