Ch9 Reasoning in Uncertain Situations

Dr. Bernard Chen Ph.D.University of Central Arkansas

Spring 2011

Outline

Reasoning in Fuzzy Sets Markov Models

Reasoning with Fuzzy Sets There are two assumptions that

are essential for the use of formal set theory: For any element and a set belonging

to some universe, the element is either a member of the set or else it is a member of the complement of that set

An element cannot belong to both a set and also to its complement

Reasoning with Fuzzy Sets Both these assumptions are violated in Lotif

Zadeh.s fuzzy set theory

Zadeh.s main contention (1983) is that, although probability theory is appropriate for measuring randomness of information, it is inappropriate for measuring the meaning of the information

Zadeh proposes possibility theory as a measure of vagueness, just like probability theory measures randomness

Reasoning with Fuzzy Sets The notation of fuzzy set can be

describes as follows: let S be a set and s a member of

that set, A fuzzy subset F od S is defined by a membership function mF(s) that measures the “degree” to which s belongs to F

Reasoning with Fuzzy Sets For example: S to be the set of positive integers and F to be the fuzzy

subset of S called small integers Now, various integer values can have a “possibility”

distribution defining their “fuzzy membership” in the set of small integers: mF(1)=1.0, mF(3)=0.9, mF(50)=0.001

Reasoning with Fuzzy Sets For the fuzzy set representation of

the set of small integers, in previous figure, each integer belongs to this set with an associated confidence measure.

In the traditional logic of “crisp” set, the confidence of an element being in a set must be either 1 or 0

Reasoning with Fuzzy Sets This figure offers a set membership function

for the concept of short, medium, and tall male humans.

Note that any one person can belong to more than one set

For example, a 5.9” male belongs to both the set of medium as well as to the set of tall males

Reasoning with Fuzzy Sets A classic in the fuzzy set literature, a control regime for

an inverted pendulum We desire to keep in balance and pointing upward We keep the pendulum in balance by moving the base

of the system to offset the force of gravity acting on the pendulum

Reasoning with Fuzzy Sets We simplify the problem by presenting it in 2D

There are two measurements are used as input values to the controller

First angle θ, the deviation of the pendulum from the vertical

Second, the speed dθ/dt, at which the pendulum is moving

Both measures are positive in the quadrant to the right and negative to the left

Reasoning with Fuzzy Sets The input value θ is partitioned into three

regions: Negative, Zero, and Positive The input value dθ/dt is also partitioned into

three regions: Negative, Zero, and Positive

Reasoning with Fuzzy Sets

This figure is the defuzzified control response, where we use middle five regions, Negative Big, Negitive, Positive, Positive Big

Note that both the original input and final output data of the controller are crisp value

Reasoning with Fuzzy Sets

How to use this??? For example, if we currently we

have the situation: θ=1 ; dθ/dt=-4

Reasoning with Fuzzy Sets For θ, the value are Zero with 0.5 and Positive with 0.5 For dθ/dt, the value are Negative with 0.8 and Zero with 0.2

The Fuzzy Associative Matrix (FAM) for the pendulum problem. The input values are on the left and top

Reasoning with Fuzzy Sets In this case, because each input value

touched on two regions of the input space, four rules must be applied

Dr. Zedah is the first to propose these combination rules for the algebra of fuzzy reasoning

In our example, all premise pairs are ANDed together, so the minimum of their measures is taken as the measure of the rule result

Reasoning with Fuzzy Sets

Reasoning with Fuzzy Sets

Reasoning with Fuzzy Sets

Outline

Reasoning in Fuzzy Sets Markov Models

Markov Models

So how is “Tomato” pronounced

A probabilistic finite state acceptor for the pronunciation of “tomato”, adapted from Jurafsky and Martin (2000).

Markov Models In section 5.3, we presented the

probabilistic finite state machine

A state machine where the next state function was represented by probability distribution on the current state

The discrete Markov process is a specialization of this approach, where the system ignores its input values

Markov Models

Markov Models

NO body understands it…

Lets take a look of an example: S1= Sun S2= Cloudy S3= Fog S4= Precipitation

Markov Models S1= Sun S2= Cloudy S3= Fog S4= Precipitation

Markov Models We now are able to ask questions

of our model. Suppose today, S1, is sunny, What is the probability of the next

five days remaining Sunny?

What is the probibility of the next five days being sunny, sunny, cloudy, cloudy, precipitation?

