Slides - Alternative Representation of Uncertainty

8/12/2019 Slides - Alternative Representation of Uncertainty

1/31


2/31

Alternative Representations of Uncertainty

Probability Theory

Evidence Theory (Dempster-Shafer Theory)

Possibility Theory

Interval Analysis

2


3/31

Probability

3

Formal definition of probability involves three components A set that contains everything that could occur in the particular universe under

consideration

A set of subsets of with the properties that (i) if , then C and (ii) if {i}is a

countable collection of elements of , then Uiiand ii are elements of

A functionp defined for elements of such that (i)p()=1, (ii) if , then 0p() 1,and (iii) if {i} is a countable collection of disjoint elements of , thenp(Ui)=ip(i)

Triple (, ,p) is called a probability space

Terminology called the sample space or universal set

Elements of are called elementary events Elements of are called events

p called a probability measure


4/31

Evidence Theory: Definition of Evidence Space

4

Formal definition an evidence theory representation of uncertainty

involves 3 components

A set that contains everything that could occur in the particular universe under

consideration

A (countable) set of subsets of

A function m defined for subsets of such that (i) m()>0 if , (ii) m()=0 if

and (iii) m()=1

Triple (, , m) is called an evidence space


Elements of are called elementary events

Elements of are called focal elements

m called a basic probability assignment (BPA)

Nature of m(): Amount of likelihood that is associated with but

cannot be further partitioned to subsets of .


5/31

5

Evidence Theory: Representation of Uncertainty

Representation of uncertainty

Belief

Plausibility

Belief: Definition:

Concept: Amount of likelihood that must be associated with .

Plausibility: Definition:

Concept: Amount of likelihood that could potentially be associated with .

=

)()( mBel

=

)()( mPl


6/31

6

Evidence Theory: Simple Example

1 2 3 4 5 6 7 8 9 10

]8,2[:

5/1)(],10,9[:5/1)(],10,5[:

5/1)(],6,5[:

5/1)(],7,3[:

5/1)(],4,1[:

55

44

33

22

11

=

==

==

==

==

==

mm

m

m

m

[ ]{ }10,1: = xx{ }54321 ,,,, =

5/2)()()()( 32 =+==

mmmBeli

i

5/4)()()()()()( 4321 =+++==

mmmmmPl ii


7/31

7

Evidence Theory: Properties

)()(

1)()(

1)()(

1)()(

PlBel

PlPl

BelBel

PlBel

C

C

C

+

+

=+

Contrast with Probability

1)()( =+ Cpp


8/31

8

Evidence Theory: Cumulative Representation

Cumulative belief function (CBF)

Cumulative plausibility function (CPF)

Complementary cumulative belief function (CCBF)

Complementary cumulative plausibility function (CCPF)

Analogous to CDF.

Plot of belief, plausibility of

being less than specified

values

Analogous to CCDF.

Plot of belief, plausibility of

being greater than specified

values

CBF, CCBF, CPF and CCPF for a variable vwith values from the interval [1, 10] and each of the

following intervals assigned a BPA of 0.1:[1, 3], [1, 4], [1, 10], [2, 4], [2, 6], [5, 8], [5, 10], [7, 8],[7, 10], [9, 10].


9/31

9

Evidence Theory: Vector-valued Quantities

x1 ,x2 ,,xn real-valued with evidence spaces (i,i,mi),

i=1,2,,n

Evidence space (,,m) for x=[x1 ,x2 ,,xn]

= 1 2 n iff = 1 2 n for ii

Belief, plausibility defined same as in one variable case

=

=

otherwise0

if)()...()()(

212211 nnnmmmm


10/31

10

Evidence Theory: Function with Uncertain Arguments

Function f(x) with evidence space (,,mx) for x

Resultant evidence space (,,my) for y=f(x)

= {y : y=f(x) , x }

= { : =f(), }

In concept, belief and plausibility defined from and my

In computational practice, belief and plausibility obtained by mapping back toevidence space (,,mx) for x

Cumulative and complementary cumulative results for y

==

otherwise0),(if)()( fmm xy

{ }( ) = )(:)( xx fBelBel xy{ }( ) = )(:)( xx fPlPl xy

{ }( )[ ] { }( )[ ]{ }( )[ ] { }( )[ ])(:,:CCPF,)(:,:CCBF

)(:,:CPF,)(:,:CBF

xxxx

xxxx

yyPlyyyBely

yyPlyyyBely

xx

xx


11/31

11

Evidence Theory: Example (1/3)

Function f(x) =f(a, b)=(a+b)a, x=[a,b]

Evidence space (,,mA ) for a

Evidence space (,,mB) for b

Evidence space (,,mX ) for x = [a, b]

Probability space (,,pX ) for x = [a, b]: Uniform distribution on each rectanglei jweighted by 1/12

