Small tooltip change and first part of feature functions script

Author: Crown421

examples/approx-prior-sample.jl

@@ -0,0 +1,6 @@
+[deps]
+AbstractGPs = "99985d1d-32ba-4be9-9821-2ec096f28918"
+AbstractGPsMakie = "7834405d-1089-4985-bd30-732a30b92057"
+CairoMakie = "13f3f980-e62b-5c42-98c6-ff1f3baf88f0"
+KernelSpectralDensities = "027d52a2-76e5-4228-9bfe-bc7e0f5a8348"
+StatsBase = "2913bbd2-ae8a-5f71-8c99-4fb6c76f3a91"

examples/feature-functions.jl

@@ -0,0 +1,89 @@
+# # Random Fourier Features
+# One of the reasons to be interested in the spectral density of a kernel is 
+# that it allows us to approximate a GP Prior.
+# 
+# In this notebook we show the two feature functions implemented in KernelSpectralDensities.jl 
+# and how to use them.
+
+# ## Load required packages
+using KernelSpectralDensities
+using AbstractGPs
+using StatsBase
+# using LinearAlgebra
+using AbstractGPsMakie
+using CairoMakie
+
+# ## Intro
+# We use the AbstractGPs package to define a stationary GP prior,
+# in other words, a GP that has not been conditioned on data yet.
+
+ker = SqExponentialKernel()
+S = SpectralDensity(ker, 1)
+
+gp = GP(ker)
+
+# We can also plot this GP using AbstractGPsMakie.jl, but 
+# we don't see very much, since we have a simple GP with 
+# zero mean and a variance of 1. 
+f = plot(0:0.1:1, gp; size=(600, 400))
+DisplayAs.PNG(f) #hide #md
+
+# ## Random Fourier Features
+# KernelSpectralDensities implements two types of random Fourier features, 
+# `ShiftedRFF` and `DoubleRFF`.
+# A feature function projects its input into a higher-dimensional "features" space.,
+
+# ### ShiftedRFF
+# The `ShiftedRFF` feature function is somewhat more common, and has 
+# been used in papers such as [Efficiently sampling functions from Gaussian process posteriors](https://proceedings.mlr.press/v119/wilson20a.html).
+#
+# It is defined as
+# ```math
+#     \sqrt{2 / l}  \cos(2  π  ((w^T  x) + b))
+# ```
+# where `w` is sampled from the spectral density `S`, 
+# `b` is uniformly sampled from `[0, 2π]` 
+# and `l` is the number of sampled frequencies, which is also 
+# the number of features.
+#
+# We generate a set of 4 feature functions, which we can evaluate 
+# at any point.
+srff = ShiftedRFF(S, 4)
+srff(1.0)
+
+# If we plot them, we see that each feature function is a harmonic 
+# with varying frequency and phase.
+x = range(0, 5; length=100)
+f = Figure(; size=(600, 400))
+ax = Axis(f[1, 1]; xlabel="x", ylabel="rff(x)", title="ShiftedRFF")
+series!(ax, x, reduce(hcat, srff.(x)); labels=["srff $i" for i in 1:4])
+axislegend(ax; position=:ct)
+f
+DisplayAs.PNG(f) #hide #md
+
+# ### DoubleRFF
+# The `DoubleRFF` feature function is less common, but is theoretically
+# equivalent to the `ShiftedRFF` feature function.
+#
+# It is defined as
+# ```math
+#     \sqrt{1 / l} [\cos(2 π w' x),  \sin(2 π w' x)]
+# ```
+# where `w` is sampled from the spectral density `S`,
+# with a total of `l/2` sampled frequencies.
+#
+# Here, each function is effectively two feature functions in one, 
+# so specifying `l` will result in `l/2` samples but an `l` dimensional 
+# feature vector.
+# 
+# We again generate a set of 4 feature functions. 
+drff = DoubleRFF(S, 4)
+drff(1.0)
+
+# We plot these features as well
+f = Figure(; size=(600, 400))
+ax = Axis(f[1, 1]; xlabel="x", ylabel="rff(x)", title="Double RFF")
+series!(ax, x, reduce(hcat, drff.(x)); labels=["drff $i" for i in 1:4])
+axislegend(ax; position=:ct)
+f
+DisplayAs.PNG(f) #hide #md

examples/features/Project.toml

@@ -6,7 +6,7 @@ abstract type AbstractRFF end
 
 """
     ShiftedRFF([rng::AbstractRNG], S::SpectralDensity, l::Int)
-Random Fourier feature function with a random shift, projecting an input x into l dimensionional feature space. 
+Random Fourier feature function with a random shift, projecting an input `x` into `l` dimensionional feature space. 
     
 # Definition
     
@@ -51,7 +51,7 @@ end
 
 """
     DoubleRFF([rng::AbstractRNG], S::SpectralDensity, l::Int)
-Random Fourier feature function with cos and sin terms, projecting an input x into l dimensionional feature space.
+Random Fourier feature function with cos and sin terms, projecting an input `x` into `l` dimensionional feature space.
 
 # Definition