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Radu C. Martin 2021-06-25 11:27:25 +02:00
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17
30_Gaussian_Processes_Background.tex
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@ -84,7 +84,7 @@ which, for the rest of the section, will be used in the abbreviated form:
"Training" a \acrshort{gp} is the process of finding the kernel parameters that
best explain the data. This is done by maximizing the probability density
function for the observations $y$i, also known as the marginal likelihood:
function for the observations $y$, also known as the marginal likelihood:
\begin{equation}\label{eq:gp_likelihood}
p(y) = \frac{1}{\sqrt{(2\pi)^{n}\det{\left(K + \sigma_n^2I\right)}}}
@ -109,6 +109,9 @@ marginal likelihood:
\subsection{Prediction}
Given the proper covariance matrices $K$ and $K_*$, predictions on new points
can be made as follows:
\begin{equation}
\begin{aligned}
\mathbf{f_*} = \mathbb{E}\left(f_*|X, \mathbf{y}, X_*\right) &=
@ -117,9 +120,9 @@ marginal likelihood:
\end{aligned}
\end{equation}
Apply the zero mean \acrshort{gp} to the \textit{difference} between the
observations and the fixed mean function:
The extensions of these predictions to a non-zero mean \acrshort{gp} comes
naturally by applying the zero mean \acrshort{gp} to the \textit{difference}
between the observations and the fixed mean function:
\begin{equation}
\bar{\mathbf{f}}_* = \mathbf{m}(X_*) + K_*\left(K +
@ -310,9 +313,9 @@ lower bound of the log probability of observations.
Systems}\label{sec:gp_dynamical_system}
In the context of Dynamical Systems Identification and Control, Gaussian
Processes are used to represent multiple different model structures, ranging
from state space and \acrshort{nfir} structures, to the more complex
\acrshort{narx}, \acrshort{noe} and \acrshort{narmax}.
Processes are used to represent different model structures, ranging from state
space and \acrshort{nfir} structures, to the more complex \acrshort{narx},
\acrshort{noe} and \acrshort{narmax}.
The general form of an \acrfull{narx} model is as follows: