[WIP]: Thesis update
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Radu C. Martin
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@ -4,14 +4,231 @@ This section will discuss and try to validate the choice of all the
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hyperparameters necessary for the training of a \acrshort{gp} model to capture
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the CARNOT building's behaviour.
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The class of black-box models is very versatile, being able to capture plant
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behaviour directly from data. This comes in contrast to white-box and grey-box
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modelling techniques, which require much more physical insight into the plant's
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behaviour.
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The advantage of black-box models lies in the lack of physical parameters to be
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fitted. On the flip side, this versatility of being able to fit much more
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complex models putely on data comes at the cost of having to properly define the
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model hyperparameters: the number of regressors, the number of autoregressive
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lags for each class of inputs, the shape of the covariance function have to be
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taken into account when designing a \acrshort{gp} model. These choices have
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direct influence on the resulting model behaviour and where it can be
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generalized, as well as indirect influence in the form of more expensive
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computations in the case of larger number of regressors and more complex kernel
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functions.
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As described in Section~\ref{sec:gp_dynamical_system}, for the purpose of this
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project the \acrlong{gp} model will be trained using the \acrshort{narx}
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structure.
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structure. This already presents an important choice in the selection of
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regressors and their respective autoregressive lags.
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\subsection{Lengthscales}
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The output of the model has been chosen as the \textit{temperature measured
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inside} the CARNOT building. This is a suitable choice for the \acrshort{ocp}
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defined in Section~\ref{sec:mpc_problem}, where the goal is tracking as close as
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possible the inside temperature of the building.
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The input of the \acrshort{gp} model conincides with the input of the CARNOT
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building, namely the \textit{power} passed to the idealized \acrshort{hvac},
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which is held constant during the complete duration of a step.
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As for the exogenous inputs the choice turned out to be more complex. The CARNOT
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WDB format (cf. Section~\ref{sec:CARNOT_WDB}) consists of information of all the
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solar angles, the different components of solar radiation, wind speed and
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direction, temperature, precipitation, etc. All of this information is required
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in order for CARNOT's proper functioning.
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Including all of this information into the \acrshort{gp}s exogenous inputs would
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come with a few downsides. First, depending on the number of lags chosen for the
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exogenous inputs, the number of inputs to the \acrshort{gp} could be very large,
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an exogenous inputs vector of 10 elements with 2 lags would yield 20 inputs for
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the \acrshort{gp} model. This is very computationally expensive both for
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training and using the model, as its algorithmic complexity is
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$\mathcal{O}(n^3)$.
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Second, this information may not always available in experimental
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implementations on real buildings, where legacy equipment might already be
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installed, or where budget restrictions call for simpler equipment. An example
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of this are the experimental datasets used for validation of the CARNOT model,
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where the only available weather information is the \acrshort{ghi} and the
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measurement of the outside temperature. This would also be a limitation when
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getting the weather predictions for the next steps during real-world
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experiments.
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Last, while very verbose information such as the solar angles and the components
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of the solar radiation is very useful for CARNOT, which simulated each node
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individually, knowing their absolute positions, this information would not
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always benefit the \acrshort{gp} model, at least not comparably to the
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additional computational complexity.
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For the exogenous inputs the choice has therefore been made to take the
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\textit{Global Solar Irradiance} and \textit{Outside Temperature Measurement}.
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\subsection{The Kernel}
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The covariance matrix is an important choice when creating the \acrshort{gp}. A
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properly chosen kernel can impose a prior desired behaviour on the
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\acrshort{gp} such as continuity of the function an its derivatives,
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periodicity, linearity, etc. On the flip side, choosing the wrong kernel can
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make computations more expensive, require more data to learn the proper
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behaviour or outright be numerically instable and/or give erroneous predictions.
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The \acrlong{se} kernel (cf. Section~\ref{sec:Kernels}) is very versatile,
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theoretically being able to fit any continuous function given enough data. When
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including the \acrshort{ard} behaviour, it also gives an insight into the
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relative importance of each regressor, though their respective lengthscales.
