Mauna Loa time series example
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In this notebook, we apply Gaussian process regression to the Mauna Loa CO₂ dataset. This showcases a rich combination of kernels, and how to handle and optimize all their parameters.
Setup
We make use of the following packages:
using CSV, DataFrames # data loading
using AbstractGPs # exact GP regression
using ParameterHandling # for nested and constrained parameters
using Optim # optimization
using Zygote # auto-diff gradient computation
using Plots # visualisationLet's load and visualize the dataset.
The let block creates a new scope, so any utility variables we define in here won't leak outside. This is particularly helpful to keep notebooks tidy! The return value of the block is given by its last expression.
(xtrain, ytrain), (xtest, ytest) = let
data = CSV.read(joinpath("/home/runner/work/AbstractGPs.jl/AbstractGPs.jl/examples/1-mauna-loa", "CO2_data.csv"), Tables.matrix; header=0)
year = data[:, 1]
co2 = data[:, 2]
# We split the data into training and testing set:
idx_train = year .< 2004
idx_test = .!idx_train
(year[idx_train], co2[idx_train]), (year[idx_test], co2[idx_test]) # block's return value
end
# The utility variables such as `idx_train` and `idx_test` are not available outside the `let` scope
function plotdata()
plot(; xlabel="year", ylabel="CO₂ [ppm]", legend=:bottomright)
scatter!(xtrain, ytrain; label="training data", ms=2, markerstrokewidth=0)
return scatter!(xtest, ytest; label="test data", ms=2, markerstrokewidth=0)
end
plotdata()Prior
We will model this dataset using a sum of several kernels which describe
- smooth trend: squared exponential kernel with long lengthscale;
- seasonal component: periodic covariance function with period of one year, multiplied with a squared exponential kernel to allow decay away from exact periodicity;
- medium-term irregularities: rational quadratic kernel;
- noise terms: squared exponential kernel with short lengthscale and uncorrelated observation noise.
For more details, see Rasmussen & Williams (2005), chapter 5.
We will use ParameterHandling.jl for handling the (hyper)parameters of our model. It provides functions such as positive with which we can put constraints on the hyperparameters, and allows us to represent all required parameters as a nested NamedTuple:
# initial values to match http://stor-i.github.io/GaussianProcesses.jl/latest/mauna_loa/
θ_init = (;
se_long = (;
σ = positive(exp(4.0)),
ℓ = positive(exp(4.0)),
),
seasonal = (;
# product kernels only need a single overall signal variance
per = (;
ℓ = positive(exp(0.0)), # relative to period!
p = fixed(1.0), # 1 year, do not optimize over
),
se = (;
σ = positive(exp(1.0)),
ℓ = positive(exp(4.0)),
),
),
rq = (;
σ = positive(exp(0.0)),
ℓ = positive(exp(0.0)),
α = positive(exp(-1.0)),
),
se_short = (;
σ = positive(exp(-2.0)),
ℓ = positive(exp(-2.0)),
),
noise_scale = positive(exp(-2.0)),
)We define a couple of helper functions to simplify the kernel construction:
SE(θ) = θ.σ^2 * with_lengthscale(SqExponentialKernel(), θ.ℓ)
Per(θ) = with_lengthscale(PeriodicKernel(; r=[θ.ℓ / 2]), θ.p) # NOTE- discrepancy with GaussianProcesses.jl
RQ(θ) = θ.σ^2 * with_lengthscale(RationalQuadraticKernel(; α=θ.α), θ.ℓ)This allows us to write a function that, given the nested tuple of parameter values, constructs the GP prior:
function build_gp_prior(θ)
k_smooth_trend = SE(θ.se_long)
k_seasonality = Per(θ.seasonal.per) * SE(θ.seasonal.se)
k_medium_term_irregularities = RQ(θ.rq)
k_noise_terms = SE(θ.se_short) + θ.noise_scale^2 * WhiteKernel()
kernel = k_smooth_trend + k_seasonality + k_medium_term_irregularities + k_noise_terms
return GP(kernel) # [`ZeroMean`](@ref) mean function by default
endPosterior
To construct the posterior, we need to first build a FiniteGP, which represents the infinite-dimensional GP at a finite number of input features:
function build_finite_gp(θ)
f = build_gp_prior(θ)
return f(xtrain)
endIn this notebook, we already included observation noise through the WhiteKernel as part of the GP prior covariance in build_gp_prior. We therefore call f(xtrain) which implies zero (additional) observation noise.
