Hybrid numbers

Post date: Jun 18, 2021 10:35:51 PM

The current work of F&PIG is concerned with unifying the possibilistic and probabilistic branches of reasoning about uncertainty, but there is a history of prior thinking and progress in this area.

Hybrid numbers marrying probability with fuzzy numbers (special kinds of possibility distributions on the real line) were introduced by Kaufmann (1986). They were originally conceived as dualities whose two components do not interact. The probabilistic parts obeyed probability theory, and the fuzzy parts obeyed fuzzy arithmetic (Kaufmann and Gupta 1985).

Ferson and Ginzburg (1995) generalized the notion of hybrid numbers to support a full hybrid arithmetic by forming the Cartesian product of the fuzzy and probabilistic components. They imagined a nesting of probability boxes at each of many levels between the broadest at the base at alpha level zero and the tightest at the top at alpha level one, forming a ridge. This conception was perhaps comparable to complex numbers which have both real and imaginary parts that are mostly separate but interact in subtle ways. These hybrid numbers can be interpreted as a fuzzily specified probability distribution or, equivalently, as a probability distribution of fuzzy numbers. See pages 40-44 of the attached PDF for details.

Neither conception of hybrid numbers was widely considered in risk analysis or uncertainty quantification except by Möller and Beer and their colleagues under the name fuzzy randomness (Möller et al. 2003, Möller 2004; Möller and Beer 2004; Beer 2009; Beer and Ferson 2011; Beer et al. 2011; Beer et al. 2013; Pannier et al. 2013) and recently under the name polymorphic uncertainty in the work of Graf and his colleagues (Graf et al. 2015; Freitag and Edler 2018; Edler et al. 2018, 2019; Götz et al. 2019; Freitag et al. 2020; Fina et al. 2020; Drieschner et al. 2021; Lahmer n.d.).

Ben-Haim (2001) described information-gap models, which are in many fundamental ways essentially a rebranding and remotivation of fuzzy arithmetic (Kaufmann and Gupta 1985). Hybrid numbers, fuzzy random variables, and polymorphically uncertain quantities are thus also info-gap models of uncertain probability distributions (Ferson and Tucker 2008).

Whatever their name, these hybrid, fuzzy-random, polymorphic, info-gappy objects are nestings of probability boxes according to a fuzzy model of how uncertainty, or, equivalently, fuzzily specified probability distributions.

Consider the multiplication of a triangular fuzzy number [1,2,3], depicted in green on the left side of the top row of visualized uncertain numbers in the figure below, with a uniform probability distribution uniform(1,3), depicted in blue as a cumulative distribution below the fuzzy number. Each of these two numbers is shown as a hybrid number in the graphs to their right. The probability distribution is a knife-like ridge coming up from the plane in the orthogonal alpha direction. The fuzzy number shown as a hybrid number is a tent, where the middle vertical line at the value 2 is at the highest alpha value and the left and right vertical lines at 0 and 3 are at the lowest alpha value. Only three isopleths are shown, but think of the tent as pulled taut. The product of these two numbers is shown in the far right graph, which is neither a pure fuzzy number nor a single p-box, but rather a set of nested p-boxes.

One major feature of hybrid numbers that they inherit from fuzzy arithmetic is their indifference to dependence. In arithmetic and logical computations, assumptions about stochastic and epistemic dependence will influence the calculations of the probabilistic parts of hybrid numbers but not fuzzy parts. <<Is this true for the Moller-Beer-Graf interpretations too?>> For example, the product of a pure probability distribution and a pure fuzzy number in the graphical example below will not differ when dependency between the factors changes from independence to perfect dependence to opposite dependence.

(click figure to embiggen)

This is the Julia code to make the graphs depicting multiplication of merged numbers:

# MERGED NUMBERS

using IntervalArithmetic

using FuzzyArithmetic

using ProbabilityBoundsAnalysis

using BivariateCopulas

#########################################################################################

Fa = Fuzzy(1,2,3)

Pb = U(1,3)

#########################################################################################

# Hose-Gray approach to Gc = Fa * Pb

Pa = makepbox(Fa)

Fb = makefuzzy(Pb)

showf(Fa,n="Fa")

showp(Pb,n="Pb")

showp(Pa,n="Pa")

showf(Fb,n="Fb")

