temperature-based-fertility-prediction-thesis/thesis/sections/background.tex

%! Author = alex
%! Date = 3/7/25


\section{Background}\label{sec:background}

\subsection{Physiological Background}\label{subsec:physiological_background}

\subsubsection{Menstrual Cycle}\label{subsec:menstrual_cycle}
The menstrual cycle describes the physiological changes in the female body that prepare it for pregnancy.
It is divided into two phases: the \textbf{follicular phase} and the \textbf{luteal phase}.
\\
During the follicular phase, the ovarian follicles mature, and the endometrium (the inner lining of the uterus) thickens
in preparation for a potential implantation of a fertilized egg.
Around day 14 of a typical cycle, ovulation occurs, marking the transition to the luteal phase.
Ovulation refers to the rupture of the mature ovarian follicle and the release of an egg cell into the fallopian tube.
Figure~\ref{fig:background_basic_female_reproductive_system} illustrates the female reproductive system,
including the ovaries and the fallopian tubes.
\\
\begin{figure}
    \centering
    \includegraphics[width=0.4\textwidth]{background_female_reproductive_organs}
    \caption{The basic female reproductive system~\cite{wikimedia_commons_basic_2019}.}
    \label{fig:background_basic_female_reproductive_system}
\end{figure}
Ovulation is triggered by a surge in \textbf{luteinizing hormone (LH)}
and \textbf{follicle-stimulating hormone (FSH)}, following a peak in estradiol levels.
As ovulation occurs, estradiol levels drop, progesterone levels begin to rise and a slight increase in body temperature
(typically around 0.5°C) can be observed.
This marks the beginning of the luteal phase.
\\
During the luteal phase, the endometrium thickens further, creating an optimal environment for embryo implantation.
LH and FSH levels decrease, while progesterone remains elevated to support endometrial maintenance.
If fertilization does not occur, progesterone levels drop, leading to the shedding of the endometrial lining along
with the unfertilized egg.
This process, known as menstruation, marks the beginning of a new cycle.
\\
The menstrual cycle typically lasts around 28 days, with ovulation occurring near the midpoint.
However, variations, particularly in the follicular phase length, are common and can be influenced by factors such as stress, diet, exercise and age~\cite{silberstein_physiology_2000}.
Figure~\ref{fig:background_menstrual_cycle_physiology} provides a detailed overview of the hormonal and physiological changes
throughout the menstrual cycle.

\begin{figure}
    \centering
    \includegraphics[width=0.6\textwidth]{background_menstrual_cycle_physiology}
    \caption{Physiological changes during the menstrual cycle~\cite{pedroso_menstrual_2022}.}
    \label{fig:background_menstrual_cycle_physiology}
\end{figure}

Not every cycle results in ovulation—a phenomenon known as anovulation—which leads to a monophasic temperature pattern.
Anovulation can have various causes, including hormonal imbalances, stress, or underlying health conditions~\cite{rosenfield_adolescent_2013}.

Anovulation is reflected in temperature data as either an absence of a clear temperature rise or a rise
that is insufficient in magnitude or duration to be considered a reliable indicator of ovulation.
Distinguishing between ovulatory and anovulatory cycles is challenging, as the only definitive confirmation of
successful ovulation in a clinical sense is a positive pregnancy test.
Even ultrasound imaging can only confirm that an egg was released from its follicle—not whether it was fertilized or successfully implanted.

%TODO: find source for this

%TODO: show plot of different cycle types

\subsubsection{Fertility Prediction}\label{subsec:fertility_prediction}
Throughout the menstrual cycle, the chance of fertilization varies significantly.
An egg cell released from the ovary during ovulation, can be fertilized for up to 24 hours.
However, since male sperm cells can survive up to 6 days inside the female reproductive tract,
the fertile window is typically defined as the five days before ovulation until one day after ovulation~\cite{dunson_day-specific_1999}.
Research by~\citeauthor{dunson_day-specific_1999} has shown that the highest chance of fertilization is around one day before ovulation,
as illustrated in Figure~\ref{fig:background_pregnancy_chance}.

