Modeling correlations and
dependencies among intervals
Scott Ferson and Vladik Kreinovich
REC’06 Savannah, Georgia, 23 February 2006
Interval analysis
Advantages
 Natural for scientists and easy to explain
 Works wherever uncertainty comes from
 Works without specifying intervariable dependencies
Disadvantages
 Ranges can grow quickly become very wide
 Cannot use information about dependence
Badmouthing interval analysis?
Probability v. intervals

Probability theory


Can handle dependence well
Has an inadequate model of ignorance
LYING: saying more than you really know

Interval analysis


Can handle epistemic uncertainty (ignorance) well
Has an inadequate model of dependence
COWARDICE: saying less than you know
I said this in Copenhagen, and nobody objected
My perspective

Elementary methods of interval analysis




Low-dimensional, usually static problems
Huge uncertainties
Verified computing
Important to be best possible

Naïve methods very easy to use

Intervals combined with probability theory

Need to be able to live with probabilists
Dependence in probability theory

Copulas fully capture arbitrary dependence
between random variables (functional, shuffles, all)
2-increasing functions onto [0,1], with four edges fixed
1
1
u
v
0
M(u,v) = min(u,v)
Perfect
1
1
u
(u,v) = uv
v
0
Independent
1
1
u
v
0
W(u,v) = max(u+v1,0)
Opposite
Dependence in the bivariate case

Any restriction on the possible pairings between
inputs (any subset of the units square)
May also require each value of u to match with at least v, and vice versa

A little simpler than a copula

The null restriction is the full unit square


Call this “nondependence” rather than independence
D denotes the set of all possible dependencies
(set of all subsets of the unit square)
Two sides of a single coin

Mechanistic dependence
Neumaier: “correlation”

Computational dependence
Neumaier: “dependent” Francisco Cháves: decorrelation



Same representations used for both
Maybe the same origin phenomenologically
I’m mostly talking about mechanistic
Three special cases
Opposite
Nondependent
Perfect
(countermonotonic)
(the Fréchet case)
(comonotonic)
1
1
1
v
v
v
0
0
0
0
u
1
0
u
1
0
u
1
Correlation

A model of dependence that’s parameterized
by a (scalar) value called the “correlation
coefficient”
 : [1, +1]  D

The correlation model is called “complete” if
(1) =
, (0) =
, (+1) =
Corner-shaving dependence
r = 1
r=0
r = +1
D(r) = { (u,v) : max(0, ur, u1+r)  v  min(1, u+1r, u+2+r)}
u  [0,1], v  [0,1]
f(A, B) = {c : c = f(u(a2 –a1)+a1, v(b2 –b1)+b1), (u,v)  D }
A+B = [env(w(A, r)+b1, a1+w(B,r)), env(a2+w(B,1+r),w(A,1+r)+b2)]
a1
if p < 0
w([a1,a2], p) = a2
if 1 < p
p(a2a1)+a1 otherwise
Other complete correlation families
r = 1
r=0
r = +1
Elliptic dependence
Elliptic dependence
r = 1

r=0
r = +1
Not complete (because r=0 isn’t nondependence)
Parabolic dependence
Parabolic dependence
r = 1
r=0
r = +1

A variable and its square or square root have
this dependence

Variables that are not related by squaring could
also have this dependence relation
e.g.,
A = [1,5], B = [1,10]
So what difference does it make?
A+B
A = [2,5]
B = [3,9]
[ 5, 14]
Perfect
[ 8, 11]
Opposite
[ 7.1, 11.9]
Corner-shaving (r = 0.7)
[ 7.27, 11.73] Elliptic (r = 0.7)
[ 5, 14]
Upper, left
[ 5, 11]
Lower, left
[ 8, 14]
Upper, right
[ 5, 14]
Lower, right
[ 6.5, 12.5]
Diamond
[ 5, 14]
Nondependent
Eliciting dependence

As hard as getting intervals (maybe a bit worse)

Theoretical or “physics-based” arguments

Inference from empirical data

Risk of loss of rigor at this step (just as there is
when we try to infer intervals from data)
Generalization to multiple dimensions

Pairwise



Multivariate



Matrix of two-dimensional dependence relations
Relatively easy to elicit
Subset of the unit hypercube
Potentially much better tightening
Computationally harder already NP-hard, so doesn’t spoil party
Computing

Sequence of binary operations



Need to deduce dependencies of intermediate
results with each other and the original inputs
Different calculation order may give different
results
Do all at once in one multivariate calculation


