Add ApproximatePeriodicKernel

Project.toml

@@ -1,10 +1,11 @@
 name = "TemporalGPs"
 uuid = "e155a3c4-0841-43e1-8b83-a0e4f03cc18f"
 authors = ["willtebbutt <[email protected]> and contributors"]
-version = "0.6.4"
+version = "0.6.5"
 
 [deps]
 AbstractGPs = "99985d1d-32ba-4be9-9821-2ec096f28918"
+Bessels = "0e736298-9ec6-45e8-9647-e4fc86a2fe38"
 BlockDiagonals = "0a1fb500-61f7-11e9-3c65-f5ef3456f9f0"
 ChainRulesCore = "d360d2e6-b24c-11e9-a2a3-2a2ae2dbcce4"
 FillArrays = "1a297f60-69ca-5386-bcde-b61e274b549b"

src/TemporalGPs.jl

@@ -1,6 +1,7 @@
 module TemporalGPs
 
     using AbstractGPs
+    using Bessels: besseli
     using BlockDiagonals
     using ChainRulesCore
     import ChainRulesCore: rrule
@@ -31,7 +32,8 @@ module TemporalGPs
         checkpointed,
         posterior,
         logpdf_and_rand,
-        Separable
+        Separable,
+        ApproxPeriodicKernel
 
     # Various bits-and-bobs. Often commiting some type piracy.
     include(joinpath("util", "harmonise.jl"))

src/gp/lti_sde.jl

@@ -241,6 +241,93 @@ function stationary_distribution(::Matern52Kernel, ::SArrayStorage{T}) where {T<
     return Gaussian(m, P)
 end
 
+# Cosine
+
+function to_sde(::CosineKernel, ::SArrayStorage{T}) where {T}
+    F = SMatrix{2, 2, T}(0, 1, -1, 0)
+    q = zero(T)
+    H = SVector{2, T}(1, 0)
+    return F, q, H
+end
+
+function stationary_distribution(::CosineKernel, ::SArrayStorage{T}) where {T<:Real}
+    m = SVector{2, T}(0, 0)
+    P = SMatrix{2, 2, T}(1, 0, 0, 1)
+    return Gaussian(m, P)
+end
+
+# Approximate Periodic Kernel
+# The periodic kernel is approximated by a sum of cosine kernels with different frequencies.
+struct ApproxPeriodicKernel{N,K<:PeriodicKernel} <: KernelFunctions.SimpleKernel
+    kernel::K
+    function ApproxPeriodicKernel{N,K}(kernel::K) where {N,K<:PeriodicKernel}
+        length(kernel.r) == 1 || error("ApproxPeriodicKernel only supports a single lengthscale")
+        return new{N,K}(kernel)
+    end
+end
+# We follow "State Space approximation of Gaussian Processes for time series forecasting"
+# by Alessio Benavoli and Giorgio Corani and use a default of 7 Cosine Kernel terms
+ApproxPeriodicKernel(;r::Real=1.0) = ApproxPeriodicKernel{7}(PeriodicKernel(;r=[r]))
+ApproxPeriodicKernel{N}(;r::Real=1.0) where {N} = ApproxPeriodicKernel{N}(PeriodicKernel(;r=[r]))
+ApproxPeriodicKernel(kernel::PeriodicKernel) = ApproxPeriodicKernel{7}(kernel)
+ApproxPeriodicKernel{N}(kernel::K) where {N,K<:PeriodicKernel} = ApproxPeriodicKernel{N,K}(kernel)
+
+KernelFunctions.kappa(k::ApproxPeriodicKernel, x) = KernelFunctions.kappa(k.kernel, x)
+KernelFunctions.metric(k::ApproxPeriodicKernel) = KernelFunctions.metric(k.kernel)
+
+function Base.show(io::IO, κ::ApproxPeriodicKernel{N}) where {N}
+    return print(io, "Approximate Periodic Kernel, (r = $(only(κ.kernel.r))) approximated with $N cosine kernels")
+end
+
+function lgssm_components(approx::ApproxPeriodicKernel{N}, t::Union{StepRangeLen, RegularSpacing}, storage::StorageType{T}) where {N,T<:Real}
+    Fs, Hs, ms, Ps = _init_periodic_kernel_lgssm(approx.kernel, storage, N)
+    nt = length(t)
+    As = map(F -> Fill(time_exp(F, T(step(t))), nt), Fs)
+    return _reduce_sum_cosine_kernel_lgssm(As, Hs, ms, Ps, N, nt, T)
+end
+function lgssm_components(approx::ApproxPeriodicKernel{N}, t::AbstractVector{<:Real}, storage::StorageType{T}) where {N,T<:Real}
+    Fs, Hs, ms, Ps = _init_periodic_kernel_lgssm(approx.kernel, storage, N)
+    t = vcat([first(t) - 1], t)
+    nt = length(diff(t))
+    As = _map(F -> _map(Δt -> time_exp(F, T(Δt)), diff(t)), Fs)
+    return _reduce_sum_cosine_kernel_lgssm(As, Hs, ms, Ps, N, nt, T)
+end
+
+function _init_periodic_kernel_lgssm(kernel::PeriodicKernel, storage, N::Int=7)
+    r = kernel.r
+    l⁻² = inv(4 * only(r)^2)
+
+    F, _, H = to_sde(CosineKernel(), storage)
+    Fs = ntuple(N) do i
+        2π * (i - 1) * F
+    end
+    Hs = Fill(H, N)
+
+    x0 = stationary_distribution(CosineKernel(), storage)
+    ms = Fill(x0.m, N)
+    P = x0.P
+    Ps = ntuple(N) do j
+        qⱼ = (1 + (j !== 1) ) * besseli(j - 1, l⁻²) / exp(l⁻²)
+        qⱼ * P
+    end    
+
+    Fs, Hs, ms, Ps
+end
+
+function _reduce_sum_cosine_kernel_lgssm(As, Hs, ms, Ps, N, nt, T)
+    as = Fill(Fill(Zeros{T}(size(first(first(As)), 1)), nt), N)
+    Qs = _map((P, A) -> _map(A -> Symmetric(P) - A * Symmetric(P) * A', A), Ps, As)
+    Hs = Fill(vcat(Hs...), nt)
+    h = Fill(zero(T), nt)
+    As = _map(block_diagonal, As...)
+    as = -map(vcat, as...)
+    Qs = _map(block_diagonal, Qs...)
+    m = reduce(vcat, ms)
+    P = block_diagonal(Ps...)
+    x0 = Gaussian(m, P)
+    return As, as, Qs, (Hs, h), x0
+end
+
 # Constant
 
