Kernels

GaussianProcesses.LinArdType
LinArd <: Kernel

ARD linear kernel (covariance)

$k(x,x') = xᵀL⁻²x'$

with length scale $ℓ = (ℓ₁, ℓ₂, …)$ and $L = diag(ℓ₁, ℓ₂, …)$.

source
GaussianProcesses.MaskedType
Masked{K<:Kernel} <: Kernel

A wrapper for kernels so that they are only applied along certain dimensions.

This is similar to the active_dims kernel attribute in the python GPy package and to the covMask function in the matlab gpml package.

The implementation is very simple: any function of the kernel that takes an X::Matrix input is delegated to the wrapped kernel along with a view of X that only includes the active dimensions.

source
GaussianProcesses.MaternMethod
Matern(ν::Real, ll::Union{Real,Vector{Real}}, lσ::Real)

Create Matérn kernel of type ν (i.e. ν = 1/2, ν = 3/2, or ν = 5/2) with length scale exp.(ll) and signal standard deviation exp(σ).

source
GaussianProcesses.Mat12ArdType
Mat12Ard <: MaternARD

ARD Matern 1/2 kernel (covariance)

$k(x,x') = σ² \exp(-|x-x'|/L)$

with length scale $ℓ = (ℓ₁, ℓ₂, …)$ and signal standard deviation $σ$ where $L = diag(ℓ₁, ℓ₂, …)$.

source
GaussianProcesses.Mat12ArdMethod

Matern 1/2 ARD covariance Function

Mat12Ard(ll::Vector{T}, lσ::T)

Arguments

• ll::Vector{Real}: vector of length scales (given on log scale)
• lσ::Real: signal standard deviation (given on log scale)
source
GaussianProcesses.Mat12IsoType
Mat12Iso <: MaternISO

Isotropic Matern 1/2 kernel (covariance)

$k(x,x') = σ^2 \exp(-|x-y|/ℓ)$

with length scale $ℓ$ and signal standard deviation $σ$.

source
GaussianProcesses.Mat12IsoMethod

Matern 1/2 isotropic covariance Function

Mat12Iso(ll::T, lσ::T)

Arguments

• ll::Real: length scale (given on log scale)
• lσ::Real: signal standard deviation (given on log scale)
source
GaussianProcesses.Mat32ArdType
Mat32Ard <: MaternARD

ARD Matern 3/2 kernel (covariance)

$k(x,x') = σ²(1 + √3|x-x'|/L)\exp(- √3|x-x'|/L)$

with length scale $ℓ = (ℓ₁, ℓ₂, …)$ and signal standard deviation $σ$ where $L = diag(ℓ₁, ℓ₂, …)$.

source
GaussianProcesses.Mat32ArdMethod

Matern 3/2 ARD covariance function

Mat32Ard(ll::Vector{T}, lσ::T)

Arguments

• ll::Vector{Real}: vector of length scales (given on log scale)
• lσ::Real: signal standard deviation (given on log scale)
source
GaussianProcesses.Mat32IsoType
Mat32Iso <: MaternIso

Isotropic Matern 3/2 kernel (covariance)

$k(x,x') = σ²(1 + √3|x-x'|/ℓ)\exp(-√3|x-x'|/ℓ)$

with length scale $ℓ$ and signal standard deviation $σ$.

source
GaussianProcesses.Mat32IsoMethod

Matern 3/2 isotropic covariance function

Mat32Iso(ll::T, lσ::T)

Arguments

• ll::Real: length scale (given on log scale)
• lσ::Real: signal standard deviation (given on log scale)
source
GaussianProcesses.Mat52ArdType
Mat52Ard <: MaternARD

ARD Matern 5/2 kernel (covariance)

$k(x,x') = σ²(1 + √5|x-x'|/L + 5|x-x'|²/(3L²))\exp(- √5|x-x'|/L)$

with length scale $ℓ = (ℓ₁, ℓ₂, …)$ and signal standard deviation $σ$ where $L = diag(ℓ₁, ℓ₂, …)$.

source
GaussianProcesses.Mat52ArdMethod

Matern 5/2 ARD covariance Function

Mat52Ard(ll::Vector{Real}, lσ::Real)

Arguments

• ll::Vector{Real}: vector of length scales (given on log scale)
• lσ::Real: signal standard deviation (given on log scale)
source
GaussianProcesses.Mat52IsoType
Mat52Iso <: MaternIso

Isotropic Matern 5/2 kernel (covariance)

$k(x,x') = σ²(1+√5|x-x'|/ℓ + 5|x-x'|²/(3ℓ²))\exp(- √5|x-x'|/ℓ)$

with length scale $ℓ$ and signal standard deviation $σ$.