{ }]6.0,1.0[],7.0,2.0[],0.1,5.0[

3/1)(,,,],0.1,1.0[

321

321

===

===

iAm

{ }].01,0.0[],7.0,1.0[],8.0,4.0[],6.0[

4/1)(,,,,],0.1,0.0[

4321

4321

=======

iBm

{ } 12/1)4/1)(3/1()(,,,,, 432111 ==== jiXm


12/31

12



13/31

13


0.0)12/1(0)5.1(

33.0)12/1(4)5.1(

5.0)12/1(6)8.0(

1)12/1(12)8.0(

===>

===>

===>

===>

yBel

yPl

yBel

yPl


14/31

14

Possibility Theory: Definition of Possibility Space

Formal definition a possibility theory representation of uncertainty

involves 2 components

A set that contains everything that could occur in the particular universe under

consideration

A function rsuch that (i) 0 r(x) 1 forx and (ii) sup{ r(x):x } = 1

Doublet (, r) is called an possibility space


ris referred to a possibility distribution function

Nature of r: Amount of likelihood or credence that can be assigned toeach element of . Analogous to membership value for elements of a

fuzzy set.


15/31

15

Possibility Theory: Representation of Uncertainty

Representation of uncertainty

Possibility

Necessity

Possibility: Definition:

Concept: Measure of amount of information that does not refute the proposition that

contains the correct value forx.

Necessity:

Definition:

Concept: Measure of amount of uncontradicted information that supports theproposition that contains the correct value forx.

{ } = xxrPos :)(sup)(

== xxrPosNec :)(sup1)(1)(


16/31


17/31

17

Possibility Theory: Properties

)()(

1)()(

1)()(

1)()(

PosNec

PosPos

NecNec

PosNec

C

C

C

+

+

=+

Contrast with Probability

1)()( =+ Cpp


18/31

18

Possibility Theory: Cumulative Representation

Cumulative necessity function (CNF)

Cumulative possibility function (CPoF)

Complementary cumulative necessity function (CCNF)

Complementary cumulative possibility function (CCPoF)

Analogous to CDF.

Plot of necessity, possibility of

being less than specified

values

Analogous to CCDF.

Plot of necessity, possibility of

being greater than specified

values

CNF, CCNF, CPoF and CCPoF for a variable vwith values from the interval [1, 10] a possibility

distribution function rvdefined as follows: rv(v) = i/5, for i= 1, 2, 3, 4, 5 and i v < i+1 and rv(v)= (10-i)/4, for i = 6, 7, 8, 9, i v < i+1, and v i+1 used instead of v < i+1 for i = 9


19/31

19

Possibility Theory: Vector-valued Quantities

x1 ,x2 ,,xn real-valued with possibility spaces (i, ri),

i=1,2,,n

Possibility space (, r) for x=[x1 ,x2 ,,xn] = 1 2 n

r(x) = min {r1(x1), r2(x2), , rn(xn) } for x=[x1 ,x2 ,,xn]

Necessity, possibility defined same as in one variable case


20/31

20

Possibility Theory: Function with Uncertain Arguments

Function f(x) with evidence space (, rx) for x

Resultant possibility space (, ry) for y=f(x)

= {y : y=f(x) , x }

ry(y) = sup{rx(x): y=f(x), x }

In concept, necessity and possibility defined from and ry

In computational practice, necessity and possibility obtained by mapping back topossibility space (, rx) for x

Cumulative and complementary cumulative results for y

{ }( ) = )(:)( xx fNecNec xy{ }( ) = )(:)( xx fPosPos xy

{ }( )[ ] { }( )[ ]{ }( )[ ] { }( )[ ])(:,:CCPoF,)(:,:CCNF

)(:,:CPoF,)(:,:CNF

xxxx

xxxx

yyPosyyyNecy

yyPosyyyNecy

xx

xx


21/31

21

Possibility Theory: Example (1/2)

Function f(x) =f(a, b)=(a+b)a, x=[a,b]

Possibility space (, rA ) for a

Possibility space (, rB) for b

Possibility space (, rX ) for x = [a, b]

Probability space (,,pX ) for x = [a, b]: Uniform distribution on each rectanglei jweighted by 1/12

[ ]( ) { })(),(min,, brarbar BAX ==

==

====

= otherwise0if1

)(where3/)()(

]6.0,1.0[],7.0,2.0[],0.1,5.0[],0.1,1.0[

3

1

321

i

iiiA

a

aaar

==

=====

= otherwise0

if1)(where4/)()(

].01,0.0[],7.0,1.0[],8.0,4.0[],.60[],0.1,0.0[

4

1

4321

i

i

i

iB

bbbbr


22/31

22

Possibility Theory: Example (2/2)

Values of distribution function rxfor possibility

space (,rx) (e.g. rx([a,b]) = 1/2 for 0.2 a 0.7

and 0.1 b 0.4

Estimated CCNF, CCDF and CCPoF for y=f(a, b)

= (a + b )a

0.00.10.1)5.1(1)5.1(

33.0)5.1(

5.05.00.1)8.0(1)8.0(

1)8.0(

=====>

==>

=====>

==>

yPosyNec

yPos

yPosyNec

yPos


23/31

23

Interval Analysis

Define range of values forx

Determine resultant range of values for y =f(x)

