Sample exam 2 SOLUTIONS

1 Given that evidence E2 adds to the original evidence El , show that

P(Di | El ÇE2) =

Proof: By Bayes Theorem

P(Di | El ÇE2) =

You should show P(A Ç B| C) = P(A| B Ç C) P(B|C) is true and use it.

P(Di | El ÇE2) = = = since P(E1) cancels out of numerator and denominator.

2 Prove that
CF(H,E) + CF(H’,E) =0

Comments on a proof.

Use the fact that CF(H,E) = MB(H,E) - MD(H,E) for any H and E, (i.e., H or H’) and use the definition for MB and MD and as many cases as is necessary (i.e. P(H) = 1, 0, or not 0.

3 Given rules
IF El AND E2 AND E3 THEN H (CF1)

IF E4 OR E5 THEN H (CF2)

where

CF1(E₁,e) = 1 CF1(E₂,e) = 0.5 CF1(E₃,e) = 0.3

CF2(E₄,e) = 0.7 CF2(E₅,e) = 0.2

CF1(H,E) =0.5 CF2(H,E) =0.9

(a) Calculate the certainty factors CF1 (H,e) and CF2(H,e).

2. Calculate CF_COMBINE [CFl(H,e),CF2(H,e)]

Solutions and/or comments

1. For example CF1(E,e) = CF(E1 Ç E2 Ç E3 | e ) = min(CF(E1 , e), CF(E2 | e), CF(E3 |e) = .3

(b) Use CF_COMBINE formula.

4 (a) Suppose Z Y X. Let

m(X) = 0.2, m(Y) = 0.3, m(Z) = 0.5

find the evidential intervals of X, Y, and Z using the Dempster-Shafer theory.

2. Suppose X and Y intersect, but neither is contained in the other. Let

m(X) = 0.4 and m(Y) = 0.6

find the evidential intervals for

x

X ÇY

Y

X ÇY’

X’ ÇY

Solutions and/or comments. To solve, note the evidential interval for a set A is [Bel(A), Pls(A)]

5 Given the rules Rule 1: IF E THEN H

Rule 2: IF E THEN H’

and assuming

Q ={H,H’}

ml(H) =0.5 ml(Q)=0.5 for Rule l

m2(H’) = 0.3 m2(Q) = 0.7 for Rule 2

(a) Write the Dempster-Shafer table showing the combination of evidence and calculate the combined belief functions.

(b) Calculate the plausibilities.

(c) Calculate the evidential intervals.

(d) Calculate the doubt of X, Dbt(X) = bel(X’).

(e) Calculate the ignorances in X, i.e., Igr(X) = pls(X) – bel(X)..

Comments on the solution. For all parts use the DS formula for combining masses and use this to compute the above quantities using the appropriate definitions.

6 Based on reports from different types of sensors, the following table gives the degrees of belief in the aircraft environment of airliner (H), bomber (B), and fighter, (F).

Focal elements Sensor 1 (ml) Sensor 2 (m2)
Q 0.15 0.2
A,B 0.3 0.1
A,F 0.1 0.05
B,F 0.1 0.1
A 0.05 03
B 0.2 0.05
F 0.1 0.2

(a) Calculate the initial belief functions, plausibility, evidential intervals, doubt, and ignorances.

2. Calculate these same parameters after the evidence is combined.

Solution and/or comments. Note: The two columns represent the masses assigned based on two sensors. Part (a) uses the first sensor, then part (b) combines the masses to get a new mass function.

7. Given the fuzzy sets,

A=.1/l +.2/2+.3/3 B = .2/1 + .3/2 + .4/3

calculate/explain the following:

(a) Are the sets equal? Explain.

(b) Set complement

(c) Set union

(d) Set intersection

(e) Does the Law of the Excluded Middle hold for each set? Explain.

(I) Set product

(h) Probabilistic sum

(i) Bounded sum

(j) Bounded product

(k) Bounded difference

(1) Concentration

(m) Dilation

(n) Intensification

(o) Normalization

Solutions and/or comments

Just use the definitions of each operation. As we remarked in class, Law of the Excluded Middle means the intersection or A and A’ should be empty, or in terms of fuzzy sets, the intersection of the fuzzy set membership functions should be 0.

8. (a) Define five linguistic values for the linguistic variable Uncertainty.

(b) Draw appropriate functions for these values and explain your choices.

(c) Draw the fuzzy sets for

Not TRUE

More Or Less TRUE

Sort Of TRUE

Pretty TRUE

Rather TRUE

TRUE

assuming TRUE is an S-function. What are the limits of TRUE? Explain.

Solution and/or comments

(a) Impossible, possible, probable, likely, certain are examples.for part (a)

(b) Use fuzzy sets with domain [0, 1]. Certain and impossible would be crisp sets defined to be 1 at 0 and 1 respectively. You could add fuzzy sets almost impossible and almost certain that would not be identically zero.

9 (a) Define at least six values for the linguistic variable Water Temperature.

(b) Draw the appropriate functions for the fuzzy set values on one graph.

(c) Give three hedged fuzzy set functions based on FREEZING.

Solutions and/or comments

1. examples are freezing, cold, cool, mild, warm, hot, burning-up

2. Use S and pi functions for example, but it could be any shape. (S and pi functions are defined in the fuzzy logic notes attached at the end of this document. As you will see by the equations they are made up of quadratic functions pieced together to form, for the S function, a function whose graph resembles an S or integral sign, or for the pi function, a mound shaped graph. I don’t really know why it was called a pi function but it was and it has stuck. It really looks more like a Gaussian distribution function. It is not of course, as there is no restriction on the area under the curve. For Gaussian distributions the area under the curve must be 1.)

3. more or less freezing = [ _freezing(x)]^0.5, sort of freezing(x) = [ _freezing(x)]^1.5, very freezing(x) = [ _freezing(x)]² There could be others of course.

10. Given numeric truth values,

x(A) = .2/.1 + .6/.5 + 1/.9

x(B) = .1/.1 + .3/.5 + 1/.9

calculate the fuzzy logic truth of the following:

(a) NOT A

2. A AND B

3. A OR B

1. A ® B

(e) B ® A

Comments on the solution

Here x(A) and x(B) just represent the fuzzy set names for condition A and B. You should just be able to use the definitions.

11. Define fuzzy functions to implement the following set of rules.

If pressure is high then turn valve lower.

If pressure is very high then turn valve much lower.

If pressure is very very high then turn valve much much lower.

If pressure is low to medium then turn valve higher.

Fuzzy Logic Notes

Definition: A fuzzy set f is a mapping from X, the universe of discourse to the unit interval {0, 1}.

There are several functions that are used quite often:

The S function which is a function of three parameters, a, b, c as follows:

S(x, a, b, c) =

The function 1 - S(x, a, b, c) will cause transition from 1 to 0 as x is increased from a to c.

The function which seems to cause confusion with the normal distribution curve. It is defined as follows as a function of two parameters a and b.

(x, a, b

The  function could be used to model the phrase “close to a.” If it was desired to use fuzzy sets to describe temperatures, one might use the S function for modeling hot temperature, a function for warm temperature, and a 1 - S function to model cold temperature.

When the domain is finite and not too large, fuzzy sets are sometimes denoted as follows:

f = {_/x₁ + _/x₂ + … + _n /x_n }

where _i= f(x_i)