Hmm Datamining

CSE5230/DMS/2004/9

Data Mining - CSE5230

Hidden Markov Models (HMMs)

CSE5230 - Data Mining, 2004 Lecture 9.1

Lecture Outline
Time- and space-varying processes
u
First-order Markov models
u
Hidden Markov models
u
Examples: coin toss experiments
u
Formal definition
u
Use of HMMs for classification
u
The three HMM problems
u
v The evaluation problem
v The Forward algorithm
v The Viterbi and Forward-Backward algorithms
HMMs for web-mining
u
References
u


Page 1

Time- and Space-varying Processes (1)
u The data mining techniques we have discussed
so far have focused on the classification,
prediction or characterization of single data
points, e.g.:
vAssign a record to one of a set of classes
» Decision trees, back-propagation neural networks,
Bayesian classifiers, etc.
vPredicting the value of a field in a record given the
values of the other fields
» Regression, back-propagation neural networks, etc.
vFinding regions of feature space where data points are
densely grouped
» Clustering, self-organizing maps



u In the methods we have considered so far, we
have assumed that each observed data point is
statistically independent from the observation
that preceded it, e.g.
vClassification: the class of data point x t (from time t) is
not influenced by the class of x_t-1 (from time t – 1), or
indeed any other data point
vPrediction: the value of a field for a record depends only
on the values of the field of that record, not on values in
any other records.
u Several important real-world data mining
problems can not be modeled in this way.


Page 2

We often encounter sequences of observations, where
u
each observation may depend on the observations which
preceded it
Examples
u
v Sequences of phonemes (fundamental sounds) in speech
(speech recognition)
v Sequences of letters or words in text (text categorization,
information retrieval, text mining)
v Sequences of web page accesses (web usage mining)
v Sequences of bases (CGAT) in DNA (genome projects
[human, fruit fly, etc.))
v Sequences of pen-strokes (hand-writing recognition)
In all these cases, the probability of observing a particular
u
value in the sequence can depend on the values which
came before it


Example: web log
Consider the following extract from a web log:
u

xxx - - [16/Sep/2002:14:50:34 +1000] quot;GET /courseware/cse5230/ HTTP/1.1quot; 200 13539
xxx - - [16/Sep/2002:14:50:42 +1000] quot;GET /courseware/cse5230/html/research_paper.html HTTP/1.1quot; 200 11118
xxx - - [16/Sep/2002:14:51:28 +1000] quot;GET /courseware/cse5230/html/tutorials.html HTTP/1.1quot; 200 7750
xxx - - [16/Sep/2002:14:51:30 +1000] quot;GET /courseware/cse5230/assets/images/citation.pdf HTTP/1.1quot; 200 32768
xxx - - [16/Sep/2002:14:51:31 +1000] quot;GET /courseware/cse5230/assets/images/citation.pdf HTTP/1.1quot; 206 146390
xxx - - [16/Sep/2002:14:51:40 +1000] quot;GET /courseware/cse5230/assets/images/clustering.pdf HTTP/1.1quot; 200 17100
xxx - - [16/Sep/2002:14:51:40 +1000] quot;GET /courseware/cse5230/assets/images/clustering.pdf HTTP/1.1quot; 206 14520
xxx - - [16/Sep/2002:14:51:56 +1000] quot;GET /courseware/cse5230/assets/images/NeuralNetworksTute.pdf HTTP/1.1quot; 200 17137
xxx - - [16/Sep/2002:14:51:56 +1000] quot;GET /courseware/cse5230/assets/images/NeuralNetworksTute.pdf HTTP/1.1quot; 206 16017
xxx - - [16/Sep/2002:14:52:03 +1000] quot;GET /courseware/cse5230/html/lectures.html HTTP/1.1quot; 200 9608
xxx - - [16/Sep/2002:14:52:05 +1000] quot;GET /courseware/cse5230/assets/images/week03.ppt HTTP/1.1quot; 200 121856
xxx - - [16/Sep/2002:14:52:24 +1000] quot;GET /courseware/cse5230/assets/images/week06.ppt HTTP/1.1quot; 200 527872

Cleary the URL which is requested depends on the URL
u
which was requested before
v If the user uses the “Back” button in his/her browser, the
requested URL may depend on earlier URLs in the sequence
too
Given a particular observed URL, we can calculate the
u
probabilities of observing all the other possible URLs next.
v Note that we may even observe the same URL next.


