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Modeling and Mining Sequential Data
Machine Learning and Data Mining
Philipp Singer
CC image courtesy of user puliarfanita on Flickr
2
What is sequential data?
3
Stock share price (Bitcoin)
Screenshot from bitcoinwisdom.com
4
Daily degrees in Cologne
Screenshot from google.com (data from weather.com)
5
Human mobility
Screenshot from maps.google.com
6
Web navigation
Austria Germany C.F. Gauss
7
Song listening sequences
Screenshots from youtube.com
8
Let us distinguish two types of sequence data
● Continuous time series
● Categorical (discrete) sequences
9
Let us distinguish two types of sequence data
● Continuous time series
– Stock share price
– Daily degree in Cologne
● Categorical (discrete) sequences (focus)
– Sunny/Rainy weather sequence
– Human mobility
– Web navigation
– Song listening sequences
10
This lecture is about...
● Modeling
● Predicting
● Pattern Mining
11
This lecture is about...
● Modeling
● Predicting
● Pattern Mining
Markov Chains
S1S1
S2S2 S3S3
1/2 1/2
1/3
2/3
1
12
Markov Chain Model
13
Markov Chain Model
● Stochastic Model
● Transitions between states
S1S1
S2S2 S3S3
1/2 1/2
1/3
2/3
1
States
Transition
probabilities
14
Markov Chain Model
● Markovian property
– The next state in a sequence only depends on the
current one, and not on a sequence of preceding ones
S1S1
S2S2 S3S3
1/2 1/2
1/3
2/3
1
States
Transition
probabilities
15
Classic weather example
0.1
Sunny Rainy
0.9
0.5
0.5
16
Formal definition
● State space
● Amounts to sequence of random variables
● Markovian memoryless property
17
Transition matrix
Rows sum to 1
Transition matrix P
Single transition
probability
18
Example
0.1
Sunny Rainy
0.9
0.5
0.5
Transition matrix
19
Likelihood
● Transition probabilities are parameters
Transition
probability
Transition
count
20
Maximum Likelihood (MLE)
● Given some sequence data, how can we
determine parameters?
● MLE estimation
Maximize!
See ref [1]
[1] http://journals.plos.org/plosone/articleid=10.1371/journal.pone.0102070
21
Prediction
● Simply derived from transition probabilities
?
One option:
Take max prob.
22
Prediction
● What about t+3?
?
23
Pattern mining
● Simply derived from (non-normalized)
transition matrix
90 2
2 1
Most common transition
Sequential pattern
24
Full example
Training sequence
25
Full example
5 2
2 1
Transition counts
5/7 2/7
2/3 1/3
Transition matrix (MLE)
26
Full example
5/7 2/7
2/3 1/3
Transition matrix (MLE) Likelihood of given sequence
We calculate the probability of the sequence with
the assumption that we start with sunny.
27
Full example
5/7 2/7
2/3 1/3
Transition matrix (MLE)
?
Prediction?
28
Full example
5/7 2/7
2/3 1/3
Transition matrix (MLE)
?
Prediction?
29
Higher order Markov Chain models
● Drop the memoryless assumption?
● Models of increasing order
– 2nd
order MC model
– 3rd
order MC model
– ...
30
Higher order Markov Chain models
● Drop the memoryless assumption?
● Models of increasing order
– 2nd
order MC model
– 3rd
order MC model
– ...
2nd
order example
depends on
31
Higher order to first order transformation
● Transform state space
●
2nd
order example – new compound states
32
2nd
order example
3 1
1 1
10
1 1
3/4 1/4
1/2 1/2
1/10
1/2 1/2
33
Reset states
R R
...R R R R
● Marking start and end of sequences
● Transformation easier (same #transitions)
34
Comparing models
●
1st
vs. 2nd
order
● Statistical model comparison necessary
● Nested models → higher order always fits better
● Account for potential overfitting
35
Model comparison
● Likelihood ratio test
– Ratio between likelihoods for order m and k
– Follows a Chi2 distribution with degrees of freedom
– Only for nested models
● Akaike Information Criterion (AIC)
–
– The lower the better
● Bayesian Information Criterion (BIC)
–
● Bayes Factors
– Ratio of evidences (marginal likelihoods)
● Cross validation
See http://journals.plos.org/plosone/articleid=10.1371/journal.pone.0102070
36
AIC example
R R
...R R R R
5/8 2/8
2/3 1/3
R
R
1/8
0/3
1/1 0/1 0/1
3/5 1/5
1/2 1/2
0
1/2 1/2
R R
R
R
R
R
R
1/5
0
1/10
0
1/1 0 0
1/1 0 0
0 00
00
0 00
0 00
1st
order parameters
2nd
order parameters
37
AIC example
5/8 2/8
2/3 1/3
R
R
1/8
0/3
1/1 0/1 0/1
3/5 1/5
1/2 1/2
0
1/2 1/2
R R
R
R
R
R
R
1/5
0
1/10
0
1/1 0 0
1/1 0 0
0 00
00
0 00
0 00
1st
order parameters
2nd
order parameters
Example on
blackboard
38
Markov Chain applications
● Google's PageRank
● DNA sequence modeling
● Web navigation
● Mobility
39
Hidden Markov Chain Model
40
Hidden Markov Models
● Extends Markov chain model
● Hidden state sequence
● Observed emissions
What is the
weather like?
