papers/hanzo-analytics-ml/sections/05-customer-segmentation.tex

\section{Customer Segmentation}


\subsection{Behavioral Embeddings}

We learn dense customer representations from behavioral sequences:

\begin{definition}[Behavioral Embedding]
For customer $c$ with event sequence $\mathcal{E}_c = (e_1, e_2, \ldots, e_n)$:
\begin{equation}
\mathbf{z}_c = encoder(\mathcal{E}_c) \in \mathbb{R}^d
\end{equation}
where $d = 64$ is the embedding dimension.
\end{definition}

\subsubsection{Sequence Encoder}

We use a bidirectional LSTM encoder:

\begin{align}
\overrightarrow{\mathbf{h}}_t &= LSTM(\mathbf{e}_t, \overrightarrow{\mathbf{h}}_{t-1}) \\
\overleftarrow{\mathbf{h}}_t &= LSTM(\mathbf{e}_t, \overleftarrow{\mathbf{h}}_{t+1}) \\
\mathbf{z}_c &= [\overrightarrow{\mathbf{h}}_n; \overleftarrow{\mathbf{h}}_1]
\end{align}

\subsection{Segmentation Algorithm}

\begin{algorithm}
\caption{Behavioral Customer Segmentation}
\begin{algorithmic}[1]
\Function{SegmentCustomers}{$\mathcal{C}$, $k$}
    \State \Comment{Compute behavioral embeddings}
    \For{$c \in \mathcal{C}$}
        \State $\mathbf{z}_c \gets$ \Call{ComputeEmbedding}{$\mathcal{E}_c$}
    \EndFor
    \State \Comment{K-means with cosine distance}
    \State $\mathcal{Z} \gets \{\mathbf{z}_c : c \in \mathcal{C}\}$
    \State $centroids \gets$ \Call{KMeansPlusPlus}{$\mathcal{Z}$, $k$}
    \Repeat
        \For{$c \in \mathcal{C}$}
            \State $segment(c) \gets \arg\min_i cosine(\mathbf{z}_c, centroids_i)$
        \EndFor
        \For{$i \in [1, k]$}
            \State $centroids_i \gets mean(\{\mathbf{z}_c : segment(c) = i\})$
        \EndFor
    \Until{converged}
    \State \Return $segment$
\EndFunction
\end{algorithmic}
\end{algorithm}

\subsection{Segment Interpretation}

Each segment is characterized by:

\begin{itemize}
    \item \textbf{Centroid features}: Average feature values
    \item \textbf{Behavioral patterns}: Common action sequences
    \item \textbf{Value metrics}: LTV, purchase frequency, engagement
\end{itemize}

\begin{table}[h]
\centering
\caption{Example Segment Profiles}
\begin{tabular}{lcccc}
\toprule
\textbf{Segment} & \textbf{Size} & \textbf{Avg LTV} & \textbf{Purchases/Yr} & \textbf{Churn Risk} \\
\midrule
Power Buyers & 8\% & \$1,247 & 12.3 & Low \\
Regular Shoppers & 24\% & \$342 & 4.1 & Medium \\
Browsers & 31\% & \$89 & 1.2 & High \\
Deal Seekers & 19\% & \$156 & 2.8 & Medium \\
Dormant & 18\% & \$45 & 0.3 & Very High \\
\bottomrule
\end{tabular}
\end{table}