Page 3

First-Order Markov Models (1)

In order to model processes such as these, we make use of
u
the idea of states. At any time t, we consider the system to
be in state q(t).
We can consider a sequence of successive states of length
u
T:

qT = (q(1), q(2), …, q(T))
We will model the production of such a sequence using
u
transition probabilities:

P ( q j (t + 1) | qi (t )) = aij
Which is the probability that the system will be in state qj at
time t+1 given that it was in state qi at time t


First-Order Markov Models (2)

uA model of states and transition probabilities,
such as the one we have just described, is called
a Markov model.
u Since we have assumed that the transition
probabilities depend only on the previous state,
this is a first-order Markov model
vHigher order Markov models are possible, but we will
not consider them here.
u For example, Markov models for human speech
could have states corresponding phonemes
vA Markov model for the word “cat” would have states for
/k/, /a/, /t/ and a final silent state


Page 4

Example: Markov model for “cat”

/a/ /t/
/k/ /silent/


Hidden Markov Models
In the preceding example, we have said that the states
u
correspond to phonemes
In a speech recognition system, however, we don’t have
u
access to phonemes – we can only measure properties of
the sound produced by a speaker
In general, our observed data does not correspond directly
u
to a state of the model: the data corresponds to the visible
states of the system
v The visible states are directly accessible for measurement.
The system can also have internal “hidden” states which
,
u
can not be observed directly
v For each hidden state, there is a probability of observing each
visible state.
This sort of model is called Hidden Markov Model (HMM)
u


Page 5

Example: coin toss experiments
u Let us imagine a scenario where we are in a
room which is divided in two by a curtain.
u We are on one side of the curtain, and on the
other is a person who will carry out a procedure
using coins resulting in a head (H) or a tail (T).
u When the person has carried out the procedure,
they call out the result, H or T, which we record.
u This system will allow us to generate a sequence
of Hs and Ts, e.g.
HHTHTHTTHTTTTTHHTHHHHTHHHTTHHHHHHTTT
TTTTTHTHHTHTTTTTHHTHTHHHTHTHHTTTTHHT
TTHHTHHTTTHTHTHTHTHHHTHHTTHT….


Example: single fair coin
Imagine that the person behind the curtain has a single fair
u
coin (i.e. it has equal probabilities of coming up heads or
tails). This generates sequences such as
THTHHHTTTTHHHHHTHHTTHHTTHHTHHHHHHHTTHTTHHHH
THTTTHHTHTTHHHHTHTHHTTHTHTTHHTHTHHHTHHTHT…
We could model the process producing the sequence of Hs
u
and Ts as a Markov model with two states, and equal
transition probabilities:
0.5

0.5 0.5
H T
0.5

Note that here the visible states correspond exactly to the
u
internal states – the model is not hidden
Note also that states can transition to themselves
u


Page 6

Example: a fair and a biased coin
Now let us imagine a more complicated scenario. The
u
person behind the curtain has three coins, two fair and
one biased (for example, P(T) = 0.9)
v One fair coin and the biased coin are used to produce output
– these are the “output coins”. The other fair coin is used to
decide whether to switch output coins.
1. The person starts by picking an output coin a random
2. The person tosses the coin, and calls out the result (H or T)
3. The person tosses the other fair coin. If the result was H, the
person switches output coins
4. Go back to step 2, and repeat.
This process generates sequences like:
u

HHHTTTTTTTHTHTTHTHTTTTTTHTTTTTTTTTTTTHHTHT
TTHHTTHTHTHTTTHTTTTTTTTHTHTTTTHTTTTHTTTHTH
HTTHTTTHTTHTTTTTTTHTTTTTHT…
Note this looks quite different from the sequence for the
u
fair coin example.

u In this scenario, the visible state no longer
corresponds exactly to the hidden state of the
system:
vVisible state: output of H or T
vHidden state: which coin was tossed
u We can model this process using a HMM:
0.5

0.5 0.5
Fair Biased

0.5 0.9
0.1
0.5 0.5
T
T H
H


Page 7

u We see from the diagram on the preceding slide
that we have extended our model
vThe visible states are shown in blue, and the emission
probabilities are shown too.
u As well as internal states qj(t) and state transition
probabilities aij, we have visible states vk(t) and
emission probabilities bjk

P (vk (t) | q j (t )) = b jk
u We now have full model such as this is called a
Hidden Markov Model


