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Travis Hinkelman

Stochastic population model in F#

In three previous posts, I wrote about different programming languages that I have considered learning. I mentioned about 15 different languages in those posts. F# was not on the list. Because my background is in R, I thought I was better off sticking to learning dynamically typed languages at this point. Moreover, I hold a longstanding bias against Microsoft and Windows and that bias was easy to transfer to F#.

A few months ago, I started a new job. My work primarily involves R, but there is a software developer at the new company who primarily uses C#. Because I'm interested in eventually shifting my role to involve more software development, I thought it might be worthwhile to invest some time in learning C#. But, after a little more reading about C# and .NET, I found myself more drawn to F# as a functional-first language with strengths in data science (relative to C#).

Around the time that I started following the F# community, I saw a call for applications to the F# Mentorship Program. I decided to apply and was assigned a mentor. I've thoroughly enjoyed the mentorship program and, although I still know very little F#, I can see my appreciation of F# growing to the level of R and Scheme.

I had previously written about implementing a stochastic population model in Racket based on an R example found in this blog post. As a first exercise in F#, I've translated the stochastic population model into a few F# versions.

I'm working with F# in Visual Studio Code with Ionide on Linux (Ubuntu), but I think that the code shared in this post will work on other platforms. We can initialize a new console app with the following line.

dotnet new console -lang "F#" -o StochasticLogistic

That line creates a folder StochasticLogistic containing the files Program.fs and StochasticLogistic.fsproj and a folder obj. The Program.fs file for the code described in this blog post is available in this gist.

We will use the MathNet.Numerics package for drawing random numbers. One way to install the package is with dotnet.

dotnet add package MathNet.Numerics --version 4.12.0

The first lines of Program.fs open the Distributions namespace of MathNet.Numerics and set the values of the parameters used in the model.

open MathNet.Numerics.Distributions

[<EntryPoint>]
let main argv =
    let yinit = 1.0            // initial population size
    let r = 1.4                // maximum population growth rate
    let k = 20.0               // carrying capacity
    let thetasd = 0.1          // standard deviation for adding noise to the population
    let t = 4                  // number of years of growth to simulate

The first version of the function, logmodFor, follows the R example by using a for loop to fill an array. In F#, we draw random numbers by initializing a distribution and then sampling from the distribution. The theta variable is the stochastic part of the model. In this version, we initialize a 1D array with random draws from a normal distribution (following the R example). Alternatively, we could have drawn one value every time through the for loop. Array.init illustrates the syntax for anonymous functions in F#, e.g., (fun x -> x + 1). F# infers types from context. In the case of Array.zeroCreate, though, we need to specify the type (float) of the zeros in the array. We initialize the 0th element of the output array ys to the initial population value y and fill the rest of the array each time through the loop.

    let logmodFor t y r k thetasd = 
        let normalDist = Normal(0.0, thetasd)
        let theta = Array.init (t - 1) (fun _ -> normalDist.Sample())
        let ys: float array = Array.zeroCreate t
        Array.set ys 0 y
        for i in 1 .. (t - 1) do
            Array.set ys i (ys.[i-1] * (r - r * (ys.[i-1] / k)) * exp(theta.[i-1]))
        ys

The next version of the function, logmodRecIf, uses recursion and is written in a style similar to how I tend to write recursive functions in Scheme. I moved the main calculation into a separate function calc to improve readability. Note the use of rec to indicate the recursive function loop. :: is the cons operator. calc acc.Head could also have been written as calc (List.head acc), but dot notation is one of the recommended object-oriented features to include in your F# code.

    let logmodRecIf t y r k thetasd =
        let normalDist = Normal(0.0, thetasd)
        let calc y = y * (r - r * (y / k)) * exp(normalDist.Sample())
        let rec loop acc t = 
            if t = 1 then List.rev acc
            else loop ((calc acc.Head) :: acc) (t - 1)
        loop [y] t

The next version, logmodRecMatch, is only slightly different than the last. In this version, we are using pattern matching instead of if then else. Even in this simple example, which doesn't require the power of match, I find the pattern matching version more readable. With match, we are first checking if the value of t is 1 and, if not, recursing through the rest of the list.

    let logmodRecMatch t y r k thetasd = 
        let normalDist = Normal(0.0, thetasd)
        let calc y = y * (r - r * (y / k)) * exp(normalDist.Sample())
        let rec loop acc t = 
            match t with 
            | 1 -> List.rev acc
            | _ -> loop ((calc acc.Head) :: acc) (t - 1)
        loop [y] t

The next version, logmodUnfold, was provided by my F# mentor. I don't fully understand it, yet. For example, I'm not sure (and don't remember what I was told) if using Some was a convenient way to make the function type check or if it represents best practice for this use case. This approach uses sequences, which are lazy. F# also provides List.unfold and List.take, but they are not drop-in replacements because lists are not lazy and calc as written here never terminates.

    let logmodUnfold t y r k thetasd = 
        let normalDist = Normal(0.0, thetasd)
        let calc y = 
            let result = y * (r - r * (y / k)) * exp(normalDist.Sample())
            Some (y, result)
        Seq.unfold calc y |> Seq.take t

Now that we have our functions we need call them in our program and display the output in the console. Because the model has a stochastic component, it is not obvious if all four versions of the function are performing the same calculation. I wanted to be able to run deterministic or stochastic versions of the functions when calling the program from the command line. I also decided to provide the number of years t as a command line argument to the program.