Markov Models

Answer for the first question: 0.4^5

Answer for the second question:

Markov Models

Use Markov Model to represent the idea

Markov Models This example follows the “first-

order” Markov assumption where weather each day is a function (only) of the weather the day before

We also observe the fact that today is sunshine !? (Fuzzy concept may be applied)

Markov Models We may also extend this example to

determine, given that we know today.s weather, the probability that the weather will be the same for exactly the next t days

O={si (today), si, …, si, sj}, where there are exactly (t+1) si, and si!=sj, then:

p(O|M)=1*aii^t*(1-aii)

Markov Models

There are many advanced Markov Models Hidden Markov Models Semi-Markov Models Markov Decision Processes

Hidden Markov Models

Markov Chains

Sunny

Rain

Cloudy

State transition matrix

Initial Distribution

Sunny Cloud Rain

1 0 0

States

Sunny Cloud Rain

Sunny 0.5 0.3 0.2

Cloud 0.4 0.2 0.4

Rain 0.2 0.5 0.3

Hidden Markov Models

Hidden states : the (TRUE) states of a system that may be described by a Markov process (e.g., the weather).

Observable states : the states of the process that are `visible. (e.g., seaweed dampness).

Components Of HMM

Initial Distribution : contains the probability of the (hidden) model being in a particular hidden state at time t = 1.

State transition matrix : holding the probability of a hidden state given the previous hidden state.

Dry Dryish Damp Soggy

Sun 0.6 0.2 0.1 0.1

Cloud 0.2 0.3 0.3 0.2

Rain 0.1 0.2 0.2 0.5

Hidden Markov Models

Question now we may ask is like:

Today is a Dryish day, what is tomorrow.s weather might be?

Hidden Markov Models

Since today is a Dryish day, we know that: Sun 20/ Cloud30/ Rain 20/

Dry Dryish Damp Soggy

Sun 0.6 0.2 0.1 0.1

Cloud 0.2 0.3 0.3 0.2

Rain 0.1 0.2 0.2 0.5

Hidden Markov Models

Today Tomorrow

Sun 20/ Sun 0.2*0.5=0.01

Cloud 0.2*0.3=0.06

Rain 0.2*0.2=0.04

Cloud 30/ Sun 0.3*0.4=0.12

Cloud 0.3*0.2=0.06

Rain 0.3*0.4=0.12

Rain 20/ Sun 0.2*0.2=0.04

Cloud 0.2*0.5=0.10

Rain 0.2*0.3=0.06

Hidden Markov Models

Therefore:The opportunity of

Sunny: 0.01+0.12+0.04=0.17Cloudy: 0.06+0.06+0.10=0.22Rain: 0.04+0.12+0.06=0.22

Tomorrow is Rain or Cloudy

Hidden Markov Models

Application of HMM HMMs are very common in

Computational Linguistics: Speech recognition (observed: acoustic

signal, hidden: words) Handwriting recognition (observed: image,

hidden: words) Part-of-speech tagging (observed: words,

hidden: part-of-speech tags) Machine translation (observed: foreign

words, hidden: words in target language)

Application of HMM

Biology Gene finding and prediction Protein-Profile Analysis Secondary Structure prediction

A HMM model for a DNA motif alignments, The transitions are shown with arrows whose thickness indicate their probability. In each state, the histogram shows the probabilities of the four bases.

ACA - - - ATG TCA ACT ATCACA C - - AGCAGA - - - ATCACC G - - ATC

Building – from an existing alignment

Transition probabilities

Output Probabilities

insertion

Building – from an existing alignment

Imagine a DNA motif like this:

A regular expression for this is [AT] [CG] [AC] [ACGT]* A [TG] [GC] ,

ACA - - - ATG TCA ACT ATCACA C - - AGCAGA - - - ATCACC G - - ATC

Building – from an existing alignment To score a sequence, we say that there

is a probability of 4/5 = 0.8 for an A in the first position and 1/5 = 0.2 for a T, because we observe that

out of 5 letters 4 are As and one is a T. Similarly in the second position the

probability of C is 4/5 and of G 1/5, and so forth.

Building – from an existing alignment

After the third position in the alignment, 3 out of 5 sequences have `insertions' of varying lengths, so we say the probability of making an insertion is 3/5 and thus 2/5 for not making one.

Building – from an existing alignment The only part that might seem tricky is the

`insertion, which is represented by the state above the other states.

The probability of each letter is found by counting all occurrences of the four nucleotides in this region of the alignment.

The total counts are one A, two Cs, one G, and one T, yielding probabilities 1/5, 2/5, 1/5, and 1/5 respectively.

Building – from an existing alignment After sequences 2, 3 and 5 have made one

insertion each, there are two more insertions (from sequence2’A sequence2’C)

and the total number of transitions back to the main line of states is 3 (all three sequences with insertions have to finish).

Therefore there are 5 transitions in total from the insert state, and the probability of making a transition to itself is 2/5 and the probability of making one to the next state is 3/5

Consensus sequence: P (ACACATC) = 0.8x1 x 0.8x1 x 0.8x0.6 x 0.4x0.6 x 1x1 x 0.8x1 x 0.8 = 4.7 x 10 -2

Suppose I have a query protein sequence, and I am interested in which family it belongs to? There can be many paths leading to the generation of this sequence. Need to find all these paths and sum the probabilities.

ACAC - - ATC

Query a new sequence