# Fc = Fa * Fb

Fc = sigmaFuzzy(Fa,Fb,op=op,C=BivariateCopulas.Pi()) # sigmaFuzzy ##################

# Pc = Pa * Pb

Pc = convIndep(Pa,Pb,op=op) # convIndep ##################

showf(Fc)

showp(Pc)

PC = makepbox(Fc)

FC = makefuzzy(Pc)

showf(Fc,n="Fuzzy")

showf(FC,n="Fuzzy")

showp(Pc,n="P-box")

showp(PC,n="P-box")

The F&PIG collaboration [read: Ander] is implementing software for constructing and computing with merged numbers. The question is whether it should also make an arithmetic package for hybrid numbers. Although Profs Beer and Graf and the polymorphic guys like hybrids, they may have too much complexity to sell as practical tools for rank-and-file engineers or risk analysts. Risk Calc offers hybrids only as a library, doesn’t compute them natively. The one really big advantage to supporting hybrids would seem to be that they close srithmetic operations, so you can, for instance, multiply a fuzzy number times a p-box. In Risk Calc, this would make hybrids the most general class of uncertain numbers. It is not clear we want that. Unless we do want that under this sort of point of view:

Modern UQ recognises three major kinds of uncertainty:

• Aleatory uncertainty arises from natural stochasticity, differences among manufactured components or individuals, environmental variation across space or through time, and varied other sources. It is also called randomness, variability, aleatoric uncertainty, objective uncertainty, statistical uncertainty, or irreducible uncertainty. When uncertainty about a variable is aleatory, it means the value is changing.

• Epistemic uncertainty arises from incomplete knowledge, including measurement uncertainty, small sample sizes, detection limits, data censoring, ignorance about the details of the mechanisms and processes involved, and other imperfections in scientific understanding. It is also called ambiguity, incertitude, reducible uncertainty, systematic uncertainty, state-of-knowledge uncertainty, Knightian or true uncertainty and subjective or personal uncertainty. Epistemic uncertainty means we don’t know what the value is. We may not even know whether it varies.

• Vagueness, often called fuzziness or sometimes information gaps, is semantic uncertainty caused by gradations or a lack of clarity in definitions. Although it is possible to define away vagueness artificially, modern UQ allows us to work with vagueness to create better designs. For instance, we can do better than defining an “endangered species” as one with fewer than N animals, because doing that would only protect some populations but abandon ones with N+1 or more individuals. Accounting for vagueness can be important in a host of other situations as well such as sorites paradoxes or when the degree of failure for some component is not binary because it can be partially working. Vagueness implies that the value can be redefined because it’s arbitrary, and could reasonably be shifted higher or lower.

I’m not saying that we should necessarily adopt this point of view, but maybe we should try to understand what it would entail. Under this point of view, uncertainty can be addressed with three fundamental approaches, which generalise deterministic calculations in three different ways and lead to the variety of uncertainty theories we see today.

(click figure to embiggen)

(click figure to embiggen)

Under this point of view, hybrid numbers would be the most general class of numbers.

(click figure to embiggen)

The resulting landscape of uncertainty methods is messy, but within this point of view we can see some structure and, importantly, some equivalences.

(click figure to embiggen)

But note well that this tripartite model for UQ is really not at all what Dominik has in mind when he talks about fuzzy numbers or possibility distributions. His structures could themselves have been called p-boxes or probability bounds, and that’s what they really are. Their similarity with Zadehan possibility is only formal, merely mathematical, and could have been developed independently of anything fuzzy or possibilistic, at least in any sense recognised by Lotfi, or Kahnemann or Gupta. Remember my bathroom conversation with Lotfi. Okay, maybe Dubois might have the same idea as Dominik, but its almost an artifact of the flexibility (mushiness?) of fuzzy theory that allows this.

Dominik, can the possibility distributions that are used in mergers be used to characterise semantic uncertainty, what philosophers call vagueness? I’m not sure I know the answer you’ll give, but I think it will be ‘no’. Certainly you’ll say that the arithmetic should not be level-wise, thus I presume you’ll agree that your conception of possibility distributions is not equivalent to, nor even consistent with, info-gap analysis.

