## Introduction

In the previous post I showed how to write your own general-purpose monadic probabilistic programming language from scratch in 50 lines of (Scala) code. That post is a pre-requisite for this one, so if you haven’t read it, go back and have a quick skim through it before proceeding. In that post I tried to keep everything as simple as possible, but at the expense of both elegance and efficiency. In this post I’ll address one problem with the implementation from that post – the memory (and computational) overhead associated with forming the Cartesian product of particle sets during monadic binding (`flatMap`). So if particle sets are typically of size $N$, then the Cartesian product is of size $N^2$, and multinomial resampling is applied to this set of size $N^2$ in order to sample back down to a set of size $N$. But this isn’t actually necessary. We can directly construct a set of size $N$, certainly saving memory, but also potentially saving computation time if the conditional distribution (on the right of the monadic bind) can be efficiently sampled. If we do this we will have a probability monad encapsulating the logic of a bootstrap particle filter, such as is often used for computing the filtering distribution of a state-space model in time series analysis. This simple change won’t solve the computational issues associated with deep monadic binding, but does solve the memory problem, and can lead to computationally efficient algorithms so long as care is taken in the formulation of probabilistic programs to ensure that deep monadic binding doesn’t occur. We’ll discuss that issue in the context of state-space models later, once we have our new SMC-based probability monad.

Materials for this post can be found in my blog repo, and a draft of this post itself can be found in the form of an executable tut document.

## An SMC-based monad

The idea behind the approach to binding used in this monad is to mimic the “predict” step of a bootstrap particle filter. Here, for each particle in the source distribution, exactly one particle is drawn from the required conditional distribution and paired with the source particle, preserving the source particle’s original weight. So, in order to operationalise this, we will need a `draw` method adding into our probability monad. It will also simplify things to add a `flatMap` method to our `Particle` type constructor.

To follow along, you can type `sbt console` from the `min-ppl2` directory of my blog repo, then paste blocks of code one at a time.

```  import breeze.stats.{distributions => bdist}
import breeze.linalg.DenseVector
import cats._
import cats.implicits._

implicit val numParticles = 2000

case class Particle[T](v: T, lw: Double) { // value and log-weight
def map[S](f: T => S): Particle[S] = Particle(f(v), lw)
def flatMap[S](f: T => Particle[S]): Particle[S] = {
val ps = f(v)
Particle(ps.v, lw + ps.lw)
}
}
```

I’ve added a dependence on cats here, so that we can use some derived methods, later. To take advantage of this, we must provide evidence that our custom types conform to standard type class interfaces. For example, we can provide evidence that `Particle[_]` is a monad as follows.

```  implicit val particleMonad = new Monad[Particle] {
def pure[T](t: T): Particle[T] = Particle(t, 0.0)
def flatMap[T,S](pt: Particle[T])(f: T => Particle[S]): Particle[S] = pt.flatMap(f)
def tailRecM[T,S](t: T)(f: T => Particle[Either[T,S]]): Particle[S] = ???
}
```

The technical details are not important for this post, but we’ll see later what this can give us.

We can now define our `Prob[_]` monad in the following way.

```  trait Prob[T] {
val particles: Vector[Particle[T]]
def draw: Particle[T]
def mapP[S](f: T => Particle[S]): Prob[S] = Empirical(particles map (_ flatMap f))
def map[S](f: T => S): Prob[S] = mapP(v => Particle(f(v), 0.0))
def flatMap[S](f: T => Prob[S]): Prob[S] = mapP(f(_).draw)
def resample(implicit N: Int): Prob[T] = {
val lw = particles map (_.lw)
val mx = lw reduce (math.max(_,_))
val rw = lw map (lwi => math.exp(lwi - mx))
val law = mx + math.log(rw.sum/(rw.length))
val ind = bdist.Multinomial(DenseVector(rw.toArray)).sample(N)
val newParticles = ind map (i => particles(i))
Empirical(newParticles.toVector map (pi => Particle(pi.v, law)))
}
def cond(ll: T => Double): Prob[T] = mapP(v => Particle(v, ll(v)))
def empirical: Vector[T] = resample.particles.map(_.v)
}

case class Empirical[T](particles: Vector[Particle[T]]) extends Prob[T] {
def draw: Particle[T] = {
val lw = particles map (_.lw)
val mx = lw reduce (math.max(_,_))
val rw = lw map (lwi => math.exp(lwi - mx))
val law = mx + math.log(rw.sum/(rw.length))
val idx = bdist.Multinomial(DenseVector(rw.toArray)).draw
Particle(particles(idx).v, law)
}
}
```

As before, if you are pasting code blocks into the REPL, you will need to use `:paste` mode to paste these two definitions together.

The essential structure is similar to that from the previous post, but with a few notable differences. Most fundamentally, we now require any concrete implementation to provide a `draw` method returning a single particle from the distribution. Like before, we are not worrying about purity of functional code here, and using a standard random number generator with a globally mutating state. We can define a `mapP` method (for “map particle”) using the new `flatMap` method on `Particle`, and then use that to define `map` and `flatMap` for `Prob[_]`. Crucially, `draw` is used to define `flatMap` without requiring a Cartesian product of distributions to be formed.

We add a `draw` method to our `Empirical[_]` implementation. This method is computationally intensive, so it will still be computationally problematic to chain several `flatMap`s together, but this will no longer be $N^2$ in memory utilisation. Note that again we carefully set the weight of the drawn particle so that its raw weight is the average of the raw weight of the empirical distribution. This is needed to propagate conditioning information correctly back through `flatMaps`. There is obviously some code duplication between the `draw` method on `Empirical` and the `resample` method on `Prob`, but I’m not sure it’s worth factoring out.

It is worth noting that neither `flatMap` nor `cond` triggers resampling, so the user of the library is now responsible for resampling when appropriate.

We can provide evidence that `Prob[_]` forms a monad just like we did `Particle[_]`.

```  implicit val probMonad = new Monad[Prob] {
def pure[T](t: T): Prob[T] = Empirical(Vector(Particle(t, 0.0)))
def flatMap[T,S](pt: Prob[T])(f: T => Prob[S]): Prob[S] = pt.flatMap(f)
def tailRecM[T,S](t: T)(f: T => Prob[Either[T,S]]): Prob[S] = ???
}
```

Again, we’ll want to be able to create a distribution from an unweighted collection of values.

```  def unweighted[T](ts: Vector[T], lw: Double = 0.0): Prob[T] =
Empirical(ts map (Particle(_, lw)))
```

We will again define an implementation for distributions with tractable likelihoods, which are therefore easy to condition on. They will typically also be easy to draw from efficiently, and we will use this fact, too.

```  trait Dist[T] extends Prob[T] {
def ll(obs: T): Double
def ll(obs: Seq[T]): Double = obs map (ll) reduce (_+_)
def fit(obs: Seq[T]): Prob[T] = mapP(v => Particle(v, ll(obs)))
def fitQ(obs: Seq[T]): Prob[T] = Empirical(Vector(Particle(obs.head, ll(obs))))
def fit(obs: T): Prob[T] = fit(List(obs))
def fitQ(obs: T): Prob[T] = fitQ(List(obs))
}
```

We can give implementations of this for a few standard distributions.

```  case class Normal(mu: Double, v: Double)(implicit N: Int) extends Dist[Double] {
lazy val particles = unweighted(bdist.Gaussian(mu, math.sqrt(v)).
sample(N).toVector).particles
def draw = Particle(bdist.Gaussian(mu, math.sqrt(v)).draw, 0.0)
def ll(obs: Double) = bdist.Gaussian(mu, math.sqrt(v)).logPdf(obs)
}

case class Gamma(a: Double, b: Double)(implicit N: Int) extends Dist[Double] {
lazy val particles = unweighted(bdist.Gamma(a, 1.0/b).
sample(N).toVector).particles
def draw = Particle(bdist.Gamma(a, 1.0/b).draw, 0.0)
def ll(obs: Double) = bdist.Gamma(a, 1.0/b).logPdf(obs)
}

case class Poisson(mu: Double)(implicit N: Int) extends Dist[Int] {
lazy val particles = unweighted(bdist.Poisson(mu).
sample(N).toVector).particles
def draw = Particle(bdist.Poisson(mu).draw, 0.0)
def ll(obs: Int) = bdist.Poisson(mu).logProbabilityOf(obs)
}
```

Note that we now have to provide an (efficient) `draw` method for each implementation, returning a single draw from the distribution as a `Particle` with a (raw) weight of 1 (log weight of 0).

