~cgeoga/HalfSpectral.jl

ae1cf33c070437a97344200eb651c8df91993a87 — Chris Geoga 2 years ago 0565ddc
Complete re-write, update examples and README.
30 files changed, 1571 insertions(+), 1019 deletions(-)

M .gitignore
A LICENSE.txt
M Manifest.toml
M Project.toml
M README.md
A example/Manifest.toml
A example/Project.toml
A example/basic_example.jl
D example/basicexample.jl
D example/basicmodel.jl
D example/fitsimulation.jl
D example/paperscripts/README.txt
D example/paperscripts/extractor.jl
D example/paperscripts/fit_trustregion.jl
D example/paperscripts/fitted_estimates_untransformed.jl
D example/paperscripts/generate_nls_init.jl
D example/paperscripts/model.jl
D example/paperscripts/model_untransformed.jl
D example/paperscripts/nsscale.jl
D example/paperscripts/untransformer.jl
D example/paperscripts/writer.jl
A example/withvecchia.jl
M src/HalfSpectral.jl
A src/methods.jl
A src/radix2.jl
A src/structstypes.jl
A src/utils.jl
A test/Manifest.toml
A test/Project.toml
A test/runtests.jl
M .gitignore => .gitignore +2 -0
@@ 4,3 4,5 @@
!simulationplot.png
plotsim.gp
cmocean_balance.pal
debug_*
*.swp

A LICENSE.txt => LICENSE.txt +336 -0
@@ 0,0 1,336 @@
GNU GENERAL PUBLIC LICENSE
                       Version 2, June 1991

 Copyright (C) 2022, Chris Geoga

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M Manifest.toml => Manifest.toml +16 -96
@@ 1,106 1,26 @@
# This file is machine-generated - editing it directly is not advised

[[AbstractFFTs]]
deps = ["LinearAlgebra"]
git-tree-sha1 = "051c95d6836228d120f5f4b984dd5aba1624f716"
uuid = "621f4979-c628-5d54-868e-fcf4e3e8185c"
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manifest_format = "2.0"

[[Base64]]
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[[Reexport]]
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[[Statistics]]
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[[UUIDs]]
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[[deps.OpenBLAS_jll]]
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[[Unicode]]
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[[deps.libblastrampoline_jll]]
deps = ["Artifacts", "Libdl", "OpenBLAS_jll"]
uuid = "8e850b90-86db-534c-a0d3-1478176c7d93"

M Project.toml => Project.toml +1 -3
@@ 1,9 1,7 @@
name = "HalfSpectral"
uuid = "28fda99a-6fd1-4789-996a-91dd14997ef0"
authors = ["Chris Geoga <cgeoga@protonmail.com>"]
version = "0.1.0"
version = "0.2.0"

[deps]
FFTW = "7a1cc6ca-52ef-59f5-83cd-3a7055c09341"
LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
Statistics = "10745b16-79ce-11e8-11f9-7d13ad32a3b2"

M README.md => README.md +17 -17
@@ 4,31 4,31 @@
A convenient interface for working with half-spectral covariance functions in
the space-time domain.  This is part of the software companion to [Flexible
nonstationary spatio-temporal modeling of high-frequency monitoring
data](http://arxiv.org/abs/2007.11418).
data](https://onlinelibrary.wiley.com/doi/10.1002/env.2670).

See the paper for a discussion of the covariance function itself, and see the
example file `./example/basicmodel.jl` for an example specification of the
model, `./example/basicexample.jl` for an example of creating a kernel function
and then simulating a process with it, and finally `./example/fitsimulation.jl`
for an example of computing an MLE for the simulated data. With very few lines
of boilerplate, you can specify  a complicated covariance function corresponding
to fields that look like this:
example file `./example/basicexample.jl` for an example of creating a kernel
that you can then call as a regular function. The example file
`./example/withvecchia.jl` gives an example of the wrapper type that you can use
to conveniently use `ForwardDiff.jl` to get derivatives of the kernel function
as well as an example of how to efficiently compose this package with
`Vecchia.jl` for scalable parameter estimation.

With very few lines of boilerplate, you can specify  a complicated covariance
function corresponding to fields that look like this:

<p align="center">
  <img src="https://git.sr.ht/~cgeoga/HalfSpectral.jl/blob/master/simulationplot.png">
</p>

For the code used to fit models in the paper, see `./examples/paperscripts/`. I
do not plan to modify these files if this code evolves, so if you want to run
those scripts, please use the appropriate tagged release.

For more information about estimation, please see another of my software
packages, [GPMaxlik.jl](https://git.sr.ht/~cgeoga/GPMaxlik.jl). All the
necessary code to try simulating and estimating is here, but several more
complete examples are provided in that repository.
**If you use this software (HalfSpectral.jl) in a manuscript or published work,
please cite the paper linked above and not the software package itself.**

**If you use this software (HalfSpectral.jl), please cite the paper and not
the software package itself.**
This release (0.2+) is a complete re-write from the version that was used in the
paper (0.1), so please go to an earlier commit if you're interested in
re-creating paper results. But unless you're doing that, I'd really strongly
suggest using the new version of the code, because it is much cleaner and has
complete AD compatibility.

# Installation


A example/Manifest.toml => example/Manifest.toml +583 -0
@@ 0,0 1,583 @@
# This file is machine-generated - editing it directly is not advised

julia_version = "1.7.1"
manifest_format = "2.0"

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version = "3.3.3"

[[deps.ArgCheck]]
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deps = ["InteractiveUtils", "Logging", "Random", "Serialization"]
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deps = ["ManualMemory"]
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uuid = "8290d209-cae3-49c0-8002-c8c24d57dab5"
version = "0.5.0"

[[deps.Transducers]]
deps = ["Adapt", "ArgCheck", "BangBang", "Baselet", "CompositionsBase", "DefineSingletons", "Distributed", "InitialValues", "Logging", "Markdown", "MicroCollections", "Requires", "Setfield", "SplittablesBase", "Tables"]
git-tree-sha1 = "c76399a3bbe6f5a88faa33c8f8a65aa631d95013"
uuid = "28d57a85-8fef-5791-bfe6-a80928e7c999"
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[[deps.UUIDs]]
deps = ["Random", "SHA"]
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version = "1.0.2"

[[deps.Unicode]]
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[[deps.Vecchia]]
deps = ["BangBang", "FLoops", "LinearAlgebra", "LoopVectorization", "MicroCollections", "NearestNeighbors", "SparseArrays", "StaticArrays"]
path = "/home/cg/Personal/Repos/Vecchia.jl/"
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deps = ["ArrayInterface", "CPUSummary", "HostCPUFeatures", "IfElse", "LayoutPointers", "Libdl", "LinearAlgebra", "SIMDTypes", "Static"]
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deps = ["Artifacts", "Libdl"]
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A example/Project.toml => example/Project.toml +5 -0
@@ 0,0 1,5 @@
[deps]
ForwardDiff = "f6369f11-7733-5829-9624-2563aa707210"
HalfSpectral = "a4cc074a-5bcd-4a73-94df-f94ee43b423f"
StaticArrays = "90137ffa-7385-5640-81b9-e52037218182"
Vecchia = "8d73829f-f4b0-474a-9580-cecc8e084068"

A example/basic_example.jl => example/basic_example.jl +28 -0
@@ 0,0 1,28 @@

using HalfSpectral, StaticArrays

# Simple SDF and a simple nontrivial coherence, which make an integrand:
sdf(f,x,p) = p[1]*(p[2]^2 + f^2)^(-p[3] - 1/2)
coh(f,x,y,p) = exp(-p[4]*(1+(p[5]*f)^2)*abs(x-y))
integrand(f,x,y,p) = coh(f,x,y,p)*sqrt(sdf(f,x,p)*sdf(f,y,p))