source
GaussianProcesses.Mat52IsoMethod

Matern 5/2 isotropic covariance function

Mat52Iso(ll::Real, lσ::Real)

Arguments

• ll::Real: length scale (given on log scale)
• lσ::Real: signal standard deviation (given on log scale)
source
GaussianProcesses.NoiseType
Noise <: Kernel

Noise kernel (covariance)

$k(x,x') = σ²δ(x-x'),$

where $δ$ is the Kronecker delta function and $σ$ is the signal standard deviation.

source
GaussianProcesses.PeriodicType
Periodic <: Isotropic{Euclidean}

Periodic kernel (covariance)

$k(x,x') = σ²\exp(-2\sin²(π|x-x'|/p)/ℓ²)$

with length scale $ℓ$, signal standard deviation $σ$, and period $p$.

source
GaussianProcesses.PeriodicMethod

Periodic kernel function

Periodic(ll::Real, lσ::Real, lp::Real)

Arguments

• ll::Real: length scale (given on log scale)
• lσ::Real: signal standard deviation (given on log scale)
• lp::Real: periodicity parameter (given on log scale)
source
GaussianProcesses.PolyType
Poly <: Kernel

Polynomial kernel (covariance)

$k(x,x') = σ²(xᵀx' + c)ᵈ$

with signal standard deviation $σ$, additive constant $c$, and degree $d$.

source
GaussianProcesses.PolyMethod

Polynomial kernel function

Poly(lc::Real, lσ::Real, deg::Int)

Arguments

• lc::Real: additive constant (given on log scale)
• lσ::Real: signal standard deviation (given on log scale)
• deg::Int: degree of polynomial
source
GaussianProcesses.RQArdType
RQArd <: StationaryARD{WeightedSqEuclidean}

$k(x,x') = σ²(1 + (x - x')ᵀL⁻²(x - x')/(2α))^{-α}$

with length scale $ℓ = (ℓ₁, ℓ₂, …)$, signal standard deviation $σ$, and shape parameter $α$ where $L = diag(ℓ₁, ℓ₂, …)$.

source
GaussianProcesses.RQArdMethod

RQArd(ll::Vector{Real}, lσ::Real, lα::Real)

Arguments

• ll::Vector{Real}: vector of length scales (given on log scale)
• lσ::Real: signal standard deviation (given on log scale)
• lα::Real: shape parameter (given on log scale)
source
GaussianProcesses.RQIsoType
RQIso <: Isotropic{SqEuclidean}

$k(x,x') = σ²(1 + (x - x')ᵀ(x - x')/(2αℓ²))^{-α}$

with length scale $ℓ$, signal standard deviation $σ$, and shape parameter $α$.

source
GaussianProcesses.RQIsoMethod

RQIso(ll:T, lσ::T, lα::T)

Arguments

• ll::Real: length scale (given on log scale)
• lσ::Real: signal standard deviation (given on log scale)
• lα::Real: shape parameter (given on log scale)
source
GaussianProcesses.SEArdType
SEArd <: StationaryARD{WeightedSqEuclidean}

ARD Squared Exponential kernel (covariance)

$k(x,x') = σ²\exp(- (x - x')ᵀL⁻²(x - x')/2)$

with length scale $ℓ = (ℓ₁, ℓ₂, …)$ and signal standard deviation $σ$ where $L = diag(ℓ₁, ℓ₂, …)$.

source
GaussianProcesses.SEArdMethod

Squared Exponential Function with ARD

SEArd(ll::Vector{Real}, lσ::Real)

Arguments

• ll::Vector{Real}: vector of length scales (given on log scale)
• lσ::Real: signal standard deviation (given on log scale)
source
GaussianProcesses.SEIsoType
SEIso <: Isotropic{SqEuclidean}

Isotropic Squared Exponential kernel (covariance)

$k(x,x') = σ²\exp(- (x - x')ᵀ(x - x')/(2ℓ²))$

with length scale $ℓ$ and signal standard deviation $σ$.

source
GaussianProcesses.SEIsoMethod

Squared Exponential kernel function

SEIso(ll::T, lσ::T)

Arguments:

• ll::Real: length scale (given on log scale)
• lσ::Real: signal standard deviation (given on log scale)
source
Statistics.covFunction
cov(k::Kernel, X::AbstractMatrix[, data::KernelData = EmptyData()])

Create covariance matrix from kernel k, matrix of observations X, where each column is an observation, and kernel data data constructed from input observations.

source
Statistics.covFunction
cov(k::Kernel, X1::AbstractMatrix, X2::AbstractMatrix)

Create covariance matrix from kernel k and matrices of observations X1 and X2, where each column is an observation.

source