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Many different kernels have been used when identifying models for building
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thermal control, such as a pure Rational Quadratic
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Kernel~\cite{pleweSupervisoryModelPredictive2020}, a combination of
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\acrshort{se}, \acrshort{rq} and a Linear
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Kernel~\cite{jainLearningControlUsing2018}, Squared Exponential Kernel and
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Kernels from the M\`atern family~\cite{massagrayThermalBuildingModelling2016}.
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For the purpose of this project the choice has been made to use the
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\textit{\acrlong{se} Kernel}, as it provides a very good balance of versatily
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and computational complexity for the modelling of the CARNOT building.
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\subsection{Lengthscales}\label{sec:lengthscales}
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The hyperparameters of the \acrshort{se} can be useful when studying the
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importance of regressors. The larger the distance between the two inputs, the
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less correlated they are. In fact, setting the kernel variance $\sigma^2$ = 1,
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we can compute the correlation of two inputs located at distance one, two and
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three lengthscales apart.
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\begin{table}[ht]
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\centering
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\begin{tabular}{||c c ||}
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\hline
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$\norm{\mathbf{x} - \mathbf{x}'}$ &
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$\exp{(-\frac{1}{2}*\frac{\norm{\mathbf{x} - \mathbf{x}'}^2}{l^2})}$ \\
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\hline \hline
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$1l$ & 0.606 \\
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$2l$ & 0.135 \\
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$3l$ & 0.011 \\
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\hline
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\end{tabular}
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\caption{Correlation of inputs relative to their distance}
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\label{tab:se_correlation}
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\end{table}
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From Table~\ref{tab:se_correlation} is can be seen that at 3 lengthscales apart,
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the inputs are already almost uncorrelated. In order to better visualize this
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difference the value of relative lengthscale importance is introduced:
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\begin{equation}
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\lambda = \frac{1}{\sqrt{l}}
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\end{equation}
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Another indicator of model behaviour is the variance of the identified
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\acrshort{se} kernel. The expected value of the variance is around the variance
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of the inputs. An extremenly high or extremely low value of the variance could
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mean a numerically unstable model.
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Table~\ref{tab:GP_hyperparameters} presents the relative lengthscale imporances
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and the variance for different combinations of the exogenous input lags ($l_w$),
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the controlled input lags ($l_u$) and the output lags ($l_y$) for a classical
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\acrshort{gp} model.
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\begin{table}[ht]
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%\vspace{-8pt}
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\centering
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\resizebox{\columnwidth}{!}{%
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\begin{tabular}{||c c c|c|c c c c c c c c c c c||}
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\hline
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\multicolumn{3}{||c|}{Lags} & Variance &\multicolumn{11}{c||}{Kernel
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lengthscales relative importance} \\
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$l_w$ & $l_u$ & $l_y$ & $\sigma^2$ &$\lambda_{w1,1}$ & $\lambda_{w1,2}$ &
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$\lambda_{w1,3}$ & $\lambda_{w2,1}$ & $\lambda_{w2,2}$ &
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$\lambda_{w2,3}$ & $\lambda_{u1,1}$ & $\lambda_{u1,2}$ &
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$\lambda_{y1,1}$ & $\lambda_{y1,2}$ & $\lambda_{y1,3}$\\