HMM: formal definition (1)
We can now give a more formal definition of a first-order
u
Hidden Markov Model (adapted from [RaJ1986]:
v There is a finite number of (internal) states, N
v At each time t, a new state is entered, based upon a transition
probability distribution which depends on the state at time t –
1. Self-transitions are allowed
v After each transition is made, a symbol is output, according to
a probability distribution which depends only on the current
state. There are thus N such probability distributions.
If we want to build an HMM to model a real sequence, we
u
have to solve several problems. We must estimate:
v the number of states N
v the transition probabilities aij
v the emission probabilities bjk


Page 8

HMM: formal definition (2)
When an HMM is “run”, it produces an sequence of
u
symbols. This called an observation sequence O of length
T:
O = (O(1), O(2), …, O(T))

In order to talk about using, building and training an HMM,
u
we need some definitions:
N — the number of states in the model
M — the number of different symbols that can observed
Q = {q1, q 2, …, qN} — the set of internal states
V = { v1 , v2, …, vM} — the set of observable symbols
A = {aij} — the set of state transition probabilities
B = {bjk} — the set of symbol emission probabilities
π = {πi = P(qi(1)} — the initial state probability distribution
λ = (A, B, π) — a particular HMM model


Generating a Sequence

To generate an observation sequence using an HMM, we
u
use the following algorithm:

Set t = 1
1.

Choose an initial state q(1) according to πi
2.
Output a symbol O(T) according to b q(t)k
3.
Choose the next state q(t + 1) according to a q(t)q(t+1)
4.
Set t = t + 1; if t < T, go to 3
5.

In applications, however, we don’t actually do this. We
u
assume that the process that generates our data does
this. The problem is to work out which HMM is
responsible for a data sequence


Page 9

Use of HMMs
We have now seen what sorts of processes can be
u
modeled using HMMs, and how an HMM is specified
mathematically.
We now consider how HMMs are actually used.
u
Consider the two H and T sequences we saw in the
u
previous examples:
v How could we decided which coin-toss system was most
likely to have produced each sequence?
To which system would you assign these sequences?
u

1: TTHHHTHHHTTTTTHTTTTTTHTHTHTTHHHHTHTH
2: TTTTTTHTHHTHTTHTTTTHHHTHHHHTTHTHTTTT
3: THTTHTTTTHTTHHHTHTTTHTHHHHTTHTHHHTHT
4: TTTHHTTTHHHTTTTTTTHTTTTTHHTHTTHTTTTH
We can answer this question using a Bayesian formulation
u
(see last week’s lecture)


Use of HMMs for classification
HMMs are often used to classify sequences
u
To do this, a separate HMM is built and trained (i.e. the
u
parameters are estimated) for each class of sequence in
which we are interested
v e.g. we might have an HMM for each word in a speech
recognition system. The hidden states would correspond to
phonemes, and the visible states to measured sound features
This gives us a set of HMMs, {λl }
u
For a given observed sequence O, we estimate the
u
probability that each HMM λl generated it:
P (O | λl ) P( λl )
P( λl | O) =
P( O )
We assign the sequence to the model with the highest
u
posterior probability
v i.e. the probability given the evidence, where the evidence is
the sequence to be classified


Page 10

The three HMM problems
u If we want to apply HMMs real problems with real
data, we must solve three problems:
vThe evaluation problem: given an observation
sequence O and an HMM model λ, compute P(O|λ), the
probability that the sequence was produced by the
model
vThe decoding problem: given an observation
sequence O and a model λ, find the most likely state
sequence q(1), q(2), … q(T) to have produced O
vThe learning problem: given a training sequence O,
find a model λ, specified by parameters A, B, π to
maximize P(O|λ) (we assume for now that Q and V are
known)
u The evaluation problem has a direct solution. The
others are harder, and involve optimization


The evaluation problem
The simplest way to solve the evaluation problem is to go
u
over all possible state sequences Ir of length T and
calculate the probability that each of them produced O:
rmax
P (O | λ ) = ∑ P(O | I r , λ ) P( I r )
r =1
where
u

P (O | I , λ ) = bi1O1 bi2O2 ...biT OT
P ( I | λ ) = π i1 ai1i 2 ai2 i3 ...aiT −1iT
While this could in principle be done, there is a problem:
u
computational complexity. Our model λ has N states. There
are thus rmax = NT possible sequences of length T. The
computational complexity is O(NT T).
v Even if we have small N and T, this is not feasible: for N = 5
and T = 100, there are ~1072 computations needed!