# Let's make some basic points. Say, 5 measurements at x=1:10 at time points
# 1:100 each. Importantly, the time has to come first.
if !(@isdefined pts)
  const xpts    = 1:30
  const timelen = 200
  const pts = reduce(vcat, map(x->[@SVector [t, x] for t in 1:timelen], xpts))
end

# Now let's make the kernel:
if !(@isdefined kernel)
  const params = ones(5)
  const kernel = HalfSpectral.HSKernel(params, xpts, integrand, timelen)
end

# Now just call it like a function! Note that you can't call it for locations
# that aren't in pts, though.
sample_call = kernel(pts[1], pts[3])




D example/basicexample.jl => example/basicexample.jl +0 -52
@@ 1,52 0,0 @@

using HalfSpectral, LinearAlgebra

# Load a basic specification of a nonstationary half-spectral model.
include("basicmodel.jl")

# Create the kernel for spatial locations 1:30 and 100 unit time measurements.
const tg     = timegrid(1:100)
const xpts   = 1:30
const params = (log(50.0),log(100.0),log(0.5),log(5.0),0.5) # sample parameter values.
kernel = tftkernel(integrand, tg, xpts, params)

# Build a second time to get a timing example.
println("Building the kernel takes this long")
@time kernel = tftkernel(integrand, tg, xpts, params)

# Simulate a realization of the field with this function. The points here should
# be ordered as a tuple (time, spatial coordinate), and the spatial coordinate
# does not need to be a real number. You can treat the kernel like any normal
# function, calling it with either two points x and y like kernel(x,y) or
# additionally with a parameter collection, and if the parameter collection
# disagrees with the internal one it will rebuild the dictionary of function
# values. In general, I suggest including the parameter vector for code safety.
# But to avoid runtime checks, you can use the unchecked kernel(x,y).
pts  = vec(reverse.(collect(Iterators.product(xpts, tg.Xt))))
K    = Symmetric([kernel(x,y) for x in pts, y in pts])
simv = cholesky(K).L*randn(length(pts))
simm = reshape(simv, length(xpts), length(tg.Xt))

# Optionally, save the simulation file.

#= If you just want a csv file to plot.
using DelimitedFiles
writedlm("sim.csv", simm, ',')
=#

# If you want to save enough output to also fit your simulated data:
using JLD
JLD.save("simulation_output.jld",
         "true_params", params,
         "pts",   pts,
         "tgrid", tg,
         "xpts",  xpts,
         "data",  simv)
# =#
# =#

#= Optionally, visualize it.
using Plots
heatmap(simm, xlabel="time", ylabel="altitude")
=#


D example/basicmodel.jl => example/basicmodel.jl +0 -31
@@ 1,31 0,0 @@

# Marginal spectral density, with the twist of making a few parameters depend on
# the spatial index.
function Sf(w,x,p)
  (scale, rate, smoothness) = (exp(p[1]), exp(p[2]), exp(p[3]))
  rate += log(x)
  smoothness *= (1+x/10)
  scale*(rate*sinpi(w)^2 + 1)^(-smoothness-1/2) 
end

# A coherence function that does happen to be stationary. See the paperscripts
# model for code implementing the Paciorek-Schervish nonstationary coherence.
function Cw(w,x,y,p)
  gamxy = exp(p[4])/(1 + (sinpi(w)/0.3)^2)^4
  exp(-abs(x-y)/gamxy)
end

# A simple phase function and it's derivative with respect to kernel parameters.
phase(w,x,y,p) = exp(2.0*pi*im*p[5]*sin(sinpi(w))*(x-y))
dphase(w,x,y,p)= 2.0*pi*im*sin(sinpi(w))*(x-y)*phase(w,x,y,p)

# The integrand in the form that HalfSpectral requests, splitting up the phase
# part so we can use AD on the real-valued parts, which seems to avoid a lot of
# finnicky issues with ForwardDiff.jl. This is not strictly necessary, but as
# you'll see in fitsimulation.jl, writing it in this slightly tricky way means
# that with just a few more lines of code, you can use very powerful
# quasi-Newton optimization methods.
integrand_nophase(w,x,y,p) = sqrt(Sf(w,x,p)*Sf(w,y,p))*Cw(w,x,y,p)
integrand(w,x,y,p) = integrand_nophase(w,x,y,p)*phase(w,x,y,p) 
integrand_dphase(w,x,y,p) = integrand_nophase(w,x,y,p)*dphase(w,x,y,p)


D example/fitsimulation.jl => example/fitsimulation.jl +0 -58
@@ 1,58 0,0 @@

using HalfSpectral, GPMaxlik, JLD, ForwardDiff

# Load in the model used to simulate the data.
include("basicmodel.jl")

# A lazy version of kernel derivatives: use AD on the integrand. This would
# obviously be much faster if you hand-wrote your derivative functions for the
# integrand, but as you'll see here this is good enough for prototyping.
function coord_deriv(f,p,j)
  ForwardDiff.derivative(z->f(vcat(p[1:(j-1)],z,p[(j+1):end])), p[j])
end
dfuns = convert(Vector{Function},
          [(w,x,y,p)->coord_deriv(z->integrand_nophase(w,x,y,z),p,j)*phase(w,y,x,p)
           for j in 1:4])
push!(dfuns, integrand_dphase)

# Load the saved data in:
const pts   = JLD.load("simulation_output.jld", "pts")
const tgrid = JLD.load("simulation_output.jld", "tgrid")
const xpts  = JLD.load("simulation_output.jld", "xpts")
const dat   = JLD.load("simulation_output.jld", "data")
const tru   = JLD.load("simulation_output.jld", "true_params")

# Choose some initial conditions, then create the kernel function and the
# derivative kernel functions.
ini    = vcat(log.([75.0, 75.0, 1.0, 4.0]), 0.75)
kfun   = tftkernel(integrand, tgrid, xpts, ini)
dkfuns = [tftkernel(integrand_dj, tgrid, xpts, ini) for integrand_dj in dfuns]  

# Create some SAA vectors for faster derivatives.
const saa = rand([-1.0, 1.0], length(pts), 100)

# Create objective only and objective+grad+fisher functions for GPMaxlik, using
# the optimized nll function it provides.
obj(p) = nll(pts, dat, kfun, dkfuns, p, saa=saa, nll=true, grad=false, fish=false).nll
function objgradfish(p)
  nll_call = nll(pts, dat, kfun, dkfuns, p, saa=saa, nll=true, grad=true, fish=true)
  (nll_call.nll, nll_call.grad, nll_call.fish)
end

# Perform the optimization using the trust region method (see GPMaxlik.jl for
# details) using the stochastic expected Fisher matrix as a Hessian
# approximation. If you want to try BFGS to compare, just change "fish=true" to
# "fish=false" in line 38 above.
maxlik_result = trustregion(obj, objgradfish, ini, vrb=true, maxit=100, rtol=1.0e-5)

# Consider the resulting MLE, std approximations from the stochastic expected
# Fisher matrix, and compare with the true parameters for the simulation.
mle   = maxlik_result.p
stds  = map(x->"("*string(x)*")", 
            round.(sqrt.(diag(inv(maxlik_result.h))).*1.96, digits=2))
table = hcat(pushfirst!(string.(round.(collect(tru), digits=4)), "true"),
             pushfirst!(string.(round.(collect(mle), digits=4)), "mle"),
             pushfirst!(stds, "±95%"))
display(table)



D example/paperscripts/README.txt => example/paperscripts/README.txt +0 -12
@@ 1,12 0,0 @@

These files are most of the code used to fit the model described in the paper.
They will not "just run" on your computer as I have removed several hard paths
for storing output and reading data files. Nonetheless, I think they are helpful
in illustrating a more advanced workflow for several data sources and a
reasonably complex model.

I will NOT be updating these files to reflect potential breaking changes in
HalfSpectral.jl, so please refer to the commit tagged "paper" to see a version
of HalfSpectral.jl that was the source code that these scripts actually ran when
generating results for the paper.