We are done. It’s a few more lines of code than that from the previous post, but this is now much closer to something that could be used in practice to solve actual inference problems using a reasonable number of particles. But to do so we will need to be careful do avoid deep monadic binding. This is easiest to explain with a concrete example.

## Using the SMC-based probability monad in practice

### Monadic binding and applicative structure

As explained in the previous post, using Scala’s `for`-expressions for monadic binding gives a natural and elegant PPL for our probability monad “for free”. This is fine, and in general there is no reason why using it should lead to inefficient code. However, for this particular probability monad implementation, it turns out that deep monadic binding comes with a huge performance penalty. For a concrete example, consider the following specification, perhaps of a prior distribution over some independent parameters.

```    val prior = for {
x <- Normal(0,1)
y <- Gamma(1,1)
z <- Poisson(10)
} yield (x,y,z)
```

Don’t paste that into the REPL – it will take an age to complete!

Again, I must emphasise that there is nothing wrong with this specification, and there is no reason in principle why such a specification can’t be computationally efficient – it’s just a problem for our particular probability monad. We can begin to understand the problem by thinking about how this will be de-sugared by the compiler. Roughly speaking, the above will de-sugar to the following nested `flatMap`s.

```    val prior2 =
Normal(0,1) flatMap {x =>
Gamma(1,1) flatMap {y =>
Poisson(10) map {z =>
(x,y,z)}}}
```

Again, beware of pasting this into the REPL.

So, although written from top to bottom, the nesting is such that the `flatMap`s collapse from the bottom-up. The second `flatMap` (the first to collapse) isn’t such a problem here, as the `Poisson` has a $O(1)$ `draw` method. But the result is an empirical distribution, which has an $O(N)$ draw method. So the first `flatMap` (the second to collapse) is an $O(N^2)$ operation. By extension, it’s easy to see that the computational cost of nested `flatMap`s will be exponential in the number of monadic binds. So, looking back at the `for` expression, the problem is that there are three `<-`. The last one isn’t a problem since it corresponds to a `map`, and the second last one isn’t a problem, since the final distribution is tractable with an $O(1)$ `draw` method. The problem is the first `<-`, corresponding to a `flatMap` of one empirical distribution with respect to another. For our particular probability monad, it’s best to avoid these if possible.

The interesting thing to note here is that because the distributions are independent, there is no need for them to be sequenced. They could be defined in any order. In this case it makes sense to use the applicative structure implied by the monad.

Now, since we have told cats that `Prob[_]` is a monad, it can provide appropriate applicative methods for us automatically. In Cats, every monad is assumed to be also an applicative functor (which is true in Cartesian closed categories, and Cats implicitly assumes that all functors and monads are defined over CCCs). So we can give an alternative specification of the above prior using applicative composition.

``` val prior3 = Applicative[Prob].tuple3(Normal(0,1), Gamma(1,1), Poisson(10))
// prior3: Wrapped.Prob[(Double, Double, Int)] = Empirical(Vector(Particle((-0.057088546468105204,0.03027578552505779,9),0.0), Particle((-0.43686658266043743,0.632210127012762,14),0.0), Particle((-0.8805715148936012,3.4799656228544706,4),0.0), Particle((-0.4371726407147289,0.0010707859994652403,12),0.0), Particle((2.0283297088320755,1.040984491158822,10),0.0), Particle((1.2971862986495886,0.189166705596747,14),0.0), Particle((-1.3111333817551083,0.01962422606642761,9),0.0), Particle((1.6573851896142737,2.4021836368401415,9),0.0), Particle((-0.909927220984726,0.019595551644771683,11),0.0), Particle((0.33888133893822464,0.2659823344145805,10),0.0), Particle((-0.3300797295729375,3.2714740256437667,10),0.0), Particle((-1.8520554352884224,0.6175322756460341,10),0.0), Particle((0.541156780497547...
```

This one is mathematically equivalent, but safe to paste into your REPL, as it does not involve deep monadic binding, and can be used whenever we want to compose together independent components of a probabilistic program. Note that “tupling” is not the only possibility – Cats provides a range of functions for manipulating applicative values.

This is one way to avoid deep monadic binding, but another strategy is to just break up a large `for` expression into separate smaller `for` expressions. We can examine this strategy in the context of state-space modelling.

### Particle filtering for a non-linear state-space model

We can now re-visit the DGLM example from the previous post. We began by declaring some observations and a prior.

```    val data = List(2,1,0,2,3,4,5,4,3,2,1)
// data: List[Int] = List(2, 1, 0, 2, 3, 4, 5, 4, 3, 2, 1)

val prior = for {
w <- Gamma(1, 1)
state0 <- Normal(0.0, 2.0)
} yield (w, List(state0))
// prior: Wrapped.Prob[(Double, List[Double])] = Empirical(Vector(Particle((4.220683377724395,List(0.37256749723762683)),0.0), Particle((0.4436668049925418,List(-1.0053578391265572)),0.0), Particle((0.9868899648436931,List(-0.6985099310193449)),0.0), Particle((0.13474375773634908,List(0.9099291736792412)),0.0), Particle((1.9654021747685184,List(-0.042127103727998175)),0.0), Particle((0.21761202474220223,List(1.1074616830012525)),0.0), Particle((0.31037163527711015,List(0.9261849914020324)),0.0), Particle((1.672438830781466,List(0.01678529855289384)),0.0), Particle((0.2257151759143097,List(2.5511304854128354)),0.0), Particle((0.3046489890769499,List(3.2918304533361398)),0.0), Particle((1.5115941814057159,List(-1.633612165168878)),0.0), Particle((1.4185906813831506,List(-0.8460922678989864))...
```

Looking carefully at the `for`-expression, there are just two `<-`, and the distribution on the RHS of the `flatMap` is tractable, so this is just $O(N)$. So far so good.

Next, let’s look at the function to add a time point, which previously looked something like the following.

```    def addTimePointSIS(current: Prob[(Double, List[Double])],
obs: Int): Prob[(Double, List[Double])] = {
println(s"Conditioning on observation: \$obs")
for {
tup <- current
(w, states) = tup
ns <- Normal(os, w)
_ <- Poisson(math.exp(ns)).fitQ(obs)
} yield (w, ns :: states)
}
// addTimePointSIS: (current: Wrapped.Prob[(Double, List[Double])], obs: Int)Wrapped.Prob[(Double, List[Double])]
```

Recall that our new probability monad does not automatically trigger resampling, so applying this function in a `fold` will lead to a simple sampling importance sampling (SIS) particle filter. Typically, the bootstrap particle filter includes resampling after each time point, giving a special case of a sampling importance resampling (SIR) particle filter, which we could instead write as follows.

```    def addTimePointSimple(current: Prob[(Double, List[Double])],
obs: Int): Prob[(Double, List[Double])] = {
println(s"Conditioning on observation: \$obs")
val updated = for {
tup <- current
(w, states) = tup
ns <- Normal(os, w)
_ <- Poisson(math.exp(ns)).fitQ(obs)
} yield (w, ns :: states)
updated.resample
}
// addTimePointSimple: (current: Wrapped.Prob[(Double, List[Double])], obs: Int)Wrapped.Prob[(Double, List[Double])]
```

This works fine, but we can see that there are three `<-` in this for expression. This leads to a `flatMap` with an empirical distribution on the RHS, and hence is $O(N^2)$. But this is simple enough to fix, by separating the updating process into separate “predict” and “update” steps, which is how people typically formulate particle filters for state-space models, anyway. Here we could write that as

```    def addTimePoint(current: Prob[(Double, List[Double])],
obs: Int): Prob[(Double, List[Double])] = {
println(s"Conditioning on observation: \$obs")
val predict = for {
tup <- current
(w, states) = tup
ns <- Normal(os, w)
}
yield (w, ns :: states)
val updated = for {
tup <- predict
(w, states) = tup
_ <- Poisson(math.exp(st)).fitQ(obs)
} yield (w, states)
updated.resample
}
// addTimePoint: (current: Wrapped.Prob[(Double, List[Double])], obs: Int)Wrapped.Prob[(Double, List[Double])]
```