Page 11

The Forward Algorithm (1)
u Luckily, there is a solution to this problem: we do
not need to do the full calculation
u We can do a recursive evaluation, using an
auxiliary variable α t(i), called the forward variable:
α t (i) = P ((O (1), O (2),..., O (t )), i(t ) = qi | λ )
this is the probability of the partial observation
sequence (up until time t) and internal state q i(t)
given the model λ
u Why does this help? Because in a first-order
HMM, the transition and emission probabilities
only depend on the current state. This makes a
recursive calculation possible.


u We can calculate a t+1(j) – the next step – using
the previous one:
N
α t +1 (i ) = b jOt+1 ∑ aijα t (i)
i =1

u This just says that the probability of the
observation sequence up to time t + 1 and being
in state qj at time t + 1 is:
the probability of observing symbol Ot+1 when in state qj,
bjOt+1, times the sum of
the probabilities of getting to state qj from state qi
times the probability of the observation
sequence up to time t and being in state qi
that we have to keep track of α t(i) for all N
u Note
possible internal states


Page 12

If we know αT (i) for all the possible states, we can
u
calculate the overall probability of the sequence given the
model (as we wanted on slide 9.22):
N
P( O | λ ) = ∑ α T (i)
i =1

We can now specify the forward algorithm, which will let
u
us calculate the αT(i):
for i = 1 to N { α1 (i) = π i biO1 } /* initialize */
for t = 1 to T – 1 {
for j = 1 to N {
N
α t +1 (i) = b jOt +1 ∑ a ijα t ( i)
i=1
}
}


u The forward algorithm allows us to calculate
P(O|λ), and has computational complexity
O(N2 T), as can be seen from the algorithm
vThis is linear in T, rather than exponential, as the direct
calculation was. This means that it is feasible
u We can use P(O|λ) in the Bayesian equation on
slide 9.20 to use a set of HMMs a classifier
vThe other terms in the equation can be estimated from
the training data, as with the Naïve Bayesian Classifier


Page 13

The Viterbi and Forward-
Backward Algorithms

u The most common solution to the decoding
problem uses the Viterbi algorithm [RaJ1986],
which also uses partial sequences and recursion
u There is no known method for finding an optimal
solution to the learning problem. The most
commonly used optimization technique is known
as the forward-backward algorithm, or the Baum-
Welch algorithm [RaJ1986,DHS2000]. It is a
generalized expectation maximization algorithm.
See references for details


HMMs for web-mining (1)

u HMMs can be used to analyse to clickstreams
the web users leave in the log files of web
servers (see slide 9.6).
u Ypma and Heskes (2002) report the application
of HMMs to:
vWeb page categorization
vUser clustering
u They applied their system to real-word web logs
from a large Dutch commercial web site
[YpH2002]


Page 14


uA mixture of HMMs is used to learn page
categories (the hidden state variables) and inter-
category transition probabilities
vThe page URLs are treated as the observations
u Web surfer types are modeled by HMMs
u A clickstream is modeled as a mixture of HMMs,
to account for several types of user being present
at once


When applied to data from a commercial web site, page
u
categories were learned as hidden states
v Inspection of the emission probabilities B showed that several
page categories were discovered:
» Shop info
» Start, customer/corporate/promotion
» Tools
» Search, download/products
Four user types were discovered too (HMMs with different
u
state prior and transition probabilities)
v Two types dominated:
» General interest users
» Shop users
v The starting state was the most important difference between
the types


Page 15

References
u [DHS2000] Richard O. Duda, Peter E. Hart and David
G. Stork, Pattern Classification (2nd Edn), Wiley, New
York, NY, 2000, pp. 128-138
u [RaJ1986] L. R. Rabiner and B. H. Juang, An
introduction to hidden Markov models, IEEE
Magazine on Acoustics, Speech and Signal
Processing, 3, 1, pp. 4 -16, January 1986.
u [YpH2002] Alexander Ypma and Tom Heskes,
Categorization of web pages and user clustering with
mixtures of hidden Markov models, In Proceedings of
the International Workshop on Web Knowledge
Discovery and Data mining (WEBKDD'02),
Edmonton, Canada, July 17 2002.


Page 16

Hmm Datamining

Recomendados

Recomendados

Mais conteúdo relacionado

Semelhante a Hmm Datamining

Semelhante a Hmm Datamining (20)

Último

Último (20)

Hmm Datamining