fix: kernel functions

src/collocation.jl

@@ -36,26 +36,26 @@
 end
 
 function calckernel(::QuarticKernel,t)
- if abs(t)>0
+ if abs(t) > 1
  return 0
  else
  return (15*(1-t^2)^2)/16
  end
 end
 
 function calckernel(::TriweightKernel,t)
- if abs(t)>0
+ if abs(t) > 1
  return 0
  else
  return (35*(1-t^2)^3)/32
  end
 end
 
 function calckernel(::TricubeKernel,t)
- if abs(t)>0
+ if abs(t) > 1
  return 0
  else
- return (70*(1-abs(t)^3)^3)/80
+ return (70*(1-abs(t)^3)^3)/81
  end
 end
 
@@ -64,7 +64,7 @@
 end
 
 function calckernel(::CosineKernel,t)
- if abs(t)>0
+ if abs(t) > 1
  return 0
  else
  return (π*cos(π*t/2))/4
@@ -92,14 +92,14 @@
 end
 
 function construct_w(t,tpoints,h,kernel)
- W = @. calckernel((kernel,),(tpoints-t)/h)/h
+ W = @. calckernel((kernel,),((tpoints-t)/(tpoints[end]-tpoints[begin]))/h)/h
  Diagonal(W)
 end
 
 
 """
 ```julia
-u′,u = collocate_data(data,tpoints,kernel=SigmoidKernel())
+u′,u = collocate_data(data,tpoints,kernel=TriangularKernel(),bandwidth=nothing)
 u′,u = collocate_data(data,tpoints,tpoints_sample,interp,args...)
 ```
 
@@ -128,24 +128,29 @@
 data from intermediate timesteps. In this case, pass any of the methods like
 `QuadraticInterpolation` as `interp`, and the timestamps to sample from as `tpoints_sample`.
 """
-function collocate_data(data,tpoints,kernel=TriangularKernel())
+function collocate_data(data, tpoints, kernel=TriangularKernel(), bandwidth=nothing)
  _one = oneunit(first(data))
  _zero = zero(first(data))
  e1 = [_one;_zero]
  e2 = [_zero;_one;_zero]
  n = length(tpoints)
- h = (n^(-1/5))*(n^(-3/35))*((log(n))^(-1/16))
+ bandwidth = isnothing(bandwidth) ? (n^(-1/5))*(n^(-3/35))*((log(n))^(-1/16)) : bandwidth
 
  Wd = similar(data, n, size(data,1))
  WT1 = similar(data, n, 2)
  WT2 = similar(data, n, 3)
+ T2WT2 = similar(data, 3, 3)
+ T1WT1 = similar(data, 2, 2)
  x = map(tpoints) do _t
  T1 = construct_t1(_t,tpoints)
  T2 = construct_t2(_t,tpoints)
- W = construct_w(_t,tpoints,h,kernel)
+ W = construct_w(_t,tpoints,bandwidth,kernel)
  mul!(Wd,W,data')
  mul!(WT1,W,T1)
  mul!(WT2,W,T2)
+ mul!(T2WT2,T2',WT2)
+ mul!(T1WT1,T1',WT1)
+ (det(T2WT2) ≈ 0.0 || det(T1WT1) ≈ 0.0) && error("Collocation failed with bandwidth $bandwidth. Please choose a higher bandwidth")
  (e2'*((T2'*WT2)\T2'))*Wd,(e1'*((T1'*WT1)\T1'))*Wd
  end
  estimated_derivative = reduce(hcat,transpose.(first.(x)))
@@ -163,7 +168,7 @@
  tpoints_sample::AbstractVector{T},interp,args...) where T
  u = zeros(T,size(data, 1),length(tpoints_sample))
  du = zeros(T,size(data, 1),length(tpoints_sample))
- for d1 in 1:size(data,1)
+ for d1 in axes(data, 1)
  interpolation = interp(data[d1,:],tpoints,args...)
  u[d1,:] .= interpolation.(tpoints_sample)
  du[d1,:] .= DataInterpolations.derivative.((interpolation,), tpoints_sample)

test/collocation.jl

@@ -0,0 +1,67 @@
+using DiffEqFlux, OrdinaryDiffEq, Test
+
+bounded_support_kernels = [
+ EpanechnikovKernel(),
+ UniformKernel(),
+ TriangularKernel(),
+ QuarticKernel(),
+ TriweightKernel(),
+ TricubeKernel(),
+ CosineKernel(),
+]
+
+unbounded_support_kernels =
+ [GaussianKernel(), LogisticKernel(), SigmoidKernel(), SilvermanKernel()]
+
+@testset "Kernel Functions" begin
+ ts = collect(-5.0:0.1:5.0)
+ @testset "Kernels with support from -1 to 1" begin
+ minus_one_index = findfirst(x -> ==(x, -1.0), ts)
+ plus_one_index = findfirst(x -> ==(x, 1.0), ts)
+ @testset "$kernel" for (kernel, x0) in zip(
+ bounded_support_kernels,
+ [0.75, 0.50, 1.0, 15.0 / 16.0, 35.0 / 32.0, 70.0 / 81.0, pi / 4.0],
+ )
+ ws = DiffEqFlux.calckernel.((kernel,), ts)
+ # t < -1
+ @test all(ws[1:minus_one_index-1] .== 0.0)
+ # t > 1
+ @test all(ws[plus_one_index+1:end] .== 0.0)
+ # -1 < t <1
+ @test all(ws[minus_one_index+1:plus_one_index-1] .> 0.0)
+ # t = 0
+ @test DiffEqFlux.calckernel(kernel, 0.0) == x0
+ end
+ end
+ @testset "Kernels with unbounded support" begin
+ @testset "$kernel" for (kernel, x0) in zip(
+ unbounded_support_kernels,
+ [1 / (sqrt(2 * pi)), 0.25, 1 / pi, 1 / (2 * sqrt(2))],
+ )
+ # t = 0
+ @test DiffEqFlux.calckernel(kernel, 0.0) == x0
+ end
+ end
+end
+
+@testset "Collocation of data" begin
+ function f(u, p, t)
+ p .* u
+ end
+ rc = 2
+ ps = repeat([-0.001], rc)
+ tspan = (0.0, 50.0)
+ u0 = 3.4 .+ ones(rc)
+ t = collect(range(minimum(tspan), stop = maximum(tspan), length = 1000))
+ prob = ODEProblem(f, u0, tspan, ps)
+ data = Array(solve(prob, Tsit5(), saveat = t, abstol = 1e-12, reltol = 1e-12))
+ @testset "$kernel" for kernel in
+ [bounded_support_kernels..., unbounded_support_kernels...]
+ u′, u = collocate_data(data, t, kernel, 0.003)
+ @test sum(abs2, u - data) < 1e-8
+ end
+ @testset "$kernel" for kernel in [bounded_support_kernels...]
+ # Errors out as the bandwidth is too low
+ @test_throws ErrorException collocate_data(data, t, kernel, 0.001)
+ end
+end

test/runtests.jl

@@ -6,8 +6,8 @@ const is_CI = haskey(ENV, "CI")
 
 @time begin
  if GROUP == "All" || GROUP == "DiffEqFlux" || GROUP == "Layers"
- @safetestset "Collocation Regression" begin
- include("collocation_regression.jl")
+ @safetestset "Collocation" begin
+ include("collocation.jl")
  end
  @safetestset "Stiff Nested AD Tests" begin
  include("stiff_nested_ad.jl")