No uncertainty structure imposed onx, only range of values

Different in spirit from probability theory, evidence theory and possibility

theory representation of uncertainty

Corresponds to degenerate evidence theory and possibility theory

representation of uncertainty

Evidence theory: sample space has BPA of 1

Possibility theory: Possibility distribution function identically equal to 1


24/31

24

Notional Example: Only Epistemic Uncertainty (1/4)

EN2: Modelf(t|eM)=Q(t|eM) for closed electrical circuit

where

0)0(d

d

,0)0(),exp(d

d

d

d02

2

===++ t

Q

QtEC

Q

t

Q

Rt

Q

L

peres).current(amd

d(volts),forceiveelectromot)exp(

(farads),ecapacitanc

(ohms),resistance

(henrys),inductance

(s),at time(coulombs)chargeelectrical)(

0

=

=

=

=

=

=

t

Q

tE

C

R

L

ttQ


25/31

25


EN3: probability space (, ,pEM) for epistemic uncertainty

eM=[eM1,eM2,eM3,eM4,eM5]=[L,R,C,E0,]

1 = {L: 0.8 L 1.2 henrys }, 2= {R: 50 R 100 ohms },

3= {C: 0.9x10-4 C 1.1x10-4 farads }, 4= {E0: 900 E0 1100 volts },

5= {: 0.4 0.8 s-1 }, = 1 x 2 x 3 x 4 x 5

Four subintervals are considered for eachof the intervals i, i=1,2,,5, defined above

i1=[ a , b (b a)/4 ],

i2

=[ a + (b a)/4, b ],

i3 =[ a + (b a)/8, b 3(b a)/8 ],

i4 = [ a + 3(b a)/8, b (b a)/8 ],

10 32 54 76 8

i1 :

i2 :

i3 :i4 :

Illustration of sets i1, i2, i3 and i4 defined

with the interval [a,b] normalized to the

interval [0,8] for representational simplicity

( ) ( ) [ ] ( )

=== otherwise0

if1 with)min()max(4

4

1

ijMi

Miij

i

ijijMiijMii

eeeed

In effect, defines dEM(e) and pEM()

Under the assumption that the four sources that provided theintervals for an element eMiof eM are equally credible


26/31

26


Evidence theory representation For each uncertain variable, set iof possible values divided into subsets i1, i2, i3, i4 as

indicated on preceding slide with i=1, 2, 3, 4, 5 corresponding L,R,C,E0,, respectively

i= {i1, i2, i3, i4 }

BPA for subset of i:

(i, i, mi) evidence space L,R,C,E0,

Evidence space (, , m) for eM=[L,R,C,E0,] results as previously described

Possibility space representation Sets i1, i2, i3, i4 same as above for i=1, 2, 3, 4, 5

For e i,

(i,ri) possibility spaces for L,R,C,E0,

Possibility space (,r) for eM=[L,R,C,E0, ] results as previously described

Interval analysis set of possible values for eM=[L,R,C,E0,]

No uncertainty structure assumed for

= otherwise0

if4/1

)( i

im

=== otherwise0if4/1)(with4/)()(

4

1

iji

i

ii eeeer


27/31

27


Uncertainty propagated with random sample of size 105

.: Time (s)

(

|

)

0.00 0.05 0.10 0.15 0.200.00

0.05

0.10

0.15

0.20

Q

t

ta,

eM

50 of 105 results


28/31

28

Notional Example: Aleatory and Epistemic Uncertainty

(1/3)

Overview: Mechanical system receiving perturbations whose occurrencefollows a stationary Poisson process with each perturbation decayingexponentially with time after its occurrence.

EN1: probability space (, ,pA) for aleatory uncertainty for time interval [0,

10 s] a = [n,t1,A1,t2,A2,,tn,An]

where

n = number of perturbations in time interval [0,10 s]

ti= occurrence time for perturbation iwith t1 < t2 < < tn

Ai= amplitude of occurrence i

Occurrence times characterized by a Poisson process with rate

AmplitudeA has triangular distribution on [a,b] with mode m

= {a: a = [n,t1,A1,t2,A2,,tn,An]}

and distribution forA in effect define (, ,pA) and dA(a)

, a, m, b epistemically uncertain dA(a|eA), eA=[, a, m, b]


29/31

29

Notional Example: Aleatory and Epistemic Uncertainty

(2/3)

EN2: ModelA(t|a,r) for accumulated perturbations at time t

Where r = epistemically uncertain perturbation decay rate (i.e., eM = [r]) EN3: Probability space (, ,pE) for epistemic uncertainty

e = [eA,eM] = [, a, m, b, r] , eA = [, a, m, b], eM = [r]

1= {: 0.5 1.5 s-1 }, 2= {a: 1.0 a 2.0 kg m/s

2 },

3 = {m: 2.0 m 4.0 kg m/s2 }, 4= {b: 4.0 b 5.0 kg m/s

2 },

1= {r: 0.2 r 1.2 s-1 }, = 1 x 2 x 3 x 4 x 1

Probability space (, ,pE), evidence space (, , mE), possibility space(, rE) and set for interval analysis defined in same manner as preceding

example

[ ]

=

Slides - Alternative Representation of Uncertainty

Documents

Transcript of Slides - Alternative Representation of Uncertainty