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Ch9 Reasoning in Uncertain Situations
Dr. Bernard Chen Ph.D.University of Central Arkansas
Spring 2011
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
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
NO body understands it…
Lets take a look of an example: 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 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
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
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