Stochastic Processes as Monad Transformers

August 3, 2010

Disclaimer: this is an old blog post from a very old wordpress blog and may contain inacuracies. I reproduced it as is for sentimental reasons. I may revisit this theme later.

I have a difficulty to understand functional programming concepts that I can’t put to some very simple and natural use (natural for me, of course). I need to find the perfect simple example to implement to finally understand something. And I’m not a computer scientist, so things like parsers and compilers have very little appeal to me (probably because I don’t understand them…). I’m a physicist, so this drives me to look for physical problems that can be implemented in Haskell so I can understand some concepts.

Monad transformers still eludes me. But I think I finally got the perfect subject were I can understand them: stochastic processes! First some book keeping:

import Control.Monad.State
import Control.Monad
import Control.Monad.Rand

Now, stochastic processes have characteristics related to two different monads. In one hand, they are dynamical processes, and the way to implement dynamics in Haskell is with state monads. For example, if I want to iterate the logistic map:

$$x_{t+1} = \alpha x_t\left(1-x_t\right)$$

$$ teste = teste $$

I could do the following:

  f :: Double -> Double
  f x = 4*x*(1-x)

  logistic :: State Double Double
  logistic = do x0 <- get
        let x1 = f x
        put x1
        return x1
  runLogistic :: State Double [Double]
  runLogistic n x0= evalState (replicateM n logistic) x0

Running this on ghci would give you, for example:

  *Main> runLogistic 5 0.2
  [0.6400000000000001,0.9215999999999999,0.28901376000000045, 0.8219392261226504,0.5854205387341]

So we can make the loose correspondence: dynamical system ↔ state monad.

On the other hand, stochastic processes are compositions of random variables, and this is done with the Rand monad (found in Control.Monad.Random). As an example, the Box-Muller formula tells us that, if I have two inpendent random variables $x$ and $y$, distributed uniformly between in the \([0, 1]\) interval, then, the expression:

$$\sqrt{-2\log(x)}\cos(2\pi y)$$

will be normally distributed. We can write then:

boxmuller :: Double -> Double -> Double
boxmuller x y = sqrt(-2*log x)*cos(2*pi*y)

normal :: Rand StdGen Double  -- normally distributed
normal = do x <- getRandom
            y <- getRandom
            return $ boxmuller x y

normals n = replicateM n normal -- n independent samples from normal

Running this function we get what we need:

*Main> (evalRand $ normals 5) (mkStdGen 0) =

So what is a stochastic process? In very rough terms: is a dynamical system with random variables. So we need a way to make the Rand monad to talk nicely with the State monad. The way to do this is to use a monad transformer, in this case, the StateT transformer. Monad transformers allows you to combine the functionalities of two different monads. In the case of the StateT monads, they allow you to add a state to any other monad you want. In our case, we want to wrap the Rand monad inside a StateT transformer and work with things of type:

foo ::  StateT s (Rand StdGen) r

This type represent a monad that can store a state with type s, like the state monad, and can generate random variables of type r, like the rand monad. In general we would have a type

foo2 ::(MonadTrans t, Monad m) => t m a

In this case, t = StateT s and m = Rand StdGen. The class MonadTrans is defined in Control.Monad.Trans, and provides the function:

lift :: (MonadTrans t, Monad m) => m a -> t m a

In this case, t is itself a monad, and can be treated like one through the code. It works like this: inside a do expression you can use the lift function to access the inner monad. Things called with lift will operate in the inner monad. Things called without lift will operate in the outer monad.

So, suppose we want to simulate this very simple process:

$$x_{t+1} = x_t + \eta_t$$

where \(\eta_t\) is drawn from a normal distribution. We would do:

randomWalk :: StateT Double (Rand StdGen) Double
randomWalk = do eta <- lift normal
                x <- get
                let x' = x + eta
                put x'
                return x'
runWalk :: Int -> Double -> StdGen -> [Double]
runWalk n x0 gen = evalRand (replicateM n $ evalStateT randomWalk x0) gen

The evalStateT function is just evalState adapted to run a StateT monad. Running this on ghci we get:

 *Main> runWalk 5 0.0 gen

This is what we can accomplish: we can easily operate simultaneously with functions that expect a state monad, like put and get, we can unwrap things with <- from the inner Rand monad by using lift , and we can return things to the state monad. We could have any monad inside the StateT transformer. For example, we could have another State monad. Here is a fancy implementation of the Fibonacci sequence using a State monad (that stores the last but one value in the sequence as its internal state) inside a StateT transfomer (that stores the last value of the sequence):

fancyFib :: StateT Int (State Int) Int
fancyFib = do old <- lift get
              new <- get
              let new' = new + old
                  old' = new
              lift $ put old'
              put new'
              return new

fancyFibs :: Int -> StateT Int (State Int) [Int]
fancyFibs n = replicateM n fancyFibs

And we can run this to get:

*Main> evalState (evalStateT (fancyFibs 10) 1) 0
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