Alternatively, we could have omitted the θ.noise_scale^2 * WhiteKernel() term and instead passed the noise variance as a second argument to the GP call in build_finite_gp, f(xtrain, θ.noise_scale^2).
These two approaches have slightly different semantics: In the first one, the WhiteKernel contributes non-zero variance to the [i, j] element of the covariance matrix of the FiniteGP if xtrain[i] == xtrain[j] (based on the values of the features). In the second one, the observation noise variance passed to FiniteGP only contributes to the diagonal elements of the covariance matrix, i.e. for i == j.
Moreover, the variance (uncertainty) of the posterior predictions includes the variance from the WhiteKernel, but does not include the variance of the observation noise passed to the FiniteGP. To include the observation noise in posterior predictions from the second approach, call fpost_opt(xtest, noise_scale^2).
For most use-cases and if in any doubt, we recommend that you pass in observation noise to the FiniteGP, and omit the explicit WhiteKernel. This is slightly faster (no need to check xtrain[i] == xtrain[j] for all pairs i, j), and WhiteKernel will not give stable gradients if you wish to compute the gradient of the log marginal likelihood logpdf(f(x), y) w.r.t. x.
We obtain the posterior, conditioned on the (finite) observations, by calling posterior:
function build_posterior_gp(θ)
fx = build_finite_gp(θ)
return posterior(fx, ytrain)
endNow we can put it all together to obtain a PosteriorGP. The call to ParameterHandling.value is required to replace the constraints (such as positive in our case) with concrete numbers:
fpost_init = build_posterior_gp(ParameterHandling.value(θ_init))AbstractGPs.PosteriorGP{AbstractGPs.GP{AbstractGPs.ZeroMean{Float64}, KernelFunctions.KernelSum{Tuple{KernelFunctions.ScaledKernel{KernelFunctions.TransformedKernel{KernelFunctions.SqExponentialKernel{Distances.Euclidean}, KernelFunctions.ScaleTransform{Float64}}, Float64}, KernelFunctions.KernelProduct{Tuple{KernelFunctions.TransformedKernel{KernelFunctions.PeriodicKernel{Float64}, KernelFunctions.ScaleTransform{Float64}}, KernelFunctions.ScaledKernel{KernelFunctions.TransformedKernel{KernelFunctions.SqExponentialKernel{Distances.Euclidean}, KernelFunctions.ScaleTransform{Float64}}, Float64}}}, KernelFunctions.ScaledKernel{KernelFunctions.TransformedKernel{KernelFunctions.RationalQuadraticKernel{Float64, Distances.Euclidean}, KernelFunctions.ScaleTransform{Float64}}, Float64}, KernelFunctions.ScaledKernel{KernelFunctions.TransformedKernel{KernelFunctions.SqExponentialKernel{Distances.Euclidean}, KernelFunctions.ScaleTransform{Float64}}, Float64}, KernelFunctions.ScaledKernel{KernelFunctions.WhiteKernel, Float64}}}}, @NamedTuple{α::Vector{Float64}, C::LinearAlgebra.Cholesky{Float64, Matrix{Float64}}, x::Vector{Float64}, δ::Vector{Float64}}}(AbstractGPs.GP{AbstractGPs.ZeroMean{Float64}, KernelFunctions.KernelSum{Tuple{KernelFunctions.ScaledKernel{KernelFunctions.TransformedKernel{KernelFunctions.SqExponentialKernel{Distances.Euclidean}, KernelFunctions.ScaleTransform{Float64}}, Float64}, KernelFunctions.KernelProduct{Tuple{KernelFunctions.TransformedKernel{KernelFunctions.PeriodicKernel{Float64}, KernelFunctions.ScaleTransform{Float64}}, KernelFunctions.ScaledKernel{KernelFunctions.TransformedKernel{KernelFunctions.SqExponentialKernel{Distances.Euclidean}, KernelFunctions.ScaleTransform{Float64}}, Float64}}}, KernelFunctions.ScaledKernel{KernelFunctions.TransformedKernel{KernelFunctions.RationalQuadraticKernel{Float64, Distances.Euclidean}, KernelFunctions.ScaleTransform{Float64}}, Float64}, KernelFunctions.ScaledKernel{KernelFunctions.TransformedKernel{KernelFunctions.SqExponentialKernel{Distances.Euclidean}, KernelFunctions.ScaleTransform{Float64}}, Float64}, KernelFunctions.ScaledKernel{KernelFunctions.WhiteKernel, Float64}}}}(AbstractGPs.ZeroMean{Float64}(), Sum of 5 kernels:
Squared Exponential Kernel (metric = Distances.Euclidean(0.0))
- Scale Transform (s = 0.018315638888734182)
- σ² = 2980.957987041728
Product of 2 kernels:
Periodic Kernel, length(r) = 1
- Scale Transform (s = 1.0)
Squared Exponential Kernel (metric = Distances.Euclidean(0.0))
- Scale Transform (s = 0.018315638888734182)
- σ² = 7.3890560989306495
Rational Quadratic Kernel (α = 0.36787944117144233, metric = Distances.Euclidean(0.0))
- Scale Transform (s = 1.0)
- σ² = 1.0
Squared Exponential Kernel (metric = Distances.Euclidean(0.0))
- Scale Transform (s = 7.3890560989306495)
- σ² = 0.018315638888734182
White Kernel
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We use the following function to plot a GP f on a specific range, using the AbstractGPs plotting recipes. By setting ribbon_scale=2 we visualize the uncertainty band with $\pm 2$ (instead of the default $\pm 1$) standard deviations.
plot_gp!(f; label) = plot!(f(1920:0.2:2030); ribbon_scale=2, linewidth=1, label)
plotdata()
plot_gp!(fpost_init; label="posterior f(⋅)")A reasonable fit to the data, but poor extrapolation away from the observations!
Hyperparameter Optimization
To improve the fit, we want to maximize the (log) marginal likelihood with respect to the hyperparameters. Optim.jl expects to minimize a loss, so we define it as the negative log marginal likelihood:
function loss(θ)
fx = build_finite_gp(θ)
lml = logpdf(fx, ytrain) # this computes the log marginal likelihood
return -lml
endIn the future, we are planning to provide the optimize_loss utility function as part of JuliaGaussianProcesses – for now, we just define it inline.
The L-BFGS parameters were chosen because they seem to work well empirically. You could also try with the defaults.
default_optimizer = LBFGS(;
alphaguess=Optim.LineSearches.InitialStatic(; scaled=true),
linesearch=Optim.LineSearches.BackTracking(),
)
function optimize_loss(loss, θ_init; optimizer=default_optimizer, maxiter=1_000)
options = Optim.Options(; iterations=maxiter, show_trace=true)
θ_flat_init, unflatten = ParameterHandling.value_flatten(θ_init)
loss_packed = loss ∘ unflatten
# https://julianlsolvers.github.io/Optim.jl/stable/#user/tipsandtricks/#avoid-repeating-computations
function fg!(F, G, x)
if F !== nothing && G !== nothing
val, grad = Zygote.withgradient(loss_packed, x)
G .= only(grad)
return val
elseif G !== nothing
grad = Zygote.gradient(loss_packed, x)
G .= only(grad)
return nothing
elseif F !== nothing
return loss_packed(x)
end
end
result = optimize(Optim.only_fg!(fg!), θ_flat_init, optimizer, options; inplace=false)
return unflatten(result.minimizer), result
endWe now run the optimization:
θ_opt, opt_result = optimize_loss(loss, θ_init)
opt_result * Status: success
* Candidate solution
Final objective value: 1.152862e+02
* Found with
Algorithm: L-BFGS
* Convergence measures
|x - x'| = 0.00e+00 ≤ 0.0e+00
|x - x'|/|x'| = 0.00e+00 ≤ 0.0e+00
|f(x) - f(x')| = 0.00e+00 ≤ 0.0e+00
|f(x) - f(x')|/|f(x')| = 0.00e+00 ≤ 0.0e+00
|g(x)| = 3.78e-02 ≰ 1.0e-08
* Work counters
Seconds run: 26 (vs limit Inf)
Iterations: 61
f(x) calls: 95
∇f(x) calls: 61
The final value of the log marginal likelihood is:
-opt_result.minimum-115.28622096603124To avoid bad local optima, we could (and should) have carried out several random restarts with different initial values for the hyperparameters, and then picked the result with the highest marginal likelihood. We omit this for simplicity. For more details on how to fit GPs in practice, check out A Practical Guide to Gaussian Processes.
