This document introduces hidden Markov models (HMMs). It defines the key components of HMMs, including states, observations, transition probabilities, and observation probabilities. It provides an example of an HMM for weather prediction with hidden states of "Rain" and "Dry" and observations of the weather. It also discusses the main problems in using HMMs, including evaluation, decoding, and learning problems. Forward-backward and Viterbi algorithms are introduced for efficiently solving the evaluation and decoding problems, respectively.
In this presentation we describe the formulation of the HMM model as consisting of states that are hidden that generate the observables. We introduce the 3 basic problems: Finding the probability of a sequence of observation given the model, the decoding problem of finding the hidden states given the observations and the model and the training problem of determining the model parameters that generate the given observations. We discuss the Forward, Backward, Viterbi and Forward-Backward algorithms.
How to make hash functions go fast inside snarks, aka a guided tour through arithmetisation friendly hash functions (useful for all cryptographic protocols where cost is dominated by multiplications -- e.g. anything using R1CS; secret sharing based multiparty computation protocols; etc)
In this presentation we describe the formulation of the HMM model as consisting of states that are hidden that generate the observables. We introduce the 3 basic problems: Finding the probability of a sequence of observation given the model, the decoding problem of finding the hidden states given the observations and the model and the training problem of determining the model parameters that generate the given observations. We discuss the Forward, Backward, Viterbi and Forward-Backward algorithms.
How to make hash functions go fast inside snarks, aka a guided tour through arithmetisation friendly hash functions (useful for all cryptographic protocols where cost is dominated by multiplications -- e.g. anything using R1CS; secret sharing based multiparty computation protocols; etc)
Shor's algorithm is for quantum computer. Using this algorithm any arbitrarily large number can be factored in polynomial time. which is not possible in classical computer
We want to identify the point(s) in time at which the rate of event occurrences changes, where the number of events is increasing or decreasing in frequency.
The Hidden Markov model (HMM) is a statistical model that was first proposed by Baum L.E. (Baum and Petrie, 1966) and uses a Markov process that contains hidden and unknown parameters. In this model, the observed parameters are used to identify the hidden parameters. These parameters are then used for further analysis. The HMM is a type of Markov chain. Its state cannot be directly observed but can be identified by observing the vector series. Since the 1980s, HMM has been successfully used for speech recognition, character recognition, and mobile communication techniques. It has also been rapidly adopted in such fields as bioinformatics and fault diagnosis. The basic principle of HMM is that the observed events have no one-to-one correspondence with states but are linked to states through the probability distribution. It is a doubly stochastic process, which includes a Markov chain as the basic stochastic process, and describes state transitions and stochastic processes that describe the statistical correspondence between the states and observed values. From the perspective of observers, only the observed value can be viewed, while the states cannot. A stochastic process is used to identify the existence of states and their characteristics. Thus, it is called a “hidden” Markov model.
Statistical methods are used to build state changes in HMM to understand the most possible trends in the surveillance data. HMM can automatically and flexibly adjust the trends, seasonal, covariant, and distributional elements. HMM has been used in many studies on time series surveillance data. For example, Le Strat and Carrat used a univariate HMM to handle influenza-like time series data in France. Additionally, Madigan indicated that HMM needed to include spatial information based on existing states.
Supervised Hidden Markov Chains.
Here, we used the paper by Rabiner as a base for the presentation. Thus, we have the following three problems:
1.- How efficiently compute the probability given a model.
2.- Given an observation to which class it belongs
3.- How to find the parameters given data for training.
The first two follow Rabiner's explanation, but in the third one I used the Lagrange Multiplier Optimization because Rabiner lacks a clear explanation about how solving the issue.
Accounting for uncertainty is a crucial component in decision making (e.g., classification) because of ambiguity in our measurements.
Probability theory is the proper mechanism for accounting for uncertainty.
Normal Labour/ Stages of Labour/ Mechanism of LabourWasim Ak
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Shor's algorithm is for quantum computer. Using this algorithm any arbitrarily large number can be factored in polynomial time. which is not possible in classical computer
We want to identify the point(s) in time at which the rate of event occurrences changes, where the number of events is increasing or decreasing in frequency.