Now note that, with the advent of mergers, we have perhaps another way to close operations that involve both p-boxes and fuzzy numbers, although it works under an entirely different conception of what fuzzy numbers are. But mind you, this conception only works, i.e., it only survives the zero problem (slide 4) because the fuzzy number and the p-box are different bounding structures on the same thing and can be translated back and forth. This is the wrinkle that the Kaufmann hybrids lacked. So for any uncertain quantity X, we can express what we know about it as a p-box, B(X). We can also express what we know about it as a fuzzy structure F(X). Suppose the original expression of a particular uncertain quantity is a probability distribution or p-box Y, and the original expression for another quantity is a possibility distribution Z, and we want to multiply these two quantities. Using the tuple scheme of slide 4 that multiplies fuzzies with fuzzies and p-boxes with p-boxes, we survive the zero problem by simply doing to multiplications:

(F(Y), Y) * (Z, B(Z)) = ( F(Y) * Z, Y * B(Z) )

which is not (0,0) unless Y or Z is zero. Of course this multiplication operation is closed for mergers because it always yields another merger. In this approach, we might argue that p-boxes are the most general (computationally feasible) uncertain number but that there is a new kind of consonant p-box, or perhaps that a second-generation p-box contains more ways of bounding credal sets than just the u- and d-edges, moments and generic credal family constraints.

Is this level of fidelity in uncertainty modelling useful. At ISIPTA 2019 someone was talking about "sets of desirable gamble sets", which to me seemed like too much at the time. Are there decisions which can't be made from a p-box or fuzzy number alone which require a fuzzy-p-box number or a fuzzy-fuzzy number?

I'm happy with whatever name. I believe the germans are the hybrid structures polymorphic uncertainty.

Your slide 14 could be viewed the other way around? You take an alpha cut of and get a p-box back. Are they two views the same? If you use your view though you might actually be able to propagate these structures with sampling quite easily ...

Not exactly sure what you mean, but, yeah, an EDF of fuzzy numbers is a hybrid, and nesting of probability distributions is a hybrid. These are equivalent (setting aside any inference issues). I don’t think there’s any loss if that’s what Dominik was suggesting. Monte Carlo methods should apply to hybrids by slicing them to create deviates in the form of fuzzy numbers. That’s what the ancient paper says.

Not sure why you call that an approximation. Isn’t it exact?

I meant its a robust outer approximation of something continuous (in both p-box and fuzzy axes).

You mean as the default for infix operators? What upper level? I presume there’d be a convolve function that can do anything, and the remaining decisions are about what would be good practice.

There's two levels of dependence at play in the hybrid numbers:

• C_lower which is the copula between the p-boxes

• C_upper between the fuzzy numbers

C_lower plays a role in the arithmetic operation between two p-boxes in each fuzzy set, and the C_upper is going to assign the masses to the cartesian product elements (the elements of which are p-boxes) in the fuzzy sigma convolution. C_lower has the usual interpretation, but what is C_upper, and how do we know it.

A p-box is already a fuzzy set. This is something more bigger-er. I guess it is to an p-box what a fuzzy number is to an interval. It is not a simple class of distributions; it’s got structure. Uncertainty of the third/fourth kind?

Sure, it's a fuzzy representation of a credal set. A fuzzy credal set perhaps. If you ask for the measure on some interval, you're going to get a fuzzy number.

I've attached a draft of a package. Activating it is a bit tricking, because fuzzy is still private. In the hybrid arithmetic the fuzzy part is levelwise for now...

If you want to play around, put it in the same directory as FuzzyArithmetic, then change your directory to HybridArithmetic.jl/src

and:

include("HybridArithmetic.jl")

Some code for mergers:

aPbox = normal(interval(0,1),

1)

a = merger(aPbox)

# Dual number

plot(a, name =

"Number a", col = "blue")

bFuzzy = Fuzzy(3,4,5)

b = merger(bFuzzy)

plot(b, name =

"Number b", col = "red")

C = Pi()

# Copula

op = +

cSigma = sigma(a, b, op = op, C= C )

plot(cSigma, name =

"sigma C", col = "green")

cTau = tau(a,b, op = op, C = C)

plot(cTau, name =

"tau C", col = "purple")

int1 = mass(cSigma,

interval(0,

6 )) # returns [0.664999, 1]

Some code for Hybrids:

steps = 10

fuzzyMean = Fuzzy(0,1,2, steps = steps)

fuzzyStd = Fuzzy(1,2,3, steps = steps)