\begin{figure}
    \centering
    \includegraphics[width=0.7\textwidth]{background_pregnancy_chance_over_time}
    \caption{Chance of fertilization depending on the day of the menstrual cycle.
    The highest chance is around one day before ovulation~\cite{dunson_day-specific_1999}.}
    \label{fig:background_pregnancy_chance}
\end{figure}

It is important to note that fertility prediction is inherently dependent on ovulation prediction.
Since the probability of conception is tightly linked to ovulation timing, the accuracy of fertility prediction methods
is constrained by the precision of ovulation detection.
This relationship underscores the necessity of developing reliable ovulation prediction models,
as even small inaccuracies can significantly impact fertility assessments.

\subsubsection{Physiological Signs of Ovulation}\label{subsec:physiological_signs}
Several physiological signs correlate with ovulation and can be used for prediction.
As shown in Figure~\ref{fig:background_menstrual_cycle_physiology}, these include hormonal fluctuations (LH and FSH surges) and
changes in body temperature.
Additionally, variations in cervical mucus consistency, salivary ferning patterns,
and electrical resistance of the skin and vaginal mucosa have been observed~\cite{silberstein_physiology_2000}.

Among these, ultrasonography provides the most accurate confirmation of ovulation by detecting follicular changes,
but it is costly and requires specialized equipment.
Hormone measurements in urine and blood are widely used and available in at-home test kits,
but they require frequent testing.
Temperature-based methods, particularly body temperature tracking, offer a non-invasive alternative by
detecting the slight temperature rise that follows ovulation.
Advances in wearable technology have further enabled continuous and automated temperature monitoring,
improving accessibility and usability~\cite{alexander_fertilitatsmonitoring_2014, luo_detection_2020, yu_tracking_2022}.

\subsection{Technical Background}\label{subsec:technological_background}

\subsubsection{Time Series Analysis}\label{subsubsec:time_series_analysis}
Time series analysis is a fundamental tool for studying sequential data that evolves over time.
Unlike other data types, time series data has an inherent temporal order, where each data point is associated
with a timestamp, capturing its dependence on past values.
Time series analysis typically serves two main goals:
Understanding the underlying mechanisms that lead to the observed data and predicting future data points based on the
historical information and potentially external factors~\cite{cryer_time_2008}
\\
Time series analysis encompasses various methods, ranging from simple statistical models to complex deep learning architectures.
Classical methods


In the following, we will introduce the two most common approaches used for machine learning on time series data,
LSTMs and transformers.

\subsubsection{RNN and LSTM Networks}\label{subsubsec:lstm_networks}
Recurrent Neural Networks (RNNs) are extensions of classical neural networks that incorporate cyclic connections between neurons.
These recurrent connections allow the network to retain information from previous inputs by feeding the hidden state
from a prior time step into the current one, enabling a form of temporal memory.

In practice, this means that input data is processed sequentially, one step at a time.
At each step \(t\), the input \(x_t\) is combined with the previous hidden state \(h_{t-1}\) to produce a new
hidden state \(h_t\), which contributes to the output \(o_t\).
This process allows the network to learn temporal dependencies and model sequential data effectively~\cite{medsker_recurrent_1999}.

However, this approach has the downside that the model cannot explicitly control how it remembers or forgets information at each step,
limiting its ability to manage long-term dependencies.
During training via backpropagation, the weights of a neural network are updated based on the partial derivatives of the loss function.
The more propagations required (e.g., in deeper networks), the more multiplications are needed to compute gradients for earlier weights.

RNNs are typically trained using \textit{backpropagation through time} (BPTT), in which gradients are propagated through many time steps,
leading to numerical instability.
If the gradients shrink exponentially, the model suffers from the \emph{vanishing gradient} problem;
if they grow exponentially, it results in \emph{exploding gradients}~\cite{hochreiter_vanishing_1998}.
In both cases, learning is significantly impaired.