 function TemporalGPs.to_sde(::ConstantKernel, ::SArrayStorage{T}) where {T<:Real}

src/models/lgssm.jl

@@ -267,8 +267,7 @@ ChainRulesCore.@non_differentiable ident_eps(args...)
 
 _collect(U::Adjoint{<:Any, <:Matrix}) = collect(U)
 _collect(U::SMatrix) = U
-
-
+_collect(U::BlockDiagonal) = U
 
 # AD stuff. No need to understand this unless you're really plumbing the depths...
 

src/util/chainrules.jl

@@ -147,6 +147,7 @@ function rrule(::Type{<:Fill}, x, sz)
     Fill_rrule(Δ::Union{Fill,Thunk}) = NoTangent(), FillArrays.getindex_value(unthunk(Δ)), NoTangent()
     Fill_rrule(Δ::Tangent{T,<:NamedTuple{(:value, :axes)}}) where {T} = NoTangent(), Δ.value, NoTangent()
     Fill_rrule(::AbstractZero) = NoTangent(), NoTangent(), NoTangent()
+    Fill_rrule(Δ::Tangent{T,<:NTuple}) where {T} = NoTangent(), sum(Δ), NoTangent()
     function Fill_rrule(Δ::AbstractArray)
         # all(==(first(Δ)), Δ) || error("Δ should be a vector of the same value")
         # sum(Δ)

test/gp/lti_sde.jl

@@ -1,5 +1,8 @@
-using TemporalGPs: build_lgssm, StorageType, is_of_storage_type
 using KernelFunctions
+using KernelFunctions: kappa
+using ChainRulesTestUtils
+using TemporalGPs: build_lgssm, StorageType, is_of_storage_type, lgssm_components
+using Test
 include("../test_util.jl")
 include("../models/model_test_utils.jl")
 _logistic(x) = 1 / (1 + exp(-x))
@@ -12,6 +15,34 @@ function _construction_tester(f_naive::GP, storage::StorageType, σ², t::Abstra
     return build_lgssm(fx)
 end
 
