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\title{LSTM Model and Stock Price Prediction}
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\date{}

\begin{document}

\maketitle

\section*{Stock Price Equation}

The price of a stock \( P(t) \) at discrete time \( t \in \{t_1, t_2, t_3, \ldots\} \) is given by:

\[
P(t) = P(t-1) + F_{\text{macro}}(t) + F_{\text{micro}}(t) + F_{\text{technical}}(t) + F_{\text{noise}}(t)
\]

\begin{itemize}
    \item \( P(t-1) \): Price of the stock at the previous time step.
    \item \( F_{\text{macro}}(t) \): Macro-level influences.
    \item \( F_{\text{micro}}(t) \): Micro-level influences.
    \item \( F_{\text{technical}}(t) \): Technical analysis factors.
    \item \( F_{\text{noise}}(t) \): Stochastic noise term.
\end{itemize}

\subsection*{Macro Influences}

\[
F_{\text{macro}}(t) = \alpha_1 G(t) + \alpha_2 I(t) + \alpha_3 R(t)
\]

\begin{itemize}
    \item \( \alpha_i \): Weights determining the strength of each factor.
    \item \( G(t) \): GDP growth/market sentiment, modeled as:
    \[
    G(t) = y \sin\left(\frac{2\pi t}{T_B}\right) + N_2 Z_2(t)
    \]
    \item \( I(t) \): Inflation rate, modeled as:
    \[
    I(t) = \Theta e^{-\lambda_0 t} + N_2 Z_2(t)
    \]
    \item \( R(t) \): Risk-free interest rate:
    \[
    R(t) = r_0 + N_3 Z_3(t)
    \]
\end{itemize}

\subsection*{Micro Influences}

\[
F_{\text{micro}}(t) = \beta_1 E(t) + \beta_2 S(t) + \beta_3 C(t)
\]

\begin{itemize}
    \item \( E(t) \): Earnings per share, \( E(t) = E_0 e^{\mu t} \left[1 + \sin\left(\frac{\pi t}{T_E}\right)\right] \)
    \item \( S(t) \): Scale growth rates:
    \[
    S(t) = \frac{\text{Max scale level}}{1 + e^{-K_0(t-t_0)}} + N_5 Z_5(t)
    \]
    \item \( C(t) \): Competition index:
    \[
    C(t) = \frac{1}{t + \text{Season growth cycle}} + N_6 Z_6(t)
    \]
\end{itemize}

\subsection*{Technical Factors}

\[
F_{\text{technical}}(t) = \delta_1 M(t) + \delta_2 V(t)
\]

\begin{itemize}
    \item \( M(t) \): Momentum, \( M(t) = P(t+1) - P(t+5) \)
    \item \( V(t) \): Volatility:
    \[
    V(t) = \sqrt{\frac{1}{W} \sum_{i=1}^N [P(t-i) - \overline{P}(t)]^2}
    \]
    where \( \overline{P}(t) = \frac{1}{N} \sum_{i=1}^N P(t-i) \).
\end{itemize}

\subsection*{Noise Term}

\[
F_{\text{noise}}(t) = \sigma Z(t)
\]

\begin{itemize}
    \item \( \sigma Z(t) \): Noise term, where \( Z(t) \sim N(0, 1) \).
\end{itemize}

\section*{LSTM Architecture}

\begin{enumerate}
    \item Feature vector \( X(t) \):
    \[
    X(t) = \begin{bmatrix}
    P(t-2) \\
    P(t-1) \\
    P(t) \\
    G(t) \\
    E(t) \\
    S(t) \\
    C(t) \\
    M(t) \\
    V(t)
    \end{bmatrix}
    \]
    \item LSTM components:
    \[
    f(t) = \sigma(W_f X(t) + U_f h(t-1) + b_f)
    \]
    \[
    i(t) = \sigma(W_i X(t) + U_i h(t-1) + b_i)
    \]
    \[
    \tilde{C}(t) = \tanh(W_c X(t) + U_c h(t-1) + b_c)
    \]
    \[
    C(t) = f(t) \cdot C(t-1) + i(t) \cdot \tilde{C}(t)
    \]
    \[
    o(t) = \sigma(W_o X(t) + U_o h(t-1) + b_o)
    \]
    \[
    h(t) = o(t) \cdot \tanh(C(t))
    \]

\end{enumerate}

\section*{Loss Function}

\[
\text{MSE} = \frac{1}{T} \sum_{t=1}^T [P(t) - \hat{P}(t)]^2
\]
\[
\text{MAE} = \frac{1}{T} \sum_{t=1}^T |P(t) - \hat{P}(t)|
\]

\end{document}

