Final version of the report
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\section{Results}\label{sec:results}
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% TODO [Results] Add info on control horizon
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This section focuses on the presentation and interpretation of the year-long
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simulation of the control schemes present previously.
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simulation of the control schemes present previously. All the control schemes
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analysed in this Section have been done with a sampling time of 15 minutes and a
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control horizon of 8 steps.
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Section~\ref{sec:GP_results} analyses the results of a conventional
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\acrlong{gp} Model trained on the first five days of gathered data. The models
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@ -25,7 +25,7 @@ is then employed for the rest of the year.
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With a sampling time of 15 minutes, the model is trained on 480 points of data.
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This size of the identification dataset is enough to learn the behaviour of the
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plant, without being too complex to solve from a numerical perspective, the
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plant, without being too complex to solve from a numerical perspective. The
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current implementation takes roughly 1.5 seconds of computation time per step.
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For reference, identifying a model on 15 days worth of experimental data (1440
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points) makes simulation time approximately 11 --- 14 seconds per step, or
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@ -39,8 +39,8 @@ $\degree$C in the stable part of the simulation. The offset becomes much larger
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once the reference temperature starts moving from the initial constant value.
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The controller becomes completely unstable around the middle of July, and can
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only regain some stability at the middle of October. It is also possible to note
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that from mid-October --- end-December the controller has very similar
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performance to that exhibited in the beginning of the year, namely January ---
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that from mid-October to end-December the controller has very similar
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performance to that exhibited in the beginning of the year, namely January to
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end-February.
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\begin{figure}[ht]
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\end{figure}
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This very large difference in performance could be explained by the change in
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weather during the year. The winter months of the beginning of the year and end
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of year exhibit similar performance, the spring months already make the
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controller less stable than at the start of the year, while the drastic
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temperature changes in the summer make the controller completely unstable.
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weather during the year. The winter months of the beginning and end of the year
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exhibit similar performance. The spring months already make the controller less
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stable than at the start of the year, while the drastic temperature changes in
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the summer make the controller completely unstable.
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\clearpage
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@ -76,14 +76,14 @@ occurring during the winter months.
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Figure~\ref{fig:GP_first_model_performance} analyses the 20-step ahead
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simulation performance of the identified model over the course of the year. At
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experimental step 250 the controller is still gathering data. It is therefore
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experimental step 250, the controller is still gathering data. It is therefore
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expected that the identified model will be capable of reproducing this data. At
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step 500, 20 steps after identification, the model correctly steers the internal
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temperature towards the reference temperature. On the flip side, already at
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experimental steps 750 and 1000, only 9 days into the simulation, the model is
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unable to properly simulate the behaviour of the plant, with the maximum
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difference at the end of the simulation reaching 0.75 and 1.5 $\degree$C
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respectively.
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difference at the end of the simulation reaching 0.75 $\degree$C and 1.5
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$\degree$C respectively.
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\begin{figure}[ht]
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\centering
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Figure~\ref{fig:SVGP_fullyear_simulation}. It can already be seen that this
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setup performs much better than the initial one. The only large deviations from
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the reference temperature are due to cold --- when the \acrshort{hvac}'s limited
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heat capacity is unable to maintain the proper temperature.
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heat capacity is unable to maintain the proper temperature. Additionnaly, the
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\acrshort{svgp} controller takes around 250-300ms of computation time for each
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simulation time, decreasing the computational cost of the original \acrshort{gp}
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by a factor of six.
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% TODO: [Results] Add info on SVGP vs GP computation speed
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\begin{figure}[ht]
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\centering
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@ -194,7 +197,7 @@ starting at 107500 and 11000 points.
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\centering
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\includegraphics[width =
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\textwidth]{Plots/1_SVGP_480pts_inf_window_12_averageYear_first_model_performance.pdf}
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\caption{GP first model performance}
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\caption{SVGP first model performance}
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\label{fig:SVGP_first_model_performance}
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\end{figure}
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@ -214,7 +217,7 @@ than the classical \acrshort{gp} model to capture the building dynamics.
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\centering
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\includegraphics[width =
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\textwidth]{Plots/1_SVGP_480pts_inf_window_12_averageYear_later_model_performance.pdf}
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\caption{GP later model performance}
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\caption{SVGP later model performance}
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\label{fig:SVGP_later_model_performance}
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\end{figure}
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@ -229,7 +232,7 @@ respectively.
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\centering
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\includegraphics[width =
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\textwidth]{Plots/1_SVGP_480pts_inf_window_12_averageYear_last_model_performance.pdf}
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\caption{GP last model performance}
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\caption{SVGP last model performance}
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\label{fig:SVGP_last_model_performance}
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\end{figure}
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closed-loop operation, will the performance deteriorate drastically if
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the first model is trained on less data?
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\item How much information can the model extract from closed-loop operation?
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Would a model trained on only the last five days of closed-loop
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operation data be able to perform correctly?
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Would a model trained on a window of only the last five days of
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closed-loop operation data be able to perform correctly?