D example/paperscripts/extractor.jl => example/paperscripts/extractor.jl +0 -24
@@ 1,24 0,0 @@

using HalfSpectral

# UNITS:
#   time (ix)
#   altitudes (ix)
#   data (m/s)
function extract_block(trange, times, gates, dopp, ints, fn)
  qj    = findall(x->trange[1]<=x<=trange[2], times)
  minr  = minimum.(eachrow(ints[:,qj]))
  snrj  = filter(x->in(x, gates), findall(x->x>1.008, minr))
  tgrd  = timegrid(times[qj])
  tgrd isa TimeFFTKernels.TimeGridFailure && return (tgrd, nothing)
  pts   = reverse.(vec(collect(Iterators.product(snrj.*30.0, tgrd.Xt))))
  dat   = vec(dopp[snrj,qj])
  if fn isa Nothing
    mldat = (pts=pts, dat=dat)
  else
    println("Transforming data...")
    mldat = (pts=pts, dat=fn(dat))
  end
  (tgrd, snrj.*30.0, mldat) # (timegrid object, altitude (m), named tuple)
end


D example/paperscripts/fit_trustregion.jl => example/paperscripts/fit_trustregion.jl +0 -107
@@ 1,107 0,0 @@

using HDF5, HalfSpectral, LinearAlgebra, DelimitedFiles, GPMaxlik, JLD, Random

# Set the seed:
Random.seed!(123456)

# At this point, do five iterations, then start using an exact gradient:
GPMaxlik.SETTINGS.GRAD_INFNORM_TOL = 0.5

include("model.jl")
include("extractor.jl")
include("writer.jl")
include("nsscale.jl")

# no build indices: re-building the integrand kernel object isn't necessary for
# these parameter indices.
const nob = (1,2,3,4,24,25)

# Create a NonstationaryScale object:
NS  = NonstationaryScale{Float64,false,0,4}((156, 311, 466, 621), (1,2,3,4))
dNS = [NonstationaryScale{Float64,true,j,4}((156, 311, 466, 621), (1,2,3,4)) for j in 1:4]

# Some simple wrapper functions:
function nll_grad_fish(pts, data, kfn, dfns, p, saa)
  nll_call = nll(pts, data, kfn, dfns, p, saa=saa, nll=true, grad=true, fish=true)
  (nll_call.nll, nll_call.grad, nll_call.fish)
end

# Convert decimal hours to minutes:
decimalhours_to_minutes(dhours) = (dhours .- 14.0).*60.0

# An extra function to print the Newton step every five iterations:
function print_ns(c, fx, xv, gx, hx, dl; kwargs...)
  if iszero(rem(c, 5))
    println("NS: $(round.(hx\gx, digits=4))")
  end
  return (false, nothing)
end

function fit_gates_trustregion(day, timewin, gates, its, ini, rt, gt, 
                               trfun=nothing, sim=false, dc=1.0e-4)
  # Load in the data:
  println("Working with day $day")
  rdata = read(h5open(__YOUR_PATH_TO_RAW_DATA__)) # Put your path here
  (times, dopp, ints) = (rdata[x] for x in ("Time", "Doppler", "Intensity"))
  (tgrid, alts, data) = extract_block(timewin, times, gates, dopp, ints, trfun)
  if unique(diff(alts)) != [30.0]
    @warn "You are missing gates in the range you've requested."
  end
  # Create the necessary objects:
  saav = rand([-1.0, 1.0], length(data.pts), 250) # was 125 for first stage of fitting
  kfun = tftkernel(integrand, tgrid, alts, ini, addfn=nugfn, mulfn=NS, nob=nob)
  dfns = vcat(
           [tftkernel(integrand, tgrid, alts, ini, mulfn=dNSj, nob=nob) for dNSj in dNS],
           [tftkernel(dk, tgrid, alts, ini, mulfn=NS, nob=nob) for dk in dfuns], 
           dnugfn1, dnugfn2
         )
  # Do the optimization:
  obj = p-> nll(data.pts, data.dat, kfun, dfns, p, saa=saav,
               nll=true, grad=false, fish=false).nll
  objgradfish = p -> nll_grad_fish(data.pts, data.dat, kfun, dfns, p, saav)
  maxlik_result = trustregion(obj, objgradfish, ini, vrb=true, maxit=its, gtol=gt,
                              rtol=rt, dcut=dc, iter_funs=(GPMaxlik.convg_info, print_ns))
  println("Estimation result: $(maxlik_result.status)")
  println()
  if sim
    L = cholesky!(Symmetric([kfun(x, y) for x in data.pts, y in data.pts])).L
    d = reshape(data.dat, length(alts), length(tgrid.Tv))
    s = reshape(L*randn(length(data.pts)), length(alts), length(tgrid.Tv))
    gnuplot_save_matrix!(__YOUR_PATH__*"sim_"*day*"_14.csv", s, alts,  # Put your path here
                         decimalhours_to_minutes(tgrid.Tv))
    gnuplot_save_matrix!(__YOUR_PATH__*"tru_"*day*"_14.csv", d, alts,  # Put your path here
                         decimalhours_to_minutes(tgrid.Tv))
  end
  return maxlik_result
end






const FITTED_DAYS = tuple() # Satisfactorily fitted days
const tw = (14.02, 14.24)

for (day, gates, mit) in zip(("02", "03", "06", "20", "24", "28"), 
                             (7:23, 7:28, 7:23, 7:28, 7:28, 7:28),
                             (30, 30, 30, 30, 30, 30))
  if in(day, FITTED_DAYS)
    println("Estimate for day $day was marked acceptable, moving on.")
    println()
    continue 
  end
  init = load(__YOUR_PATH__*"fit_"*day*"_results.jld")["result"].p
  try
    result = fit_gates_trustregion(day, tw, gates, mit, init, 
                                   1.0e-6, 1.0e-3, nothing, true, 1.0e-7)
    # Deal with results:
    GPMaxlik.reset_settings!()
    JLD.save(__YOUR_PATH__*"fit_"*day*"_results.jld", "result", result)
  catch
    println()
    println("Day $day broke somehow. Check out the logs.")
    println()
  end
end


D example/paperscripts/fitted_estimates_untransformed.jl => example/paperscripts/fitted_estimates_untransformed.jl +0 -65
@@ 1,65 0,0 @@

using HDF5, TimeFFTKernels, LinearAlgebra, DelimitedFiles, GPMaxlik, JLD, Random

# Set the seed:
Random.seed!(123456)

# At this point, do five iterations, then start using an exact gradient:
GPMaxlik.SETTINGS.GRAD_INFNORM_TOL = 0.5

include("model_untransformed.jl")
include("extractor.jl")
include("writer.jl")
include("nsscale.jl")
include("untransformer.jl")

# no build indices: re-building the integrand kernel object isn't necessary for
# these parameter indices.
const nob = (1,2,3,4,23,24,25)

# Create a NonstationaryScale object:
NS  = NonstationaryScale{Float64,false,0,4}((156, 311, 466, 621), (1,2,3,4))
dNS = [NonstationaryScale{Float64,true,j,4}((156, 311, 466, 621), (1,2,3,4)) for j in 1:4]