By breaking the `for` expression into two: the first for the “predict” step and the second for the “update” (conditioning on the observation), we get two $O(N)$ operations, which for large $N$ is clearly much faster. We can then run the filter by folding over the observations.

```  import breeze.stats.{meanAndVariance => meanVar}
// import breeze.stats.{meanAndVariance=>meanVar}

val mod = data.foldLeft(prior)(addTimePoint(_,_)).empirical
// Conditioning on observation: 2
// Conditioning on observation: 1
// Conditioning on observation: 0
// Conditioning on observation: 2
// Conditioning on observation: 3
// Conditioning on observation: 4
// Conditioning on observation: 5
// Conditioning on observation: 4
// Conditioning on observation: 3
// Conditioning on observation: 2
// Conditioning on observation: 1
// mod: Vector[(Double, List[Double])] = Vector((0.24822528144246606,List(0.06290285371838457, 0.01633338109272575, 0.8997103339551227, 1.5058726341571411, 1.0579925693609091, 1.1616536515200064, 0.48325623593870665, 0.8457351097543767, -0.1988290999293708, -0.4787511341321954, -0.23212497417019512, -0.15327432440577277)), (1.111430233331792,List(0.6709342824443849, 0.009092797044165657, -0.13203367846117453, 0.4599952735399485, 1.3779288637042504, 0.6176597963402879, 0.6680455419800753, 0.48289163013446945, -0.5994001698510807, 0.4860969602653898, 0.10291798193078927, 1.2878325765987266)), (0.6118925941009055,List(0.6421161986636132, 0.679470360928868, 1.0552459559203342, 1.200835166087372, 1.3690372269589233, 1.8036766847282912, 0.6229883551656629, 0.14872642198313774, -0.122700856878725...

meanVar(mod map (_._1)) // w
// res0: breeze.stats.meanAndVariance.MeanAndVariance = MeanAndVariance(0.2839184023932576,0.07391602428256917,2000)

meanVar(mod map (_._2.reverse.head)) // initial state
// res1: breeze.stats.meanAndVariance.MeanAndVariance = MeanAndVariance(0.26057368528422714,0.4802810202354611,2000)

meanVar(mod map (_._2.head)) // final state
// res2: breeze.stats.meanAndVariance.MeanAndVariance = MeanAndVariance(0.5448036669181697,0.28293080584600894,2000)
```

## Summary and conclusions

Here we have just done some minor tidying up of the rather naive probability monad from the previous post to produce an SMC-based probability monad with improved performance characteristics. Again, we get an embedded probabilistic programming language “for free”. Although the language itself is very flexible, allowing us to construct more-or-less arbitrary probabilistic programs for Bayesian inference problems, we saw that a bug/feature of this particular inference algorithm is that care must be taken to avoid deep monadic binding if reasonable performance is to be obtained. In most cases this is simple to achieve by using applicative composition or by breaking up large `for` expressions.

There are still many issues and inefficiencies associated with this PPL. In particular, if the main intended application is to state-space models, it would make more sense to tailor the algorithms and implementations to exactly that case. OTOH, if the main concern is a generic PPL, then it would make sense to make the PPL independent of the particular inference algorithm. These are both potential topics for future posts.

### Software

• min-ppl2 – code associated with this blog post
• Rainier – a more efficient PPL with similar syntax
• monad-bayes – a Haskell library exploring related ideas

## Introduction

There is a fairly large literature on reaction-diffusion modelling using partial differential equations (PDEs). There is also a fairly large literature on stochastic modelling of coupled chemical reactions, which account for the discreteness of reacting species at low concentrations. There is some literature on combining the two, to form stochastic reaction-diffusion systems, but much less.

In this post we will look at one approach to the stochastic reaction-diffusion problem, based on an underlying stochastic process often described by the reaction diffusion master equation (RDME). We will start by generating exact realisations from this process using the spatial Gillespie algorithm, before switching to a continuous stochastic approximation often known as the spatial chemical Langevin equation (spatial CLE). For fine discretisations, this spatial CLE is just an explicit numerical scheme for an associated reaction-diffusion stochastic partial differential equation (SPDE), and we can easily contrast such SPDE dynamics with their deterministic PDE approximation. We will investigate using simulation, based on my Scala library, scala-smfsb, which accompanies the third edition of my textbook, Stochastic modelling for systems biology, as discussed in previous posts.

All of the code used to generate the plots and movies in this post is available in my blog repo, and is very simple to build and run.

## The Lotka-Volterra reaction network

### Exact simulation from the RDME

My favourite toy coupled chemical reaction network is the Lotka-Volterra predator-prey system, presented as the three reactions

$X \longrightarrow 2X$
$X + Y \longrightarrow 2Y$
$Y \longrightarrow \emptyset$

with $X$ representing the prey species and $Y$ the predator. I showed how to simulate realisations from this process using the Scala library in the previous post. Here we will consider simulation of this model in 2d, and simulate exact realisation from the appropriate RDME using the spatial Gillespie algorithm. Full runnable code for this simulation is here, but the key lines are:

```val r = 100; val c = 120
val model = SpnModels.lv[IntState]()
val step = Spatial.gillespie2d(model, DenseVector(0.6, 0.6), maxH=1e12)
val x00 = DenseVector(0, 0)
val x0 = DenseVector(50, 100)
val xx00 = PMatrix(r, c, Vector.fill(r*c)(x00))
val xx0 = xx00.updated(c/2, r/2, x0)
val s = Stream.iterate(xx0)(step(_,0.0,0.1))
```

which sets up an infinite lazy `Stream` of states on a 100×120 grid over time steps of 0.1 units with diffusion rates of 0.6 for both species. We can then map this to a stream of images and visualise it using my scala-view library (described in this post). Running gives the following output:

The above image is the final frame of a movie which can be viewed by clicking on the image. In the simulation, blue represents the prey species, $X$, and red represents the predator, $Y$. The simulation is initialised with a few prey and predators in the central pixel. At each time step of the simulation, either a reaction or a diffusion event may occur. If diffusion occurs, an individual moves from its current location to one of the four adjacent pixels. This algorithm is extremely computationally intensive, however well it is implemented. The implementation used here (using the function `Spatial.gillespie2d` in the `scala-smfsb` library) is quite inefficient. A more efficient implementation would use the next subvolume method or similar algorithm. But since every reaction event is simulated sequentially, this algorithm is always going to be intolerably slow for most interesting problems.

### The spatial CLE

The spatial CLE effectively approximates the true RDME dynamics with a set of coupled stochastic differential equations (SDEs) on the spatial grid. This can be interpreted as an explicit scheme for numerically integrating an SPDE. But this numerical scheme is much more efficient, allowing sensible time-stepping of the process, and vectorises and parallelises nicely. The details are in my book, but the Scala implementation is here. Diffusion is implemented efficiently and in parallel using the comonadic approach that I’ve described previously. We can quickly and easily generate large simulations using the spatial CLE. Here is a movie generated on a 250×300 grid.

Again, clicking on the frame should give the movie. We see that although the quantitative details are slightly different to the exact algorithm, the essential qualitative behaviour of the system is captured well by the spatial CLE. Full code for this simulation is here.

### Reaction-diffusion PDE

If we remove all of the noise terms from the spatial CLE, we get a set of coupled ODEs, which again, may be interpreted as a numerical scheme for integrating a reaction-diffusion PDE model. Below are the dynamics on the same 250×300 grid.

It seems a bit harsh to describe a reaction-diffusion PDE as “boring”, but it certainly isn’t as interesting as the stochastic dynamics. Also, it has qualitatively quite different behaviour to the stochastic models, with wavefronts being less pronounced and less well separated. The code for this one is here.

### Other initialisations

Instead of just seeding the simulation with some individuals in the central pixel, we can initialise 3 pixels. We can look first at a spatial CLE simulation.

Code here.

We can look at the same problem, but now using a PDE.

Code here.

Alternatively, we can initialise every pixel independently with random numbers of predator and prey. A movie for this is given below, following a short warm-up.

Code here.

Again, we can look at the corresponding deterministic integration.

Code here.

## The SIR model

Let’s now turn attention to a spatial epidemic process model, the spatial susceptible-infectious-recovered model. Again, we’ll start from the discrete reaction formulation.

$S + I \longrightarrow 2I$
$I \longrightarrow R$

I’ll add this model to the next release of `scala-smfsb`, but in the meantime we can easily define it ourselves with:

```def sir[S: State](p: DenseVector[Double] = DenseVector(0.1, 0.5)): Spn[S] =
UnmarkedSpn[S](
List("S", "I", "R"),
DenseMatrix((1, 1, 0), (0, 1, 0)),
DenseMatrix((0, 2, 0), (0, 0, 1)),
(x, t) => {
val xd = x.toDvd
DenseVector(
xd(0) * xd(1) * p(0), xd(1) * p(1)
)}
)
```

We can seed a simulation with a few infectious individuals in the centre of a roughly homogeneous population of susceptibles.

## Spatial CLE

This time we’ll skip the exact simulation, since it’s very slow, and go straight to the spatial CLE. A simulation on a 250×300 grid is given below.

Here, green represents $S$, red $I$ and blue $R$. In this simulation, $I$ diffuses more slowly than $S$, and $R$ doesn’t diffuse at all.
Code here.

## PDE model

If we ditch the noise to get a reaction-diffusion PDE model, the dynamics are as follows.

Again, we see that the deterministic model is quite different to the stochastic version, and kind-of boring. Code here.

## Further information

All of the code used to generate the plots and movies in this post is available in an easily runnable form in my blog repo. It is very easy to adapt the examples to vary parameters and initial conditions, and to study other reaction systems. Further details relating to stochastic reaction-diffusion modelling based on the RDME can be found in Chapter 9 of my textbook, Stochastic modelling for systems biology, third edition.

## The scala-smfsb library

In the previous post I gave a very quick introduction to the `smfsb` R package. As mentioned in that post, although good for teaching and learning, R isn’t a great language for serious scientific computing or computational statistics. So for the publication of the third edition of my textbook, Stochastic modelling for systems biology, I have created a library in the Scala programming language replicating the functionality provided by the R package. Here I will give a very quick introduction to the scala-smfsb library. Some familiarity with both Scala and the `smfsb` R package will be helpful, but is not strictly necessary. Note that the library relies on the Scala Breeze library for linear algebra and probability distributions, so some familiarity with that library can also be helpful.

## Setup

To follow the along you need to have Sbt installed, and this in turn requires a recent JDK. If you are new to Scala, you may find the setup page for my Scala course to be useful, but note that on many Linux systems it can be as simple as installing the packages `openjdk-8-jdk` and `sbt`.

Once you have Sbt installed, you should be able to run it by entering `sbt` at your OS command line. You now need to use Sbt to create a Scala REPL with a dependency on the `scala-smfsb` library. There are many ways to do this, but if you are new to Scala, the simplest way is probably to start up Sbt from an empty or temporary directory (which doesn’t contain any Scala code), and then paste the following into the Sbt prompt:

```set libraryDependencies += "com.github.darrenjw" %% "scala-smfsb" % "0.6"
set libraryDependencies += "org.scalanlp" %% "breeze-viz" % "0.13.2"
set scalaVersion := "2.12.6"
set scalacOptions += "-Yrepl-class-based"
console
```

The first time you run this it will take a little while to download and cache various library dependencies. But everything is cached, so it should be much quicker in future. When it is finished, you should have a Scala REPL ready to enter Scala code.

## An introduction to `scala-smfsb`

It should be possible to type or copy-and-paste the commands below one-at-a-time into the Scala REPL. We need to start with a few imports.

```import smfsb._
import breeze.linalg.{Vector => BVec, _}
import breeze.numerics._
import breeze.plot._
```

Note that I’ve renamed Breeze’s `Vector` type to `BVec` to avoid clashing with that in the Scala standard library. We are now ready to go.

### Simulating models

Let’s begin by instantiating a Lotka-Volterra model, simulating a single realisation of the process, and then plotting it.

```// Simulate LV with Gillespie
val model = SpnModels.lv[IntState]()
val step = Step.gillespie(model)
val ts = Sim.ts(DenseVector(50, 100), 0.0, 20.0, 0.05, step)
Sim.plotTs(ts, "Gillespie simulation of LV model with default parameters")
```

The library comes with a few other models. There’s a Michaelis-Menten enzyme kinetics model:

```// Simulate other models with Gillespie
val stepMM = Step.gillespie(SpnModels.mm[IntState]())
val tsMM = Sim.ts(DenseVector(301,120,0,0), 0.0, 100.0, 0.5, stepMM)
Sim.plotTs(tsMM, "Gillespie simulation of the MM model")
```

and an auto-regulatory genetic network model, for example.

```val stepAR = Step.gillespie(SpnModels.ar[IntState]())
val tsAR = Sim.ts(DenseVector(10, 0, 0, 0, 0), 0.0, 500.0, 0.5, stepAR)
Sim.plotTs(tsAR, "Gillespie simulation of the AR model")
```

If you know the book and/or the R package, these models should all be familiar.
We are not restricted to exact stochastic simulation using the Gillespie algorithm. We can use an approximate Poisson time-stepping algorithm.

```// Simulate LV with other algorithms
val stepPts = Step.pts(model)
val tsPts = Sim.ts(DenseVector(50, 100), 0.0, 20.0, 0.05, stepPts)
Sim.plotTs(tsPts, "Poisson time-step simulation of the LV model")
```

Alternatively, we can instantiate the example models using a continuous state rather than a discrete state, and then simulate using algorithms based on continous approximations, such as Euler-Maruyama simulation of a chemical Langevin equation (CLE) approximation.

```val stepCle = Step.cle(SpnModels.lv[DoubleState]())
val tsCle = Sim.ts(DenseVector(50.0, 100.0), 0.0, 20.0, 0.05, stepCle)
Sim.plotTs(tsCle, "Euler-Maruyama/CLE simulation of the LV model")
```

If we want to ignore noise temporarily, there’s also a simple continuous deterministic Euler integrator built-in.

```val stepE = Step.euler(SpnModels.lv[DoubleState]())
val tsE = Sim.ts(DenseVector(50.0, 100.0), 0.0, 20.0, 0.05, stepE)
Sim.plotTs(tsE, "Continuous-deterministic Euler simulation of the LV model")
```

### Spatial stochastic reaction-diffusion simulation

We can do 1d reaction-diffusion simulation with something like:

```val N = 50; val T = 40.0
val model = SpnModels.lv[IntState]()
val step = Spatial.gillespie1d(model,DenseVector(0.8, 0.8))
val x00 = DenseVector(0, 0)
val x0 = DenseVector(50, 100)
val xx00 = Vector.fill(N)(x00)
val xx0 = xx00.updated(N/2,x0)
val output = Sim.ts(xx0, 0.0, T, 0.2, step)
Spatial.plotTs1d(output)
```

For 2d simulation, we use `PMatrix`, a comonadic matrix/image type defined within the library, with parallelised `map` and `coflatMap` (cobind) operations. See my post on comonads for scientific computing for further details on the concepts underpinning this, though note that it isn’t necessary to understand comonads to use the library.

```val r = 20; val c = 30
val model = SpnModels.lv[DoubleState]()
val step = Spatial.cle2d(model, DenseVector(0.6, 0.6), 0.05)
val x00 = DenseVector(0.0, 0.0)
val x0 = DenseVector(50.0, 100.0)
val xx00 = PMatrix(r, c, Vector.fill(r*c)(x00))
val xx0 = xx00.updated(c/2, r/2, x0)
val output = step(xx0, 0.0, 8.0)
val f = Figure("2d LV reaction-diffusion simulation")
val p0 = f.subplot(2, 1, 0)
p0 += image(PMatrix.toBDM(output map (_.data(0))))
val p1 = f.subplot(2, 1, 1)
p1 += image(PMatrix.toBDM(output map (_.data(1))))
```

### Bayesian parameter inference

The library also includes functions for carrying out parameter inference for stochastic dynamical systems models, using particle MCMC, ABC and ABC-SMC. See the examples directory for further details.

## Next steps

Having worked through this post, the next step is to work through the tutorial. There is some overlap of content with this blog post, but the tutorial goes into more detail regarding the basics. It also finishes with suggestions for how to proceed further.

## Source

This post started out as a tut document (the Scala equivalent of an RMarkdown document). The source can be found here.