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\hline \hline
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1 & 1 & 1 & 0.11 & 0.721 & & & 2.633 & & & 0.569 & & 2.645 & & \\
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1 & 1 & 2 & 22.68 & 0.222 & & & 0.751 & & & 0.134 & & 3.154 & 3.073
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& \\ 1 & 1 & 3 & 0.29 & 0.294 & & & 1.303 & & & 0.356 & & 2.352
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& 1.361 & 2.045 \\ 1 & 2 & 1 & 7.55 & 0.157 & & & 0.779 & & &
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0.180 & 0.188 & 0.538 & & \\ 1 & 2 & 3 & 22925.40 & 0.018 & & &
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0.053 & & & 0.080 & 0.393 & 0.665 & 0.668 & 0.018 \\ 2 & 1 & 2 & 31.53
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& 0.010 & 0.219 & & 0.070 & 0.719 & & 0.123 & & 3.125 & 3.044 &
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\\ 2 & 1 & 3 & 0.44 & 0.007 & 0.251 & & 0.279 & 1.229 & & 0.319
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& & 2.705 & 1.120 & 2.510 \\ 3 & 1 & 3 & 0.56 & 0.046 &
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0.064 & 0.243 & 0.288 & 1.151 & 0.233 & 0.302 & & 2.809 & 1.086 &
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2.689 \\ 3 & 2 & 2 & 1.65 & 0.512 & 0.074 & 0.201 & 0.161 & 1.225
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& 0.141 & 0.231 & 0.331 & 0.684 & 0.064 & \\
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\hline
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\end{tabular}%
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}
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\caption{GP hyperparameter values for different autoregressive lags}
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\label{tab:GP_hyperparameters}
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\end{table}
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In general, the results of Table~\ref{tab:GP_hyperparameters} show that the
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past outputs are important when predicting future values. Of importance is also
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the past inputs, with the exception of the models with very high variance, where
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the relative importances stay almost constant accross all the inputs. For the
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exogenous inputs, the outside temperature ($w2$) is generally more important
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than the solar irradiation ($w1$). In the case of more autoregressive lags for
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the exogenous inputs, the more recent information is usually more important,
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with a few exceptions {\color{red} Continue this sentence after considering the
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2/1/3 classical GP model}
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% TODO: [Hyperparameters] Classical GP parameters choice
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As for the case of the \acrlong{svgp}, the results for the classical
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\acrshort{gp} (cf. Table~\ref{tab:GP_hyperparameters}) are not necessarily
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representative of the relationships between the regressors of the
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\acrshort{svgp} model due to the fact that the dataset used for training is
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composed of the \textit{inducing variables}, which are not the real data, but a
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set of parameters chosen by the training algorithm in a way to best generate the
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original data.
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Therefore to better understand the behaviour of the \acrshort{svgp} models, the
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same computations as in Table~\ref{tab:GP_hyperparameters} have been made,
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presented in Table~\ref{tab:SVGP_hyperparameters}:
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\begin{table}[ht]
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%\vspace{-8pt}
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\centering
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\resizebox{\columnwidth}{!}{%
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\begin{tabular}{||c c c|c|c c c c c c c c c c c||}
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\hline
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\multicolumn{3}{||c|}{Lags} & Variance &\multicolumn{11}{c||}{Kernel
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lengthscales relative importance} \\
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$l_w$ & $l_u$ & $l_y$ & $\sigma^2$ &$\lambda_{w1,1}$ & $\lambda_{w1,2}$ &
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$\lambda_{w1,3}$ & $\lambda_{w2,1}$ & $\lambda_{w2,2}$ &
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$\lambda_{w2,3}$ & $\lambda_{u1,1}$ & $\lambda_{u1,2}$ &
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$\lambda_{y1,1}$ & $\lambda_{y1,2}$ & $\lambda_{y1,3}$\\
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\hline \hline
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1 & 1 & 1 & 0.2970 & 0.415 & & & 0.748 & & & 0.675 & & 0.680 & & \\