# Some simple wrapper functions:
function nll_grad_fish(pts, data, kfn, dfns, p, saa)
  nll_call = nll(pts, data, kfn, dfns, p, saa=saa, nll=true, grad=true, fish=true)
  (nll_call.nll, nll_call.grad, nll_call.fish)
end

function untransformed_uncertainty(day, timewin, gates, trfun=nothing)
  # Load in the data:
  println("Working with day $day")
  rdata = read(h5open(__YOUR_PATH_TO_RAW_DATA)) # your path to the raw files
  (times, dopp, ints) = (rdata[x] for x in ("Time", "Doppler", "Intensity"))
  (tgrid, alts, data) = extract_block(timewin, times, gates, dopp, ints, trfun)
  # Load in the MLE and un-transform it:
  mle_result = load(__YOUR_PATH__*"fit_"*day*"_results.jld")["result"]
  mle = transf(mle_result.p)
  # Create the necessary objects:
  saav = rand([-1.0, 1.0], length(data.pts), 125)
  kfun = tftkernel(integrand, tgrid, alts, mle, addfn=nugfn, mulfn=NS, nob=nob)
  dfns = vcat(
           [tftkernel(integrand, tgrid, alts, mle, mulfn=dNSj, nob=nob) for dNSj in dNS],
           [tftkernel(dk, tgrid, alts, mle, mulfn=NS, nob=nob) for dk in dfuns], 
           dnugfn1, dnugfn2
         )
  # Obtain the fisher information matrix for the untransformed model at the MLE:
  out = nll_grad_fish(data.pts, data.dat, kfun, dfns, mle, saav)
  println("Likelihood at MLE for un-transformed data: $(round(out[1], digits=5))")
  (mle, out)
end

const tw    = (14.02, 14.24)
const gater = (7:23, 7:28, 7:23, 7:28, 7:28, 7:28)
for (day, gates) in zip(("02", "03", "06", "20", "24", "28"), gater)
  # Obtain the un-transformed MLE and uncertainty:
  untransf_result = untransformed_uncertainty(day, tw, gates, nothing)
  # Deal with results:
  GPMaxlik.reset_settings!()
  JLD.save(__YOUR_PATH__*"fit_"*day*"_results_untransformed.jld", 
           "mle",       untransf_result[1],
           "nllresult", untransf_result[2])
end



D example/paperscripts/generate_nls_init.jl => example/paperscripts/generate_nls_init.jl +0 -169
@@ 1,169 0,0 @@

using DelimitedFiles, Statistics, HDF5, SMultitaper, Optim, ForwardDiff, HalfSpectral

# Load in the model and the extractor function.
include("model.jl")
include("extractor.jl")

# A function that reads in the data and parses it into the format I want. In
# this case, a Dict, indexed by gate _number_, not altitude, since indexing a
# hashmap by a floating point number is bad style (to m
function extract_day_slice(day, timewin, gates)
  # Load in the data:
  rdata =  read(h5open("../../data/raw/Vert_2015_06_"*day*"_14.h5")) 
  (times, dopp, ints) = (rdata[x] for x in ("Time", "Doppler", "Intensity"))
  (tgrid, alts, data) = extract_block(timewin, times, gates, dopp, ints, 
                                      nothing, asmatrix=true)
  if unique(diff(alts)) != [30.0]
    @warn "You are missing gates in the range you've requested."
  end
  # Reshape to a dict for easier reading in the fitting:
  datadict = Dict{Int64, Vector{Float64}}()
  for (j, gatej) in enumerate(gates)
    datadict[gatej] = data[2][j,:]
  end
  datadict
end

function cross_specs(datadict; coherence=true)
  out_type = coherence ? Complex{Float64} : Float64
  crosses = Dict{Tuple{Int64, Int64}, Vector{out_type}}()
  gates   = sort(collect(keys(datadict)))
  for gj in gates
    for gk in (gj+1):gates[end]
      if coherence
        cross  = real(multitaper(datadict[gj], datadict[gk], nw=15.0, k=25,
                                 coherence=true, shift=true, center=true))
      else
        cross = multitaper(datadict[gj], datadict[gk], nw=5.0, k=8,
                           aweight=true, shift=true, center=true)
      end
      crosses[(gj, gk)] = cross
    end
  end
  crosses
end

function marginal_specs(datadict)
  specs   = Dict{Int64, Vector{Float64}}()
  gates   = sort(collect(keys(datadict)))
  for gj in gates
    specs[gj] = multitaper(datadict[gj], nw=5.0, k=8,
                           aweight=true, shift=true, center=true)
  end
  specs
end

# For coherences. A very simple sum of squares. But honestly, these parameters
# are hard enough to estimate even for maximum likelihood that you might be
# better off just plugging in random values that don't make obviously wrong
# curves.
function cross_objective(parms, crosses, gates)
  out = 0.0
  len = length(crosses[(gates[1], gates[2])])
  fgd = collect(range(-0.5,0.5,length=len+1)[1:len])
  for gj in gates
    for gk in (gj+1):gates[end]
      model_cross = [Cw(f, gj*30, gk*30, parms) for f in fgd]
      out += sum(abs2, model_cross .- crosses[(gj,gk)])
    end
  end
  out
end

function marginal_objective(parms, specs, gates)
  out = 0.0
  len = length(specs[gates[1]])
  fgd = collect(range(-0.5,0.5,length=len+1)[1:len])
  for gj in gates
    model_spec = [Sf(f,gj*30.0,parms) + parms[24]^2 for f in fgd]
    out += sum(log.(model_spec) .+ specs[gj]./model_spec)
  end
  out
end


# So, the INIT here can't be truly naive, like just a bunch of ones or
# something, because then things like \beta will never come through. But it CAN
# be pretty naive, as you can see here. Here is some justification for the
# things that aren't just one:
#
# (i) 13.0: this is a range parameter for the coherence, which is exponentiated.
# They spatial locations are coded as 30,60,90,etc, so if I set that to one the
# coherence functions will be zero for x \neq y. No optimization software could
# overcome that. This value is chosen so that the coherence at the zero
# frequency for adjacent gates is about 0.95.
#
# (ii, iii) -5.0, 0.0: again an exponentiated parameter for the coherence.
# These are chosen to be what they are just to make the initial coherence
# function have an even vaguely reasonable shape based on how the model is
# paramterized. They're set to the same above and below the boundary layer.
#
# (iv) 500: boundary layer height. If this were one, every altitude would be
# completely below the boundary layer and so the gradient of this objective with
# respect to beta would be zero. 
#
# (v) 0.0: a phase term. I just don't try to capture this at all here. Zero is a
# more natural naive init than one for a parameter like this.
#
# (vi) 1.0e-3: a pretty small but nontrivial nugget. 1.0 is obviously too large for
# this data and 0 is not a good init. 
#
const INIT = vcat(ones(4),    # ns scale parameters. Not using those here because of the extra low-frequency multiplier, and because maxlik is very good at picking these parms up.
                  ones(4),    # sdf parameters, will come through.
                  13.0,       # first coherence parameter (scale-dependent). (i)
                 -5.0,        # second coherence parameter (ii)
                  0.0,        # third coherence parameter (iii)
                  13.0,       # first coherence parameter (scale-dependent). (i)
                 -5.0,        # second coherence parameter (ii)
                  0.0,        # third coherence parameter (iii)
                  500.0,      # beta height (iv)
                  1.0,        # tau parameter 
                  ones(4),    # extra low frequency power, will come through
                  ones(2),    # scale decay above boundary layer
                  0.0,        # phase term, won't come through. (v)
                  ones(2).*1.0e-3 # nuggets, won't come through. (vi)
                 )


