Fixed unconsistent use of acronyms
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7 changed files with 49 additions and 47 deletions
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@ -144,7 +144,7 @@ choices~\cite{kocijanModellingControlDynamic2016}:
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\subsubsection*{Squared Exponential Kernel}
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This kernel is used when the system to be modelled is assumed to be smooth and
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continuous. The basic version of the \acrshort{se} kernel has the following form:
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continuous. The basic version of the \acrfull{se} kernel has the following form:
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\begin{equation}
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k(\mathbf{x}, \mathbf{x'}) = \sigma^2 \exp{\left(- \frac{1}{2}\frac{\norm{\mathbf{x} -
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@ -182,7 +182,7 @@ value of the hyperparameters. This is the \acrfull{ard} property.
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The \acrfull{rq} Kernel can be interpreted as an infinite sum of \acrshort{se}
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kernels with different lengthscales. It has the same smooth behaviour as the
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\acrlong{se} Kernel, but can take into account the difference in function
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\acrshort{se} Kernel, but can take into account the difference in function
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behaviour for large scale vs small scale variations.
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\begin{equation}
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@ -207,11 +207,11 @@ without inquiring the penalty of inverting the covariance matrix. An overview
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and comparison of multiple methods is given
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at~\cite{liuUnderstandingComparingScalable2019}.
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For the scope of this project, the choice of using the \acrfull{svgp} models has
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been made, since it provides a very good balance of scalability, capability,
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For the scope of this project, the choice of using the \acrshort{svgp} models
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has been made, since it provides a very good balance of scalability, capability,
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robustness and controllability~\cite{liuUnderstandingComparingScalable2019}.
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The \acrlong{svgp} has been first introduced
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The \acrshort{svgp} has been first introduced
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by~\textcite{hensmanGaussianProcessesBig2013} as a way to scale the use of
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\acrshort{gp}s to large datasets. A detailed explanation on the mathematics of
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\acrshort{svgp}s and reasoning behind it is given
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@ -264,7 +264,7 @@ In order to solve this problem, the log likelihood equation
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classical \acrshort{gp} is replaced with an approximate value, that is
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computationally tractable on larger sets of data.
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The following derivation of the \acrshort{elbo} is based on the one presented
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The following derivation of the \acrfull{elbo} is based on the one presented
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in~\cite{yangUnderstandingVariationalLower}.
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Assume $X$ to be the observations, and $Z$ the set parameters of the
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@ -300,7 +300,7 @@ divergence, which for variational inference takes the following form:
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\end{equation}
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\vspace{5pt}
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where L is the \acrfull{elbo}. Rearranging this equation we get:
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where L is the \acrshort{elbo}. Rearranging this equation we get:
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\begin{equation}
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L = \log{\left(p(X)\right)} - KL\left[q(Z)||p(Z|X)\right]
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@ -312,13 +312,13 @@ lower bound of the log probability of observations.
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\subsection{Gaussian Process Models for Dynamical
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Systems}\label{sec:gp_dynamical_system}
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In the context of Dynamical Systems Identification and Control, Gaussian
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Processes are used to represent different model structures, ranging from state
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space and \acrshort{nfir} structures, to the more complex \acrshort{narx},
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\acrshort{noe} and \acrshort{narmax}.
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In the context of Dynamical Systems Identification and Control, \acrshort{gp}s
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are used to represent different model structures, ranging from state
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space and \acrfull{nfir} structures, to the more complex \acrfull{narx},
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\acrfull{noe} and \acrfull{narmax}.
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The general form of an \acrfull{narx} model is as follows:
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The general form of an \acrshort{narx} model is as follows:
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\begin{equation}
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\hat{y}(k) =
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