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\end{itemize}
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These questions will be further analysed in the Sections~\ref{sec:svgp_window}
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and~\ref{sec:svgp_96pts} respectively.
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These questions will be further analysed in the Sections~\ref{sec:svgp_96pts}
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and~\ref{sec:svgp_window} respectively.
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\clearpage
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@ -299,8 +302,8 @@ months. This might be due to the fact that during the colder months, the
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additional heat to the system.
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A similar trend can be observed for the evolution of the input's
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hyperparameters, with the exception that the first lag of the controlled input,
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$u1,1$ remains the most important over the course of the year.
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hyperparameters, with the exception that the first lag of the controlled input
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($u1,1$) remains the most important over the course of the year.
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For the lags of the measured output it can be seen that, over the course of the
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year, the importance of the first lag decreases, while that of the second and
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@ -342,7 +345,7 @@ refinements being done as data is added to the system.
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One question that could be addressed given these mostly linear kernel terms is
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how well would an \acrshort{svgp} model perform with a linear kernel.
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Intuition would hint that it should still be able to track the reference
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temperature, albeit not as precisely due to the correlation that diminished much
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temperature, albeit not as precisely due to the correlation that diminishes much
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slower when the two points are closer together in the \acrshort{se} kernel. This
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will be further investigated in Section~\ref{sec:svgp_linear}.
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@ -351,7 +354,7 @@ will be further investigated in Section~\ref{sec:svgp_linear}.
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\subsection{SVGP with one day of starting data}\label{sec:svgp_96pts}
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As previously discussed in Section~\ref{sec:SVGP_results}, the \acrshort{svgp}
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model is able to properly adapt given new information, overtime refining it's
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model is able to properly adapt given new information, overtime refining its
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understanding of the plant's dynamics.
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Analyzing the results of a simulation done on only one day's worth of initial
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\subsection{SVGP with a five days moving window}\label{sec:svgp_window}
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This section presents the result of running a different control scheme. Here,
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as the base \acrshort{svgp} model, it is first trained on 5 days worth of data,
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with the difference being that each new model is only identified using the last
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five days' worth of data. This should provide an insight on whether the
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\acrshort{svgp} model is able to understand model dynamics only based on
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closed-loop operation.
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This section presents the result of running a different control scheme. Here, as
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was the case for the base \acrshort{svgp} model, it is first trained on 5 days
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worth of data, with the difference being that each new model is only identified
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using the last five days' worth of data. This should provide an insight on
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whether the \acrshort{svgp} model is able to understand model dynamics only
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based on closed-loop operation.
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\begin{figure}[ht]
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\centering
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\includegraphics[width =
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\textwidth]{Plots/5_SVGP_480pts_480pts_window_12_averageYear_fullyear.pdf}
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\caption{SVGP full year simulation}
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\caption{Windowed SVGP full year simulation}
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\label{fig:SVGP_480window_fullyear_simulation}
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\end{figure}
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As it can be seen in Figure~\ref{fig:SVGP_480window_fullyear_simulation}, this
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model is unable to properly track the reference temperature. In fact, five days
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after the identification the model forgets all the initial data and becomes
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after the identification, the model forgets all the initial data and becomes
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unstable. This instability then generates enough excitation of the plant for the
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model to again learn its behaviour. This cycle repeats every five days, when the
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model, to again learn its behaviour. This cycle repeats every five days, when the
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controller becomes unstable. In the stable regions, however, the controller is
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able to track the reference temperature.
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\centering
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\includegraphics[width =
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\textwidth]{Plots/10_SVGP_480pts_inf_window_12_averageYear_LinearKernel_fullyear.pdf}
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\caption{SVGP full year simulation}
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\caption{Linear SVGP full year simulation}
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\label{fig:SVGP_linear_fullyear_simulation}
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\end{figure}
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\subsection{Comparative analysis}
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This section will compare all the results presented in the previous Sections and
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try to analyze the differences and their origin.
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Presented in Table~\ref{tab:Model_comparations} are the Mean Error, Error
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Variance and Mean Absolute Error for the full year simulation for the three
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stable \acrshort{svgp} models, as well as the classical \acrshort{gp} model.
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have a comparable performance, with very small differences in Mean Absolute
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Error and Error variance. This leads to the conclusion that the \acrshort{svgp}
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models can be deployed with less explicit identification data, but they will
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continue to improve over the course of the year as the building passes through
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continue to improve over the course of the year, as the building passes through
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different regions of the state space and more data is collected.
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These results do not, however, discredit the use of \acrlong{gp} for use in a
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multi-seasonal situation. As shown before, given the same amount of data and
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ignoring the computational cost, they perform better than the alternative
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These results do not, however, discredit the use of \acrlong{gp} for employment
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in a multi-seasonal situation. As shown before, given the same amount of data
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and ignoring the computational cost, they perform better than the alternative
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\acrshort{svgp} models. The bad initial performance could be mitigated by
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sampling the identification data at different points in time during multiple
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experiments, updating a fixed-size dataset based on the gained information, as
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