The arguments provided to the program are collected in a string list argv. We will pass t as the first argument and convert the string to an integer with int. If the t argument can't be converted to an integer, then the program will produce a runtime error.

[<EntryPoint>]
let main argv =
    let yinit = 1.0            // initial population size
    let r = 1.4                // maximum population growth rate
    let k = 20.0               // carrying capacity
    let thetasd = 0.1          // standard deviation for adding noise to the population
    let t = int argv.[0]       // number of years of growth to simulate
    let seedType = argv.[1]    // "fixed" or "random" for setting deterministic behavior

The second argv allows for specifying fixed or random behavior. To use seedType, I added two new lines to each function and changed the Normal function to use the RandomSource parameter.

        let random = System.Random()
        let seed = if seedType = "fixed" then 2005 else random.Next()
        let normalDist = Normal(0.0, thetasd, RandomSource = System.Random(seed))

The following block of code runs our four function versions and pipes |> the output into a printfn. Note, the |> operator pipes the previous output into the last argument of the next function. "%A" is used to print any F# object.

    printfn "for loop"
    logmodFor t yinit r k thetasd seedType |> printfn "%A"
    printfn "---"
    printfn "recursion (if)"
    logmodRecIf t yinit r k thetasd seedType |> printfn "%A"
    printfn "---"
    printfn "recursion (match)"
    logmodRecMatch t yinit r k thetasd seedType |> printfn "%A"
    printfn "---"
    printfn "unfold"
    logmodUnfold t yinit r k thetasd seedType |> printfn "%A"

We can run our program with dotnet run and the required arguments.

$ dotnet run 4 fixed
for loop
[|1.0; 1.305926731; 1.454837341; 1.798817442|]
---
recursion (if)
[1.0; 1.305926731; 1.454837341; 1.798817442]
---
recursion (match)
[1.0; 1.305926731; 1.454837341; 1.798817442]
---
unfold
seq [1.0; 1.305926731; 1.454837341; 1.798817442]

dotnet run compiles the program before running it. If you want to re-run the program with new arguments, but no changes to the program, you can use dotnet run 4 fixed --no-build. Alternatively, you can call the previously built dll directly with dotnet bin/Debug/netcoreapp3.1/StochasticLogistic.dll. The last approach seemed to run the program with the least lag time. It's also easy to create cross-platform executables of your program (described here).

Next, we want to run each logmod function many times. This repeat function repeats an anonymous function, f, n times. The anonymous function is called with f() and the output is cons'd onto the accumulator.

    let repeat n f =
        let rec loop n acc =
            match n with
            | 0 -> acc
            | _ -> loop (n - 1) (f() :: acc)
        loop n []

Because I wanted to conduct some informal performance comparisons among the functions, I added a couple of more arguments to the top of the program.

    let funType = argv.[2]     // "for", "recif", "recmatch", "unfold"
    let reps = int argv.[3]    // number of replications

And added this code to the bottom of the program. Because we are not interested in the actual output (only the execution time), the output is piped to ignore.

    if funType = "for" then
        repeat reps (fun () -> logmodFor t yinit r k thetasd seedType) |> ignore
    elif funType = "recif" then
        repeat reps (fun () -> logmodRecIf t yinit r k thetasd seedType) |> ignore
    elif funType = "recmatch" then
        repeat reps (fun () -> logmodRecMatch t yinit r k thetasd seedType) |> ignore
    elif funType = "unfold" then
        repeat reps (fun () -> logmodUnfold t yinit r k thetasd seedType) |> ignore
    else
        invalidArg "funType" "must be one of for, recif, recmatch, unfold"

First, we will run the program with a small number of years and reps to compile it.

dotnet run 4 random for 2

And then we can do some informal timing tests.

$ time dotnet bin/Debug/netcoreapp3.1/StochasticLogistic.dll 100 random for 50000

real	0m1.355s
user	0m1.280s
sys	0m0.073s

$ time dotnet bin/Debug/netcoreapp3.1/StochasticLogistic.dll 100 random recif 50000

real	0m2.244s
user	0m2.211s
sys	0m0.111s

$ time dotnet bin/Debug/netcoreapp3.1/StochasticLogistic.dll 100 random recmatch 50000

real	0m2.237s
user	0m2.231s
sys	0m0.108s

$ time dotnet bin/Debug/netcoreapp3.1/StochasticLogistic.dll 100 random unfold 50000

real	0m0.516s
user	0m0.492s
sys	0m0.028s

I was not surprised to see that using arrays in a for loop was nearly 2x as fast as recursion. But I was surprised that using sequences with unfold was over 2x as fast as the for. I'm wondering if sequences are so lazy that this code is only timing the creation of a lists of sequences to be evaluated later. When I explicitly convert the sequence to a list in logmodUnfold,

        Seq.unfold calc y |> Seq.take t |> List.ofSeq 

the function is 2x as slow as recursion.

$ time dotnet bin/Debug/netcoreapp3.1/StochasticLogistic.dll 100 random unfold 50000

real	0m4.049s
user	0m4.055s
sys	0m0.185s

I don't know how much of the extra time is related to list conversion versus executing the lazy sequences. When I learn more about sequences, I will update this post with an answer.