papers/hanzo-analytics-ml/sections/03-feature-engineering.tex

\section{Feature Engineering}


\subsection{Feature Categories}

We extract 847 features organized into categories:

\begin{table}[h]
\centering
\caption{Feature Categories}
\begin{tabular}{lcp{6cm}}
\toprule
\textbf{Category} & \textbf{Count} & \textbf{Description} \\
\midrule
Recency & 42 & Time since last action by type \\
Frequency & 89 & Action counts over time windows \\
Monetary & 67 & Purchase amounts and patterns \\
Engagement & 156 & Session depth, page views, dwell time \\
Product Affinity & 234 & Category preferences, brand loyalty \\
Temporal & 78 & Day/time patterns, seasonality \\
Device/Channel & 45 & Platform usage, referral sources \\
Social & 34 & Reviews, shares, referrals \\
Lifecycle & 102 & Customer age, milestone events \\
\bottomrule
\end{tabular}
\end{table}

\subsection{Recency-Frequency-Monetary (RFM) Features}

Classical RFM features are computed over multiple time windows:

\begin{definition}[Windowed RFM]
For customer $c$ and time window $w$:
\begin{align}
R_w(c) &= t_{now} - \max\{t : (t, c, PURCHASE, \cdot, \cdot) \in \mathcal{E}, t \in w\} \\
F_w(c) &= |\{e \in \mathcal{E} : e.c = c \land e.a = PURCHASE \land e.t \in w\}| \\
M_w(c) &= \sum_{\substack{e \in \mathcal{E} \\ e.c = c, e.a = PURCHASE, e.t \in w}} e.amount
\end{align}
\end{definition}

Time windows: $w \in \{7d, 30d, 90d, 365d, lifetime\}$

\subsection{Engagement Features}

\begin{definition}[Session Engagement Score]
For session $s$ with events $\mathcal{E}_s$:
\begin{equation}
engagement(s) = \alpha \cdot pages(s) + \beta \cdot duration(s) + \gamma \cdot depth(s)
\end{equation}
where:
\begin{itemize}
    \item $pages(s)$: unique pages viewed
    \item $duration(s)$: session length in seconds
    \item $depth(s)$: maximum funnel stage reached
\end{itemize}
\end{definition}

\subsection{Product Affinity Features}

We compute category and brand affinity scores:

\begin{equation}
affinity(c, category) = \frac{\sum_{p \in category} interactions(c, p)}{\sum_{p \in \mathcal{P}} interactions(c, p)}
\end{equation}

\begin{algorithm}
\caption{Real-Time Feature Computation}
\begin{algorithmic}[1]
\Function{UpdateFeatures}{$event$, $features$}
    \State $c \gets event.customer$
    \State $features[c].last\_seen \gets event.t$
    \If{$event.action = PURCHASE$}
        \State $features[c].purchase\_count \gets features[c].purchase\_count + 1$
        \State $features[c].total\_spend \gets features[c].total\_spend + event.amount$
        \State $features[c].last\_purchase \gets event.t$
    \ElsIf{$event.action = VIEW$}
        \State $features[c].view\_count \gets features[c].view\_count + 1$
        \State \Call{UpdateAffinity}{$c$, $event.product.category$}
    \EndIf
    \State \Call{UpdateWindowed}{$features[c]$, $event$}
    \State \Return $features[c]$
\EndFunction
\end{algorithmic}
\end{algorithm}