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September, 13th
gR2002, Vienna
PAOLO GIUDICI
Faculty of Economics, University of Pavia
Research carried out within the laboratory:
Statistical models for data mining (SMDM)
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A small sample of web
clickstream data
(from a logfile)
C, “10908”, 10908
V, 1108
V, 1017
C, “10909”, 10909
V, 1113
V, 1009
V, 1034
C, “10910”, 10910
V, 1026
V, 1017
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Analysis of web clickstream data
1. In data matrix form (Giudici and Castelo, 2001; Blanc
and Giudici, 2001):
- Association measures
- Association models (graphical association models)
2. In transactional data form (in this talk)
- Association and sequence rules
- Statistical models for sequences
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Association measures and models
Based on data arranged in contingency table form
FOR INSTANCE:
Odds ratios
Graphical loglinear models
Recursive logistic regression models
For a review, see Giudici, Applied data mining, Wiley, 2003
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Association and sequence rules
Implemented in main Data Mining softwares
Based on transactional databases
Such databases arise for instance in
- Market basket analysis (order does not matter)
- Web clickstream analysis (order matters)
Aim: search for itemsets (groups of events) that occurr
simultaneously with a high frequency
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Formally:
• A1, .., Ap: p binary random variables. Itemset: logical
expression such as A = (Aj1 = 1 ,...,.Ajk =1), k< p.
Association rule: logical relationship between two itemsets:
e.g. if A, then B  A  B
Example:A= (Milk, Coffee) B=(Bread, Biscuits)

Sequence rule: the relationship is determined by a temporal
order.
Example: A= (Home, Register) B=(P_info)
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Interestingness of a rule
• Support
 A  B =
• Confidence  A  B =
N A B
N
N A B
NA
=
support
support
A  B
 A
• Lift  A  B=Confidence  A  B / Support (B)
A priori search algorithm (Agrawal et al., 1995):
based on the support.
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Application to real data
Data set from a logfile of an e-commerce site, kindly supplied by SAS.
Contains the userid (C_VALUE), the time of connection (C_TIME) and
the page visualised (C_CALLER).
Number of clicks: 21889; Number of visitors (sessions): 1240.
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Exploratory step
(data selected from a cluster of visitors, N. 3)
Cluster
mean
Overall
mean
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Cluster
N.obs
Variables
1
8802
CLICKS
LENGTH
start
%PURCH
8
6 min
h. 18
0.034
2
2859
CLICKS
LENGTH
start
%PURCH
22
17 min
h. 15
0.241
3
1240
CLICKS
LENGTH
start
%PURCH
18
59min
h. 13
0.194
4
9251
CLICKS
LENGTH
start
%PURCH
8
6 min
h. 10
0.039
10
10 min
14 h
0.072
Remark
Data could have been transformed from transactional to data matrix format.
Doing so information on the order of the visited pages would have been lost
Data matrix format for the considered data:
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Application of the apriori
algorithm
Most frequent indirect sequences of order 2
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Most frequent indirect sequences of any order
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Proposal: direct sequences
• Only “subsequent” visits are being considered
• We have inserted two fictitious (deterministic) pages:
(start_session; end_session)
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Most frequent direct sequences of order 2
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Towards a global model:
graphical representation of direct
association rules
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Link analysis representation
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Global models for web mining
Sequence rules are an instance of a local model (or pattern, see Hand et al, 2001) of
data mining.
A local model draws statistical conclusions on parts of the dataset, rather than on the
whole.
Link analysis is an example of a global descriptive model.
We have considered two global inferential models:
- probabilistic expert systems
- Markov chains
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Probabilistic expert systems
Graphical models that allow to describe (recursive) dependencies
between (binary) random variables
Can be described by a directed conditional independence graph, that
specifies the factorisation of the joint probability distribution.
They ARE NOT directly comparable with sequence rules, that are local
indexes to study dependencies between events (itemsets)
They are built from contingency table data, thus DO NOT model order of
visit to pages.
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Probabilistic expert systems:
structural learning
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Probabilistic expert systems:
quantitative learning
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Markov Chains for web mining
Ideal to model dependencies between events. Order of the
chain parallels order of a sequence rule.
Data have been structured in the following form:
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Results from Markov chains
(entrance to the site- start session)
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Exit from the site
(end session)
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Most likely paths
17,80%
45,81%
Start_session
Home
70,18%
Progra
m
Product
26,73%
P_info
Markov chains ARE DIRECTLY comparable with direct sequence rules.
E.g. for the most likely path:
from start_session, the highest confidence is with home (45,81%), then program
(20.39,), product ( 78,09% ) and addcart (28,79%).
There are small differences, due to the fact that apriori algorithm considers
only rules with support higher than a fixed threshold (e.g. 5%).
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Essential references
Agrawal, R., Manilla, H., Srikant, R., Toivonen, H. and Verkamo, A.I. (1995) Fast
discovery of association rules, in: Advances in knowledge discovery and data
mining, AAAI/MIT Press, Cambridge.
Giudici, P. (2003) Applied Data mining. Wiley, London.
Giudici, P. and Castelo, R. (2001) Association models for web mining. Journal
of Knowledge discovery and data mining, 5, pp. 183-196.
Trevor Hastie, Robert Tibshirani and Jerome Friedman (2001).The elements of
statistical learning: data mining, inference and prediction. Springer-Verlag.
Hand, D.J., Mannilla, H. and Smyth, P (2001) Principles of Data Mining, MIT
Press, New York.
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THANKS FOR THE
ATTENTION !
Comments to:
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www.baystat.it/giudici/index.htm
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