Commit 6311774e authored by Azat Garifullin's avatar Azat Garifullin
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docs fixes

parent fc20063a
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......@@ -54,13 +54,25 @@
\section{Sampling Mat\'{e}rn fields}
Let's consider a model:
\begin{gather}
\mathbf{y} = \mathbf{z}(\mathbf{x}\ + \epsilon), \\
\mathbf{y} = \mathbf{z}(\mathbf{x}) + \epsilon, \\
\epsilon \sim \mathcal{N}(0, \sigma_{\epsilon}^2), \\
\mathbf{z} \sim \mathcal{GP}\left(0, \mathcal{K}_{\theta}\right),
\label{eq:basic-model}
\end{gather}
where \(\mathbf{y}\) is an observed field, \(\mathbf{z}\) is a hidden field, and
\(\mathcal{K}\) is a Mat\'{e}rn covariance function with parameters \(\theta\).
\(\mathcal{K}\) is a Mat\'{e}rn covariance function with parameters
\(\theta = (\sigma, l, \sigma)\):
\begin{equation}
\mathcal{K}(\rho) =
\frac{\sigma^2 2^{1 - \nu}}{\Gamma(\nu)}
\left(
\frac{\sqrt{2\nu} \rho}{l}
\right)^{\nu}
K_{\nu} \left(
\frac{\sqrt{2\nu} \rho}{l}
\right)^{\nu},
\end{equation}
where \(\Gamma\) is the gamma function and \(K_{\nu}\) is the modified Bessel function of the second kind.
\begin{figure}
\centerline{
......@@ -173,11 +185,15 @@ The joint sampling then consists of the following steps:
\left( \mathbf{C}_{\theta} + \sigma_{\epsilon}^2 \mathbf{I} \right)^{-1}
\mathbf{C}_{\theta},
\\
\mathbf{L}_{\mathbf{z}}\mathbf{L}_{\mathbf{z}}^{T} &= \mathbf{C}_{\mathbf{z}}
\end{align},
\mathbf{L}_{\mathbf{z}}\mathbf{L}_{\mathbf{z}}^{T} &= \mathbf{C}_{\mathbf{z}},
\\
\mathbf{n} &\sim \mathcal{N}(0, \mathbf{I}),
\end{align}
\item Evaluate posterior according to \eqref{eq:theta-post} and \eqref{eq:theta-nl-post}
\item Accept the parameters according to Metropolis formula.
\end{itemize}
Optionally (essential for me), one can change sampling the field to deterministically
considering only the posterior mean \(\mathbf{z}^{t + 1} = \mu_{\mathbf{z}}\).
The trivial extension for cloud modeling affects \eqref{eq:basic-model}:
\begin{equation}
......
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