Model-Based Clustering and Visualization of Navigation Patterns on a Web Site

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Model-Based Clustering and Visualization of
Navigation Patterns on a Web Site

• I. Cadez, D. Heckerman, C. Meek, P. Smyth, and S.
  White

Presented by Motaz El Saban
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Outline of the talk

• Introduction and problem definition.
• Model-based clustering.
• Model learning.
• Application to Msnbc.com IIS log data.
• Data Visualization.
• Scalability.
• Why mixtures of first-order Markov models?
• Conclusions.
• Future work.

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Introduction

• New methodology for analyzing web navigation
  patterns. (a form of human behavior in digital
  environments)
• Patterns sequence of URL categories traversed by
  users, stored in web-server logs for a duration
  of 24 hours on msnbc.com.
• Functionality
• Clustering users based on navigation patterns.
• Visualization (WebCANVAS tool).

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Web data analysis approach

• Clustering
• Partition users into clusters with users having
  similar dynamic behavior in the same cluster.
• Visualization
• Display the behavior of the users within that
  cluster.

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Related Work

• Most Previous work on Web Navigation patterns and
  visualization uses non-probabilistic methods
  YAN96 CHE98, mostly finding rules that govern
  navigation patterns.
• Other work used probabilistic methods for
  predicting the behavior of users on Web pages,
  but not for clustering purposes using random walk
  models HUB 97, Markov models for pre-fetching
  pages PAD96, modeling next probable link use
  kth order Markov model BOR00.
• These approaches use a single Markov model for
  all users as opposed to a clustering of users
  first.

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Related Work

• On the clustering side, FU00 applied BIRCH to
  cluster user web navigation patterns.
• For Web navigation sequence-based clustering, and
  visualization no previously known work has been
  done using probabilistic clustering.
• Rather, user history has been visualized using
  visual metaphors of maps, paths, and signposts
  WEX99.
• MIN99 use planar graphs to visualize crowds of
  users at particular web pages.

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What do we mean by pattern?
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Challenges

• Web navigation patterns are dynamic. No static
  techniques could capture its patterns, such as
  histograms ?Markov models.
• Different users have heterogeneous dynamic
  behavior ? Mixture of models.
• Large data size.
• The proposed algorithm for learning the mixture
  of 1st order Markov models has runtime
  O(KNLKM2).
• K clusters.
• N sequences.
• L average length of sequence.
• M of Web page categories.
• For typically small M, the algorithm scales
  linearly with N and K.
• Hierarchical clustering methods scale as O(N2)

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Model-Based Clustering

• Assuming data is generated as follows
• A user arrives at the web site and is assigned to
  one of K clusters with some probability, and
• given that a user is in this cluster, his
  behavior is generated from some statistical model
  specific to that cluster.
• let X be a multivariate random variable taking on
  values corresponding to the behavior of
  individual users.
• Let C be a discrete-valued variable taking on
  values
• c1 ...,cK, corresponding to the unknown cluster
  assignment for a user.

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Model-Based Clustering

• A mixture model for X with K components has the
  form

Where is the marginal probability of the kth
cluster, is the statistical model
describing the distribution for the variables for
users in the kth cluster, and denotes the
parameters of the model
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Model-Based Clustering

• In our case X (X1,,XL) is a sequence of
  variables describing the users path through the
  website.
• Xi takes on some value xi from the M different
  page categories.
• Each component in the model obeys the 1st order
  Markov model

• where denotes the parameters of the
  probability distribution over the initial
  page-category request among users in cluster k,
• and denotes the parameters of the probability
  distributions over transitions from one category
  to the next by a user in cluster k .
• Both distributions are taken to be multinomial
  distribution.

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Model-Based Clustering

• EM algorithm is used to learn the model
  parameters.
• Once learned, we can use the model to assign
  users to clusters by finding the class K that
  maximizes the membership probabilities
• The user class assignment may be soft or hard.