H1 = normal(fuzzyMean, fuzzyStd)

# Hybrid number

plot(H1, name =

"Hybrid 1", col = "blue")

minFuzzy = Fuzzy(1,2,3, steps = steps)

H2 = uniform(minFuzzy,

interval(4))

plot(H2, name =

"Hybrid 2", col = "red")

C = Pi()

# Copula

op = +

H3 = sigma(H1, H2, op = op, C= C )

# Upper level is levelwise

plot(H3, name =

"Hybrid sigma", col = "green")

H4 = tau(H1, H2, op = op, C= C )

# Upper level is levelwise

plot(H4, name =

"Hybrid tau", col = "purple")

probMass = mass(H3,

interval(2,4))

# Fuzzy Number

FuzzyArithmetic.plot(probMass, name =

"prob mass", col =

"black")

cuts = cut(H3,

0.5) # Fuzzy Number

FuzzyArithmetic.plot(cuts, name =

"cuts", col =

"orange")

Cheers,

Ander

From: Ferson, Scott

Sent: 17 June 2021 00:50:04

To: Gray, Ander; 'Hose, Dominik'; De Angelis, Marco

Subject: RE: two kinds of hybrid numbers

Ander:

I also really like the quote from Hamilton. It's about Quaternions right?

I think so. According to https://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Letters/BroomeBridge.html, it’s paraphrased slightly over Hamilton’s original wordiness:

Every morning in the early part of the above-cited month, on my coming down to breakfast, your (then) little brother William Edwin, and yourself, used to ask me, ``Well, Papa, can you multiply triplets''? Whereto I was always obliged to reply, with a sad shake of the head: ``No, I can only add and subtract them.''

Is a hybrid number a nested set of p-boxes, or the merge of a fuzzy number and a p-box (one for tails, one for internal)?

I presume this is rhetorical. When we first conceived it, it was the former. The merge is only know possible. I think that the word hybrid should stay with the nested p-boxes, as it would only really be sensical in that context, hmm, maybe

They're fundamentally different, with different arithmetic rules. I believe the original plan was for the merged object, but the hybrid object is interesting, especially since Dominik's inference gives π distributions on distribution parameters, and would lead to these objects naturally.

Really? I don’t think I was aware of this. A possibility distribution over distribution parameters would certainly be a Kaufmannesque hybrid. Is this in a paper he’s sent me multiple times?

Maybe we call the merger a dual number since it's two representations of the same uncertain number, whilst the hybrid number is a fuzzy characterisation of a p-box.

I don’t know about that name, but it does seem that we might want to preserve both the merged and the nested structures, right? But if we do that, there should be a reason that we do it. Right now, I don’t know what that justification would be. They both use the same information, yet produce really different answers. Unless those answers are somehow equivalent, or perhaps are expressing different assumptions or something, I don’t think preserving both could be legitimate.

Are moments also included in the dual, or is that implicit in the p-box arithmetic?

Well, they’re integrated into PBA of course, although only as computational afterthoughts…they’re not truly integrated-integrated, and the results are almost surely not optimal. The mathematical job of finding best possible inferences on tail risks and moments in the general case beyond some easy special cases would be well beyond my capabilities. Is it soluble at all? I guess it would be a generalisation of the Karlin & Studden’s theory of Chebyshev systems.

It’s not clear that it would be useful to add moments to the possibilistic calculations. It certainly might, and I’d think it would be an exciting wrinkle if it did. Is there something like the Markov inequality or the Chebyshev inequality for possibility distributions? I would suspect there should be. What are the bounds on a possibility distribution given that you know the min, max, mean and standard deviation? What does a possibility distribution imply about the min, max, mean and standard deviation? Ander and Dominik should be able to answer these questions in your sleep, right? If that’s not described already, that would be a very nice paper.

Assuming that the calculations do not proceed by converting the possibility to a p-box, I would suppose that you could end up with there being an advantage to using moments with fuzzy structures, just as there is with p-boxes. Maybe information about the mode could also be used with fuzzy structures. Right?

In terms of code structure, you've essentially done that on slide 17.

• For the dual, which I think is the right side of slide 17, you have structure with both a p-box and fuzzy as properties, and call the appropriate convolution on both, doing whatever constraining and transforming in between operations. Any probabilistic query on the object will return the intersection of both.