## Stochastic Modelling for Systems Biology, third edition

The third edition of my textbook, Stochastic Modelling for Systems Biology has recently been published by Chapman & Hall/CRC Press. The book has ISBN-10 113854928-2 and ISBN-13 978-113854928-9. It can be ordered from CRC Press, Amazon.com, Amazon.co.uk and similar book sellers.

I was fairly happy with the way that the second edition, published in 2011, turned out, and so I haven’t substantially re-written any of the text for the third edition. Instead, I’ve concentrated on adding in new material and improving the associated on-line resources. Those on-line resources are all free and open source, and hence available to everyone, irrespective of whether you have a copy of the new edition. I’ll give an introduction to those resources below (and in subsequent posts). The new material can be briefly summarised as follows:

• New chapter on spatially extended systems, covering the spatial Gillespie algorithm for reaction diffusion master equation (RDME) models in 1- and 2-d, the next subvolume method, spatial CLE, scaling issues, etc.
• Significantly expanded chapter on inference for stochastic kinetic models from data, covering approximate methods of inference (ABC), including ABC-SMC. The material relating to particle MCMC has also been improved and extended.
• Updated R package, including code relating to all of the new material
• New R package for parsing SBML models into simulatable stochastic Petri net models
• New software library, written in Scala, replicating most of the functionality of the R packages in a fast, compiled, strongly typed, functional language

## New content

Although some minor edits and improvements have been made throughout the text, there are two substantial new additions to the text in this new edition. The first is an entirely new chapter on spatially extended systems. The first two editions of the text focused on the implications of discreteness and stochasticity in chemical reaction systems, but maintained the well-mixed assumption throughout. This is a reasonable first approach, since discreteness and stochasticity are most pronounced in very small volumes where diffusion should be rapid. In any case, even these non-spatial models have very interesting behaviour, and become computationally challenging very quickly for non-trivial reaction networks. However, we know that, in fact, the cell is a very crowded environment, and so even at small spatial scales, many interesting processes are diffusion limited. It therefore seems appropriate to dedicate one chapter (the new Chapter 9) to studying some of the implications of relaxing the well-mixed assumption. Entire books can be written on stochastic reaction-diffusion systems, so here only a brief introduction is provided, based mainly around models in the reaction-diffusion master equation (RDME) style. Exact stochastic simulation algorithms are discussed, and implementations provided in the 1- and 2-d cases, and an appropriate Langevin approximation is examined, the spatial CLE.

The second major addition is to the chapter on inference for stochastic kinetic models from data (now Chapter 11). The second edition of the book included a discussion of “likelihood free” Bayesian MCMC methods for inference, and provided a working implementation of likelihood free particle marginal Metropolis-Hastings (PMMH) for stochastic kinetic models. The third edition improves on that implementation, and discusses approximate Bayesian computation (ABC) as an alternative to MCMC for likelihood free inference. Implementation issues are discussed, and sequential ABC approaches are examined, concentrating in particular on the method known as ABC-SMC.

## New software and on-line resources

Accompanying the text are new and improved on-line resources, all well-documented, free, and open source.

### New website/GitHub repo

Information and materials relating to the previous editions were kept on my University website. All materials relating to this new edition are kept in a public GitHub repo: darrenjw/smfsb. This will be simpler to maintain, and will make it much easier for people to make copies of the material for use and studying off-line.

### Updated R package(s)

Along with the second edition of the book I released an accompanying R package, “smfsb”, published on CRAN. This was a very popular feature, allowing anyone with R to trivially experiment with all of the models and algorithms discussed in the text. This R package has been updated, and a new version has been published to CRAN. The updates are all backwards-compatible with the version associated with the second edition of the text, so owners of that edition can still upgrade safely. I’ll give a proper introduction to the package, including the new features, in a subsequent post, but in the meantime, you can install/upgrade the package from a running R session with

```install.packages("smfsb")
```

and then pop up a tutorial vignette with:

```vignette("smfsb")
```

This should be enough to get you started.

In addition to the main R package, there is an additional R package for parsing SBML models into models that can be simulated within R. This package is not on CRAN, due to its dependency on a non-CRAN package. See the repo for further details.

There are also Python scripts available for converting SBML models to and from the shorthand SBML notation used in the text.

### New Scala library

Another major new resource associated with the third edition of the text is a software library written in the Scala programming language. This library provides Scala implementations of all of the algorithms discussed in the book and implemented in the associated R packages. This then provides example implementations in a fast, efficient, compiled language, and is likely to be most useful for people wanting to use the methods in the book for research. Again, I’ll provide a tutorial introduction to this library in a subsequent post, but it is well-documented, with all necessary information needed to get started available at the scala-smfsb repo/website, including a step-by-step tutorial and some additional examples.

## Introduction

In the previous post I discussed how comonads can be useful for structuring certain kinds of scientific and statistical computations. Two of the examples I gave were concerned with the time-evolution of 2-d images. In that post I used Breeze to animate the sequence of computed images. In this post I want to describe an alternative that is better suited to animating an image sequence.

Scala-view is a small Scala library for animating a Stream of Images on-screen in a separate window managed by your window manager. It works with both ScalaFX `Images` (recommended) and Scala Swing/AWT `BufferedImages` (legacy). The stream of images is animated in a window with some simple controls to start and stop the animation, and to turn on and off the saving of image frames to disk (typically for the purpose of turning the image sequence into a movie). An example of what a window might look like is given below.

More comprehensive documentation is available from the scala-view github repo, but here I give a quick introduction to the library to outline its capabilities.

## A Scala-view tutorial

This brief tutorial gives a quick introduction to using the Scala-view library for viewing a ScalaFX Image Stream. It assumes only that you have SBT installed, and that you run SBT from an empty directory.

## An SBT REPL

Start by running SBT from an empty or temporary directory to get an SBT prompt:

```\$ sbt
>
```

Now we need to configure SBT to use the Scala-view library, and start a console. From the SBT prompt:

```set libraryDependencies += "com.github.darrenjw" %% "scala-view" % "0.5"
set scalaVersion := "2.12.4"
console
```

The should result in a `scala>` REPL prompt. We can now use Scala and the Scala-view library interactively.

## An example REPL session

You should be able to paste the code snippets below directly into the REPL. You may find `:paste` mode helpful.

We will replicate the heat equation example from the examples-sfx directory, which is loosely based on the example from my blog post on comonads. We will start by defining a simple parallel Image and corresponding comonadic pointed image `PImage` type. If you aren’t familiar with comonads, you may find it helpful to read through that post.

```import scala.collection.parallel.immutable.ParVector
case class Image[T](w: Int, h: Int, data: ParVector[T]) {
def apply(x: Int, y: Int): T = data(x * h + y)
def map[S](f: T => S): Image[S] = Image(w, h, data map f)
def updated(x: Int, y: Int, value: T): Image[T] =
Image(w, h, data.updated(x * h + y, value))
}

case class PImage[T](x: Int, y: Int, image: Image[T]) {
def extract: T = image(x, y)
def map[S](f: T => S): PImage[S] = PImage(x, y, image map f)
def coflatMap[S](f: PImage[T] => S): PImage[S] = PImage(
x, y, Image(image.w, image.h,
(0 until (image.w * image.h)).toVector.par.map(i => {
val xx = i / image.h
val yy = i % image.h
f(PImage(xx, yy, image))
})))
def up: PImage[T] = {
val py = y - 1
val ny = if (py >= 0) py else (py + image.h)
PImage(x, ny, image)
}
def down: PImage[T] = {
val py = y + 1
val ny = if (py < image.h) py else (py - image.h)
PImage(x, ny, image)
}
def left: PImage[T] = {
val px = x - 1
val nx = if (px >= 0) px else (px + image.w)
PImage(nx, y, image)
}
def right: PImage[T] = {
val px = x + 1
val nx = if (px < image.w) px else (px - image.w)
PImage(nx, y, image)
}
}
```

We will need a function to convert this image into a ScalaFX `WritableImage`.

```import scalafx.scene.image.WritableImage
import scalafx.scene.paint._
def toSfxI(im: Image[Double]): WritableImage = {
val wi = new WritableImage(im.w, im.h)
val pw = wi.pixelWriter
(0 until im.w) foreach (i =>
(0 until im.h) foreach (j =>
pw.setColor(i, j, Color.gray(im(i,j)))
))
wi
}
```

We will need a starting image representing the initial condition for the heat equation.

```val w = 600
val h = 500
val pim0 = PImage(0, 0, Image(w, h,
((0 until w*h).toVector map {i: Int => {
val x = i / h
val y = i % h
0.1*math.cos(0.1*math.sqrt((x*x+y*y))) + 0.1 + 0.8*math.random
}}).par
))
```

We can define a kernel associated with the update of a single image pixel based on a single time step of a finite difference solution of the heat equation.

```def kernel(pi: PImage[Double]): Double = (2*pi.extract+
pi.up.extract+pi.down.extract+pi.left.extract+pi.right.extract)/6.0
```

We can now create a `Stream` of `PImage` with

```def pims = Stream.iterate(pim0)(_.coflatMap(kernel))
```

We can turn this into a `Stream[WritableImage]` with

```def sfxis = pims map (im => toSfxI(im.image))
```

Note that we are essentially finished at this point, but so far everything we have done has been purely functional with no side effects. We haven’t even computed our solution to the heat equation. All we have constructed are lazy infinite streams representing the solution of the heat equation.