41
Forward-Backward algorithm
● Given emission sequence
● Probability of emission sequence?
● Probable sequence of hidden states?
Hidden seq.Obs. seq.
Check out YouTube tutorial: https://www.youtube.com/watch?v=7zDARfKVm7s
Further material: cs229.stanford.edu/section/cs229-hmm.pdf
42
Setup
0.7 0.3
0.6 0.4
R 0.5 0.5
0.9
0.2
0.1
0.8
R
0.5
0.5
Note: Literature usually uses a start
probability and uniform end probability
for the forward-backward algorithm.
43
Forward
0.7 0.3
0.6 0.4
R 0.5 0.5
0.9
0.2
0.1
0.8
R
0.4
0.1
R
0.5
0.5
44
Forward
0.7 0.3
0.6 0.4
R 0.5 0.5
0.9
0.2
0.1
0.8
R
0.4
0.1
0.034
0.144
R
0.5
0.5
What is the probability of
going to each possible
state at t2 given t1?
45
Forward
0.7 0.3
0.6 0.4
R 0.5 0.5
0.9
0.2
0.1
0.8
R
0.4
0.1
0.034
0.144
0.011
0.061
R
0.5
0.5
46
Forward
0.7 0.3
0.6 0.4
R 0.5 0.5
0.9
0.2
0.1
0.8
R
0.4
0.1
0.034
0.144
0.011
0.061
0.035
0.006
R
0.5
0.5
forward
R
0.5
0.5
reset transition
47
Backwards
0.7 0.3
0.6 0.4
R 0.5 0.5
0.9
0.2
0.1
0.8
R
R
0.5
0.5
0.5
0.5
48
Backwards
0.7 0.3
0.6 0.4
R 0.5 0.5
0.9
0.2
0.1
0.8
0.31
0.28
R
0.5
0.5
What is the probability of
arriving at t4 given each
possible state at t3?
R
0.5
0.5
49
Backwards
0.7 0.3
0.6 0.4
R 0.5 0.5
0.9
0.2
0.1
0.8
R
0.5
0.5
0.010
0.12
R
0.31
0.28
0.5
0.5
50
Backwards
0.7 0.3
0.6 0.4
R 0.5 0.5
0.9
0.2
0.1
0.8
R
0.5
0.5
0.039
0.049
R
0.097
0.12
0.31
0.28
0.5
0.5
R
backward
reset
emission
51
Forward-Backward
Most likely state at t2
0.4
0.1
0.034
0.144
0.011
0.061
0.035
0.006
0.039
0.049
0.097
0.12
0.31
0.28
0.5
0.5
52
Forward-Backward
● Posterior decoding
● Most likely state at each t
● For most likely sequence: Viterbi algorithm
53
Learning parameters
● Train parameters of HMM
● No tractable solution for MLE known
● Baum-Welch algorithm
– Special case of EM algorithm
– Uses Forward-Backward
54
HMM applications
● Speech recognition
● POS tagging
● Translation
● Gene prediction
55
Other related methods
56
Sequential Pattern Mining
● PrefixSpan
● Apriori Algorithm
● GSP Algorithm
● SPADE
Reference: rakesh.agrawal-family.com/papers/icde95seq.pdf
57
Graphical models
● Bayesian networks
– Random variables
– Conditional dependence
– Directed acyclic graph
● Markov random fields
– Random variables
– Markov property
– Undirected graph
58
Questions?
Philipp Singer
philipp.singer@gesis.org

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Modeling and Mining Sequential Data