Can be much more difficult computationally
Can produce much better tightening
Living (in sin) with probabilists
Probability box (p-box)
Cumulative probability
Interval bounds on an cumulative distribution function
1
0
0.0
1.0
X
2.0
3.0
Generalizes intervals and probability
Cumulative probability
Probability
distribution
Probability
box
Interval
1
1
1
0
0
0
0
10
20
30
40
10
20
30
40
10
20
Not a uniform
distribution
30
1
Cumulative Probability
Cumulative Probability
Probability bounds arithmetic
A
0
0
1
2
3
4
5
6
1
B
0
0
2
4
6
What’s the sum of A+B?
8
10
12
14
Cartesian product
A+B
A[1,3]
p1 = 1/3
A[2,4]
p2 = 1/3
A[3,5]
p3 = 1/3
B[2,8]
q1 = 1/3
A+B[3,11]
prob=1/9
A+B[4,12]
prob=1/9
A+B[5,13]
prob=1/9
B[6,10]
q2 = 1/3
A+B[7,13]
prob=1/9
A+B[8,14]
prob=1/9
A+B[9,15]
prob=1/9
B[8,12]
q3 = 1/3
A+B[9,15]
prob=1/9
A+B[10,16]
prob=1/9
A+B[11,17]
prob=1/9
independence
nondependent
Cumulative probability
A+B, independent/nondependent
1.00
0.75
0.50
0.25
0.00
0
3
6
9
A+B
12
15
18
Opposite/nondependent
A+B
A[1,3]
p1 = 1/3
A[2,4]
p2 = 1/3
A[3,5]
p3 = 1/3
B[2,8]
q1 = 1/3
A+B[3,11]
prob=0
A+B[4,12]
prob=0
A+B[5,13]
prob=1/3
B[6,10]
q2 = 1/3
A+B[7,13]
prob=0
A+B[8,14]
prob=1/3
A+B[9,15]
prob=0
B[8,12]
q3 = 1/3
A+B[9,15]
prob= 1/3
A+B[10,16]
prob=0
A+B[11,17]
prob=0
opposite
nondependent
A+B, opposite / nondependent
Cumulative probability
1
0
0
3
6
9
A+B
12
15
18
Opposite / opposite
A+B
A[1,3]
p1 = 1/3
A[2,4]
p2 = 1/3
A[3,5]
p3 = 1/3
B[2,8]
q1 = 1/3
A+B[5,9]
prob=0
A+B[6,10]
prob=0
A+B[7,11]
prob=1/3
B[6,10]
q2 = 1/3
A+B[9,11]
prob=0
A+B[10,12]
prob=1/3
A+B[11,13]
prob=0
B[8,12]
q3 = 1/3
A+B[11,13]
prob= 1/3
A+B[12,14]
prob=0
A+B[13,15]
prob=0
opposite
opposite
Cumulative probability
A+B, opposite / opposite
1.00
0.75
0.50
0.25
0.00
0
3
6
9
A+B
12
15
18
Cumulative probability
Three answers say different things
1.00
0.75
0.50
0.25
0.00
0
3
6
9
A+B
12
15
18
Conclusions

Interval analysis automatically accounts for
all possible dependencies

Unlike probability theory, where the default
assumption often underestimates uncertainty

Information about dependencies isn’t usually
used to tighten results, but it can be

Variable repetition is just a special kind of
dependence
End
Success
Good
engineering
Dumb luck
Wishful
thinking
Prudent
analysis
Honorable
failure
Negligence
Failure
Independence

In the context of precise probabilities, there
was a unique notion of independence

In the context of imprecise probabilities,
however, this notion disintegrates into several
distinct concepts

The different kinds of independence behave
differently in computations
Equivalent
Several definitions
definitionsofofindependence
independence





H(x,y) = F(x) G(y) , for all values x and y
P(XI, YJ) = P(XI) P(YJ), for any I, J  R
h(x,y) = f(x) g(y) , for all values x and y
E(w(X) z(Y)) = E(w(X)) E(z(Y)), for arbitrary w, z
X,Y(t,s) = X(t) Y(s), for arbitrary t and s
P(X  x) = F(x), P(Y  y) = G(y) and P(X  x, Y  y) = H(x, y);
f, g and h are the density analogs of F,G and H; and
 denotes the Fourier transform
For precise probabilities, all these definitions
are equivalent, so there’s a single concept
Imprecise probability independence






Random-set independence
Epistemic irrelevance (asymmetric)
Epistemic independence
Strong independence
Repetition independence
Others?
Which should be called ‘independence’?
Notation



