Custom Affine Transformations

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This page explains how to implement your own affine transformation operations by way of example. You can add your own custom affine transformations into Stheno using the same mechanism as all of the existing transformations (addition, multiplication, composition, etc). First, load up the relevant packages.

using AbstractGPs
using LinearAlgebra
using Stheno

The Affine Transformation

Suppose that, for some reason, you wish to implement the affine transformation of a single process f given by (Af)(x) = f(x) + f(x + 3) - 2. In order to define this transformation, first create a function which accepts f and returns a DerivedGP:

using Stheno: AbstractGP, DerivedGP, SthenoAbstractGP

A(f::SthenoAbstractGP) = DerivedGP((A, f), f.gpc)

A (generic function with 1 method)

The first argument to DerivedGP contains A itself and any data needed to fully specify the process results from this transformation. In this case the only piece of information required is f, but really any data can be put in this argument. For example, if we wished to replace the translation of -3 by a parameter, we could do so, and make it a part of this first argument.

The second argument is the book-keeping object that needs to be passed around in order to know how to compute covariances properly. It is because of this argument that we restrict the accepted GPs to be SthenoAbstractGP, as we can safely assume that these have a gpc field.

We'll now define a type alias in order to simplify some methods later on:

const A_args = Tuple{typeof(A), SthenoAbstractGP};

Most Important Methods

We must now define methods of three functions on A_args: mean, cov, and var. First the mean – this method should accept both an A_args and an AbstracVector, and return the mean vector of A(f) at x. Some textbook calculations reveal that this is

Stheno.mean((A, f)::A_args, x::AbstractVector) = mean(f, x) .+ mean(f, x .+ 3) .- 2

The first argument here is always going to be precisely the tuple of arguments passed into the DerivedGP constructor above. You can assume that you can compute any statistics of f that the AbstractGPs API provides.

We now turn our attention to cov. The first method we consider is cov(args::A_args, x::AbstractVector, y::AbstractVector), which should return the cross-covariance matrix between all pairs of points in x and y under the transformed process, A(f). Again, some standard manipulations reveal that this covariance is given by

function Stheno.cov((A, f)::A_args, x::AbstractVector, y::AbstractVector)
    return cov(f, x, y) + cov(f, x, y .+ 3) + cov(f, x .+ 3, y) + cov(f, x .+ 3, y .+ 3)
end

The last substantially new method to implement is cov(args::A_args, g::AbstractGP, x::AbstractVector, y::AbstractVector), which should return the cross-covariance matrix between A(f) at x and g at y. When implementing this method, you can assume you have access to functions like cov(f, g, x, y) etc:

function Stheno.cov((A, f)::A_args, g::AbstractGP, x::AbstractVector, y::AbstractVector)
    return cov(f, g, x, y) + cov(f, g, x .+ 3, y)
end

Additional (Required) Methods

There are a number of other methods that you should implement. These are all just special cases or slight modifications of the three methods above, and should be straightforward to implement given that you've implemented the above methods.

First, lets build a GPPP containing an instance of our transformation so that some properties can be verified. The definition of the methods being implemented is demonstrated by checking an equality after defining each method.

gppp = @gppp let
    f = GP(SEKernel())
    Af = A(f)
end

Stheno.GaussianProcessProbabilisticProgramme{NamedTuple{(:f, :Af), Tuple{Stheno.AtomicGP{AbstractGPs.GP{AbstractGPs.ZeroMean{Float64}, KernelFunctions.SqExponentialKernel{Distances.Euclidean}}}, Stheno.DerivedGP{Tuple{typeof(Main.##274.A), Stheno.AtomicGP{AbstractGPs.GP{AbstractGPs.ZeroMean{Float64}, KernelFunctions.SqExponentialKernel{Distances.Euclidean}}}}}}}}((f = Stheno.AtomicGP{AbstractGPs.GP{AbstractGPs.ZeroMean{Float64}, KernelFunctions.SqExponentialKernel{Distances.Euclidean}}}(AbstractGPs.GP{AbstractGPs.ZeroMean{Float64}, KernelFunctions.SqExponentialKernel{Distances.Euclidean}}(AbstractGPs.ZeroMean{Float64}(), Squared Exponential Kernel (metric = Distances.Euclidean(0.0))), 1, Stheno.GPC(2)), Af = Stheno.DerivedGP{Tuple{typeof(Main.##274.A), Stheno.AtomicGP{AbstractGPs.GP{AbstractGPs.ZeroMean{Float64}, KernelFunctions.SqExponentialKernel{Distances.Euclidean}}}}}((Main.##274.A, Stheno.AtomicGP{AbstractGPs.GP{AbstractGPs.ZeroMean{Float64}, KernelFunctions.SqExponentialKernel{Distances.Euclidean}}}(AbstractGPs.GP{AbstractGPs.ZeroMean{Float64}, KernelFunctions.SqExponentialKernel{Distances.Euclidean}}(AbstractGPs.ZeroMean{Float64}(), Squared Exponential Kernel (metric = Distances.Euclidean(0.0))), 1, Stheno.GPC(2))), 2, Stheno.GPC(2))), Stheno.GPC(2))

