Mo documentation (#371)

* First commit

* Streamline MO kernel descriptions

* Missed a few

* Fix referencing issue

* Update src/mokernels/lmm.jl

Co-authored-by: willtebbutt <wt0881@my.bristol.ac.uk>

* Bump version

Co-authored-by: willtebbutt <wt0881@my.bristol.ac.uk>

Author: Crown421

Project.toml

@@ -1,6 +1,6 @@
 name = "KernelFunctions"
 uuid = "ec8451be-7e33-11e9-00cf-bbf324bd1392"
-version = "0.10.18"
+version = "0.10.19"
 
 [deps]
 ChainRulesCore = "d360d2e6-b24c-11e9-a2a3-2a2ae2dbcce4"

docs/src/kernels.md

@@ -128,6 +128,9 @@ NormalizedKernel
 ```
 
 ## Multi-output Kernels
+Kernelfunctions implements multi-output kernels as scalar kernels on an extended output domain. For more details on this read [the section on inputs for multi-output GPs](@ref Inputs-for-Multiple-Outputs).
+
+For a function ``f(x) \\rightarrow y`` denote the inputs as ``x, x'``, such that we compute the covariance between output components ``y_{p}`` and ``y_{p'}``. The total number of outputs is ``m``.
 
 ```@docs
 MOKernel

src/mokernels/independent.jl

@@ -5,11 +5,11 @@ Kernel for multiple independent outputs with kernel `k` each.
 
 # Definition
 
-For inputs ``x, x'`` and output dimensions ``p_x, p_{x'}'``, the kernel ``\\widetilde{k}``
+For inputs ``x, x'`` and output dimensions ``p, p'``, the kernel ``\\widetilde{k}``
 for independent outputs with kernel ``k`` each is defined as
 ```math
-\\widetilde{k}\\big((x, p_x), (x', p_{x'})\\big) = \\begin{cases}
-    k(x, x') & \\text{if } p_x = p_{x'}, \\\\
+\\widetilde{k}\\big((x, p), (x', p')\\big) = \\begin{cases}
+    k(x, x') & \\text{if } p = p', \\\\
     0 & \\text{otherwise}.
 \\end{cases}
 ```

src/mokernels/lmm.jl

@@ -2,19 +2,19 @@
     LinearMixingModelKernel(k::Kernel, H::AbstractMatrix)
     LinearMixingModelKernel(Tk::AbstractVector{<:Kernel},Th::AbstractMatrix)
 
-Kernel associated with the linear mixing model, taking a vector of `m` kernels and a `m × p` matrix H for a function with `p` outputs. Also accepts a single kernel `k` for use across all `m` basis vectors. 
+Kernel associated with the linear mixing model, taking a vector of `Q` kernels and a `Q × m` mixing matrix H for a function with `m` outputs. Also accepts a single kernel `k` for use across all `Q` basis vectors. 
 
 # Definition
 
-For inputs ``x, x'`` and output dimensions ``p_x, p_{x'}'``, the kernel is defined as[^BPTHST]
+For inputs ``x, x'`` and output dimensions ``p, p'``, the kernel is defined as[^BPTHST]
 ```math
-k\big((x, p_x), (x, p_{x'})\big) = H_{:,p_{x}}K(x, x')H_{:,p_{x'}}
+k\big((x, p), (x, p')\big) = H_{:,p}K(x, x')H_{:,p'}
 ```
-where ``K(x, x') = Diag(k_1(x, x'), ..., k_m(x, x'))`` with zero off-diagonal entries.
-``H_{:,p_{x}}`` is the ``p_x``-th column (`p_x`-th output) of ``H \in \mathbb{R}^{m \times p}``
-representing ``m`` basis vectors for the ``p`` dimensional output space of ``f``.
-``k_1, \ldots, k_m`` are ``m`` kernels, one for each latent process, ``H`` is a
-mixing matrix of ``m`` basis vectors spanning the output space.
+where ``K(x, x') = Diag(k_1(x, x'), ..., k_Q(x, x'))`` with zero off-diagonal entries.
+``H_{:,p}`` is the ``p``-th column (`p`-th output) of ``H \in \mathbb{R}^{Q \times m}``
+representing ``Q`` basis vectors for the ``m`` dimensional output space of ``f``.
+``k_1, \ldots, k_Q`` are ``Q`` kernels, one for each latent process, ``H`` is a
+mixing matrix of ``Q`` basis vectors spanning the output space.
 
 [^BPTHST]: Wessel P. Bruinsma, Eric Perim, Will Tebbutt, J. Scott Hosking, Arno Solin, Richard E. Turner (2020). [Scalable Exact Inference in Multi-Output Gaussian Processes](https://arxiv.org/pdf/1911.06287.pdf).
 """