# An incremental approach to the incremental generation of initializations for
# the maximum likelihood estimation: fit marginal spectra first, then fit
# cross-spectra. You need to use real derivatives here or else the generic first
# order methods will drive the irrelevant parameters up to ridiculous values.
# This is not particularly speedy on my box. But it does serve as nice
# reproducible proof that you could start from an impossibly stupid
# initialization and bootstrap yourself to a good MLE for the half-spectral model.
function fit_day(day, timewin, gates, init=INIT)
  datadict = extract_day_slice(day, timewin, gates)
  out      = Dict{String, Vector{Float64}}()
  out["init"] = deepcopy(init)
  for (objfn, spfn, nam) in ((marginal_objective, marginal_specs, "marginal"),
                             (cross_objective,    cross_specs,    "coherence"))
    println("Fitting $nam spectra...")
    sp   = spfn(datadict)
    obj  = p -> objfn(p, sp, gates)
    gr   = (g,p) -> ForwardDiff.gradient!(g, obj, p)
    hs   = (h,p) -> ForwardDiff.hessian!(h, obj, p)
    ini  = (nam == "marginal") ? out["init"] : out["marginal"]
    opts = Optim.Options(iterations=3_000, g_tol=1.0e-5, 
                         show_trace=true, show_every=10)
    est  = optimize(obj, gr, hs, ini, NewtonTrustRegion(), opts)
    display(est)
    out[nam] = deepcopy(est.minimizer)
  end
  out
end

for (day, gates) in zip(("02", "03", "06", "20", "24", "28"),
                        (7:23, 7:28, 7:23, 7:28, 7:28, 7:28))
  mle_init = fit_day(day, (14.02, 14.24), gates)
  println("Working with day $day...")
  outfilename = string("../../data/initialization/init_", day, "_14.csv")
  if haskey(mle_init, "coherence")
    writedlm(outfilename, mle_init["coherence"], ',')
  else
    writedlm(outfilename, mle_init["marginal"], ',')
  end
  println("Finished with day $day.\n\n\n")
end


D example/paperscripts/model.jl => example/paperscripts/model.jl +0 -88
@@ 1,88 0,0 @@

using SpecialFunctions, ForwardDiff, DelimitedFiles

# For the derivatives of the kernel:
function coord_deriv(f, p, j)
  ForwardDiff.derivative(z->f(vcat(p[1:(j-1)], z, p[(j+1):end])), p[j])
end

# The matern correlation for smoothness 1/2 and 3/2:
mtn_cor_12(x) = exp(-abs(x))
mtn_cor_32(x) = exp(-abs(x))*(1.0 + abs(x))
function mtn_cor(nu, x)
  abs(x)<1.0e-8 && return 1.0
  out  = (sqrt(2.0*nu)*x)^nu
  out *= besselk(nu, sqrt(2.0*nu)*x)
  out /= gamma(nu)*2.0^(nu-1.0)
  out
end

# The butterworth-like function. Several tweaks for numerical reasons to avoid
# breaking AD.
function butter(w,p) 
  iszero(w) && return p[1]  # tweak one: a branch for w=0.
  arg = abs2(w/p[2])^p[3]
  arg > 1.0e18 && return 0.0 # tweak two: a branch for the arg being too large.
  p[1]/(1.0 + arg)
end

function _logistic(x, p_0, p_1, shape, center)
  alpha = 1.0/(1.0 + exp(-shape*(x-center)))
  (1-alpha)*p_0 + alpha*p_1
end

function _logistic_multiple(x, shape, center, a::NTuple, b::NTuple)
  map(z->_logistic(x, z[1], z[2], shape, center), zip(a,b))
end

# Marginal spectral density:
function Sf(w,x,p)
  (_i, _v) = _logistic_multiple(x, p[16], p[15], (exp(p[5]), p[6]), (exp(p[7]), p[8]))
  extra_parm = _logistic(x, exp(p[17]), exp(p[18]), p[16], p[15])
  extra = 1.0+butter(sinpi(w), (extra_parm, exp(p[20]), exp(p[19])))
  scale_x = x <= p[15] ? 1.0 : butter(x-p[15], (1.0, exp(p[21]), exp(p[22])))
  scale_x*extra*(_i*sinpi(w)^2 + 1)^(-_v-1/2) 
end

function Cw(w,x,y,p)
  x == y && return 1.0
  (gp1x, gp1y) = (_logistic(z, exp(p[9]),  exp(p[12]), p[16], p[15]) for z in (x,y))
  (gp2x, gp2y) = (_logistic(z, exp(p[10]), exp(p[13]), p[16], p[15]) for z in (x,y))
  (gp3x, gp3y) = (_logistic(z, exp(p[11]), exp(p[14]), p[16], p[15]) for z in (x,y))
  nux  = x <= p[15] ? 0.5 : 0.75
  nuy  = y <= p[15] ? 0.5 : 0.75
  gamx = butter(sinpi(w), (gp1x, gp2x, gp3x))
  gamy = butter(sinpi(w), (gp1y, gp2y, gp3y))
  gamxy  = sqrt(0.5*(gamx+gamy))
  gamx_y = (gamx^0.25)*(gamy^0.25)
  return (gamx_y/gamxy)*mtn_cor(0.5*(nux+nuy), abs(x-y)/gamxy)
end

# Phase term:
phase(w,x,y,p) = exp(2.0*pi*im*p[23]*sinpi(w)*(x-y))
function phased1(w,x,y,p)
  2.0*pi*im*sinpi(w)*(x-y)*phase(w,x,y,p)
end

# Whole integrand, and manual derivative with respect to phase parameter:
integrand(w,x,y,p) = sqrt(Sf(w,x,p)*Sf(w,y,p))*Cw(w,x,y,p)*phase(w,x,y,p) 
integrand_dp1(w,x,y,p) = sqrt(Sf(w,x,p)*Sf(w,y,p))*Cw(w,x,y,p)*phased1(w,x,y,p)

# Integrand with no phase for autodiff:
integrand_np(w,x,y,p) = sqrt(Sf(w,x,p)*Sf(w,y,p))*Cw(w,x,y,p)

# Derivatives of the kernel:
dfuns =convert(Vector{Function}, 
               [(w,x,y,p)->coord_deriv(z->integrand_np(w,x,y,z), p, j)*phase(w,x,y,p)
                for j in 5:22])

# do the phase functions derivative by hand because ForwardDiff doesn't 
# support complex numbers....
push!(dfuns, integrand_dp1)

# The nugget function, added as a post-call. This derivative function gets 
# added to the list of kernel derivatives in the space-time domain.
nugfn(x,y,p)   = Float64(x==y)*p[24]^2 + Float64(x[1]==y[1])*p[25]^2
dnugfn1(x,y,p) = Float64(x==y)*2.0*p[24]
dnugfn2(x,y,p) = Float64(x[1]==y[1])*2.0*p[25]


D example/paperscripts/model_untransformed.jl => example/paperscripts/model_untransformed.jl +0 -88
@@ 1,88 0,0 @@

using SpecialFunctions, ForwardDiff, DelimitedFiles

# For the derivatives of the kernel:
function coord_deriv(f, p, j)
  ForwardDiff.derivative(z->f(vcat(p[1:(j-1)], z, p[(j+1):end])), p[j])
end

# The matern correlation for smoothness 1/2 and 3/2:
mtn_cor_12(x) = exp(-abs(x))
mtn_cor_32(x) = exp(-abs(x))*(1.0 + abs(x))
function mtn_cor(nu, x)
  abs(x)<1.0e-8 && return 1.0
  out  = (sqrt(2.0*nu)*x)^nu
  out *= besselk(nu, sqrt(2.0*nu)*x)
  out /= gamma(nu)*2.0^(nu-1.0)
  out
end

# The butterworth-like function. Several tweaks for numerical reasons to avoid
# breaking AD.
function butter(w,p) 
  iszero(w) && return p[1]  # tweak one: a branch for w=0.
  arg = abs2(w/p[2])^p[3]
  arg > 1.0e18 && return 0.0 # tweak two: a branch for the arg being too large.
  p[1]/(1.0 + arg)
end

function _logistic(x, p_0, p_1, shape, center)
  alpha = 1.0/(1.0 + exp(-shape*(x-center)))
  (1-alpha)*p_0 + alpha*p_1
end

function _logistic_multiple(x, shape, center, a::NTuple, b::NTuple)
  map(z->_logistic(x, z[1], z[2], shape, center), zip(a,b))
end

