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dynamic_dirichlet_deep_gp.pdf
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dynamic_dirichlet_deep_gp.tex
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] | |||

\begin{abstract} | |||

The data association problem is concearned with separating data coming from different generating processes, for example when data come from different data sources, contain significant noise, or exhibit multimodality. | |||

The data association problem is concerned with separating data coming from different generating processes, for example when data come from different data sources, contain significant noise, or exhibit multimodality. | |||

We present a fully Bayesian approach to this problem. | |||

Our model is capable of simultaneously solving the data association problem and the induced supervised learning problems. | |||

Underpinning our approach is the use of Gaussian process priors to encode the structure of both the data and the data associations. | |||

@@ -50,10 +50,10 @@ | |||

\section{Introduction} | |||

\label{sec:introduction} | |||

Real-world data often include multiple operational regimes of the considered system, for example a wind turbine or gas turbine~\parencite{hein_benchmark_2017}. | |||

As an example, consider a model describing the lift resulting from airflow around a wing profile as a function of attack angle. | |||

As an example, consider a model describing the lift resulting from airflow around the wing profile of an airplane as a function of attack angle. | |||

At a low angle the lift increases linearly with attack angle until the wing stalls and the characteristic of the airflow fundamentally changes. | |||

Building a truthful model of such data requires learning two separate models and correctly associating the observed data to each of the dynamical regimes. | |||

A similar example would be if our sensors that measure the lift are faulty in a manner such that we either get a accurate reading or a noisy one. | |||

A similar example would be if our sensors that measure the lift are faulty in a manner such that we either get an accurate reading or a noisy one. | |||

Estimating a model in this scenario is often referred to as a \emph{data association problem}~\parencite{Bar-Shalom:1987, Cox93areview}, where we consider the data to have been generated by a mixture of processes and we are interested in factorising the data into these components. | |||

\Cref{fig:choicenet_data} shows an example of faulty sensor data, where sensor readings are disturbed by uncorrelated and asymmetric noise. | |||

@@ -316,7 +316,7 @@ We collect the latent multi-layer function values as $\mat{F^\prime} = \Set{\mat | |||

\begin{align} | |||

\begin{split} | |||

\label{eq:deep_variational_distribution} | |||

\Variat*{\mat{F^\prime}, \mat{\alpha}, \mat{U^\prime}} | |||

\MoveEqLeft\Variat*{\mat{F^\prime}, \mat{\alpha}, \mat{U^\prime}} = \\ | |||

= &\Variat*{\mat{\alpha}, \Set*{\mat{u_\alpha^{\pix{k}}}}_{k=1}^K, \Set*{\mat{F_l^{\prime\pix{k}}}, \mat{u_l^{\prime\pix{k}}}}_{k=1,l=1}^{K,L}} \\ | |||

= &\prod_{k=1}^K\prod_{n=1}^N \Prob*{\mat{\alpha_n^{\pix{k}}} \given \mat{u_\alpha^{\pix{k}}}, \mat{x_n}}\Variat*{\mat{u_\alpha^{\pix{k}}}} \\ | |||

\MoveEqLeft\prod_{k=1}^K \prod_{l=1}^L \prod_{n=1}^N \Prob*{\mat{f_{n,l}^{\prime\pix{k}}} \given \mat{u_l^{\prime\pix{k}}}, \mat{x_n}}\Variat*{\mat{u_l^{\prime\pix{k}}}}, | |||

@@ -366,7 +366,8 @@ This extended bound thus has complexity $\Fun*{\Oh}{NM^2LK}$ to evaluate in the | |||

\newcolumntype{Z}{>{\columncolor{sStone!33}\centering\arraybackslash}X}% | |||

\begin{tabularx}{\linewidth}{rY|YZZZZZZ} | |||

\toprule | |||

Outliers & DAGP (MLL) & DAGP (RMSE) & CN & MDN & MLP & GPR & LGPR & RGPR \\ | |||

Outliers & DAGP & DAGP & CN & MDN & MLP & GPR & LGPR & RGPR \\ | |||

& \scriptsize MLL & \scriptsize RMSE & \scriptsize RMSE & \scriptsize RMSE & \scriptsize RMSE & \scriptsize RMSE & \scriptsize RMSE & \scriptsize RMSE \\ | |||