Exploding gradients can often be mitigated using techniques such as \emph{gradient clipping},
where the magnitude of the gradient is capped---typically within a range of \([-1, 1]\)---to stabilize training.

Figure~\ref{fig:rnn_unfolded} illustrates the unfolded structure of an RNN across three time steps.
This technique, known as \emph{unfolding}, clarifies how sequential inputs update the hidden state and generate
outputs at each time step.

\begin{figure}
    \centering
    \includegraphics[width=0.7\textwidth]{recurrent_neural_network_unfold}
    \caption{Schematic diagram of the unfolded structure of a recurrent neural network~\cite{fdeloche_english_2017}}
    \label{fig:rnn_unfolded}
\end{figure}

To address the vanishing gradient problem and enable better long-term memory,
\emph{Long Short-Term Memory} (LSTM) networks were introduced by \citeauthor{hochreiter_long_1997}~\cite{hochreiter_long_1997}.
LSTMs extend the RNN architecture by incorporating a memory cell and a series of gates that regulate the flow of
information: the \emph{forget gate}, the \emph{input gate}, and the \emph{output gate}.

\begin{itemize}
    \item The \emph{forget gate} determines which information from the previous cell state should be discarded.
    \item The \emph{input gate} controls what new information is added to the cell state.
    \item The \emph{output gate} selects relevant parts of the current cell state to produce the output and the next hidden state.
\end{itemize}

Each gate employs a sigmoid activation function to regulate the flow of information,
allowing LSTMs to preserve and update memory over long sequences.


\begin{figure}
    \centering
    \includegraphics[width=0.6\textwidth]{lstm_cell_diagram}
    \caption{Diagram of a LSTM cell showing the flow of information \cite{chevalier_english_2018}}
    \label{fig:lstm_architecture}
\end{figure}

Figure~\ref{fig:lstm_architecture} illustrates the architecture of an LSTM cell.
On the left, the previous cell state \(C_{t-1}\) represents the long-term memory,
while the hidden state \(h_{t-1}\) encodes the short-term memory from the preceding time step.
The current input \(x_t\) is processed together with these states to update the cell.
The resulting new cell state \(C_t\) and hidden state \(h_t\) are passed forward to the next time step or used to produce the model’s output.
Within the cell, the forget gate determines how much of the previous cell state \(C_{t-1}\) is retained.
The input gate updates the cell state with new information derived from the current input and previous hidden state.
Finally, the output gate controls how much of the updated cell state contributes to the hidden state \(h_t\),
which is passed on to the next time step or used for prediction.

Mathematically, the core LSTM operations are given by:

\[
    \begin{aligned}
        f_t &= \sigma(W_f [h_{t-1}, x_t] + b_f) \\
        i_t &= \sigma(W_i [h_{t-1}, x_t] + b_i) \\
        \tilde{c}_t &= \tanh(W_c [h_{t-1}, x_t] + b_c) \\
        c_t &= f_t \odot c_{t-1} + i_t \odot \tilde{c}_t \\
        o_t &= \sigma(W_o [h_{t-1}, x_t] + b_o) \\
        h_t &= o_t \odot \tanh(c_t)
    \end{aligned}
\]

Here, \( \odot \) denotes element-wise multiplication, and \( \sigma \) is the sigmoid function.
These equations allow for more stable training and long-range temporal modeling.
\\

LSTMs are widely used in biomedical applications due to their capacity to handle sequences of variable length and complexity.
In the context of ovulation prediction, where hormonal patterns exhibit periodicity but also irregularity,
LSTMs are well-suited to learn relevant time-dependent signals from sequential physiological measurements.

While powerful, LSTMs can be computationally intensive and sensitive to hyperparameter tuning.
Therefore, they are often compared with alternative architectures,
including simpler feedforward networks and more recent attention-based models,
to evaluate trade-offs in performance, interpretability, and computational cost.

The next section introduce the \emph{Transformer} architecture, a more recent alternative that forgoes
recurrence in favor of attention mechanisms

\subsubsection{Transformer Models}\label{subsubsec:transformer_models}