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1 & 1 & 2 & 0.2717 & 0.430 & & & 0.640 & & & 0.687 & & 0.559 & 0.584 & \\
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1 & 1 & 3 & 0.2454 & 0.455 & & & 0.589 & & & 0.671 & & 0.522 & 0.512 & 0.529 \\
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1 & 2 & 1 & 0.2593 & 0.310 & & & 0.344 & & & 0.534 & 0.509 & 0.597 & & \\
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1 & 2 & 3 & 0.2139 & 0.330 & & & 0.368 & & & 0.537 & 0.447 & 0.563 & 0.410 & 0.363 \\
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2 & 1 & 2 & 0.2108 & 0.421 & 0.414 & & 0.519 & 0.559 & & 0.680 & & 0.525 & 0.568 & \\
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2 & 1 & 3 & 0.1795 & 0.456 & 0.390 & & 0.503 & 0.519 & & 0.666 & & 0.508 & 0.496 & 0.516 \\
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3 & 1 & 3 & 0.1322 & 0.432 & 0.370 & 0.389 & 0.463 & 0.484 & 0.491 & 0.666 & & 0.511 & 0.501 & 0.526 \\
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3 & 2 & 2 & 0.1228 & 0.329 & 0.317 & 0.325 & 0.334 & 0.337 & 0.331 &
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0.527 & 0.441 & 0.579 & 0.435 & \\
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\hline
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\end{tabular}%
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}
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\caption{SVGP hyperparameter values for different autoregressive lags}
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\label{tab:SVGP_hyperparameters}
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\end{table}
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The results of Table~\ref{tab:SVGP_hyperparameters} are not very suprising, even
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if very different from the classical \acrshort{gp} case. The kernel variance is
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always of a reasonable value, and the relative importance of the lengthscales is
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relatively constant accross the board. It is certainly harder to interpret these
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results as pertaining to the relevance of the chosen regressors. For the
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\acrshort{svgp} model, the choice of the autoregressive lags has been made
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purely on the values of the loss functions, presented in
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Table~\ref{tab:SVGP_loss_functions}.
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\subsection{Loss functions}
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The most important metric for measuring the performance of a model is the value
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@ -84,14 +301,18 @@ mean of the obtained result:
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The \acrshort{msll} is approximately zero for simple models and negative for
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better ones.
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% TODO: [Hyperparameters] Explain loss table, difference in lags, etc.
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Table~\ref{tab:GP_loss_functions} and Table~\ref{tab:SVGP_loss_functions}
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present the values of the different loss functions for the same lag combinations
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as the ones analyzed in Section~\ref{sec:lengthscales} for the classical
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\acrshort{gp} and the \acrshort{svgp} models respectively:
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\begin{table}[ht]
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%\vspace{-8pt}
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\centering
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\begin{tabular}{||c c c|c c c c||}
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\hline
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\multicolumn{3}{||c|}{Lags} & \multicolumn{4}{c||}{Loss functions}\\
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w & u & y & RMSE & SMSE & MSLL & LPD\\
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$l_w$ & $l_u$ & $l_y$ & RMSE & SMSE & MSLL & LPD\\
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\hline \hline
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1 & 1 & 1 & 0.3464 & 0.36394 & 20.74 & 21.70 \\
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1 & 1 & 2 & 0.1415 & 0.06179 & -9.62 & -8.67 \\
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@ -108,39 +329,24 @@ better ones.
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\label{tab:GP_loss_functions}
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\end{table}
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\begin{table}[ht]
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%\vspace{-8pt}
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\centering
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\resizebox{\columnwidth}{!}{%
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\begin{tabular}{||c c c|c|c c c c c c c c c c c||}
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\hline
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\multicolumn{3}{||c|}{Lags} & Variance &\multicolumn{11}{c||}{Kernel
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lengthscales relative importance} \\