The Hidden Markov model (HMM) is a statistical model that was first proposed by Baum L.E. (Baum and Petrie, 1966) and uses a Markov process that contains hidden and unknown parameters. In this model, the observed parameters are used to identify the hidden parameters. These parameters are then used for further analysis. The HMM is a type of Markov chain. Its state cannot be directly observed but can be identified by observing the vector series. Since the 1980s, HMM has been successfully used for speech recognition, character recognition, and mobile communication techniques. It has also been rapidly adopted in such fields as bioinformatics and fault diagnosis. The basic principle of HMM is that the observed events have no one-to-one correspondence with states but are linked to states through the probability distribution. It is a doubly stochastic process, which includes a Markov chain as the basic stochastic process, and describes state transitions and stochastic processes that describe the statistical correspondence between the states and observed values. From the perspective of observers, only the observed value can be viewed, while the states cannot. A stochastic process is used to identify the existence of states and their characteristics. Thus, it is called a “hidden” Markov model.
Statistical methods are used to build state changes in HMM to understand the most possible trends in the surveillance data. HMM can automatically and flexibly adjust the trends, seasonal, covariant, and distributional elements. HMM has been used in many studies on time series surveillance data. For example, Le Strat and Carrat used a univariate HMM to handle influenza-like time series data in France. Additionally, Madigan indicated that HMM needed to include spatial information based on existing states.
Supervised Hidden Markov Chains.
Here, we used the paper by Rabiner as a base for the presentation. Thus, we have the following three problems:
1.- How efficiently compute the probability given a model.
2.- Given an observation to which class it belongs
3.- How to find the parameters given data for training.
The first two follow Rabiner's explanation, but in the third one I used the Lagrange Multiplier Optimization because Rabiner lacks a clear explanation about how solving the issue.
Accounting for uncertainty is a crucial component in decision making (e.g., classification) because of ambiguity in our measurements.
Probability theory is the proper mechanism for accounting for uncertainty.
Normal Labour/ Stages of Labour/ Mechanism of LabourWasim Ak
Normal labor is also termed spontaneous labor, defined as the natural physiological process through which the fetus, placenta, and membranes are expelled from the uterus through the birth canal at term (37 to 42 weeks
Model Attribute Check Company Auto PropertyCeline George
In Odoo, the multi-company feature allows you to manage multiple companies within a single Odoo database instance. Each company can have its own configurations while still sharing common resources such as products, customers, and suppliers.
The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
For more information, visit-www.vavaclasses.com
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2. • Set of states:
• Process moves from one state to another generating a
sequence of states :
• Markov chain property: probability of each subsequent state
depends only on what was the previous state:
• To define Markov model, the following probabilities have to be
specified: transition probabilities and initial
probabilities
Markov Models
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• Two states : ‘Rain’ and ‘Dry’.
• Transition probabilities: P(‘Rain’|‘Rain’)=0.3 ,
P(‘Dry’|‘Rain’)=0.7 , P(‘Rain’|‘Dry’)=0.2, P(‘Dry’|‘Dry’)=0.8
• Initial probabilities: say P(‘Rain’)=0.4 , P(‘Dry’)=0.6 .
Example of Markov Model
4. • By Markov chain property, probability of state sequence can be
found by the formula:
• Suppose we want to calculate a probability of a sequence of
states in our example, {‘Dry’,’Dry’,’Rain’,Rain’}.
P({‘Dry’,’Dry’,’Rain’,Rain’} ) =
P(‘Rain’|’Rain’) P(‘Rain’|’Dry’) P(‘Dry’|’Dry’) P(‘Dry’)=
= 0.3*0.2*0.8*0.6
Calculation of sequence probability
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5. Hidden Markov models.