\midrule | |||

0\,\% & 2.86 & \textbf{0.008} & 0.034 & 0.028 & 0.039 & \textbf{0.008} & 0.022 & 0.017 \\ | |||

20\,\% & 2.71 & \textbf{0.008} & 0.022 & 0.087 & 0.413 & 0.280 & 0.206 & 0.013 \\ | |||

@@ -453,7 +454,8 @@ This extended bound thus has complexity $\Fun*{\Oh}{NM^2LK}$ to evaluate in the | |||

\label{fig:semi_bimodal:c} | |||

Normalized samples from the assignment process $\mat{\alpha}$ of the model shown in \cref{fig:semi_bimodal}. | |||

The assignment process is used to weigh the predictive distributions of the different modes depending on the position in the input space. | |||

The model has learned that the mode $k = 2$ is irrelevant, that the mode $k = 1$ is only relevant around the interval $[0, 5]$ and the outside this interval, the mode $k = 3$ is twice as likely as the mode $k = 4$. | |||

The model has learned that the mode $k = 2$ is irrelevant, that the mode $k = 1$ is only relevant around the interval $[0, 5]$. | |||

Outside this interval, the mode $k = 3$ is twice as likely as the mode $k = 4$. | |||

} | |||

\end{figure} | |||

% | |||

@@ -481,7 +483,7 @@ To avoid pathological solutions for high outlier ratios, we add a prior to the l | |||

The model proposed in~\parencite{choi_choicenet_2018}, called ChoiceNet (CN), is a specific neural network structure and inference algorithm to deal with corrupted data. | |||

In their work, they compare their approach to a standard multi-layer perceptron (MLP), a mixture density network (MDN), standard Gaussian process regression (GPR), leveraged Gaussian process regression (LGPR)~\parencite{choi_robust_2016}, and infinite mixtures of Gaussian processes (RGPR)~\parencite{rasmussen_infinite_2002}. | |||

\Cref{tab:choicenet} shows results for outlier rates varied from 0\,\% to 80\,\%. | |||

Besides the root mean squared error (RMSE), we also report the mean test log likelihood (MLL) of the process representing the target function in our model. | |||

Besides the root mean squared error (RMSE), we also report the mean test log likelihood (MLL) of the process representing the signal in our model. | |||

Up to an outlier rate of 40\,\%, our model correctly identifies the outliers and ignores them, resulting in a predictive posterior of the signal equivalent to standard GP regression without outliers. | |||

In the special case of 0\,\% outliers, DAGP correctly identifies that the process modelling the noise is not necessary and disables it, thereby simplifying itself to standard GP regression. | |||

@@ -553,7 +555,7 @@ Our third experiment is based on the cart-pole benchmark for reinforcement learn | |||

In this benchmark, the objective is to apply forces to a cart moving on a frictionless track to keep a pole, which is attached to the cart via a joint, in an upright position. | |||

We consider the regression problem of predicting the change of the pole's angle given the current state of the cart and the action applied. | |||

The current state of the cart consists of the cart's position and velocity and the pole's angular position and velocity. | |||

To simulate a dynamical system with changing states of operation our experimental setup is to sample trajectories from two different cart-pole systems and merging the resulting data into one training set. | |||

To simulate a dynamical system with changing system characteristics our experimental setup is to sample trajectories from two different cart-pole systems and merging the resulting data into one training set. | |||

The task is not only to learn a model which explains this data well, but to solve the association problem introduced by the different system configurations. | |||

This task is important in reinforcement learning settings where we study systems with multiple operational regimes. | |||

@@ -606,7 +608,7 @@ The data association problem is inherently ill-constrained and requires signific | |||

In this paper, we make use of interpretable Gaussian process priors allowing global a priori information to be included into the model. | |||

Importantly, our model is able to exploit information both about the underlying functions and the association structure. | |||

We have derived a principled approximation to the marginal likelihood which allows us to perform inference for flexible hierarchical processes. | |||

In future work, we would like to incorporate the proposed model in a reinforcement learning scenario where we study a dynamical system with state changes. | |||

In future work, we would like to incorporate the proposed model in a reinforcement learning scenario where we study a dynamical system with different operational regimes. | |||

\printbibliography |

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