+@testset "ApproxPeriodicKernel" begin
+    k = ApproxPeriodicKernel()
+    @test k isa ApproxPeriodicKernel{7}
+    # Test that it behaves like a normal PeriodicKernel
+    k_base = PeriodicKernel()
+    x = rand()
+    @test kappa(k, x) == kappa(k_base, x)
+    x = rand(3)
+    @test kernelmatrix(k, x) ≈ kernelmatrix(k_base, x)
+    # Test dimensionality of LGSSM components
+    Nt = 10
+    @testset "$(typeof(t)), $storage, $N" for t in (
+            sort(rand(Nt)), RegularSpacing(0.0, 0.1, Nt)
+        ),
+        storage in (SArrayStorage{Float64}(), ArrayStorage{Float64}()),
+        N in (5, 8)
+
+        k = ApproxPeriodicKernel{N}()
+        As, as, Qs, emission_projections, x0 = lgssm_components(k, t, storage)
+        @test length(As) == Nt
+        @test all(x -> size(x) == (N * 2, N * 2), As)
+        @test length(as) == Nt
+        @test all(x -> size(x) == (N * 2,), as)
+        @test length(Qs) == Nt
+        @test all(x -> size(x) == (N * 2, N * 2), Qs)
+    end
+end
+
 println("lti_sde:")
 @testset "lti_sde" begin
     @testset "block_diagonal" begin
@@ -37,7 +68,11 @@ println("lti_sde:")
         )
 
         kernels = [
-            Matern12Kernel(), Matern32Kernel(), Matern52Kernel(), ConstantKernel(; c=1.5)
+            Matern12Kernel(),
+            Matern32Kernel(),
+            Matern52Kernel(),
+            ConstantKernel(; c=1.5),
+            CosineKernel(),
         ]
 
         @testset "$kernel, $(storage.name)" for kernel in kernels, storage in storages
@@ -56,53 +91,60 @@ println("lti_sde:")
         N = 13
         kernels = vcat(
             # Base kernels.
-            (name="base-Matern12Kernel", val=Matern12Kernel()),
+            (name="base-Matern12Kernel", val=Matern12Kernel(), to_vec_grad=false),
             map([Matern32Kernel, Matern52Kernel]) do k
-                (; name="base-$k", val=k())
+                (; name="base-$k", val=k(), to_vec_grad=false)
             end,
 
             # Scaled kernels.
             map([1e-1, 1.0, 10.0, 100.0]) do σ²
-                (; name="scaled-σ²=$σ²", val=σ² * Matern32Kernel())
+                (; name="scaled-σ²=$σ²", val=σ² * Matern32Kernel(), to_vec_grad=false)
             end,
 
             # Stretched kernels.
             map([1e-2, 0.1, 1.0, 10.0, 100.0]) do λ
-                (; name="stretched-λ=$λ", val=Matern32Kernel() ∘ ScaleTransform(λ))
+                (; name="stretched-λ=$λ", val=Matern32Kernel() ∘ ScaleTransform(λ), to_vec_grad=false)
             end,
 
+            # Approx periodic kernels
+            map([7, 11]) do N
+                (
+                    name="approx-periodic-N=$N",
+                    val=ApproxPeriodicKernel{N}(; r=1.0),
+                    to_vec_grad=true,
+                )
+            end,
+            # TEST_TOFIX
             # Gradients should be fixed on those composites.
             # Error is mostly due do an incompatibility of Tangents
             # between Zygote and FiniteDifferences.
 