# Marginal spectral density:
function Sf(w,x,p)
  (_i, _v) = _logistic_multiple(x, p[16], p[15], (exp(p[5]), p[6]), (exp(p[7]), p[8]))
  extra_parm = _logistic(x, p[17], p[18], p[16], p[15])
  extra = 1.0+butter(sinpi(w), (extra_parm, p[20], p[19]))
  scale_x = x <= p[15] ? 1.0 : butter(x-p[15], (1.0, p[21], p[22]))
  scale_x*extra*(_i*sinpi(w)^2 + 1)^(-_v-1/2) 
end

function Cw(w,x,y,p)
  x == y && return 1.0
  (gp1x, gp1y) = (_logistic(z, exp(p[9]),  exp(p[12]), p[16], p[15]) for z in (x,y))
  (gp2x, gp2y) = (_logistic(z, p[10], p[13], p[16], p[15]) for z in (x,y))
  (gp3x, gp3y) = (_logistic(z, p[11], p[14], p[16], p[15]) for z in (x,y))
  nux  = x <= p[15] ? 0.5 : 0.75
  nuy  = y <= p[15] ? 0.5 : 0.75
  gamx = butter(sinpi(w), (gp1x, gp2x, gp3x))
  gamy = butter(sinpi(w), (gp1y, gp2y, gp3y))
  gamxy  = sqrt(0.5*(gamx+gamy))
  gamx_y = (gamx^0.25)*(gamy^0.25)
  return (gamx_y/gamxy)*mtn_cor(0.5*(nux+nuy), abs(x-y)/gamxy)
end

# Phase term:
phase(w,x,y,p) = exp(2.0*pi*im*p[23]*sinpi(w)*(x-y))
function phased1(w,x,y,p)
  2.0*pi*im*sinpi(w)*(x-y)*phase(w,x,y,p)
end

# Whole integrand, and manual derivative with respect to phase parameter:
integrand(w,x,y,p) = sqrt(Sf(w,x,p)*Sf(w,y,p))*Cw(w,x,y,p)*phase(w,x,y,p) 
integrand_dp1(w,x,y,p) = sqrt(Sf(w,x,p)*Sf(w,y,p))*Cw(w,x,y,p)*phased1(w,x,y,p)

# Integrand with no phase for autodiff:
integrand_np(w,x,y,p) = sqrt(Sf(w,x,p)*Sf(w,y,p))*Cw(w,x,y,p)

# Derivatives of the kernel:
dfuns =convert(Vector{Function}, 
               [(w,x,y,p)->coord_deriv(z->integrand_np(w,x,y,z), p, j)*phase(w,x,y,p)
                for j in 5:22])

# do the phase functions derivative by hand because ForwardDiff doesn't 
# support complex numbers....
push!(dfuns, integrand_dp1)

# The nugget function, added as a post-call. This derivative function gets 
# added to the list of kernel derivatives in the space-time domain.
nugfn(x,y,p)   = Float64(x==y)*p[24]^2 + Float64(x[1]==y[1])*p[25]^2
dnugfn1(x,y,p) = Float64(x==y)*2.0*p[24]
dnugfn2(x,y,p) = Float64(x[1]==y[1])*2.0*p[25]


D example/paperscripts/nsscale.jl => example/paperscripts/nsscale.jl +0 -35
@@ 1,35 0,0 @@

using LinearAlgebra

mutable struct NonstationaryScale{T,B,F,N} <: Function
  locs::NTuple{N,T}
  ix::NTuple{N,Int64}
end

function nwts(pt, locs::NTuple{N,T})::NTuple{N, Float64} where{N,T}
  tmp = map(z->exp(-abs(pt-z)/50), locs) 
  tmp ./ sum(tmp)
end

# Nonstationary scale function evaluate at space-time location
function (NS::NonstationaryScale{T,false,0,N})(x, y, p) where{T,N}
  out = 1.0
  for _x in (x,y)
    (time, space) = _x 
    wts = nwts(time, NS.locs)
    out *= sum(z->p[z[1]]*z[2], zip(NS.ix, wts))
  end
  out
end

# Efficient derivatives---but need the chain rule for two applications!
function (NS::NonstationaryScale{T,true,J,N})(x, y, p) where{T,J,N}
  (wx, wy) = map(pt->nwts(pt[1], NS.locs), (x,y))
  # Evaluations for x and y:
  (nx, ny) = map(w->sum(z->p[z[1]]*z[2], zip(NS.ix, w)), (wx,wy))
  # Derivatives for x and y:
  (dx, dy) = map(w->w[J]*2.0*p[NS.ix[J]], (wx,wy))
  # Return chain rule output:
  nx*dy + dx*ny
end


D example/paperscripts/untransformer.jl => example/paperscripts/untransformer.jl +0 -29
@@ 1,29 0,0 @@

function transf(v)
  [v[1],
   v[2],
   v[3],
   v[4],
   v[5],
   v[6],
   v[7],
   v[8],
   v[9],
   exp(v[10]),
   exp(v[11]),
   v[12],
   exp(v[13]),
   exp(v[14]),
   v[15],
   v[16],
   exp(v[17]),
   exp(v[18]),
   exp(v[19]),
   exp(v[20]),
   exp(v[21]),
   exp(v[22]),
   v[23],
   v[24],
   v[25]]
end


D example/paperscripts/writer.jl => example/paperscripts/writer.jl +0 -31
@@ 1,31 0,0 @@
using DelimitedFiles

function tablesave!(name, M::Matrix; rlabels=nothing, clabels=nothing)
  out = string.(M)
  if !isnothing(rlabels)
    out = hcat(string.(rlabels), out)
  end
  if !isnothing(clabels)
    is_row_nothing = isnothing(rlabels)
    rw1 = string.(clabels)
    if !is_row_nothing
      pushfirst!(rw1, "")
    end
    rw1[end] = rw1[end] * "\\\\"
    writedlm("header_"*name, reshape(rw1, 1, length(rw1)), '&')
  end
  out[:,end] .= out[:,end] .* repeat(["\\\\"], size(out,1))
  writedlm(name, out, '&')
end

function gnuplot_save_matrix!(name, M::Matrix{Float64}, row_pts, col_pts)
  out = zeros(size(M,1)+1, size(M,2)+1)
  out[1,2:end] .= col_pts
  out[2:end,1] .= row_pts
  out[2:end, 2:end] .= M
  writedlm(name, out, ',')
end

function gnuplot_save_vector!(name, M, pts)
  writedlm(name, hcat(pts, M), ',')
end

A example/withvecchia.jl => example/withvecchia.jl +57 -0
@@ 0,0 1,57 @@

using Vecchia, ForwardDiff
include("basic_example.jl")

# This is a struct that is itself callable. But the point here is that sometimes
# the parameters will be of different types, so this wraps a Dict{Type,
# HSKernel}. It makes it possible to squeeze out some redundant computations for
# AD gradients/Hessians.
const KWrapper = HSKernelADWrapper{Int64}(Dict{Type,HSKernel}())

# See the README for Vecchia.jl for more details, but this function is the only
# place in the Vecchia.jl codebase where the kernel is evaluated to assemble
# matrix blocks. So if you write your own method for this here, you can use your
# own exotic types for the kernel. 
#
# Here, you see that there's some extra logic about making sure that we have a
# kernel that accepts parameters of type eltype(params). 
function Vecchia.updatebuf!(buf, pts1, pts2, 
                            kfn::HSKernelADWrapper,
                            params; skipltri=false)
  # Check if the wrapper has a kernel of the right type:
  if !haskey(kfn.kernels, eltype(params))
    newkernel = HSKernel(params, xpts, integrand, timelen)
    kfn.kernels[eltype(params)] = newkernel
    kernel = newkernel
  else
    kernel = kfn.kernels[eltype(params)]
  end
  # Now check if the parameters agree. If they don't, rebuild the kernel:
  if !HalfSpectral.check_hskernel_params(kernel, params)
    newkernel = HSKernel(params, xpts, integrand, timelen)
    kfn.kernels[eltype(params)] = newkernel
    kernel = newkernel
  end
  # From here, we know that the kernel is the right type and that the parameters
  # agree. So we can really blaze through the rest of this:
  for k in 1:length(pts2)
    ptk = pts2[k]
    for j in 1:length(pts1)
      buf[j,k] = kernel(pts1[j], ptk)
    end
  end
  nothing
end

# The point of the fancy wrapper struct and the new method is that now the
# kernel composes nicely with the approximate Vecchia likelihood, which gives
# O(n) complexity approximations that are in general pretty accurate. Here's an
# nll, gradient, and Hessian that you can now plug in to your favorite
# optimizer. An expressive kernel, a linear-cost approximation, second order
# optimization....what's not to love?
const rdata = randn(length(pts))
const cfg   = Vecchia.kdtreeconfig(rdata, pts, 20, 3, KWrapper)
nll(p)  = Vecchia.nll(cfg, p)
grad(p) = ForwardDiff.gradient(smart_nll, p)
hess(p) = ForwardDiff.hessian(smart_nll, p)


M src/HalfSpectral.jl => src/HalfSpectral.jl +7 -114
@@ 1,125 1,18 @@

module HalfSpectral

  export timegrid, tftkernel
  using LinearAlgebra

  using LinearAlgebra, Statistics, FFTW
  export HSKernel, HSKernelADWrapper

  abstract type TimeGridResult end
  include("structstypes.jl")

  struct ZeroFunction <: Function
  end
  (zrf::ZeroFunction)(x,y,p) = 0.0
  include("radix2.jl")

  struct IdFunction <: Function
  end
  (idf::IdFunction)(x,p) = 1.0
  include("utils.jl")

  struct TimeGrid <: TimeGridResult
    Tv  :: Vector{Float64}
    Xt  :: Vector{Int64} 
    dt  :: Float64
    tol :: Float64
  end
  include("methods.jl")

  struct TimeGridFailure <: TimeGridResult 
    Tv  ::Vector{Float64}
    tol ::Float64
  end

  mutable struct TimeFFTKernel{T,F<:Function,N,G<:Function,H<:Function,M} <: Function
    V  :: TimeGrid
    Fn :: F
    D  :: Dict{Tuple{Int64, T, T}, Float64}
    p  :: NTuple{N, Float64}
    X  :: AbstractVector{T}
    addfn :: G
    mulfn :: H
    nobuild_ix :: NTuple{M, Int64}
    padsz::Int64
    longmemory::Bool
  end

  function timegrid(X; tol=0.075, warn_tol=2*tol)::TimeGridResult
    dt, dtmax  = extrema(abs.(diff(X)))
    d_reldiff  = rem(dtmax-dt, dt)/dt
    if d_reldiff <= warn_tol
      d_reldiff > tol && (@info "Only the acceptable tol was acheived.")
      Xd     = (X .- X[1])./dt
      Xd_int = Int64.(round.(Xd))
      minimum(diff(Xd_int)) > 0 || throw(error("time grid construction failed."))
      return TimeGrid(X, Xd_int, dt, d_reldiff)
    end
    return TimeGridFailure(X, d_reldiff)
  end

  function tftkernel(integrand::F, tg::TimeGrid, X::AbstractVector{T}, p;
                     addfn::G=ZeroFunction(), 
                     mulfn::H=IdFunction(),
                     nob=NTuple{0, Int64}(),
                     padsz::Int64=7,
                     longmemory=false) where{T,F,G,H}
    dict = Dict{Tuple{Int64, T, T}, Float64}()
    lp   = length(p)
    nnb  = length(nob)
    out  = TimeFFTKernel{T,F,lp,G,H,nnb}(tg, integrand, dict, tuple(p...), 
                                         X, addfn, mulfn, nob, padsz, longmemory)
    build!(out, p)
    return out
  end

  function build!(K::TimeFFTKernel{T,F,N,G,H,M}, p) where{T,F,N,G,H,M}
    ftn   = nextprod([2,3,5,7], K.padsz*K.V.Xt[end])
    (K.longmemory && iseven(ftn)) && (ftn += 1) 
    wgd   = collect(range(-0.5, 0.5, length=ftn+1)[1:ftn])
    if K.longmemory
      @assert !(in(0.0, wgd)) "debug, this should be impossible."
    end
    ix    = sort(unique(map(z->abs(z[2]-z[1]), Iterators.product(K.V.Xt, K.V.Xt))))
    plan  = plan_ifft(wgd)
    for (x,y) in Iterators.product(K.X, K.X)
      covxy = real(plan*(fftshift([K.Fn(w,x,y,p) for w in wgd])))[ix.+1]
      newv  = Dict(zip(zip(ix, fill(x, length(ix)), fill(y, length(ix))), covxy))
      merge!(K.D, newv)
    end
    K.p = convert(NTuple{N, Float64}, Tuple(p))
    return nothing
  end

  # Avoid the runtime cost of checking the parameters AND the post-function:
  function (K::TimeFFTKernel{T,F,N,ZeroFunction,IdFunction,0})(x, y) where{T,F,N} 
    opt1 = (abs(x[1]-y[1]), x[2], y[2])
    haskey(K.D, opt1) && return K.D[opt1]
    throw(error("No computed covariance for these two points: $x, $y"))
  end

  # Avoid the runtime cost of checking the parameters, but call the post
  # computation function:
  function (K::TimeFFTKernel{T,F,N,G,H,M})(x, y) where{T,F,N,G,H,M} 
    opt1 = (abs(x[1]-y[1]), x[2], y[2])
    haskey(K.D, opt1) && return K.mulfn(x,y,K.p)*K.D[opt1] + K.addfn(x,y,K.p)
    throw(error("No computed covariance for these two points: $x, $y"))
  end

  # Safer checked version:
  function (K::TimeFFTKernel{T,F,N,M})(x, y, p) where{T,F,N,M} 
    update_p_flag = false
    for (j, (Kpj, pj)) in enumerate(zip(K.p, p))
      if Kpj != pj 
        if !in(j, K.nobuild_ix)
          build!(K, p) 
          update_p_flag = false
          break
        else
          update_p_flag = true
        end
      end
    end
    if update_p_flag
      K.p = tuple(p...)
    end
    K(x, y)
  end
  include("interp.jl")

end


A src/methods.jl => src/methods.jl +45 -0
@@ 0,0 1,45 @@

# TODO (cg 2022/02/24 11:50): Make sure that this doesn't allocate. Also, get
# rid of this if thing about x[2:end] being a singleton or not.
function (K::HSKernel{T,Q})(x, y) where{T,Q}
  (tx, ty) = (x[1], y[1])
  if length(x) == 2
    (_x, _y) = (x[2], y[2])
  else
    (_x, _y) = (x[2:end], y[2:end])
  end
  key = (_x, _y)
  haskey(K.store, key) ||throw(error("No key for location pair $key."))
  ix = round(Int64, abs(tx-ty)+1)
  K.store[key][ix]
end

function (K::HSKernel{T,Q})(tdif::Int64, x, y) where{T,Q}
  key = (x, y)
  haskey(K.store, key) ||throw(error("No key for location pair $key."))
  K.store[key][tdif+1]
end

function (K::HSKernel{T,Q})(tdif::Float64, x, y) where{T,Q}
  key = (x, y)
  haskey(K.store, key) ||throw(error("No key for location pair $key."))
  K.store[key][round(Int64, tdif)+1]
end

# Hopefully in this format the compiler will understand that it should hoist the
# check outside of a hot loop, for example.
function (K::HSKernel{T,Q})(x, y, p) where{T,Q}
  check_hskernel_params(K, p)
  K(x,y)
end

function Base.display(K::HSKernel)
  println("Half-spectral kernel with:")
  println("  - parameters $(round.(K.params, digits=3))")
end

