\begin{algorithm}[htb]
    \caption{Training procedure of the moment-based neural Hawkes}
    \label{algo:pinn_training}
    \begin{enumerate}
        \item
        Estimate the first order statistics $\Lambda$ and second order statistics $G$ given by Equation \eqref{eq:def_G_statistics} over the time and mark domains.

        \item
        For $i=1$ to $D$:
        \begin{enumerate}
            \item
            Represent the solution of the $i$-th characterization equation of Theorem \ref{th:characterization_theorem} by a DGM neural network $u_\theta^i$ and initialize its weights $\theta$ with a Glorot scheme.
            
            \item
            Set the number of epochs $E$, the size $N^c$ of the training set $\mathcal{E}$, the batch size $B$ such that $N^c=0\,[B]$, a decreasing sequence of learning rates $(\gamma_e)_{e\geq 1}$ and a random sampling scheme, \textit{e.g.} the procedure described above. We use $\gamma_e=\gamma_0100^{-\frac{e}{E}}$.
            
            \item
            For $e=1$ to $E$:
            \begin{enumerate}
                \item Sample a training set $\mathcal{E}=\{(t_n,x_n), 1\leq n\leq N^c\}\}$ and a validation set $\bar{\mathcal{E}}$ . 
                \item Apply a standardization and log-scaling to the data points.
                \item Over the elements of $\mathcal{E}$: compute the equation residuals $(\varepsilon_n^{ij}(\theta))_{1\leq j\leq D}$ and the temporal weights $(\omega_n^{ij}(\theta))_{1\leq j\leq D}$.
                \item For $s = 1$ to $\frac{N^c}{B}$:
                \begin{itemize}
                    \item Using Equation \eqref{eq:weighted_loss_function_nn_i}, update the weights via a mini-batch gradient descent
                    
                    \begin{equation}
                      \theta \leftarrow \theta - \gamma_e\frac{1}{B}\sum_{n=(s-1)B+1}^{sB}\nabla_\theta\mathcal{L}_{i,n}^{\omega}(\theta).
                    \end{equation}
                \end{itemize}
                \item Using Equation \eqref{eq:unweighted_loss_function_nn_i}, compute the unweighted validation loss $\mathcal{L}_i(\theta, \bar{\mathcal{E}})$.
            \end{enumerate}
            
            \item \textbf{Outputs:} An estimate $\theta^i$ of the optimal weights for the $i$-th model and a list of validation losses over the epochs.
        \end{enumerate}
        \item \textbf{Outputs:} $D$ models $(u_{\theta^i}^i)_{1\leq i\leq D}$, the $i$-th model representing the $i$-th row of the kernel matrix.
    \end{enumerate}
\end{algorithm}

\begin{algorithm}
[htb]
\caption{Training procedure of the moment-based neural Hawkes}
\begin{enumerate}
\item
Estimate the first order statistics $\Lambda$ and second order statistics $G$ given by Equation \eqref{eq:def_G_statistics} over the time and mark domains.
\item
For $i=1$ to $D$:
\begin{enumerate}
\item
Represent the solution of the $i$-th characterization equation of Theorem \ref{th:characterization_theorem} by a DGM neural network $u_\theta^i$ and initialize its weights $\theta$ with a Glorot scheme.
\item
Set the number of epochs $E$, the size $N^c$ of the training set $\mathcal{E}$, the batch size $B$ such that $N^c=0\,[B]$, a decreasing sequence of learning rates $(\gamma_e)_{e\geq 1}$ and a random sampling scheme, \textit{e.g.} the procedure described above. We use $\gamma_e=\gamma_0100^{-\frac{e}{E}}$.
\item
For $e=1$ to $E$:
\begin{enumerate}
\item Sample a training set $\mathcal{E}=\{(t_n,x_n), 1\leq n\leq N^c\}\}$ and a validation set $\bar{\mathcal{E}}$ .
\item Apply a standardization and log-scaling to the data points.
\item Over the elements of $\mathcal{E}$: compute the equation residuals $(\varepsilon_n^{ij}(\theta))_{1\leq j\leq D}$ and the temporal weights $(\omega_n^{ij}(\theta))_{1\leq j\leq D}$.
\item For $s = 1$ to $\frac{N^c}{B}$:
\begin{itemize}
\item Using Equation \eqref{eq:weighted_loss_function_nn_i}, update the weights via a mini-batch gradient descent
\begin{equation*}
\theta \leftarrow \theta - \gamma_e\frac{1}{B}\sum_{n=(s-1)B+1}^{sB}\nabla_\theta\mathcal{L}_{i,n}^{\omega}(\theta).
\end{equation*}
\end{itemize}
\item Using Equation \eqref{eq:unweighted_loss_function_nn_i}, compute the unweighted validation loss $\mathcal{L}_i(\theta, \bar{\mathcal{E}})$.
\end{enumerate}
\item \textbf{Outputs:} An estimate $\theta^i$ of the optimal weights for the $i$-th model and a list of validation losses over the epochs.
\end{enumerate}
\item \textbf{Outputs:} $D$ models $(u_{\theta^i}^i)_{1\leq i\leq D}$, the $i$-th model representing the $i$-th row of the kernel matrix.
\end{enumerate}
\end{algorithm}
{
    "title": "neural hawkes: non-parametric estimation in high dimension and causality   analysis in cryptocurrency markets",
    "id": "2401.09361",
    "abstract": "we propose a novel approach to marked hawkes kernel inference which we name the moment-based neural hawkes estimation method. hawkes processes are fully characterized by their first and second order statistics through a fredholm integral equation of the second kind. using recent advances in solving partial differential equations with physics-informed neural networks, we provide a numerical procedure to solve this integral equation in high dimension. together with an adapted training pipeline, we give a generic set of hyperparameters that produces robust results across a wide range of kernel shapes. we conduct an extensive numerical validation on simulated data. we finally propose two applications of the method to the analysis of the microstructure of cryptocurrency markets. in a first application we extract the influence of volume on the arrival rate of btc-usd trades and in a second application we analyze the causality relationships and their directions amongst a universe of 15 cryptocurrency pairs in a centralized exchange.",
    "categories": "q-fin.tr q-fin.mf",
    "doi": "",
    "created": "2024-01-17",
    "updated": "2024-01-18",
    "authors": [
        "timoth\u00e9e fabre",
        "ioane muni toke"
    ],
    "affiliation": [],
    "url": "https://arxiv.org/abs/2401.09361"
}
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