Spectral Densities
A kernel $k(x,y)$ is described as stationary or shift-invariant, if it can be written as $k(τ) = k(x-y)$, which means that it only depends on the difference between $x$ and $y$ but not their absolute values.
For a given stationary kernel $k(τ)$, the spectral density is its Fourier transform
\[S(\omega) = \int_{-\infty}^{\infty} k(τ) e^{-2 \pi \omega^T \tau} d\tau\]
The exact form of the Fourier transform may change slightly between fields (see Details and Options). This package uses the "signal processing" form above, as done in this presentation by Markus Heinonen. However, Rahimi & Recht used the "classical physics" form, for example. All options are equivalent, if used consistently.
Shared interface
KernelSpectralDensities.SpectralDensity — TypeSpectralDensity{K<:Kernel}(k::Kernel, dim::Int)Spectral density for the kernel K for dim dimensional frequency space.
Definition
Given a stationary kernel $k(x, x')$ the spectral density is the Fourier transform of $k(τ) = k(x-x')$. It can be seen as a probablity density function over the frequency domain, and can be evaluated at any frequency w.
Examples
julia> k = SqExponentialKernel();
julia> S = SpectralDensity(k, 1);
julia> S(0.0)
2.5066282746310007
julia> S = SpectralDensity(k, 2);
julia> S(zeros(2))
6.2831853071795845Base.rand — Methodrand(S::AbstractSpectralDensity, [n::Int])Generate a sample [n samples] from the spectral density S. For a scalar spectral density over $ω \in R$, computing n samples results in a n dimensional vector. For a multi-dimensional spectral density over $ω \in R^d, d>1$, computing n samples results in a d,n matrix.
Examples
julia> k = SqExponentialKernel();
julia> S = SpectralDensity(k, 1);
julia> rand(S, 1);Supported Kernel
- Squared Exponential
- Any Matern Kernel
Supported Transformations
- with_lengthscale