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w & u & y & $\sigma^2$ &$\lambda_{w1,1}$ & $\lambda_{w1,2}$ &
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$\lambda_{w1,3}$ & $\lambda_{w2,1}$ & $\lambda_{w2,2}$ &
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$\lambda_{w2,3}$ & $\lambda_{u1,1}$ & $\lambda_{u1,2}$ &
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$\lambda_{y1,1}$ & $\lambda_{y1,2}$ & $\lambda_{y1,3}$\\
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\hline \hline
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1 & 1 & 1 & 0.11 & 0.721 & & & 2.633 & & & 0.569 & & 2.645 & & \\
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1 & 1 & 2 & 22.68 & 0.222 & & & 0.751 & & & 0.134 & & 3.154 & 3.073
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& \\ 1 & 1 & 3 & 0.29 & 0.294 & & & 1.303 & & & 0.356 & & 2.352
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& 1.361 & 2.045 \\ 1 & 2 & 1 & 7.55 & 0.157 & & & 0.779 & & &
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0.180 & 0.188 & 0.538 & & \\ 1 & 2 & 3 & 22925.40 & 0.018 & & &
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0.053 & & & 0.080 & 0.393 & 0.665 & 0.668 & 0.018 \\ 2 & 1 & 2 & 31.53
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& 0.010 & 0.219 & & 0.070 & 0.719 & & 0.123 & & 3.125 & 3.044 &
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\\ 2 & 1 & 3 & 0.44 & 0.007 & 0.251 & & 0.279 & 1.229 & & 0.319
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& & 2.705 & 1.120 & 2.510 \\ 3 & 1 & 3 & 0.56 & 0.046 &
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0.064 & 0.243 & 0.288 & 1.151 & 0.233 & 0.302 & & 2.809 & 1.086 &
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2.689 \\ 3 & 2 & 2 & 1.65 & 0.512 & 0.074 & 0.201 & 0.161 & 1.225
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& 0.141 & 0.231 & 0.331 & 0.684 & 0.064 & \\
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\hline
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\end{tabular}%
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}
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\caption{GP hyperparameter values for different autoregressive lags}
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\label{tab:GP_hyperparameters}
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\end{table}
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For the classical \acrshort{gp} model (cf. Table~\ref{tab:GP_loss_functions}) a
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number of different lag combinations give rise to models with very large
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\acrshort{msll}/\acrshort{lpd} values. This might indicate that those models are
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overconfident, either due to the very large kernel variance parameter, or the
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specific lengthscales combinations. The model with the best
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\acrshort{rmse}/\acrshort{smse} metrics $\mathcal{M}$($l_w = 1$, $l_u = 2$, $l_y
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= 3$) had very bad \acrshort{msll} and \acrshort{lpd} metrics, as well as by far
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the largest variance of all the combinations. On the contrary the
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$\mathcal{M}$($l_w = 3$, $l_u = 1$, $l_y = 3$) model has the best
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\acrshort{msll} and \acrshort{lpd} performance, while still maintaining small
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\acrshort{rmse} and \acrshort{smse} values. The inconvenience of this set of
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lags is the large number of regressors, which leads to much more expensive
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computations. Other good choices for the combinations of lags are
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$\mathcal{M}$($l_w = 2$, $l_u = 1$, $l_y = 3$) and $\mathcal{M}$($l_w = 1$, $l_u
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= 1$, $l_y = 3$), which have good performance on all four metrics, as well as
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being cheaper from a computational perspective. In order to make a more informed
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choice for the best hyperparamerers, the performance of all three combinations
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has been analysed.
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\begin{table}[ht]
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%\vspace{-8pt}
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@ -148,7 +354,7 @@ better ones.
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\begin{tabular}{||c c c|c c c c||}
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\hline
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\multicolumn{3}{||c|}{Lags} & \multicolumn{4}{c||}{Loss functions}\\
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w & u & y & RMSE & SMSE & MSLL & LPD\\
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$l_w$ & $l_u$ & $l_y$ & RMSE & SMSE & MSLL & LPD\\
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\hline \hline
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1 & 1 & 1 & 0.3253 & 0.3203 & 228.0278 & 228.9843 \\
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1 & 1 & 2 & 0.2507 & 0.1903 & 175.5525 & 176.5075 \\
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@ -166,42 +372,21 @@ better ones.