• Set of states:
•Process moves from one state to another generating a
sequence of states :
• Markov chain property: probability of each subsequent state
depends only on what was the previous state:
• States are not visible, but each state randomly generates one of M
observations (or visible states)
• To define hidden Markov model, the following probabilities
have to be specified: matrix of transition probabilities A=(aij),
aij= P(si | sj) , matrix of observation probabilities B=(bi (vm )),
bi(vm )= P(vm | si) and a vector of initial probabilities =(i),
i = P(si) . Model is represented by M=(A, B, ).
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7. • Two states : ‘Low’ and ‘High’ atmospheric pressure.
• Two observations : ‘Rain’ and ‘Dry’.
• Transition probabilities: P(‘Low’|‘Low’)=0.3 ,
P(‘High’|‘Low’)=0.7 , P(‘Low’|‘High’)=0.2,
P(‘High’|‘High’)=0.8
• Observation probabilities : P(‘Rain’|‘Low’)=0.6 ,
P(‘Dry’|‘Low’)=0.4 , P(‘Rain’|‘High’)=0.4 ,
P(‘Dry’|‘High’)=0.3 .
• Initial probabilities: say P(‘Low’)=0.4 , P(‘High’)=0.6 .
Example of Hidden Markov Model
8. •Suppose we want to calculate a probability of a sequence of
observations in our example, {‘Dry’,’Rain’}.
•Consider all possible hidden state sequences:
P({‘Dry’,’Rain’} ) = P({‘Dry’,’Rain’} , {‘Low’,’Low’}) +
P({‘Dry’,’Rain’} , {‘Low’,’High’}) + P({‘Dry’,’Rain’} ,
{‘High’,’Low’}) + P({‘Dry’,’Rain’} , {‘High’,’High’})
where first term is :
P({‘Dry’,’Rain’} , {‘Low’,’Low’})=
P({‘Dry’,’Rain’} | {‘Low’,’Low’}) P({‘Low’,’Low’}) =
P(‘Dry’|’Low’)P(‘Rain’|’Low’) P(‘Low’)P(‘Low’|’Low)
= 0.4*0.4*0.6*0.4*0.3
Calculation of observation sequence probability
9. Evaluation problem. Given the HMM M=(A, B, ) and the
observation sequence O=o1 o2 ... oK , calculate the probability that
model M has generated sequence O .
• Decoding problem. Given the HMM M=(A, B, ) and the
observation sequence O=o1 o2 ... oK , calculate the most likely
sequence of hidden states si that produced this observation sequence
O.
• Learning problem. Given some training observation sequences
O=o1 o2 ... oK and general structure of HMM (numbers of hidden
and visible states), determine HMM parameters M=(A, B, )
that best fit training data.
O=o1...oK denotes a sequence of observations ok{v1,…,vM}.
Main issues using HMMs :
10. • Typed word recognition, assume all characters are separated.
• Character recognizer outputs probability of the image being
particular character, P(image|character).
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Word recognition example(1).
Hidden state Observation
11. • Hidden states of HMM = characters.
• Observations = typed images of characters segmented from the
image . Note that there is an infinite number of
observations
• Observation probabilities = character recognizer scores.
•Transition probabilities will be defined differently in two
subsequent models.
Word recognition example(2).
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12. • If lexicon is given, we can construct separate HMM models
for each lexicon word.
Amherst a m h e r s t
Buffalo b u f f a l o
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• Here recognition of word image is equivalent to the problem
of evaluating few HMM models.
•This is an application of Evaluation problem.
Word recognition example(3).
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13. • We can construct a single HMM for all words.
• Hidden states = all characters in the alphabet.
• Transition probabilities and initial probabilities are calculated
from language model.
• Observations and observation probabilities are as before.
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• Here we have to determine the best sequence of hidden states,
the one that most likely produced word image.
• This is an application of Decoding problem.
Word recognition example(4).
14. • The structure of hidden states is chosen.
• Observations are feature vectors extracted from vertical slices.
• Probabilistic mapping from hidden state to feature vectors:
1. use mixture of Gaussian models
2. Quantize feature vector space.
Character recognition with HMM example.