You think? Well, we need to package it, don’t you think?

You think the data structure we eventually encode in the software should just be a p-box with an extra vector for the possibility distribution? Recall that the data structure for a p-box is usually the regularly discretised percentiles (x-values for each percent level from 0% to 100%) for the left and right edges of the p-box. Should the new part’s data structure be a sequence of (x,pi) pairs going from (x_0,0), …, (x_maxleft, 1), (x_maxright,1), …, (x_omega, 0)? Or should we use interval cuts of the possibility part? If so, could the alpha levels be the discretization levels that the u- and d-vectors use? Maybe it is too soon to worry about data structures. I see now with the Membership method (attribute?) that you are using stacked intervals, apparently at discretizations similar or identical to the p-boxes, right? Does Dominik agree with this structure? Okay, so packaging can be something trivial like

struct UncNumber

b::pbox

p::FuzzyNumber

end

or maybe we would prefer something more closely integrated?

Which of the three graphs on the right side of slide 19 is the right way to depict a merged object? Or should it look like something else altogether? Maybe the user can flip back and forth between a cumulative view to see the p-boxy nature, and a consonance view to see the possibilistic nature. The other side functions in operations but isn’t normally seen. If we had a third we’d need to work on a Nicene Creed to explain how they are united yet separate, different but the same.

In the lower graph on the far right side of the slide, I would say that we should intersect the two p-boxes. In the depicted case, the p-box that came from the fuzzy result is totally inside the other. Will this always happen? If not, then obviously this is another avenue where merging the two approaches might yield improved results. I’m not sure we envisioned this kind of improvement when we started, but I don’t know whether it can ever happen. I presume we should likewise synthesize the upper plot in green. How should the synthesis work? Do we want the maximum with the flat top, or the minimum with the more rounded top?

• For the hybrid you could use a vector of p-boxes as an outer approximation of it's membership.

Not sure why you call that an approximation. Isn’t it exact?

• Funnily enough, with some Julia magic you might be able to use the exact same code in the interval-based and p-box based fuzzy arithmetic. It worked with interval-unions with minimum changes. Only part that will break is the IP-π-transformation ... so a sigma convolution wouldn't work out of the box. (To make it work we'd need a complement and an is_intersect_empty functions for p-boxes). I don't know if you'd want to do a sigma convolution anyway, it might be extremely expensive.

Well, they get what they pay for. The question I worry about is whether level-wise arithmetic ever makes sense.

What dependence would use on the upper level? Level-wise? Fréchet?

You mean as the default for infix operators? What upper level? I presume there’d be a convolve function that can do anything, and the remaining decisions are about what would be good practice.

Where does the hybrid number sit in the IP landscape? It's not a DSS.

Good point! Not everything should be I guess.

You would get a fuzzy number for the probability mass on some interval. It's a fuzzy credal set?

I don’t think so. A p-box is already a fuzzy set. This is something more bigger-er. I guess it is to an p-box what a fuzzy number is to an interval. It is not a simple class of distributions; it’s got structure. Uncertainty of the third/fourth kind?

Cheers,

Scott

From: Ferson, Scott

Sent: 13 June 2021 23:27:39

To: Gray, Ander; 'Hose, Dominik'; De Angelis, Marco

Subject: RE: two kinds of hybrid numbers

Again with the attachment.

From: Ferson, Scott

Sent: 13 June 2021 11:27 PM

To: Gray, Ander <Ander.Gray@liverpool.ac.uk>; 'Hose, Dominik' <dominik.hose@itm.uni-stuttgart.de>; De Angelis, Marco <mda@liverpool.ac.uk>

Subject: two kinds of hybrid numbers

It occurs to me that the notion of merging p-boxes and possibility distributions should have made me recall the “hybrid number” stuff we briefly thought about before any of you were born. A hybrid number is a nested set of p-boxes from zero (broadest) to one (tightest). They are equivalent to an info-gap structure on a probability distribution.

I had assumed that the new-fangled merged objects would displace those things. But maybe that’s not true.

I used Julia to try to compare the merged versus nested objects in computing the product between a purely fuzzy object and a purely probabilistic object. See slides 17, 18 and 19. I am not really sure how to depict (or code the data structure for) the merged object.

Could you tell me what I’m doing or thinking wrong?

Scott

IP-p-Transform

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