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Add A3I neuro-probabilistic model + survival analysis

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Tian Cheng (Riccardo) Xia
2024-11-01 11:29:58 +01:00
parent e9b9b4835c
commit 7dbab460a8
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src/year2/artificial-intelligence-in-industry/img/_rul_censoring.pdf Normal file
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src/year2/artificial-intelligence-in-industry/sections/_arrivals_predicition.tex
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@ -1,4 +1,4 @@
\chapter{Arrivals prediction: Hospital emergency room}
\chapter{Arrivals prediction: Hospital emergency room} \label{ch:ap_hospital}
\section{Data}
@ -22,7 +22,7 @@ Each row of the dataset represents a patient and the features are:
\section{Approaches}
\begin{remark}
MSE assumes that the conditional distribution of the predictions follows a normal distribution.
MSE assumes that the conditional distribution $\prob{y | x; \theta}$ of the predictions follows a normal distribution (i.e., $y \sim \mathcal{N}(\mu(x; \theta), \sigma)$).
\end{remark}
\begin{remark}
@ -107,7 +107,7 @@ Each row of the dataset represents a patient and the features are:
Some considerations must be made:
\begin{descriptionlist}
\item[Only positive rates]
As $\hat{\lambda}$ must be positive, it is possible to combine a logarithm and an exponentiation to achieve this:
As $\hat{\lambda}$ must be positive, it is possible to combine a logarithm (i.e., assume that the MLP outputs a log-rate) and an exponentiation to achieve this:
\[
\begin{split}
\log(\hat{\lambda}) &= \texttt{MLP}(x) \\
@ -115,6 +115,10 @@ Each row of the dataset represents a patient and the features are:
\end{split}
\]
\begin{remark}
A strictly positive activation function can also be used.
\end{remark}
\item[Standardization]
The input of the network can be standardized. On the other hand, standardizing the output is wrong as the Poisson distribution is discrete.
@ -133,4 +137,59 @@ Each row of the dataset represents a patient and the features are:
\end{remark}
\end{descriptionlist}
\end{description}
\begin{remark}
With one-hot encoding, a linear model can be non-linearized into a lookup table.
\indenttbox
\begin{example}
\phantom{}
\begin{minipage}[t]{0.3\linewidth}
Consider the dataset:
\begin{table}[H]
\centering
\footnotesize
\begin{tabular}{c|c}
\toprule
$x$ & $y$ \\
\midrule
$0$ & $1$ \\
$1$ & $4$ \\
$2$ & $2$ \\
\bottomrule
\end{tabular}
\end{table}
\end{minipage}
\begin{minipage}[t]{0.65\linewidth}
A linear model learns a straight line:
\[ f(x) = \alpha x \]
\end{minipage}\\[1em]
\begin{minipage}[t]{0.3\linewidth}
With the dataset:
\begin{table}[H]
\centering
\footnotesize
\begin{tabular}{ccc|c}
\toprule
$x_0$ & $x_1$ & $x_2$ & $y$ \\
\midrule
$1$ & $0$ & $0$ & $1$ \\
$0$ & $1$ & $0$ & $4$ \\
$0$ & $0$ & $1$ & $2$ \\
\bottomrule
\end{tabular}
\end{table}
\end{minipage}
\begin{minipage}[t]{0.65\linewidth}
A linear model learns a linear combination:
\[ f(x) = \alpha x_0 + \beta x_1 + \gamma x_2 \]
Which allows to easily learn the dataset with $\alpha=1$, $\beta=4$, and $\gamma=2$
\end{minipage}
\end{example}
\end{remark}
\begin{remark}
Having access to the distribution allows to compute all possible statistics. For instance, it is possible to plot mean and standard deviation.
\end{remark}
\end{description}

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src/year2/artificial-intelligence-in-industry/sections/_remaining_useful_life.tex
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@ -253,4 +253,104 @@ Predict RUL with a classifier $f_\varepsilon$ (for a chosen $\varepsilon$) that
\end{enumerate}
\end{enumerate}
\end{description}
\end{description}
\end{description}
\subsection{Survival analysis model}
\begin{remark}
Chronologically, this approach has been presented after \Cref{ch:ap_hospital}.
\end{remark}
\begin{description}
\item[Survival analysis model] \marginnote{Survival analysis model}
Probabilistic model to estimate the survival time of an entity.