X and Y are random variables
FX and FY are their probability distributions
FX and FY aren’t known precisely, but we know
they’re within classes MX and MY
X ~ FX  MX
Y ~ FY  MY
Repetition independence





X and Y are random variables
X and Y are independent (in the traditional sense)
X and Y are identically distributed according to F
F is unknown, but we know that F  M
X and Y are repetition independent
Analog of iid (independent and identically distributed)
MX,Y = {H : H(x, y) = F(x) F(y), F  M}
Strong independence




X ~ FX  MX and Y ~ FY  MY
X and Y are stochastically independent
All possible combinations of distributions from
MX and MY are allowed
X and Y are strongly independent
Complete absence of any relationship between X, Y
MX,Y = {H : H(x, y) = FX(x) FY(y),
FX  MX, FY  MY}
Epistemic independence



X ~ FX  MX and Y ~ FY  MY
E(f(X)|Y) = E(f(X)) and
E(f(Y)|X) = E(f(Y)) for all functions f
where E is the smallest mean over all possible
probability distributions
X and Y are epistemically independent
Lower bounds on expectations generalize the conditions
P(X|Y) = P(X) and P(Y|X) = P(Y)
Random-set independence

Embodied in Cartesian products

X and Y with mass functions mX and mY are
random-set independent if the DempsterShafer structure for their joint distribution has
mass function m(A1A2) = mX (A1) mY (A2)
whenever A1 is a focal element of X and A2 is a
focal element of Y, and m(A) = 0 otherwise

Often easiest to compute
These cases of
independence
are nested.
(Uncorrelated)
Random-set
Epistemic
Strong
Repetition
(Nondependent)
Interesting example

X = [1, +1],
1
Y ={([1, 0], ½), ([0, 1], ½)}
1
X
0

1
0 +1
X
Y
0
1
0 +1
Y
If X and Y are “independent”, what is Z = XY ?
Compute via Yager’s convolution
Y
X ([1, +1], 1)
([1, 0], ½)
([0, 1], ½)
([1, +1], ½)
([1, +1], ½)
The Cartesian product with
one row and two columns
produces this p-box
1
XY
0
1
0 +1
XY
But consider the means



Clearly, EX = [1,+1] and EY=[½, +½].
Therefore, E(XY) = [½, +½].
But if this is the mean of the product, and its
range is [1,+1], then we know better bounds
on the CDF.
1
XY
0
1
0 +1
XY
And consider the quantity signs







What’s the probability PZ that Z < 0?
Z < 0 only if X < 0 or Y < 0 (but not both)
PZ = PX(1PY) + PY(1PX), where
PX = P(X < 0), PY = P(Y < 0)
But PY is ½ by construction
1
XY
So PZ = ½PX + ½(1PX) = ½
Thus, zero is the median of Z
0
1 0 +1
Knowing median and range improves bounds
XY
Best possible

These bounds are realized by solutions
If X = 0, then Z=0
If X = Y = B = {(1, ½),(+1, ½)}, then Z = B
1
1
Z=0

B
1
0
0
So0 these
are also
possible
+1
0 +1
1 0bounds
1best
XY
1
0 +1
XY
So which is correct?
Random-set
independence
1
Moment
independence
1
1
XY
0
1
0 +1
XY
Strong
independence
XY
0
1
0 +1
XY
XY
0
1
0 +1
XY
The answer depends on what one meant by “independent”.
So what?

The example illustrates a practical difference between
random-set independence and strong independence

It disproves the conjecture that the convolution of
uncertain numbers is not affected by dependence
assumptions if at least one of them is an interval

It tempers the claim about the best-possible nature of
convolutions with probability boxes and DempsterShafer structures
Strategy for risk analysts

Random-set independence is conservative

Using the Cartesian product approach is
always rigorous, though may not be optimal

Convenient methods to obtain tighter bounds
under other kinds of independence await
discovery
Uncertainty algebra for convolutions
Operands and operation
Answers under different dependence assumptions
1) One interval 
random-set = unknown
2) Two intervals 
strong = epistemic = random-set = unknown
3) Interval and a function of an interval  strong = epistemic = random-set = unknown
4) One interval and monotone operation  strong = epistemic = random-set = unknown
5) Monotone operation 
strong = epistemic = random-set
6) Two precise distributions 
strong = epistemic = random-set
7) All cases 
repetition  strong  epistemic  random-set  unknown
(after Fetz and Oberguggenberger 2004)
(Colored words denote the set of distributions
that result from binary operations invoking the
specified assumption about dependence)