Collocation is not timescale independent

[JTaets]

All the collocation methods only work when the timescale is correct.
This is easily fixed by artificially "normalising" the time-scale.
But when the data has multiple dimensions, with multiple orders of size, this method will also fail. I think the code of the kernels need to be adapted to do something like this automatically for each row in the data.

tsteps = 0:0.01:1
data = reshape( sin.(15*tsteps),1,:)

# bad
du,u = collocate_data(data,tsteps,EpanechnikovKernel())
scatter(tsteps,data')
plot!(tsteps,u',lw=5) 

# good
scale = 10 # higher scale -> better fit
du,u = collocate_data(data,scale*tsteps,EpanechnikovKernel()) # scaled tsteps
du.*= scale # scaled du
scatter(tsteps,data')
plot!(tsteps,u',lw=5) 

[ChrisRackauckas]

I agree with you. Can you open a PR for this?

[JTaets]

Apologies for the late response.

The problem is what the value of the parameter scale should be.
Taking a high value will let you have good fits, but too high will let us fit to the noise.

I see 3 solutions:

1. Let the user pass the scale, maybe something like EpanechnikovKernel(scale). A good choice of scale can be easily checked by a plot like i did above.
2. Automatically detection of the scale. But I don't know how easy this is with noise.
3. Implement signal filters such that the user needs to filter the noise out him/herself (DSP.jl?), then do a fit with high scale.

My preference is for 3, since this is way more tangible.

[ChrisRackauckas]

We should probably do 1 and 3, but 1 is low hanging fruit.

[sathvikbhagavan]

Scaling the timeseries does work but it can easily break:

tsteps = 0:0.1:1
data = reshape( sin.(15*tsteps),1,:)
scale = 10
du,u = collocate_data(data,scale*tsteps,EpanechnikovKernel())

Error:

ERROR: LinearAlgebra.SingularException(2)
Stacktrace:
  [1] ldiv!(B::Matrix{Float64}, D::LinearAlgebra.Diagonal{Float64, Vector{Float64}}, A::LinearAlgebra.Adjoint{Float64, Matrix{Float64}})
    @ LinearAlgebra ~/.julia/juliaup/julia-1.9.3+0.x64.linux.gnu/share/julia/stdlib/v1.9/LinearAlgebra/src/diagonal.jl:425
  [2] \
    @ ~/.julia/juliaup/julia-1.9.3+0.x64.linux.gnu/share/julia/stdlib/v1.9/LinearAlgebra/src/diagonal.jl:412 [inlined]
  [3] \(A::Matrix{Float64}, B::LinearAlgebra.Adjoint{Float64, Matrix{Float64}})
    @ LinearAlgebra ~/.julia/juliaup/julia-1.9.3+0.x64.linux.gnu/share/julia/stdlib/v1.9/LinearAlgebra/src/generic.jl:1107
  [4] (::DiffEqFlux.var"#179#180"{Matrix{Float64}, StepRangeLen{Float64, Base.TwicePrecision{Float64}, Base.TwicePrecision{Float64}, Int64}, EpanechnikovKernel, Matrix{Float64}, Matrix{Float64}, Matrix{Float64}, Float64, Vector{Float64}, Vector{Float64}})(_t::Float64)
    @ DiffEqFlux ~/.julia/packages/DiffEqFlux/5uK1j/src/collocation.jl:149
  [5] iterate
    @ ./generator.jl:47 [inlined]
  [6] _collect(c::StepRangeLen{Float64, Base.TwicePrecision{Float64}, Base.TwicePrecision{Float64}, Int64}, itr::Base.Generator{StepRangeLen{Float64, Base.TwicePrecision{Float64}, Base.TwicePrecision{Float64}, Int64}, DiffEqFlux.var"#179#180"{Matrix{Float64}, StepRangeLen{Float64, Base.TwicePrecision{Float64}, Base.TwicePrecision{Float64}, Int64}, EpanechnikovKernel, Matrix{Float64}, Matrix{Float64}, Matrix{Float64}, Float64, Vector{Float64}, Vector{Float64}}}, #unused#::Base.EltypeUnknown, isz::Base.HasShape{1})
    @ Base ./array.jl:802
  [7] collect_similar
    @ ./array.jl:711 [inlined]
  [8] map
    @ ./abstractarray.jl:3263 [inlined]
  [9] collocate_data(data::Matrix{Float64}, tpoints::StepRangeLen{Float64, Base.TwicePrecision{Float64}, Base.TwicePrecision{Float64}, Int64}, kernel::EpanechnikovKernel)
    @ DiffEqFlux ~/.julia/packages/DiffEqFlux/5uK1j/src/collocation.jl:142

This happens when for any point, the kernel weights becomes zero for the neighbouring points. This happens when the time points are far from each other (which happens when they are scaled) as the support for kernel function is between 1 and -1.

[sathvikbhagavan]

@ChrisRackauckas can this issue be closed as #853 addresses bandwidth being passed by the user or should we also implement point 3 mentioned above: "Implement signal filters such that the user needs to filter the noise out him/herself (DSP.jl?), then do a fit with high scale."?