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Fixed unconsistent use of acronyms

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Radu C. Martin 2021-07-22 22:13:51 +02:00
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24
Sections/30_Gaussian_Processes_Background.tex
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@ -144,7 +144,7 @@ choices~\cite{kocijanModellingControlDynamic2016}:
\subsubsection*{Squared Exponential Kernel}
This kernel is used when the system to be modelled is assumed to be smooth and
continuous. The basic version of the \acrshort{se} kernel has the following form:
continuous. The basic version of the \acrfull{se} kernel has the following form:
\begin{equation}
k(\mathbf{x}, \mathbf{x'}) = \sigma^2 \exp{\left(- \frac{1}{2}\frac{\norm{\mathbf{x} -
@ -182,7 +182,7 @@ value of the hyperparameters. This is the \acrfull{ard} property.
The \acrfull{rq} Kernel can be interpreted as an infinite sum of \acrshort{se}
kernels with different lengthscales. It has the same smooth behaviour as the
\acrlong{se} Kernel, but can take into account the difference in function
\acrshort{se} Kernel, but can take into account the difference in function
behaviour for large scale vs small scale variations.
\begin{equation}
@ -207,11 +207,11 @@ without inquiring the penalty of inverting the covariance matrix. An overview
and comparison of multiple methods is given
at~\cite{liuUnderstandingComparingScalable2019}.
For the scope of this project, the choice of using the \acrfull{svgp} models has
been made, since it provides a very good balance of scalability, capability,
For the scope of this project, the choice of using the \acrshort{svgp} models
has been made, since it provides a very good balance of scalability, capability,
robustness and controllability~\cite{liuUnderstandingComparingScalable2019}.
The \acrlong{svgp} has been first introduced
The \acrshort{svgp} has been first introduced
by~\textcite{hensmanGaussianProcessesBig2013} as a way to scale the use of
\acrshort{gp}s to large datasets. A detailed explanation on the mathematics of
\acrshort{svgp}s and reasoning behind it is given
@ -264,7 +264,7 @@ In order to solve this problem, the log likelihood equation
classical \acrshort{gp} is replaced with an approximate value, that is
computationally tractable on larger sets of data.
The following derivation of the \acrshort{elbo} is based on the one presented
The following derivation of the \acrfull{elbo} is based on the one presented
in~\cite{yangUnderstandingVariationalLower}.
Assume $X$ to be the observations, and $Z$ the set parameters of the
@ -300,7 +300,7 @@ divergence, which for variational inference takes the following form:
\end{equation}
\vspace{5pt}
where L is the \acrfull{elbo}. Rearranging this equation we get:
where L is the \acrshort{elbo}. Rearranging this equation we get:
\begin{equation}
L = \log{\left(p(X)\right)} - KL\left[q(Z)||p(Z|X)\right]
@ -312,13 +312,13 @@ lower bound of the log probability of observations.
\subsection{Gaussian Process Models for Dynamical
Systems}\label{sec:gp_dynamical_system}
In the context of Dynamical Systems Identification and Control, Gaussian
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}.
In the context of Dynamical Systems Identification and Control, \acrshort{gp}s
are used to represent different model structures, ranging from state
space and \acrfull{nfir} structures, to the more complex \acrfull{narx},
\acrfull{noe} and \acrfull{narmax}.
The general form of an \acrfull{narx} model is as follows:
The general form of an \acrshort{narx} model is as follows:
\begin{equation}
\hat{y}(k) =