#function updatewrapper(W::HSKernelWrapper{Q}, hsk::HSKernel{T,Q}) where{T,Q}
#  W.kernels[T] = hsk
#end



A src/radix2.jl => src/radix2.jl +83 -0
@@ 0,0 1,83 @@

# Very midly changed derived product from:
#   https://github.com/JuliaApproximation/FastTransforms.jl.
#
# copyright Richard Mikael Slevinsky and other contributors
# originally distributed under MIT "Expat" license

function generic_fft_pow2!(x::Vector{T}) where{T}
  n,big2=length(x),2one(T)
  nn,j=n÷2,1
  for i=1:2:n-1
    if j>i
      x[j], x[i] = x[i], x[j]
      x[j+1], x[i+1] = x[i+1], x[j+1]
    end
    m = nn
    while m ≥ 2 && j > m
      j -= m
      m = m÷2
    end
    j += m
  end
  logn = 2
  while logn < n
    θ=-big2/logn
    wtemp = sinpi(θ/2)
    wpr, wpi = -2wtemp^2, sinpi(θ)
    wr, wi = one(T), zero(T)
    for m=1:2:logn-1
      for i=m:2logn:n
        j=i+logn
        mixr, mixi = wr*x[j]-wi*x[j+1], wr*x[j+1]+wi*x[j]
        x[j], x[j+1] = x[i]-mixr, x[i+1]-mixi
        x[i], x[i+1] = x[i]+mixr, x[i+1]+mixi
      end
      wr = (wtemp=wr)*wpr-wi*wpi+wr
      wi = wi*wpr+wtemp*wpi+wi
    end
    logn = logn << 1
  end
  return x
end

function rradix2(x)
  @assert isinteger(log2(length(x))) "This only applies to vectors of length 2^k."
  _x = zeros(eltype(x), 2*length(x))
  ix  = 1
  @inbounds @simd for j in 1:2:length(_x)
    _x[j] = x[ix] 
    ix += 1
  end
  generic_fft_pow2!(_x)
  _x[1:2:end]
end

function irradix2(x)
  @assert isinteger(log2(length(x))) "This only applies to vectors of length 2^k."
  _x = zeros(eltype(x), 2*length(x))
  ix  = 1
  @inbounds @simd for j in 1:2:length(_x)
    _x[j] = x[ix] 
    ix += 1
  end
  generic_fft_pow2!(_x)
  _x[1:2:end]./length(x)
end

# General purpose.
function radix2(x, iflag)
  @assert isinteger(log2(length(x))) "This only applies to vectors of length 2^k."
  _x = zeros(eltype(x), 2*length(x))
  ix  = 1
  @inbounds @simd for j in 1:2:length(_x)
    _x[j] = x[ix] 
    ix += 1
  end
  generic_fft_pow2!(_x)
  if iflag
    return complex.(_x[1:2:end], _x[2:2:end])./length(x)
  else
    return complex.(_x[1:2:end], _x[2:2:end])
  end
end

A src/structstypes.jl => src/structstypes.jl +35 -0
@@ 0,0 1,35 @@

# (H)alf-(S)pectral Kernel. Defaults to long memory-compatible gridding.
# Unlike the original implementation, I'm going to leave the nugget and
# nonstationary scale outside this kernel, or if somebody wants they'll have to
# put it into the integrand function.
#
# As another simplification, I'm just going to assume that the data put in is
# equally spaced and with unit spacing and that dt=1. So all of that rescaling
# is just going to go into the parameters.
#
# In general, this version is built with NOT updating in mind, instead just
# building a new kernel.
struct HSKernel{T,Q}
  params::Vector{T}
  store::Dict{Tuple{Q, Q}, Vector{T}}
end

# Integrand needs to be of the signature (f, x, y, params) ->
# C_f(x,y)*sqrt(S_x(f)*S_y(f))*... and stuff.
function HSKernel(params, locs, integrand, timelen; padsize=7, use_sinpi=false)
  ftsz = nextprod([2], max(512, padsize*timelen))
  sfgd = _fftshift(collect(range(-0.5, 0.5, length=ftsz+1)[1:ftsz]))
  lprs = Iterators.product(locs, locs)
  Dv   = map(xy->_subdict(params, xy, sfgd, integrand, timelen, use_sinpi), lprs)
  HSKernel(params, Dict(Dv))
end

struct HSKernelADWrapper{Q}
  kernels::Dict{Type, HSKernel}
end

#function HSKernelADWrapper(Q::Type)
#  HSKernelWrapper{Q}(Dict{Type, HSKernel}())
#end


A src/utils.jl => src/utils.jl +26 -0
@@ 0,0 1,26 @@

_fftshift(v)  = circshift(v, div(length(v),   2))
_ifftshift(v) = circshift(v, div(length(v)+1, 2))

function _subdict(params, xy, sfgd, integrand, timelen, use_sinpi)
  (x,y) = xy
  if use_sinpi 
    integrand_grid = [integrand(sinpi(f),x,y,params) for f in sfgd]
  else
    integrand_grid = [integrand(f,x,y,params) for f in sfgd]
  end
  covxy = irradix2(integrand_grid)[1:(timelen+1)]
  xy => covxy
end

# Put this at the top of an assembly block and then you don't have to check the
# parameters with every call.
function check_hskernel_params(K::HSKernel{T,Q}, p) where{T,Q}
  eltype(p) == T || return false
  length(p) == length(K.params) || return false
  @inbounds for j in eachindex(K.params)
    K.params[j] == p[j] || return false
  end
  true
end


A test/Manifest.toml => test/Manifest.toml +280 -0
@@ 0,0 1,280 @@
# This file is machine-generated - editing it directly is not advised

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A test/Project.toml => test/Project.toml +7 -0
@@ 0,0 1,7 @@
[deps]
FiniteDifferences = "26cc04aa-876d-5657-8c51-4c34ba976000"
ForwardDiff = "f6369f11-7733-5829-9624-2563aa707210"
LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
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Test = "8dfed614-e22c-5e08-85e1-65c5234f0b40"

A test/runtests.jl => test/runtests.jl +43 -0
@@ 0,0 1,43 @@

using Test, HalfSpectral, StaticArrays, LinearAlgebra, QuadGK
using FiniteDifferences, ForwardDiff

# Simple SDF and a simple nontrivial coherence, which make an integrand:
sdf(f,x,p) = p[1]*(p[2]^2 + f^2)^(-p[3] - 1/2)
coh(f,x,y,p) = exp(-p[4]*(1+(p[5]*f)^2)*abs(x-y))
integrand(f,x,y,p) = coh(f,x,y,p)*sqrt(sdf(f,x,p)*sdf(f,y,p))

const pts = reduce(vcat, map(x->[@SVector [t, x] for t in 1:100], 1:5))
const params = ones(5)
const kernel = HalfSpectral.HSKernel(params, 1:5, integrand, 100)
const x = randn(length(pts))

function qform(p)
  Kf = cholesky!(Symmetric([kernel(x,y) for x in pts, y in pts]))
  sum(_x->_x^2, Kf.U'\x)
end

# test 1: integral accuracy.
@testset "accuracy" begin
  for (x1,x2) in ((1,1), (1,2), (3,5))
    for k in 1:5:30
      qk = quadgk(f->cos(2*pi*(k-1)*f)*integrand(f,x1,x2,params), -1/2, 1/2)[1]
      @test isapprox(qk, kernel.store[(x1,x2)][k], rtol=1e-2)
    end
  end
end

# test 2: positive definite.
@testset "positive definite" begin
  K = Symmetric([kernel(x,y) for x in pts, y in pts])
  @test isposdef(K)
end

# test3: Accurate AD:
@testset "autodiff" begin
  fdd = central_fdm(10,1)
  fdg = FiniteDifferences.grad(fdd, qform, params)[1]
  adg = ForwardDiff.gradient(qform, params)
  @test isapprox(fdg, adg, atol=1e-8)
end