Finally, we can render our Stream of Images on screen with

```scalaview.SfxImageViewer(sfxis,1e7.toInt)
```

which has a delay of 1e7 nanoseconds (10 milliseconds) between frames.

This should pop up a window on your display containing the initial image. Click on the Start button to animate the solution of the heat equation. See the API docs for SfxImageViewer for additional options. The ScalaFX API docs may also be useful, especially the docs for Image and WritableImage.

## Introduction

In a previous post I’ve given a brief introduction to monads in Scala, aimed at people interested in scientific and statistical computing. Monads are a concept from category theory which turn out to be exceptionally useful for solving many problems in functional programming. But most categorical concepts have a dual, usually prefixed with “co”, so the dual of a monad is the comonad. Comonads turn out to be especially useful for formulating algorithms from scientific and statistical computing in an elegant way. In this post I’ll illustrate their use in signal processing, image processing, numerical integration of PDEs, and Gibbs sampling (of an Ising model). Comonads enable the extension of a local computation to a global computation, and this pattern crops up all over the place in statistical computing.

Simplifying massively, from the viewpoint of a Scala programmer, a monad is a mappable (functor) type class augmented with the methods `pure` and `flatMap`:

```trait Monad[M[_]] extends Functor[M] {
def pure[T](v: T): M[T]
def flatMap[T,S](v: M[T])(f: T => M[S]): M[S]
}
```

In category theory, the dual of a concept is typically obtained by “reversing the arrows”. Here that means reversing the direction of the methods `pure` and `flatMap` to get `extract` and `coflatMap`, respectively.

```trait Comonad[W[_]] extends Functor[W] {
def extract[T](v: W[T]): T
def coflatMap[T,S](v: W[T])(f: W[T] => S): W[S]
}
```

So, while `pure` allows you to wrap plain values in a monad, `extract` allows you to get a value out of a comonad. So you can always get a value out of a comonad (unlike a monad). Similarly, while `flatMap` allows you to transform a monad using a function returning a monad, `coflatMap` allows you to transform a comonad using a function which collapses a comonad to a single value. It is `coflatMap` (sometimes called `extend`) which can extend a local computation (producing a single value) to the entire comonad. We’ll look at how that works in the context of some familiar examples.

## Applying a linear filter to a data stream

One of the simplest examples of a comonad is an infinite stream of data. I’ve discussed streams in a previous post. By focusing on infinite streams we know the stream will never be empty, so there will always be a value that we can `extract`. Which value does `extract` give? For a `Stream` encoded as some kind of lazy list, the only value we actually know is the value at the head of the stream, with subsequent values to be lazily computed as required. So the head of the list is the only reasonable value for `extract` to return.

Understanding `coflatMap` is a bit more tricky, but it is `coflatMap` that provides us with the power to apply a non-trivial statistical computation to the stream. The input is a function which transforms a stream into a value. In our example, that will be a function which computes a weighted average of the first few values and returns that weighted average as the result. But the return type of `coflatMap` must be a stream of such computations. Following the types, a few minutes thought reveals that the only reasonable thing to do is to return the stream formed by applying the weighted average function to all sub-streams, recursively. So, for a `Stream` `s` (of type `Stream[T]`) and an input function `f: W[T] => S`, we form a stream whose head is `f(s)` and whose tail is `coflatMap(f)` applied to `s.tail`. Again, since we are working with an infinite stream, we don’t have to worry about whether or not the `tail` is empty. This gives us our comonadic `Stream`, and it is exactly what we need for applying a linear filter to the data stream.

In Scala, Cats is a library providing type classes from Category theory, and instances of those type classes for parametrised types in the standard library. In particular, it provides us with comonadic functionality for the standard Scala `Stream`. Let’s start by defining a stream corresponding to the logistic map.

```import cats._
import cats.implicits._

val lam = 3.7
def s = Stream.iterate(0.5)(x => lam*x*(1-x))
s.take(10).toList
// res0: List[Double] = List(0.5, 0.925, 0.25668749999999985,
//  0.7059564011718747, 0.7680532550204203, 0.6591455741499428, ...
```

Let us now suppose that we want to apply a linear filter to this stream, in order to smooth the values. The idea behind using comonads is that you figure out how to generate one desired value, and let `coflatMap` take care of applying the same logic to the rest of the structure. So here, we need a function to generate the first filtered value (since `extract` is focused on the head of the stream). A simple first attempt a function to do this might look like the following.

```  def linearFilterS(weights: Stream[Double])(s: Stream[Double]): Double =
(weights, s).parMapN(_*_).sum
```

This aligns each weight in parallel with a corresponding value from the stream, and combines them using multiplication. The resulting (hopefully finite length) stream is then summed (with addition). We can test this with

```linearFilterS(Stream(0.25,0.5,0.25))(s)
// res1: Double = 0.651671875
```

and let `coflatMap` extend this computation to the rest of the stream with something like:

```s.coflatMap(linearFilterS(Stream(0.25,0.5,0.25))).take(5).toList
// res2: List[Double] = List(0.651671875, 0.5360828502929686, ...
```

This is all completely fine, but our `linearFilterS` function is specific to the `Stream` comonad, despite the fact that all we’ve used about it in the function is that it is a parallelly composable and foldable. We can make this much more generic as follows:

```  def linearFilter[F[_]: Foldable, G[_]](
weights: F[Double], s: F[Double]
)(implicit ev: NonEmptyParallel[F, G]): Double =
(weights, s).parMapN(_*_).fold
```

This uses some fairly advanced Scala concepts which I don’t want to get into right now (I should also acknowledge that I had trouble getting the syntax right for this, and got help from Fabio Labella (@SystemFw) on the Cats gitter channel). But this version is more generic, and can be used to linearly filter other data structures than `Stream`. We can use this for regular `Streams` as follows:

```s.coflatMap(s => linearFilter(Stream(0.25,0.5,0.25),s))
// res3: scala.collection.immutable.Stream[Double] = Stream(0.651671875, ?)
```

But we can apply this new filter to other collections. This could be other, more sophisticated, streams such as provided by FS2, Monix or Akka streams. But it could also be a non-stream collection, such as `List`:

```val sl = s.take(10).toList
sl.coflatMap(sl => linearFilter(List(0.25,0.5,0.25),sl))
// res4: List[Double] = List(0.651671875, 0.5360828502929686, ...
```

Assuming that we have the Breeze scientific library available, we can plot the raw and smoothed trajectories.

```def myFilter(s: Stream[Double]): Double =
linearFilter(Stream(0.25, 0.5, 0.25),s)
val n = 500
import breeze.plot._
import breeze.linalg._
val fig = Figure(s"The (smoothed) logistic map (lambda=\$lam)")
val p0 = fig.subplot(3,1,0)
p0 += plot(linspace(1,n,n),s.take(n))
p0.ylim = (0.0,1.0)
p0.title = s"The logistic map (lambda=\$lam)"
val p1 = fig.subplot(3,1,1)
p1 += plot(linspace(1,n,n),s.coflatMap(myFilter).take(n))
p1.ylim = (0.0,1.0)
p1.title = "Smoothed by a simple linear filter"
val p2 = fig.subplot(3,1,2)
p2 += plot(linspace(1,n,n),s.coflatMap(myFilter).coflatMap(myFilter).coflatMap(myFilter).coflatMap(myFilter).coflatMap(myFilter).take(n))
p2.ylim = (0.0,1.0)
p2.title = "Smoothed with 5 applications of the linear filter"
fig.refresh
```

## Image processing and the heat equation

Streaming data is in no way the only context in which a comonadic approach facilitates an elegant approach to scientific and statistical computing. Comonads crop up anywhere where we want to extend a computation that is local to a small part of a data structure to the full data structure. Another commonly cited area of application of comonadic approaches is image processing (I should acknowledge that this section of the post is very much influenced by a blog post on comonadic image processing in Haskell). However, the kinds of operations used in image processing are in many cases very similar to the operations used in finite difference approaches to numerical integration of partial differential equations (PDEs) such as the heat equation, so in this section I will blur (sic) the distinction between the two, and numerically integrate the 2D heat equation in order to Gaussian blur a noisy image.