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\label{tab:SVGP_loss_functions}
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\end{table}
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\begin{table}[ht]
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%\vspace{-8pt}
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\centering
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\resizebox{\columnwidth}{!}{%
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\begin{tabular}{||c c c|c|c c c c c c c c c c c||}
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\hline
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\multicolumn{3}{||c|}{Lags} & Variance &\multicolumn{11}{c||}{Kernel
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lengthscales relative importance} \\
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w & u & y & $\sigma^2$ &$\lambda_{w1,1}$ & $\lambda_{w1,2}$ &
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$\lambda_{w1,3}$ & $\lambda_{w2,1}$ & $\lambda_{w2,2}$ &
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$\lambda_{w2,3}$ & $\lambda_{u1,1}$ & $\lambda_{u1,2}$ &
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$\lambda_{y1,1}$ & $\lambda_{y1,2}$ & $\lambda_{y1,3}$\\
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\hline \hline
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1 & 1 & 1 & 0.2970 & 0.415 & & & 0.748 & & & 0.675 & & 0.680 & & \\
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1 & 1 & 2 & 0.2717 & 0.430 & & & 0.640 & & & 0.687 & & 0.559 & 0.584 & \\
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1 & 1 & 3 & 0.2454 & 0.455 & & & 0.589 & & & 0.671 & & 0.522 & 0.512 & 0.529 \\
|
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1 & 2 & 1 & 0.2593 & 0.310 & & & 0.344 & & & 0.534 & 0.509 & 0.597 & & \\
|
||||
1 & 2 & 3 & 0.2139 & 0.330 & & & 0.368 & & & 0.537 & 0.447 & 0.563 & 0.410 & 0.363 \\
|
||||
2 & 1 & 2 & 0.2108 & 0.421 & 0.414 & & 0.519 & 0.559 & & 0.680 & & 0.525 & 0.568 & \\
|
||||
2 & 1 & 3 & 0.1795 & 0.456 & 0.390 & & 0.503 & 0.519 & & 0.666 & & 0.508 & 0.496 & 0.516 \\
|
||||
3 & 1 & 3 & 0.1322 & 0.432 & 0.370 & 0.389 & 0.463 & 0.484 & 0.491 & 0.666 & & 0.511 & 0.501 & 0.526 \\
|
||||
3 & 2 & 2 & 0.1228 & 0.329 & 0.317 & 0.325 & 0.334 & 0.337 & 0.331 &
|
||||
0.527 & 0.441 & 0.579 & 0.435 & \\
|
||||
\hline
|
||||
\end{tabular}%
|
||||
}
|
||||
\caption{SVGP hyperparameter values for different autoregressive lags}
|
||||
\label{tab:SVGP_hyperparameters}
|
||||
\end{table}
|
||||
The results for the \acrshort{svgp} model, presented in
|
||||
Table~\ref{tab:SVGP_loss_functions} are much less ambiguous. The
|
||||
$\mathcal{M}$($l_w = 1$, $l_u = 2$, $l_y = 3$) model has the best performance
|
||||
according to all four metrics, with most of the other combinations scoring much
|
||||
worse on the \acrshort{msll} and \acrshort{lpd} loss functions. This has
|
||||
therefore been chosen as the model for the full year simulations.
|
||||
|
||||
\clearpage
|
||||
|
||||
\subsection{Validation of hyperparameters}
|
||||
|
||||
% TODO: [Hyperparameters] Validation of hyperparameters
|
||||
|
||||
The validation of model parameters has the dual purpose of
|
||||
|
||||
|
||||
\subsubsection{Conventional Gaussian Process}
|
||||
|
||||
\begin{figure}[ht]
|
||||
|
|
@ -222,7 +407,7 @@ better ones.
|
|||
\begin{figure}[ht]
|
||||
\centering
|
||||
\includegraphics[width =
|
||||
\textwidth]{Plots/GP_113_-1pts_test_prediction_20_steps.png}
|
||||
\textwidth]{Plots/GP_113_-1pts_test_prediction_20_steps.pdf}
|
||||
\caption{}
|
||||
\label{fig:GP_multistep_validation}
|
||||
\end{figure}
|
||||
|
|
@ -248,7 +433,7 @@ better ones.
|
|||
\begin{figure}[ht]
|
||||
\centering
|
||||
\includegraphics[width =
|
||||
\textwidth]{Plots/SVGP_123_test_prediction_20_steps.png}
|
||||
\textwidth]{Plots/SVGP_123_test_prediction_20_steps.pdf}
|
||||
\caption{}
|
||||
\label{fig:SVGP_multistep_validation}
|
||||
\end{figure}
|
||||
|
|
|
|||
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