Update readme + example (#131)

* Improve readme + example

* Remove outdated section of readme

* Fix link in readme

* Fix typo in readme

README.md

@@ -15,20 +15,11 @@ TemporalGPs.jl is registered, so simply type the following at the REPL:
 ```
 While you can install TemporalGPs without AbstractGPs and KernelFunctions, in practice the latter are needed for all common tasks in TemporalGPs.
 
-## Note !!!
-
-This package is currently not guaranteed to work with all current versions of dependencies. If something is not working on the current release of TemporalGPs,
-please try out v0.6.7, which pins some dependencies in order to circumvent some of the problems. You can do so by typing instead:
-```julia
-] add AbstractGPs KernelFunctions TemporalGPs@0.6.7
-```
-Please report an issue if this work-around fails.
-
 # Example Usage
 
 Most examples can be found in the [examples](https://github.com/JuliaGaussianProcesses/TemporalGPs.jl/tree/master/examples) directory. In particular see the associated [README](https://github.com/JuliaGaussianProcesses/TemporalGPs.jl/tree/master/examples/README.md).
 
-This is a small problem by TemporalGPs' standard. See timing results below for expected performance on larger problems.
+The following is a small problem by TemporalGPs' standard. See timing results below for expected performance on larger problems.
 
 ```julia
 using AbstractGPs, KernelFunctions, TemporalGPs
@@ -66,72 +57,11 @@ logpdf(f_post(x), y)
 ## Learning kernel parameters with [Optim.jl](https://github.com/JuliaNLSolvers/Optim.jl), [ParameterHandling.jl](https://github.com/invenia/ParameterHandling.jl), and [Mooncake.jl](https://github.com/compintell/Mooncake.jl/)
 
 TemporalGPs.jl doesn't provide scikit-learn-like functionality to train your model (find good kernel parameter settings).
-Instead, we offer the functionality needed to easily implement your own training functionality using standard tools from the Julia ecosystem, as shown below.
-```julia
-# Load our GP-related packages.
-using AbstractGPs
-using KernelFunctions
-using TemporalGPs
-
-# Load standard packages from the Julia ecosystem
-using Optim # Standard optimisation algorithms.
-using ParameterHandling # Helper functionality for dealing with model parameters.
-using Mooncake # Algorithmic Differentiation
-
-using ParameterHandling: flatten
-
-# Declare model parameters using `ParameterHandling.jl` types.
-flat_initial_params, unflatten = flatten((
-    var_kernel = positive(0.6),
-    λ = positive(2.5),
-    var_noise = positive(0.1),
-))
-
-# Construct a function to unpack flattened parameters and pull out the raw values.
-unpack = ParameterHandling.value ∘ unflatten
-params = unpack(flat_initial_params)
-
-function build_gp(params)
-    f_naive = GP(params.var_kernel * Matern52Kernel() ∘ ScaleTransform(params.λ))
-    return to_sde(f_naive, SArrayStorage(Float64))
-end
-
-# Generate some synthetic data from the prior.
-const x = RegularSpacing(0.0, 0.1, 10_000)
-const y = rand(build_gp(params)(x, params.var_noise))
-
-# Specify an objective function for Optim to minimise in terms of x and y.
-# We choose the usual negative log marginal likelihood (NLML).
-function objective(params)
-    f = build_gp(params)
-    return -logpdf(f(x, params.var_noise), y)
-end
-
-# Check that the objective function works:
-objective(params)
-
-# Optimise using Optim. This optimiser often works fairly well in practice,
-# but it's not going to be the best choice in all situations. Consult
-# Optim.jl for more info on available optimisers and their properties.
-training_results = Optim.optimize(
-    objective ∘ unpack,
-    θ -> only(Mooncake.gradient(objective ∘ unpack, θ)),
-    flat_initial_params + randn(3), # Add some noise to make learning non-trivial
-    BFGS(
-        alphaguess = Optim.LineSearches.InitialStatic(scaled=true),
-        linesearch = Optim.LineSearches.BackTracking(),
-    ),
-    Optim.Options(show_trace = true);
-    inplace=false,
-)
-
-# Extracting the final values of the parameters.
-# Should be close to truth.
-final_params = unpack(training_results.minimizer)
-```
-Once you've learned the parameters, you can use `posterior`, `marginals`, and `rand` to make posterior-predictions with the optimal parameters.
+Instead, we offer the functionality needed to easily implement your own training functionality using standard tools from the Julia ecosystem.
+See [exact_time_learning.jl](https://github.com/JuliaGaussianProcesses/TemporalGPs.jl/blob/master/examples/exact_time_learning.jl).
 