15. • The structure of hidden states:
• Observation = number of islands in the vertical slice.
s1 s2 s3
•HMM for character ‘A’ :
Transition probabilities: {aij}=
Observation probabilities: {bjk}=
.8 .2 0
0 .8 .2
0 0 1
.9 .1 0
.1 .8 .1
.9 .1 0
•HMM for character ‘B’ :
Transition probabilities: {aij}=
Observation probabilities: {bjk}=
.8 .2 0
0 .8 .2
0 0 1
.9 .1 0
0 .2 .8
.6 .4 0
Exercise: character recognition with HMM(1)
16. • Suppose that after character image segmentation the following
sequence of island numbers in 4 slices was observed:
{ 1, 3, 2, 1}
• What HMM is more likely to generate this observation
sequence , HMM for ‘A’ or HMM for ‘B’ ?
Exercise: character recognition with HMM(2)
17. Consider likelihood of generating given observation for each
possible sequence of hidden states:
• HMM for character ‘A’:
Hidden state sequence Transition probabilities Observation probabilities
s1 s1 s2s3 .8 .2 .2 .9 0 .8 .9 = 0
s1 s2 s2s3 .2 .8 .2 .9 .1 .8 .9 = 0.0020736
s1 s2 s3s3 .2 .2 1 .9 .1 .1 .9 = 0.000324
Total = 0.0023976
• HMM for character ‘B’:
Hidden state sequence Transition probabilities Observation probabilities
s1 s1 s2s3 .8 .2 .2 .9 0 .2 .6 = 0
s1 s2 s2s3 .2 .8 .2 .9 .8 .2 .6 = 0.0027648
s1 s2 s3s3 .2 .2 1 .9 .8 .4 .6 = 0.006912
Total = 0.0096768
Exercise: character recognition with HMM(3)
18. •Evaluation problem. Given the HMM M=(A, B, ) and the
observation sequence O=o1 o2 ... oK , calculate the probability that
model M has generated sequence O .
• Trying to find probability of observations O=o1 o2 ... oK by
means of considering all hidden state sequences (as was done in
example) is impractical:
NK hidden state sequences - exponential complexity.
• Use Forward-Backward HMM algorithms for efficient
calculations.
• Define the forward variable k(i) as the joint probability of the
partial observation sequence o1 o2 ... ok and that the hidden state at
time k is si : k(i)= P(o1 o2 ... ok , qk= si )
Evaluation Problem.
21. • Define the forward variable k(i) as the joint probability of the
partial observation sequence ok+1 ok+2 ... oK given that the hidden
state at time k is si : k(i)= P(ok+1 ok+2 ... oK |qk= si )
• Initialization:
K(i)= 1 , 1<=i<=N.
• Backward recursion:
k(j)= P(ok+1 ok+2 ... oK |qk= sj ) =
i P(ok+1 ok+2 ... oK , qk+1= si |qk= sj ) =
i P(ok+2 ok+3 ... oK |qk+1= si) aji bi (ok+1 ) =
i k+1(i) aji bi (ok+1 ) , 1<=j<=N, 1<=k<=K-1.
• Termination:
P(o1 o2 ... oK) = i P(o1 o2 ... oK , q1= si) =
i P(o1 o2 ... oK |q1= si) P(q1= si) = i 1(i) bi (o1) i
Backward recursion for HMM
22. •Decoding problem. Given the HMM M=(A, B, ) and the
observation sequence O=o1 o2 ... oK , calculate the most likely
sequence of hidden states si that produced this observation sequence.
• We want to find the state sequence Q= q1…qK which maximizes
P(Q | o1 o2 ... oK ) , or equivalently P(Q , o1 o2 ... oK ) .
• Brute force consideration of all paths takes exponential time. Use
efficient Viterbi algorithm instead.
• Define variable k(i) as the maximum probability of producing
observation sequence o1 o2 ... ok when moving along any hidden
state sequence q1… qk-1 and getting into qk= si .
k(i) = max P(q1… qk-1 ,qk= si , o1 o2 ... ok)
where max is taken over all possible paths q1… qk-1 .