\item[Survival analysis formalization]
Consider a random variable $T$ to model the survival time. The simplest model can be defined as:
\[ t \sim \prob{T} \]
Remaining survival time depends on the time $\bar{t}$, therefore we have that:
\[ t \sim \prob{T \mid \bar{t}} \]
Remaining survival time also depends on the past sensor readings $X_{\leq \bar{t}}$ and future readings $X_{> \bar{t}}$ (at this stage, we want to capture what affects the survival time, even if we do not have access to some variables), therefore we have that:
\[ t \sim \prob{T \mid \bar{t}, X_{\leq \bar{t}}, X_{> \bar{t}}} \]
\begin{description}
\item[Marginalization]
Average over all the possible outcomes of a random variable to cancel it out.
For this problem, we do not have access to the future sensor readings, so we can marginalize them:
\[ t = \underset{X_{> \bar{t}} \sim \prob{X_{> \bar{t}}}}{\mathbb{E}} \left[ \prob{T \mid \bar{t}, X_{\leq \bar{t}}} \right] \]
\end{description}
\begin{remark}
In probabilistic terms, the regression approach previously taken can be modelled as:
\[ t \sim \mathcal{N}(\mu(X_{\bar{t}}), \sigma) \]
where $\mu(\cdot)$ is the regressor.
Compared to the survival analysis model, there are the following differences:
\begin{itemize}
\item Regressor only reasons on the sensor readings $X_{\bar{t}}$ at a single time step.
\item Regressor does not consider the current time.
\item Regressor assumes normal distribution with constant variance.
\end{itemize}
\end{remark}
\item[Neuro-probabilistic model]
We can assume that the model follows a normal distribution parametrized on both the mean and standard deviation:
\[ t \sim \mathcal{N}(\mu(X_{\bar{t}}, \bar{t}), \sigma(X_{\bar{t}}, \bar{t})) \]
\begin{remark}
The readings of a single time step are used as it can be shown that using multiple time steps does not yield significant improvements for this dataset.
\end{remark}
\begin{description}
\item[Architecture]
Use a neural network that output both $\mu$ and $\sigma$ that are passed to a distribution head.
\end{description}
\begin{figure}[H]
\centering
\includegraphics[width=0.7\linewidth]{./img/_rul_neuroprobabilistic.pdf}
\end{figure}
\end{description}
% \subsection{Survival function}
% \begin{description}
% \item[Censoring] \marginnote{Censoring}
% Hide from the dataset key events.
% \begin{remark}
% For this dataset, it is more realistic to use partial runs as run-to-failure experiments are expensive to obtain.
% \begin{figure}[H]
% \centering
% \includegraphics[width=0.7\linewidth]{./img/_rul_censoring.pdf}
% \caption{Example of partial runs}
% \end{figure}
% \end{remark}
% \item[Survival function] \marginnote{Survival function}
% Given the random variable $T$ to model survival time, the survival function $S$ is defined as:
% \[ S(\bar{t}) = \prob{T > \bar{t}} \]
% In other words, it is the probability of surviving at least until time $\bar{t}$.
% For this problem, the survival function can account for past sensor readings:
% \[ S(\bar{t}, X_{\leq \bar{t}}) = \prob{T > \bar{t} \mid X_{\leq \bar{t}}} \]
% \item[Hazard function] \marginnote{Hazard function}
% Given the random variable $T$ to model survival time, the hazard function $\lambda$ is defined as:
% \[ \lambda(\bar{t}) = \prob{T > \bar{t} \mid T > \bar{t}-1} \]
% In other words, it is the conditional probability of not surviving at time $\bar{t}$ knowing that the entity survived until time $\bar{t}-1$.
% With discrete time, the survival function can be factorized using the hazard function:
% \[ S(\bar{t}) = (1-\lambda(\bar{t})) \cdot (1 - \lambda(\bar{t}-1)) \cdot \dots \]
% For this problem, the hazard function is the following:
% \[
% \begin{gathered}
% \lambda(\bar{t}, X_{\bar{t}}) = \prob{T > \bar{t} \mid T > \bar{t}-1, X_{\bar{t}}} \\
% S(\bar{t}, X_{\bar{t}}) = (1-\lambda(\bar{t}, X_{\bar{t}})) \cdot (1 - \lambda(\bar{t}-1, X_{\bar{t}-1})) \cdot \dots
% \end{gathered}
% \]
% \end{description}