Homework 1 Reminder Due date: 21.11.2011 (till 23:59) Submission: – [email protected]...
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Transcript of Homework 1 Reminder Due date: 21.11.2011 (till 23:59) Submission: – [email protected]...
Homework 1 Reminder
Due date: 21.11.2011 (till 23:59) Submission:
– [email protected]– Write the names of students in your team– Send only one e-mail
Midterm 1 Reminder
Date: 24.11.211 14:30 – 16:20 Place: 1029, 2009
Hidden Markov Models
Hidden Markow Models:– A hidden Markov model (HMM) is a statistical
model,– in which the system being modeled is
assumed to be a Markov process (Memoryless process: its future and past are independent ),
– with hidden states.
Hidden Markov Models
Hidden Markow Models:– Has a set of states each of which has limited
number of transitions and emissions,– Each transition between states has an
assisgned probability,– Each model strarts from start state and ends
in end state,
Hidden Markov Models
Hidden Markow Models parameters:– A set of finite number of states, Si,– The transition probability from state Si to Sj, aij,– The emission probability density of a symbol ω– in state Si
Hidden Markov Models
Hidden Markow Models parameters:– Firstly discuss:
Morkov Models, Markov Assumption
Hidden Markov Models
Markow Models and Assumption (cont.):– To understand HMMs:
Talk about weather, Assume there are three types of weather:
– Sunny,
– Rainy,
– Foggy.
Assume weather does not change during the day (if it is sunny it will sunny all the day)
Hidden Markov Models
Markow Models and Assumption (cont.): Weather prediction is about the what would be the weather tomorrow,– Based on the observations on the past.
Hidden Markov Models
Markow Models and Assumption (cont.): Weather at day n is
– qn depends on the known weathers of the past days (qn-1, qn-2,…)
},,{ foggyrainysunnyqn
Hidden Markov Models
Markow Models and Assumption (cont.): We want to find that:
– means given the past weathers what is the probability of any possible weather of today.
Hidden Markov Models
Markow Models and Assumption (cont.): For example:
if we knew the weather for last three days was:
the probability that tomorrow would be is:
P(q4 = | q3 = , q2 = , q1 = )
Hidden Markov Models
Markow Models and Assumption (cont.):– For example:
this probability could be infered from the statistics of past observations
the problem: the larger n is, the more observations we must collect.
– for example: if n = 6 we need 3(6-1) = 243 past observations.
Hidden Markov Models
Markow Models and Assumption (cont.):– Therefore, make a simplifying assumption Markov
assumption: For sequence:
the weather of tomorrow only depends on today (first order Markov model)
Hidden Markov Models
Markow Models and Assumption (cont.): Examples:
The probabilities table:
Hidden Markov Models
Markow Models and Assumption (cont.): Examples:
HMM:
Hidden Markov Models
Markow Models and Assumption (cont.): Examples:
Given that day the weather is sunny, what is the probability that tomorrow is sunny and the next day is rainy ?
Markov assumption
Hidden Markov Models
Markow Models and Assumption (cont.): Examples:
If the weather yesterday was rainy and today is foggy what is the probability that tomorrow it will be sunny?
Hidden Markov Models
Markow Models and Assumption (cont.):– Examples:
If the weather yesterday was rainy and today is foggy what is the probability that tomorrow it will be sunny?
Markov assumption
Hidden Markov Models
Hidden Markov Models (HMMs):– What is HMM:
Suppose that you are locked in a room for several days, you try to predict the weather outside, The only piece of evidence you have is whether the
person who comes into the room bringing your daily meal is carrying an umbrella or not.
Hidden Markov Models
Hidden Markov Models (HMMs):– What is HMM (cont.):
assume probabilities as seen in the table:
Hidden Markov Models
Hidden Markov Models (HMMs):– What is HMM (cont.):
Now the actual weather is hidden from you. You can not directly see what is the weather.
Hidden Markov Models
Hidden Markov Models (HMMs):– What is HMM (cont.):
Finding the probability of a certain weather
is based on the observations xi:
},,{ foggyrainysunnyqn
Hidden Markov Models
Hidden Markov Models (HMMs):– What is HMM (cont.):
Using Bayes rule:
For n days:
Hidden Markov Models
Hidden Markov Models (HMMs):– What is HMM (cont.):
We can omit So:
With Markov assumptions:
Hidden Markov Models
Hidden Markov Models (HMMs):– Examples:
Suppose the day you were locked in it was sunny. The next day, the caretaker carried an umbrella into the room.
You would like to know, what the weather was like on this second day.
Hidden Markov Models
Hidden Markov Models (HMMs):– Examples:
Calculate 3 probabilities:
Hidden Markov Models
Hidden Markov Models (HMMs):– Examples:
Consider the event with highest value. It is most likely to happen.
Hidden Markov Models
Hidden Markov Models (HMMs):– Another Examples:
Suppose you do not know how the weather was when your were locked in. The following three days the caretaker always comes without an umbrella. Calculate the likelihood for the weather on these three days to have been
Hidden Markov Models
Hidden Markov Models (HMMs):– Another Examples:
As you do not know how the weather is on the first day, you assume the 3 weather situations are equi-probable on this day and the prior probability for sun on day one is therefore
Hidden Markov Models
Hidden Markov Models (HMMs):– Another Examples:
Assumption:
Hidden Markov Models
Hidden Markov Models:– Another Examples:
33Discrete Markov Processes
(Markov Chains)
34
First-Order Markov Models
35
First-Order Markov Models
36
First-Order Markov Models
37First-Order Markov Model
Examples
38First-Order Markov Model
Examples
39First-Order Markov Model
Examples
40
First-Order Markov Models
41
First-Order Markov Models
42
First-Order Markov Models
43
First-Order Markov Models
44
First-Order HMM Examples
45
First-Order HMM Examples
46
First-Order HMM Examples
47
First-Order HMM Examples
48Three Fundamental Problems for
HMMs
49
HMM Evaluation Problem
50
HMM Evaluation Problem
51
HMM Evaluation Problem
52
HMM Evaluation Problem
53
HMM Evaluation Problem
54
HMM Decoding Problem
55
HMM Decoding Problem
56
HMM Decoding Problem
57
HMM Learning Problem
58
HMM Learning Problem
59
HMM Learning Problem
60
HMM Learning Problem
References
R.O. Duda, P.E. Hart, and D.G. Stork, Pattern Classification, New York: John Wiley, 2001. Selim Aksoy, “Pattern Recognition Course Materials”, Bilkent University, 2011.