First we need a simple image type which can have pixels of arbitrary type `T` (this is very important – all functors must be fully type polymorphic).

```  import scala.collection.parallel.immutable.ParVector
case class Image[T](w: Int, h: Int, data: ParVector[T]) {
def apply(x: Int, y: Int): T = data(x*h+y)
def map[S](f: T => S): Image[S] = Image(w, h, data map f)
def updated(x: Int, y: Int, value: T): Image[T] =
Image(w,h,data.updated(x*h+y,value))
}
```

Here I’ve chosen to back the image with a parallel immutable vector. This wasn’t necessary, but since this type has a `map` operation which automatically parallelises over multiple cores, any `map` operations applied to the image will be automatically parallelised. This will ultimately lead to all of our statistical computations being automatically parallelised without us having to think about it.

As it stands, this image isn’t comonadic, since it doesn’t implement `extract` or `coflatMap`. Unlike the case of `Stream`, there isn’t really a uniquely privileged pixel, so it’s not clear what `extract` should return. For many data structures of this type, we make them comonadic by adding a “cursor” pointing to a “current” element of interest, and use this as the focus for computations applied with `coflatMap`. This is simplest to explain by example. We can define our “pointed” image type as follows:

```  case class PImage[T](x: Int, y: Int, image: Image[T]) {
def extract: T = image(x, y)
def map[S](f: T => S): PImage[S] = PImage(x, y, image map f)
def coflatMap[S](f: PImage[T] => S): PImage[S] = PImage(
x, y, Image(image.w, image.h,
(0 until (image.w * image.h)).toVector.par.map(i => {
val xx = i / image.h
val yy = i % image.h
f(PImage(xx, yy, image))
})))
```

There is missing a closing brace, as I’m not quite finished. Here `x` and `y` represent the location of our cursor, so `extract` returns the value of the pixel indexed by our cursor. Similarly, `coflatMap` forms an image where the value of the image at each location is the result of applying the function `f` to the image which had the cursor set to that location. Clearly `f` should use the cursor in some way, otherwise the image will have the same value at every pixel location. Note that `map` and `coflatMap` operations will be automatically parallelised. The intuitive idea behind `coflatMap` is that it extends local computations. For the stream example, the local computation was a linear combination of nearby values. Similarly, in image analysis problems, we often want to apply a linear filter to nearby pixels. We can get at the pixel at the cursor location using `extract`, but we probably also want to be able to move the cursor around to nearby locations. We can do that by adding some appropriate methods to complete the class definition.

```    def up: PImage[T] = {
val py = y-1
val ny = if (py >= 0) py else (py + image.h)
PImage(x,ny,image)
}
def down: PImage[T] = {
val py = y+1
val ny = if (py < image.h) py else (py - image.h)
PImage(x,ny,image)
}
def left: PImage[T] = {
val px = x-1
val nx = if (px >= 0) px else (px + image.w)
PImage(nx,y,image)
}
def right: PImage[T] = {
val px = x+1
val nx = if (px < image.w) px else (px - image.w)
PImage(nx,y,image)
}
}
```

Here each method returns a new pointed image with the cursor shifted by one pixel in the appropriate direction. Note that I’ve used periodic boundary conditions here, which often makes sense for numerical integration of PDEs, but makes less sense for real image analysis problems. Note that we have embedded all “indexing” issues inside the definition of our classes. Now that we have it, none of the statistical algorithms that we develop will involve any explicit indexing. This makes it much less likely to develop algorithms containing bugs corresponding to “off-by-one” or flipped axis errors.

This class is now fine for our requirements. But if we wanted Cats to understand that this structure is really a comonad (perhaps because we wanted to use derived methods, such as `coflatten`), we would need to provide evidence for this. The details aren’t especially important for this post, but we can do it simply as follows:

```  implicit val pimageComonad = new Comonad[PImage] {
def extract[A](wa: PImage[A]) = wa.extract
def coflatMap[A,B](wa: PImage[A])(f: PImage[A] => B): PImage[B] =
wa.coflatMap(f)
def map[A,B](wa: PImage[A])(f: A => B): PImage[B] = wa.map(f)
}
```

It’s handy to have some functions for converting Breeze dense matrices back and forth with our image class.

```  import breeze.linalg.{Vector => BVec, _}
def BDM2I[T](m: DenseMatrix[T]): Image[T] =
Image(m.cols, m.rows, m.data.toVector.par)
def I2BDM(im: Image[Double]): DenseMatrix[Double] =
new DenseMatrix(im.h,im.w,im.data.toArray)
```

Now we are ready to see how to use this in practice. Let’s start by defining a very simple linear filter.

```def fil(pi: PImage[Double]): Double = (2*pi.extract+
pi.up.extract+pi.down.extract+pi.left.extract+pi.right.extract)/6.0
```

This simple filter can be used to “smooth” or “blur” an image. However, from a more sophisticated viewpoint, exactly this type of filter can be used to represent one time step of a numerical method for time integration of the 2D heat equation. Now we can simulate a noisy image and apply our filter to it using `coflatMap`:

```import breeze.stats.distributions.Gaussian
val bdm = DenseMatrix.tabulate(200,250){case (i,j) => math.cos(
0.1*math.sqrt((i*i+j*j))) + Gaussian(0.0,2.0).draw}
val pim0 = PImage(0,0,BDM2I(bdm))
def pims = Stream.iterate(pim0)(_.coflatMap(fil))
```

Note that here, rather than just applying the filter once, I’ve generated an infinite stream of pointed images, each one representing an additional application of the linear filter. Thus the sequence represents the time solution of the heat equation with initial condition corresponding to our simulated noisy image.

We can render the first few frames to check that it seems to be working.

```import breeze.plot._
val fig = Figure("Diffusing a noisy image")
pims.take(25).zipWithIndex.foreach{case (pim,i) => {
val p = fig.subplot(5,5,i)
p += image(I2BDM(pim.image))
}}
```

Note that the numerical integration is carried out in parallel on all available cores automatically. Other image filters can be applied, and other (parabolic) PDEs can be numerically integrated in an essentially similar way.

## Gibbs sampling the Ising model

Another place where the concept of extending a local computation to a global computation crops up is in the context of Gibbs sampling a high-dimensional probability distribution by cycling through the sampling of each variable in turn from its full-conditional distribution. I’ll illustrate this here using the Ising model, so that I can reuse the pointed image class from above, but the principles apply to any Gibbs sampling problem. In particular, the Ising model that we consider has a conditional independence structure corresponding to a graph of a square lattice. As above, we will use the comonadic structure of the square lattice to construct a Gibbs sampler. However, we can construct a Gibbs sampler for arbitrary graphical models in an essentially identical way by using a graph comonad.

