User behaviour modelling from web log (Cont.)

Summary

The lecture explains how to model website navigation from access logs as a discrete-time Markov process, then use the model to measure customer behavior and compare groups of sessions.

1. Representing Website Navigation

• Treat each page or group of related pages as a state.
• Add two fictitious states:
  • Input/entry state: Represents a visitor arriving at the site.
  • Output/exit state: Represents a visitor leaving the site.
• Count how often users move from one state to another. These counts describe the navigation graph and are used to estimate transition probabilities.
• The resulting transition probability matrix records the probability of moving from each state to each other state. For each state with outgoing transitions, its probabilities are normalized by the total number of departures from that state.

2. Modeling Assumptions and Matrix Structure

The model makes assumptions that shape the transition matrix:

• Visitors enter through the fictitious input state; ordinary page states do not transition back to it.
• A visitor must visit at least one actual site state before reaching the output state, so the model disallows a direct input-to-output transition.
• The basic access-log model does not identify returning visitors. Cookie or other historical data could potentially support an extended model that accounts for repeat visits.
• The output state is treated as an absorbing state: once visitors reach it, they have left the site.

The lecture’s spoken explanation is inconsistent about how to represent the exit-state row in a row-stochastic matrix. It says transitions from the exit state are impossible, but also discusses assigning a transition so that the row sums to one. It also corrects its notation for whether the exit state is numbered (n) or (n+1). The intended model is an absorbing exit state, with the exact indexing and matrix convention to be made consistent in the slides.

3. Building a Residence-Time Matrix

Alongside transition probabilities, construct a residence-time matrix:

• Record the accumulated time associated with transitions between states.
• Divide accumulated time by the number of corresponding transitions to obtain an average time.
• Use these averages to estimate time spent on a page or state, as well as time associated with movement between states.

The lecture refers to these average values as residence or dwell times.

4. Behavioral Measures Derived from the Markov Model

Once the transition and time matrices are available, the lecture describes several useful measures:

• Average time spent in a state: Estimated from the average residence times associated with that state.
• Marginal state probabilities: Obtained by repeatedly multiplying the transition matrix by itself. The lecture says that, as the power increases, the matrix eventually stabilizes and its rows become alike. These values describe state probabilities under the model.
• Expected number of transitions: Calculated using the transient-state portion of the transition matrix and an identity-matrix-based calculation. The exit state is absorbing; the remaining states are transient.
• Average session length: Estimated from the expected number of transitions through the states.
• Average time per session: Estimated by combining expected transitions with the corresponding average residence times.
• Cart abandonment or migration rate: Measures how often users leave after reaching the shopping-cart state without proceeding to payment.
• Purchase-to-visit ratio: Compares completed purchases with site visits.

5. Example Navigation Model

The lecture proposes an example with five substantive website states—such as product viewing or catalog, search, selection, cart, and payment—plus the input and output states. Under this setup, both the transition-probability matrix and the average-time matrix are (7 \times 7).

A navigation graph accompanies the example:

• Nodes represent states.
• Branch labels represent transition counts.
• Accumulated time values can be divided by those counts to estimate average time.
• Some links are drawn as dangling for readability, but are meant to connect to the exit state.

Students are encouraged to calculate the transition matrix, residence-time matrix, state-time measures, and session-duration measures from the example.

6. Segmenting Visitors with Clustering

The lecture extends the analysis by clustering sessions—for example, with K-means—and comparing the resulting groups using the behavioral measures.

Cluster comparisons can reveal patterns such as:

• Groups with a higher purchase-to-visit ratio, whose visitors appear more likely to buy.
• Groups with long sessions but low purchase rates, which may represent visitors who browse extensively but are less experienced or less ready to purchase.
• Groups that add items to the cart frequently but do not consistently complete purchases.
• Groups that browse or search little and move more directly toward buying.

These patterns can help businesses identify audiences for further study or targeted marketing. The lecture presents targeting as a possible application, not as a guaranteed way to convert visitors.

7. Overall Lesson

Access logs may look like raw, difficult-to-interpret records. Modeling them as a Markov process turns navigation and timing information into measures that can describe customer behavior. Comparing those measures across session clusters can reveal differences in browsing, cart use, and purchasing, providing potential business insights.

Speakers and Sources Featured

• Speaker: An instructor or lecturer presenting the customer-navigation modeling lesson; no personal name is given.
• Data source discussed: Website access logs.
• Potential supplementary data source discussed: Cookies or other records of past visits, for modeling returning visitors.
• Reference mentioned: Hiller–Lieberman, cited as an example of a standard textbook where the relevant stochastic-process formula can be found.