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Learning Mixture Models from Data

• For a known number of K clusters.
• Training data dtrain x1,,xN, with iid
  assumption.
• MAP Estimate of

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EM learning algorithm (briefly)

• An iterative method to find local maxima for the
  MAP problem of .
• Problem at hand involves two sub-problems
• Compute user class assignment (membership
  probabilities).
• Compute class parameters.
• Chicken-egg problem!

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EM learning algorithm (briefly)

• EM approach
• E-step given a current value of the parameters
  , assign a user with behavior X to cluster Ck
  using the membership probabilities.
• M-step pretend that these assignments correspond
  to real data, and reassign to be the MAP
  estimate given this fictitious data.
• Stop iteration when two consecutive iterations
  produce log likelihoods on the training data that
  differ by less than p (0.01 in the paper).

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How to choose K?

• Let the site administrator try several K values
  and choose the convenient one for visualization ?
  too time consuming. Rather,
• Choose K by finding the model that accurately
  predicts Nt new test cases dtest XN1 ,...,XN
  Nt. That is, choose a model with K clusters
  that minimizes the out-of-sample predictive log
  score

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Application to Msnbc.com

• Each sequence in the data set corresponds to page
  views of a user during a twenty-four hour period.
  
• Each event in the sequence corresponds to a user
  request for a page. The event denotes a page
  category rather than a URL.
• Example categories are frontpage, news, tech,
• The number of URLs per category ranges from 10 to
  5000.
• Modeling only the order in which the pages are
  requested (no duration is modeled) .
• Page requests served via a caching mechanism were
  not recorded in the server logs and, hence, not
  present in the data.

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Application to Msnbc.com

• The full data set consists of approximately one
  million sequences (users),with an average of 5.7
  events per sequence.
• Model learning for various cluster sizes K is
  done with a training set size of 100,023.
• Model evaluation was done using the out-of-sample
  predictive log score on a different sample of
  98,687 sequences drawn from the original data.

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Observation on the model components

• Some of the individual model components encode
  two or more clusters.
• Example consider two clusters a cluster of
  users who initially request category a and then
  choose between categories b and c ,and a cluster
  of users who initially request category d and
  then choose between categories e and f .
• These two clusters can be encoded in a single
  component of the mixture model, although the
  sequences for the separate clusters do not
  contain common elements.
• The presence of multi-cluster components does not
  affect the out-of-sample predictive log score of
  a model.
• However, it is problematic for visualization
  purposes.

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Observation on the model components

• Solutions
• One method is to run the EM algorithm and then
  post-process the resulting model, separating any
  multi-cluster components found.
• A second method is to allow only one state
  (category) to have a non-zero probability of
  being the initial state in each of the 1st-order
  Markov models.
• Using the second method has the drawback that a
  cluster of users that have different initial
  states but similar paths after the initial state
  are divided into separate clusters.
• Nonetheless,this potential problem was fairly
  insignificant for the Msnbc.com data.

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Constrained models

• Experimentally, constrained models have a
  predictive power almost equal to that of the
  unconstrained models.
• However, introducing this constraint,more
  components are needed to represent the data than
  in the unconstrained case.
• For this particular data,the constrained
  1st-order Markov models reach limit in predictive
  accuracy around K 100,
• as compared to the unconstrained models,which
  reach their limit around K 60.

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Out of sample results
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Data VisualizationWebCANVAS tool

• Display of twenty four hour period using 100
  clusters.
• Each window corresponds to a cluster.
• Each row of squares in a cluster corresponds to a
  user sequence.
• WebCANVAS uses hard clustering, assigning each
  user to a single cluster.
• Each square in a row encodes a page request in a
  particular category encoded by the color of the
  square.
• Note that the use of color to encode URL category
  limits the utility of this tool to domains where
  the number of categories can be limited to fifty
  or so.

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WebCANVAS Display
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Discovering unexpected facts

• Large groups of people enter msnbc.com on tech
  and local pages
• Large group of people navigating from on-air to
  local
• Little navigation between tech and business
  sections
• and large number of hits to the weather pages.