Let’s begin by simulating a random image containing +/-1s:

```import breeze.stats.distributions.{Binomial,Bernoulli}
val beta = 0.4
val bdm = DenseMatrix.tabulate(500,600){
case (i,j) => (new Binomial(1,0.2)).draw
}.map(_*2 - 1) // random matrix of +/-1s
val pim0 = PImage(0,0,BDM2I(bdm))
```

We can use this to initialise our Gibbs sampler. We now need a Gibbs kernel representing the update of each pixel.

```def gibbsKernel(pi: PImage[Int]): Int = {
val sum = pi.up.extract+pi.down.extract+pi.left.extract+pi.right.extract
val p1 = math.exp(beta*sum)
val p2 = math.exp(-beta*sum)
val probplus = p1/(p1+p2)
if (new Bernoulli(probplus).draw) 1 else -1
}
```

So far so good, but there a couple of issues that we need to consider before we plough ahead and start coflatMapping. The first is that pure functional programmers will object to the fact that this function is not pure. It is a stochastic function which has the side-effect of mutating the random number state. I’m just going to duck that issue here, as I’ve previously discussed how to fix it using probability monads, and I don’t want it to distract us here.

However, there is a more fundamental problem here relating to parallel versus sequential application of Gibbs kernels. `coflatMap` is conceptually parallel (irrespective of how it is implemented) in that all computations used to build the new comonad are based solely on the information available in the starting comonad. OTOH, detailed balance of the Markov chain will only be preserved if the kernels for each pixel are applied sequentially. So if we `coflatMap` this kernel over the image we will break detailed balance. I should emphasise that this has nothing to do with the fact that I’ve implemented the pointed image using a parallel vector. Exactly the same issue would arise if we switched to backing the image with a regular (sequential) immutable `Vector`.

The trick here is to recognise that if we coloured alternate pixels black and white using a chequerboard pattern, then all of the black pixels are conditionally independent given the white pixels and vice-versa. Conditionally independent pixels can be updated by parallel application of a Gibbs kernel. So we just need separate kernels for updating odd and even pixels.

```def oddKernel(pi: PImage[Int]): Int =
if ((pi.x+pi.y) % 2 != 0) pi.extract else gibbsKernel(pi)
def evenKernel(pi: PImage[Int]): Int =
if ((pi.x+pi.y) % 2 == 0) pi.extract else gibbsKernel(pi)
```

Each of these kernels can be coflatMapped over the image preserving detailed balance of the chain. So we can now construct an infinite stream of MCMC iterations as follows.

```def pims = Stream.iterate(pim0)(_.coflatMap(oddKernel).
coflatMap(evenKernel))
```

We can animate the first few iterations with:

```import breeze.plot._
val fig = Figure("Ising model Gibbs sampler")
fig.width = 1000
fig.height = 800
pims.take(50).zipWithIndex.foreach{case (pim,i) => {
print(s"\$i ")
fig.clear
val p = fig.subplot(1,1,0)
p.title = s"Ising model: frame \$i"
p += image(I2BDM(pim.image.map{_.toDouble}))
fig.refresh
}}
println
```

Here I have a movie showing the first 1000 iterations. Note that youtube seems to have over-compressed it, but you should get the basic idea.

Again, note that this MCMC sampler runs in parallel on all available cores, automatically. This issue of odd/even pixel updating emphasises another issue that crops up a lot in functional programming: very often, thinking about how to express an algorithm functionally leads to an algorithm which parallelises naturally. For general graphs, figuring out which groups of nodes can be updated in parallel is essentially the graph colouring problem. I’ve discussed this previously in relation to parallel MCMC in:

Wilkinson, D. J. (2005) Parallel Bayesian Computation, Chapter 16 in E. J. Kontoghiorghes (ed.) Handbook of Parallel Computing and Statistics, Marcel Dekker/CRC Press, 481-512.

There are quite a few blog posts discussing comonads in the context of Haskell. In particular, the post on comonads for image analysis I mentioned previously, and this one on cellular automata. Bartosz’s post on comonads gives some connection back to the mathematical origins. Runar’s Scala comonad tutorial is the best source I know for comonads in Scala.

Full runnable code corresponding to this blog post is available from my blog repo.

## Introduction

As discussed in the previous post, I’ve recently constructed and delivered a short course on statistical computing with Scala. Much of the course is concerned with writing statistical algorithms in Scala, typically making use of the scientific and numerical computing library, Breeze. Breeze has all of the essential tools necessary for building statistical algorithms, but doesn’t contain any higher level modelling functionality. As part of the course, I walked through how to build a small library for regression modelling on top of Breeze, including all of the usual regression diagnostics (such as standard errors, t-statistics, p-values, F-statistics, etc.). While preparing the course materials it occurred to me that it would be useful to package and document this code properly for general use. In advance of the course I packaged the code up into a bare-bones library, but since then I’ve fleshed it out, tidied it up and documented it properly, so it’s now ready for people to use.

The library covers PCA, linear regression modelling and simple one-parameter GLMs (including logistic and Poisson regression). The underlying algorithms are fairly efficient and numerically stable (eg. linear regression uses the QR decomposition of the model matrix, and the GLM fitting uses QR within each IRLS step), though they are optimised more for clarity than speed. The library also includes a few utility functions and procedures, including a pairs plot (scatter-plot matrix).

## A linear regression example

Plenty of documentation is available from the scala-glm github repo which I won’t repeat here. But to give a rough idea of how things work, I’ll run through an interactive session for the linear regression example.

First, download a dataset from the UCI ML Repository to disk for subsequent analysis (caching the file on disk is good practice, as it avoids unnecessary load on the UCI server, and allows running the code off-line):

```import scalaglm._
import breeze.linalg._

val url = "http://archive.ics.uci.edu/ml/machine-learning-databases/00291/airfoil_self_noise.dat"
val fileName = "self-noise.csv"

// download the file to disk if it hasn't been already
val file = new java.io.File(fileName)
if (!file.exists) {
val s = new java.io.PrintWriter(file)
val data = scala.io.Source.fromURL(url).getLines
data.foreach(l => s.write(l.trim.
split('\t').filter(_ != "").
mkString("", ",", "\n")))
s.close
}
```

Once we have a CSV file on disk, we can load it up and look at it.

```val mat = csvread(new java.io.File(fileName))
// mat: breeze.linalg.DenseMatrix[Double] =
// 800.0    0.0  0.3048  71.3  0.00266337  126.201
// 1000.0   0.0  0.3048  71.3  0.00266337  125.201
// 1250.0   0.0  0.3048  71.3  0.00266337  125.951
// ...
println("Dim: " + mat.rows + " " + mat.cols)
// Dim: 1503 6
val figp = Utils.pairs(mat, List("Freq", "Angle", "Chord", "Velo", "Thick", "Sound"))
// figp: breeze.plot.Figure = breeze.plot.Figure@37718125
```

We can then regress the response in the final column on the other variables.

```val y = mat(::, 5) // response is the final column
// y: DenseVector[Double] = DenseVector(126.201, 125.201, ...
val X = mat(::, 0 to 4)
// X: breeze.linalg.DenseMatrix[Double] =
// 800.0    0.0  0.3048  71.3  0.00266337
// 1000.0   0.0  0.3048  71.3  0.00266337
// 1250.0   0.0  0.3048  71.3  0.00266337
// ...
val mod = Lm(y, X, List("Freq", "Angle", "Chord", "Velo", "Thick"))
// mod: scalaglm.Lm =
// Lm(DenseVector(126.201, 125.201, ...
mod.summary
// Estimate	 S.E.	 t-stat	p-value		Variable
// ---------------------------------------------------------
// 132.8338	 0.545	243.866	0.0000 *	(Intercept)
//  -0.0013	 0.000	-30.452	0.0000 *	Freq
//  -0.4219	 0.039	-10.847	0.0000 *	Angle
// -35.6880	 1.630	-21.889	0.0000 *	Chord
//   0.0999	 0.008	12.279	0.0000 *	Velo
// -147.3005	15.015	-9.810	0.0000 *	Thick
// Residual standard error:   4.8089 on 1497 degrees of freedom
// Multiple R-squared: 0.5157, Adjusted R-squared: 0.5141
// F-statistic: 318.8243 on 5 and 1497 DF, p-value: 0.00000
val fig = mod.plots
// fig: breeze.plot.Figure = breeze.plot.Figure@60d7ebb0
```

There is a `.predict` method for generating point predictions (and standard errors) given a new model matrix, and fitting GLMs is very similar – these things are covered in the quickstart guide for the library.

## Summary

scala-glm is a small Scala library built on top of the Breeze numerical library which enables simple and convenient regression modelling in Scala. It is reasonably well documented and usable in its current form, but I intend to gradually add additional features according to demand as time permits.