papers/hanzo-analytics-ml/sections/04-predictive-models.tex

\section{Predictive Models}


\subsection{Purchase Prediction}

Predict probability of purchase within time horizon $h$:

\begin{equation}
P(purchase | \mathbf{f}_c, h) = \sigma(\mathbf{w}^T \phi(\mathbf{f}_c) + b)
\end{equation}

where $\mathbf{f}_c$ is the feature vector for customer $c$ and $\phi$ is a feature transformation.

\subsubsection{Model Architecture}

We employ a gradient boosted decision tree ensemble (XGBoost):

\begin{equation}
\hat{y}_c = \sum_{k=1}^{K} f_k(\mathbf{f}_c), \quad f_k \in \mathcal{F}
\end{equation}

where $\mathcal{F}$ is the space of regression trees.

\begin{table}[h]
\centering
\caption{Purchase Prediction Model Parameters}
\begin{tabular}{ll}
\toprule
\textbf{Parameter} & \textbf{Value} \\
\midrule
Number of trees & 500 \\
Max depth & 8 \\
Learning rate & 0.05 \\
Subsample & 0.8 \\
Column subsample & 0.7 \\
Min child weight & 10 \\
\bottomrule
\end{tabular}
\end{table}

\subsubsection{Feature Importance}

Top predictive features:

\begin{enumerate}
    \item Days since last purchase ($R_{lifetime}$)
    \item 30-day purchase frequency ($F_{30d}$)
    \item Cart abandonment rate (90-day)
    \item Average session engagement score
    \item Category affinity entropy
\end{enumerate}

\subsection{Churn Prediction}

\begin{definition}[Customer Churn]
Customer $c$ has churned if:
\begin{equation}
churned(c) \iff t_{now} - last\_activity(c) > \tau_{churn}
\end{equation}
where $\tau_{churn}$ is the merchant-specific churn threshold (typically 90 days).
\end{definition}

We predict churn probability using a survival analysis approach:

\begin{equation}
S(t | \mathbf{f}_c) = P(T > t | \mathbf{f}_c)
\end{equation}

where $T$ is the time until churn.

\subsubsection{Cox Proportional Hazards Model}

\begin{equation}
h(t | \mathbf{f}_c) = h_0(t) \exp(\boldsymbol{\beta}^T \mathbf{f}_c)
\end{equation}

The hazard ratio for feature $j$:

\begin{equation}
HR_j = \exp(\beta_j)
\end{equation}

\subsubsection{Risk Stratification}

Customers are stratified into risk tiers:

\begin{equation}
risk\_tier(c) = \begin{cases}
    HIGH & \text{if } P(churn | \mathbf{f}_c) > 0.7 \\
    MEDIUM & \text{if } 0.3 < P(churn | \mathbf{f}_c) \leq 0.7 \\
    LOW & \text{if } P(churn | \mathbf{f}_c) \leq 0.3
\end{cases}
\end{equation}