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\title{Arithmetic Mean}
\author{algorembrant}
\date{\today}

\begin{document}
\maketitle







The average of total lookback $T$ is defined as:
\[
\text{average} = \frac{1}{T} \sum_{t=0}^{T-1} x_{0-t}, \quad \text{where $t$ is $timestamp$ $indexing$ and $0$ as current time}
\]
which expands as
\[
    \frac{x_0 + x_{-1} + x_{-2} + x_{-3} + \ldots + x_{-(T-1)}}{T}, \quad \text{like} \quad
    \frac{x_0 + x_{-1} + x_{-2} + x_{-3}}{4}
\]
and other variants

\[
\text{average} = 
\begin{cases}
    \text{if lookback}, & \displaystyle \frac{1}{T} \sum_{t=0}^{T-1} x_{0-t} = \frac{x_0 + x_{-1} + x_{-2} + \ldots + x_{-(T-1)}}{T} \\[10pt]
    \text{if lookahead}, & \displaystyle \frac{1}{T} \sum_{t=0}^{T-1} x_{0+t} = \frac{x_0 + x_{1} + x_{2} + x_{3} + \ldots + x_{T-1}}{T}
\end{cases}
\]
or starts indexing at 1
\[
\text{average} = 
\begin{cases}
    \text{if lookback}, & \displaystyle \frac{1}{T} \sum_{t=1}^{T} x_{-t} = \frac{x_{-1} + x_{-2} + x_{-3} + \ldots + x_{-(T)}}{T} \\[10pt]
    \text{if lookahead}, & \displaystyle \frac{1}{T} \sum_{t=1}^{T} x_{t} = \frac{x_{1} + x_{2} + x_{3} + \ldots + x_{T}}{T}
\end{cases}
\]
\end{document}