-In the above example we optimised the parameters, but we could just as easily have utilised e.g. [AdvancedHMC.jl](https://github.com/TuringLang/AdvancedHMC.jl) in conjunction with a prior over the parameters to perform approximate Bayesian inference in them -- indeed, [this is often a very good idea](http://proceedings.mlr.press/v118/lalchand20a/lalchand20a.pdf). We leave this as an exercise for the interested user (see e.g. the examples in [Stheno.jl](https://github.com/willtebbutt/Stheno.jl/) for inspiration).
+In this example we optimised the parameters, but we could just as easily have utilised e.g. [AdvancedHMC.jl](https://github.com/TuringLang/AdvancedHMC.jl) in conjunction with a prior over the parameters to perform approximate Bayesian inference in them -- indeed, [this is often a very good idea](http://proceedings.mlr.press/v118/lalchand20a/lalchand20a.pdf).
+We leave this as an exercise for the interested user (see e.g. the examples in [Stheno.jl](https://github.com/willtebbutt/Stheno.jl/) for inspiration).
 
 Moreover, it should be possible to plug this into probabilistic programming framework such as `Turing` and `Soss` with minimal effort, since `f(x, params.var_noise)` is a plain old `Distributions.MultivariateDistribution`.
 
@@ -155,20 +85,6 @@ This tells TemporalGPs that you want all parameters of `f` and anything derived
 Gradient computations use Mooncake. Custom adjoints have been implemented to achieve this level of performance.
 
 
-
-# On-going Work
-
-- Optimisation
-    + in-place implementation with `ArrayStorage` to reduce allocations
-    + input data types for posterior inference - the `RegularSpacing` type is great for expressing that the inputs are regularly spaced. A carefully constructed data type to let the user build regularly-spaced data when working with posteriors would also be very beneficial.
-- Interfacing with other packages
-    + When [Stheno.jl](https://github.com/willtebbutt/Stheno.jl/) moves over to the AbstractGPs interface, it should be possible to get some interesting process decomposition functionality in this package.
-- Approximate inference under non-Gaussian observation models
-
-If you're interested in helping out with this stuff, please get in touch by opening an issue, commenting on an open one, or messaging me on the Julia Slack.
-
-
-
 # Relevant literature
 
 See chapter 12 of [1] for the basics.

examples/README.md

@@ -2,7 +2,7 @@
 
 Ideally, you would have worked through a few examples involving AbstractGPs.jl, as the code
 in TemporalGPs.jl implements the (primary) interface specified there.
-Equally, these examples stand alone, so if you're not familiar with AbstractGPS.jl, don't
+Equally, these examples stand alone, so if you're not familiar with AbstractGPs.jl, don't
 worry too much.
 
 The examples in this directory are best worked through in the following order:

examples/exact_time_learning.jl

@@ -6,9 +6,6 @@
 using AbstractGPs
 using TemporalGPs
 
-# Load up the separable kernel from TemporalGPs.
-using TemporalGPs: RegularSpacing
-
 # Load standard packages from the Julia ecosystem
 using Optim # Standard optimisation algorithms.
 using ParameterHandling # Helper functionality for dealing with model parameters.
@@ -27,14 +24,17 @@ flat_initial_params, unpack = ParameterHandling.value_flatten((
 # Pull out the raw values.
 params = unpack(flat_initial_params);
 
+# Functionality to load build a TemporalGPs.jl GP given the model parameters.
+# Specifying SArrayStorage ensures that StaticArrays.jl is used to represent model
+# parameters under the hood, which enables very strong peformance.
 function build_gp(params)
     k = params.var_kernel * Matern52Kernel() ∘ ScaleTransform(params.λ)
     return to_sde(GP(params.mean, k), SArrayStorage(Float64))
 end
 
 # Specify a collection of inputs. Must be increasing.
 T = 1_000_000;
-x = RegularSpacing(0.0, 1e-4, T);
+x = TemporalGPs.RegularSpacing(0.0, 1e-4, T);
 
 # Generate some noisy synthetic data from the GP.
 f = build_gp(params)
@@ -48,6 +48,7 @@ function objective(flat_params)
     return -logpdf(f(x, params.var_noise), y)
 end
 
+# A helper function to get the gradient.
 function objective_grad(rule, flat_params)
     return Mooncake.value_and_gradient!!(rule, objective, flat_params)[2][2]
 end
@@ -74,7 +75,7 @@ f_post = posterior(f_final(x, final_params.var_noise), y);
 
 # Specify some locations at which to make predictions.
 T_pr = 1_200_000;
-x_pr = RegularSpacing(0.0, 1e-4, T_pr);
+x_pr = TemporalGPs.RegularSpacing(0.0, 1e-4, T_pr);
 
 # Compute the exact posterior marginals at `x_pr`.
 f_post_marginals = marginals(f_post(x_pr));