Also create some input vectors.

x_f = GPPPInput(:f, randn(3))
y_f = GPPPInput(:f, randn(6))
x_Af = GPPPInput(:Af, randn(3))
y_Af = GPPPInput(:Af, randn(6))
z_Af = GPPPInput(:Af, randn(3))

3-element Stheno.GPPPInput{Symbol, Float64, Vector{Float64}}:
 (:Af, -0.48091401784914256)
 (:Af, 0.7684945796886042)
 (:Af, 0.2813939947666527)

The covariance matrix at a single pair of inputs:

function Stheno.cov((A, f)::A_args, x::AbstractVector)
    return cov(f, x) + cov(f, x, x .+ 3) + cov(f, x .+ 3, x) + cov(f, x .+ 3)
end

cov(gppp, x_Af, x_Af) ≈ cov(gppp, x_Af)

true

The diagonal of the covariance matrix at a single pair of inputs:

function Stheno.var((A, f)::A_args, x::AbstractVector)
    return var(f, x) + var(f, x .+ 3) + var(f, x, x .+ 3) + var(f, x .+ 3, x)
end

var(gppp, x_Af) ≈ diag(cov(gppp, x_Af))

true

The diagonal of the cross-covariance matrix for equal-length inputs:

function Stheno.var((A, f)::A_args, x::AbstractVector, y::AbstractVector)
    return var(f, x, y) + var(f, x, y .+ 3) + var(f, x .+ 3, y) + var(f, x .+ 3, y .+ 3)
end

var(gppp, x_Af, z_Af) ≈ diag(cov(gppp, x_Af, z_Af))

true

The diagonal of the cross-covariance between different processes for equal-length inputs:

function Stheno.var((A, f)::A_args, g::AbstractGP, x::AbstractVector, y::AbstractVector)
    return var(f, g, x, y) + var(f, g, x .+ 3, y)
end

var(gppp, x_Af, x_f) ≈ diag(cov(gppp, x_Af, x_f))

true

cov and var between processes when Af's arguments are the second argument, rather than the first:

function Stheno.cov(g::AbstractGP, (A, f)::A_args, x::AbstractVector, y::AbstractVector)
    return cov(g, f, x, y) + cov(g, f, x, y .+ 3)
end

cov(gppp, x_f, x_Af) ≈ cov(gppp, x_Af, x_f)'

function Stheno.var(g::AbstractGP, (A, f)::A_args, x::AbstractVector, y::AbstractVector)
    return var(g, f, x, y) + var(g, f, x, y .+ 3)
end

var(gppp, x_f, x_Af) ≈ var(gppp, x_Af, x_f)

true

Checking Your Implementation

Given the numerous methods above, it's a really good idea to utilise the functionality provided by AbstractGPs.jl to check that you've implemented them all consistently with one another.

using AbstractGPs.TestUtils: test_internal_abstractgps_interface
using Random

rng = MersenneTwister(123456);
test_internal_abstractgps_interface(rng, gppp, x_Af, y_Af);
test_internal_abstractgps_interface(rng, gppp, x_Af, y_f);
test_internal_abstractgps_interface(rng, gppp, x_f, y_Af);

Roughly speaking, provided that you've implemented the first three methods correctly, this test ought to catch any glaring problems if you've made a mistake in the rest. If course, it won't check that your implementations of the first three methods correctly implement the desired affine transformation, so you should write whatever tests you need in order to convince yourself of that.

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