             # Product kernels
             (
                 name="prod-Matern12Kernel-Matern32Kernel",
-                val=1.5 * Matern12Kernel() ∘ ScaleTransform(0.1) *
-                    Matern32Kernel() ∘ ScaleTransform(1.1),
-                skip_grad=true,
-                ), 
-                (
+                val=1.5 * Matern12Kernel() ∘ ScaleTransform(0.1) * Matern32Kernel() ∘
+                    ScaleTransform(1.1),
+                to_vec_grad=nothing,
+            ),
+            (
                 name="prod-Matern32Kernel-Matern52Kernel-ConstantKernel",
-                val = 3.0 * Matern32Kernel() *
-                    Matern52Kernel() *
-                    ConstantKernel(),
-                skip_grad=true,
+                val=3.0 * Matern32Kernel() * Matern52Kernel() * ConstantKernel(),
+                to_vec_grad=nothing,
             ),
 
             # Summed kernels.
             (
                 name="sum-Matern12Kernel-Matern32Kernel",
                 val=1.5 * Matern12Kernel() ∘ ScaleTransform(0.1) +
                     0.3 * Matern32Kernel() ∘ ScaleTransform(1.1),
-                skip_grad=true,
-            ), 
+                to_vec_grad=nothing,
+            ),
             (
                 name="sum-Matern32Kernel-Matern52Kernel-ConstantKernel",
-                val = 2.0 * Matern32Kernel() +
+                val=2.0 * Matern32Kernel() +
                     0.5 * Matern52Kernel() +
                     1.0 * ConstantKernel(),
-                skip_grad=true,
+                to_vec_grad=nothing,
             ),
         )
 
@@ -126,14 +168,14 @@ println("lti_sde:")
             (name="Custom Mean", val=CustomMean(x -> 2x)),
         )
 
-        @testset "$(kernel.name), $(m.name), $(storage.name), $(t.name), $(σ².name)" for
-            kernel in kernels,
+        @testset "$(kernel.name), $(m.name), $(storage.name), $(t.name), $(σ².name)" for kernel in
+                                                                                         kernels,
             m in means,
             storage in storages,
             t in ts,
             σ² in σ²s
 
-            println("$(kernel.name), $(storage.name), $(t.name), $(σ².name)")
+            println("$(kernel.name), $(storage.name), $(m.name), $(t.name), $(σ².name)")
 
             # Construct Gauss-Markov model.
             f_naive = GP(m.val, kernel.val)
@@ -174,7 +216,21 @@ println("lti_sde:")
             end
 
             # Just need to ensure we can differentiate through construction properly.
-            if !(hasfield(typeof(kernel), :skip_grad) && kernel.skip_grad)
+            if isnothing(kernel.to_vec_grad)
+                @test_broken "Gradient tests are not passing"
+                continue
+            elseif kernel.to_vec_grad
+                test_zygote_grad_finite_differences_compatible(
+                    _construction_tester,
+                    f_naive,
+                    storage.val,
+                    σ².val,
+                    t.val;
+                    check_inferred=false,
+                    rtol=1e-6,
+                    atol=1e-6,
+                )
+            else
                 test_zygote_grad(
                     _construction_tester,
                     f_naive,

test/test_util.jl

@@ -46,7 +46,7 @@ test_zygote_grad(f, args...; check_inferred=false, kwargs...) = test_rrule(Zygot
 function test_zygote_grad_finite_differences_compatible(f, args...; kwargs...)
     x_vec, from_vec = to_vec(args)
     function finite_diff_compatible_f(x::AbstractVector)
-        return @ignore_derivatives(f)(from_vec(x)...)
+        return @ignore_derivatives(f)(@ignore_derivatives(from_vec)(x)...)
     end
     test_zygote_grad(finite_diff_compatible_f ⊢ NoTangent(), x_vec; testset_name="test_rrule: $(f) on $(typeof.(args))", kwargs...)
 end
@@ -134,6 +134,14 @@ function ChainRulesTestUtils.test_approx(actual::Tangent{<:Fill}, expected, msg=
     test_approx(actual.value, expected.value, msg; kwargs...)
 end
 
+function to_vec(x::PeriodicKernel)
+    x, to_r = to_vec(x.r)
+    function PeriodicKernel_from_vec(x)
+        return PeriodicKernel(;r=exp.(to_r(x)))
+    end
+    log.(x), PeriodicKernel_from_vec
+end
+
 to_vec(x::T) where {T} = generic_struct_to_vec(x)
 
 # This is a copy from FiniteDifferences.jl without the try catch