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WebCANVAS tool (model-direct sampling)

• WebCANVAS display performed better subjectively
  than two other methods
• Showing the 0th-order and 1st-order Markov models
  of a cluster.
• traffic flow movie by Microsoft Site Server
  3.0.
• Advantage of model-directed sampling over
  displaying the models themselves is that the
  former approach is not as sensitive to errors in
  modeling.
• That is, by displaying sampled raw data,
  behaviors in the data not consistent with the
  model used can still be seen and appreciated.

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Alternative Displaying models themselves
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Scalability

• Memory requirements of the algorithm are
• O(NLKM2KM), which typically reduces to
• O (NL) - i.e. the data size- for data sets where
  M is relatively small.
• The runtime of the algorithm per iteration is
  linear in N and K.

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Scalability in K
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Scalability in N
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Mixtures of 1st order Markov Models Too simple
model?

• Sen and Hansen (2001), Deshpande and Karypis
  (2001) have shown that the 1st-order Markov model
  to be an inadequate model for empirically-observed
  page-request sequences.
• It is not surprising, because for example
• if a user visits a particular page,there tends to
  be a greater chance of he returning to that same
  page at a later time.
• 1st order Markov model cannot capture this type
  of long-term memory.
• However
• Though the mixture model is 1st order Markov
  within a cluster, the overall unconditional model
  is NOT 1st order Markov.
• Msnbc data is different from typical raw
  page-request sequences. Namely, URL categories
  result in a relatively small alphabet size as
  compared to working with uncategorized URLs.

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Mixtures of 1st order Markov Models Too simple
model?

• The combined effects of clustering and a small
  alphabet tend to produce low-entropy clusters in
  the sense that a few (two or three) categories
  often dominate the sequences within each cluster.
• Thus, the tendency to return to a specific page
  that was visited earlier in a session can be well
  approximated by the simple mixture of 1st order
  Markov models.

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Mixture of 1st order Markov Models vs 1st order
Markov Models

• Mixture Model
• Looking at the predictive distribution for the
  next symbol under the mixture model, i.e
  
• Thus the probability of the next symbol is a
  weighted combination of the transition
  probabilities from
  each of the individual 1st order component
  models.

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Mixture of 1st order Markov Models vs 1st order
Markov Models

• The weights are determined by the partial
  membership probabilities of the
  prefix (history) subsequence
• .
• These weights are in turn a function of the
  history of the sequence (via Bayes rule), and
  typically depend strongly on the pattern of
  behavior before .
• This prediction behavior of is opposed to
  the simple prediction distribution of the 1st
  order Markov model

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Empirical proof of 1st order Markov Model

• Diagnostic check empirically calculate the run
  lengths of page categories for several of the
  most likely clusters.
• If the data are being generated by a 1st order
  Markov model, then the distribution of these run
  lengths will obey a geometric distribution.
• Results are shown in each cluster for the three
  most frequently visited categories that had at
  least one run length of four or
  greater.(Categories that have run lengths of
  three or fewer provide relatively uninformative
  diagnostic plots.)

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Empirical proof of 1st order Markov Model

• Asterisks mark the empirically observed counts.
• The center dotted line on each plot is the
  expected count as a function of run length under
  a geometric model using the empirically estimated
  self-transition probability of the Markov chain
  for the corresponding cluster.
• The upper and lower dotted lines represent the
  plus and minus three-sigma sampling deviations
  for each count under the model.

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Conclusions

• Using a model-based clustering approach to
  cluster users based on web navigation patterns.
• Develop a visualization tool that enables web
  administrators to better understand user
  behavior.
• Using mixture of 1st order Markov models for
  clustering taking into account the order of page
  requests pages.
• Experiments suggest that 1st order Markov model
  mixture components are appropriate for the
  msnbc.com data.
• The algorithm learning time scales linearly with
  sample size. In contrast,agglomerative
  distance-based methods scale quadratically with
  sample size.

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Future Work

• Modeling the duration of each visit.
• Avoiding the limitation of the proposed method to
  small M , modeling page visits at the URL level.
• In one such extension,we can use Markov models to
  characterize both the transitions among
  categories and the transitions among pages within
  a given category.
• Alternatively,we can use a hidden-Markov mixture
  model to learn categories and category
  transitions simultaneously.