Let's construct the posterior GP with the optimized hyperparameters:
fpost_opt = build_posterior_gp(ParameterHandling.value(θ_opt))This is the kernel with the point-estimated hyperparameters:
fpost_opt.prior.kernelSum of 5 kernels:
Squared Exponential Kernel (metric = Distances.Euclidean(0.0))
- Scale Transform (s = 0.006456502042458794)
- σ² = 191273.47839039602
Product of 2 kernels:
Periodic Kernel, length(r) = 1
- Scale Transform (s = 1.0)
Squared Exponential Kernel (metric = Distances.Euclidean(0.0))
- Scale Transform (s = 0.012139119857609751)
- σ² = 5.99003503589132
Rational Quadratic Kernel (α = 0.0007411653885252467, metric = Distances.Euclidean(0.0))
- Scale Transform (s = 0.04622167413324385)
- σ² = 178.92175101210103
Squared Exponential Kernel (metric = Distances.Euclidean(0.0))
- Scale Transform (s = 8.752621585494843)
- σ² = 0.028315070269881937
White Kernel
- σ² = 0.03576077215040964Let's print the optimized values of the hyperparameters in a more helpful format:
This is another utility function we would eventually like to move out of this notebook:
using Printf
show_params(nt::Union{Dict,NamedTuple}) = String(take!(show_params(IOBuffer(), nt)))
function show_params(io, nt::Union{Dict,NamedTuple}, indent::Int=0)
for (s, v) in pairs(nt)
if v isa Union{Dict,NamedTuple}
println(io, " "^indent, s, ":")
show_params(io, v, indent + 4)
else
println(io, " "^indent, s, " = ", @sprintf("%.3f", v))
end
end
return io
end
print(show_params(ParameterHandling.value(θ_opt)))se_long:
σ = 437.348
ℓ = 154.883
seasonal:
per:
ℓ = 1.456
p = 1.000
se:
σ = 2.447
ℓ = 82.378
rq:
σ = 13.376
ℓ = 21.635
α = 0.001
se_short:
σ = 0.168
ℓ = 0.114
noise_scale = 0.189
And, finally, we can visualize our optimized posterior GP:
plotdata()
plot_gp!(fpost_opt; label="optimized posterior f(⋅)")Package and system information
Package information (click to expand)
Status `~/work/AbstractGPs.jl/AbstractGPs.jl/examples/1-mauna-loa/Project.toml` [99985d1d] AbstractGPs v0.5.21 `/home/runner/work/AbstractGPs.jl/AbstractGPs.jl#8969233` [336ed68f] CSV v0.10.13 [a93c6f00] DataFrames v1.6.1 [98b081ad] Literate v2.16.1 [429524aa] Optim v1.9.2 [2412ca09] ParameterHandling v0.5.0 [91a5bcdd] Plots v1.40.2 [e88e6eb3] Zygote v0.6.69To reproduce this notebook's package environment, you can download the full Manifest.toml.
System information (click to expand)
Julia Version 1.10.2 Commit bd47eca2c8a (2024-03-01 10:14 UTC) Build Info: Official https://julialang.org/ release Platform Info: OS: Linux (x86_64-linux-gnu) CPU: 4 × AMD EPYC 7763 64-Core Processor WORD_SIZE: 64 LIBM: libopenlibm LLVM: libLLVM-15.0.7 (ORCJIT, znver3) Threads: 1 default, 0 interactive, 1 GC (on 4 virtual cores) Environment: JULIA_DEBUG = Documenter JULIA_LOAD_PATH = :/home/runner/.julia/packages/JuliaGPsDocs/7M86H/src
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