Decoding problem
23. • General idea:
if best path ending in qk= sj goes through qk-1= si then it
should coincide with best path ending in qk-1= si .
s1
si
sN
sj
aij
aNj
a1j
qk-1 qk
• k(i) = max P(q1… qk-1 ,qk= sj , o1 o2 ... ok) =
maxi [ aij bj (ok ) max P(q1… qk-1= si , o1 o2 ... ok-1) ]
• To backtrack best path keep info that predecessor of sj was si.
Viterbi algorithm (1)
24. • Initialization:
1(i) = max P(q1= si , o1) = i bi (o1) , 1<=i<=N.
•Forward recursion:
k(j) = max P(q1… qk-1 ,qk= sj , o1 o2 ... ok) =
maxi [ aij bj (ok ) max P(q1… qk-1= si , o1 o2 ... ok-1) ] =
maxi [ aij bj (ok ) k-1(i) ] , 1<=j<=N, 2<=k<=K.
•Termination: choose best path ending at time K
maxi [ K(i) ]
• Backtrack best path.
This algorithm is similar to the forward recursion of evaluation
problem, with replaced by max and additional backtracking.
Viterbi algorithm (2)
25. •Learning problem. Given some training observation sequences
O=o1 o2 ... oK and general structure of HMM (numbers of
hidden and visible states), determine HMM parameters M=(A,
B, ) that best fit training data, that is maximizes P(O |M) .
• There is no algorithm producing optimal parameter values.
• Use iterative expectation-maximization algorithm to find local
maximum of P(O |M) - Baum-Welch algorithm.
Learning problem (1)
26. • If training data has information about sequence of hidden states
(as in word recognition example), then use maximum likelihood
estimation of parameters:
aij= P(si | sj) =
Number of transitions from state sj to state si
Number of transitions out of state sj
bi(vm )= P(vm | si)=
Number of times observation vm occurs in state si
Number of times in state si
Learning problem (2)
27. General idea:
aij= P(si | sj) =
Expected number of transitions from state sj to state si
Expected number of transitions out of state sj
bi(vm )= P(vm | si)=
Expected number of times observation vm occurs in state si
Expected number of times in state si
i = P(si) = Expected frequency in state si at time k=1.
Baum-Welch algorithm
28. • Define variable k(i,j) as the probability of being in state si at
time k and in state sj at time k+1, given the observation
sequence o1 o2 ... oK .
k(i,j)= P(qk= si ,qk+1= sj |o1 o2 ... oK)
k(i,j)=
P(qk= si , qk+1= sj , o1 o2 ... ok)
P(o1 o2 ... ok) =
P(qk= si , o1 o2 ... ok) aij bj (ok+1 ) P(ok+2 ... oK | qk+1= sj )
P(o1 o2 ... ok) =
k(i) aij bj (ok+1 ) k+1(j)
i j k(i) aij bj (ok+1 ) k+1(j)
Baum-Welch algorithm: expectation step(1)
29. • Define variable k(i) as the probability of being in state si at
time k, given the observation sequence o1 o2 ... oK .
k(i)= P(qk= si |o1 o2 ... oK)
k(i)=
P(qk= si , o1 o2 ... ok)
P(o1 o2 ... ok) =
k(i) k(i)
i k(i) k(i)
Baum-Welch algorithm: expectation step(2)
30. •We calculated k(i,j) = P(qk= si ,qk+1= sj |o1 o2 ... oK)
and k(i)= P(qk= si |o1 o2 ... oK)
• Expected number of transitions from state si to state sj =
= k k(i,j)
• Expected number of transitions out of state si = k k(i)
• Expected number of times observation vm occurs in state si =
= k k(i) , k is such that ok= vm
• Expected frequency in state si at time k=1 : 1(i) .
Baum-Welch algorithm: expectation step(3)
31. aij =
Expected number of transitions from state sj to state si
Expected number of transitions out of state sj
bi(vm ) =
Expected number of times observation vm occurs in state si
Expected number of times in state si
i = (Expected frequency in state si at time k=1) = 1(i).
=
k k(i,j)
k k(i)
=
k k(i,j)
k,ok= vm k(i)
Baum-Welch algorithm: maximization step