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Backcalculation is an operation that untangles a numerical expression involving uncertainty. It provides, for instance, a solution B given A and C such that the sum A+B is sure to be constrained by C. Backcalculation generalises 'tolerance solutions' from interval analysis and is related to but different from deconvolution and simple inverse operations such as subtraction.
From: Ferson, Scott
Sent: 03 February 2021 7:14 PM
To: 'corrado.mencar@niba.it' <corrado.mencar@niba.it>
Cc: De Angelis, Marco <mda@liverpool.ac.uk>; 'Peter Hristov' <peter.hristov24@gmail.com>
Subject: backcalculation in engineering design
Dear Corrado,
I was interested in your chat comment. (I hope you feel comfortable speaking during the sessions. The whole point is to meet each other and discuss.)
Attached are some text and a slide presentation on the Vladik’s topic of tolerance and control solutions (what I call backcalculations) from a now ancient NASA report.
I’d be interested in any thoughts you have on the topic.
Best regards,
Scott
Attachments:
https://sites.google.com/site/fuzzypossrisk/backcalculation/NASA_InterimQ7Report_11-18-2008.doc?attredirects=0&d=1
https://sites.google.com/site/fuzzypossrisk/backcalculation/backcalc%20for%20nasa%20example.doc?attredirects=0&d=1
https://sites.google.com/site/fuzzypossrisk/backcalculation/untangling%20uncertain%20equations.ppt?attredirects=0&d=1
From: Corrado Mencar <corrado.mencar@uniba.it>
Sent: 04 February 2021 10:24 AM
To: Ferson, Scott <ferson@liverpool.ac.uk>
Cc: corrado.mencar@uniba.it
Subject: Re: backcalculation in engineering design
Dear Scott,
thanks a lot for the material: I will go through it and get back to you. I am currently working on a kind "granular counting", based on possibility distributions, and I am eager to make connections with the different methods for handling uncertainty. In particular, Valdik's argument seems to me as related to interactive possibility distributions, but this is just a thought and I want to study the matter more deeply.
The VICE cycle of seminars is revealing as a treasure to me (alas, I have to fight with other academic duties which prevent me to be online from the beginning to the end); I know there are many questions for the participants and time is limited, so I prefer to write mine on the chat to give the chairs the possibility of selecting the most relevant, and intervene if asked.
If you are curious about my work on granular counting, you may have a look at https://doi.org/10.1016/j.fss.2019.04.018. In that paper granular count is defined from a set of possibility distributions: I think I can prove that the counting formula is a consequence of some weaker assumptions that can be made on the distributions (but I have to put down this idea in formulas). Of course feedback is welcome!
Best regards,
Corrado
From: Corrado Mencar <corrado.mencar@uniba.it>
Sent: 15 March 2021 6:48 PM
To: Ferson, Scott <ferson@liverpool.ac.uk>
Subject: Re: backcalculation in engineering design
Dear Scott,
I read with much interest the documents that you sent me. I'd like to offer an alternative view, based on Possibility Theory, which is just a formal reinterpretation of the reported results but it may also give a different viewpoint and, perhaps, other insights. Attached is a rough description of this interpretation. Hope you will find it interesting.
Best regards,
Corrado
Attachments:
https://sites.google.com/site/fuzzypossrisk/backcalculation/esercizio.pdf?attredirects=0&d=1
https://sites.google.com/site/fuzzypossrisk/backcalculation/esercizio2.pdf?attredirects=0&d=1
From: Ferson, Scott
Sent: 15 March 2021 21:33:17
To: 'Corrado Mencar'
Cc: Gray, Ander; Alex; De Angelis, Marco; 'dominik.hose@itm.uni-stuttgart.de'
Subject: RE: backcalculation in engineering design
Dear Corrado,
I do find the reinterpretation of backcalculation within possibility theory to be very interesting.
After a long period of quiescence, we’ve lately been revisiting fuzzy and possibility theory in weekly meetings of our interest group on the topic. You are more than welcome to attend. Perhaps you’d like to give the group a presentation about your work sometime. The next meeting will be
Fuzzy and Possibility Interest Group
Fridays, 5 pm Paris, 4 pm Liverpool, 12 noon New York, 10 am Santa Fe, 9 am Los Angeles
https://liverpool-ac-uk.zoom.us/j/7555947629
hosted by Ander Gray (Ander.Gray@liverpool.ac.uk)
(Note that times may change when Britain observes daylight savings time in a couple of weeks.)
My old software did backcalculations on fuzzy numbers (unimodal possibility distributions on the reals that reach unity) using the Kaufmann and Gupta arithmetic operations, which are essentially level-wise interval analysis. In this arithmetic, solutions don’t always exist, but the operations easily generalise to possibility distributions other than intervals. Example calculations (see below) show the backcalculation solutions B, when propagated in the forward calculation, yield results identical to the original constraint C.
However, our discussions at the interest group meetings have convinced me that Kaufman and Gupta’s fuzzy arithmetic is not the correct calculus for arithmetic operations on possibility distributions, even though almost everyone uses it. So it will be very interesting to understand how backcalculation should work with possibilities.
Cheers,
Scott
procedure go(A,C) begin
B = decon(A,C) // backcalculate
D = A+B // forward propagation
say B
end
show C in red
show D
go( [1,2], [4,7] ) // intervals
[ 3, 5]
go( [1,1.5,2], [4,5.5,7] ) // triangular fuzzy numbers
[ 3, 4.249999, 5]
for i = 4 to 7 do go( [1,1,2], [4,i,7] )
[ 3, 3, 5]
[ 3, 4, 5]
[ 3, 5, 5]
Math Problem: non-fuzzy number result
for i = 4 to 17 do go( [1,1,2], [4,i,17] )
[ 3, 3, 15]
[ 3, 4, 15]
[ 3, 5, 15]
[ 3, 6, 15]
[ 3, 7, 15]
[ 3, 8, 15]
[ 3, 9, 15]
[ 3, 10, 15]
[ 3, 11, 15]
[ 3, 12, 15]
[ 3, 13, 15]
[ 3, 14, 15]
Math Problem: non-fuzzy number result
From: Gray, Ander
Sent: 15 March 2021 11:46 PM
To: Ferson, Scott <ferson@liverpool.ac.uk>; 'Corrado Mencar' <corrado.mencar@uniba.it>
Cc: Alex <alexanderpwimbush@gmail.com>; De Angelis, Marco <mda@liverpool.ac.uk>; 'dominik.hose@itm.uni-stuttgart.de' <dominik.hose@itm.uni-stuttgart.de>
Subject: Re: backcalculation in engineering design
Hi all,
You're more than welcome to join us Corrado!
What do you guys think about the backcalculation problem being a dependence problem?
If Z = X + Y, then sure Y = Z - X, but you have to do something with the copulas/dependencies to make the uncertain number fit.
I guess if you know Czx, then you have Y by convolution. But if you only have Fz, Fx and Cxy, then you would have to do some sort of calculation to get Czx.
Also weirdly, X + Y - Z = 0...
Maybe you could find the joint p-box/possibility distribution which fits that constraint, starting from Fz, Fx and Cxy?
Ander
From: Corrado Mencar <corrado.mencar@uniba.it>
Sent: 16 March 2021 11:22 AM
To: Gray, Ander <Ander.Gray@liverpool.ac.uk>
Cc: Ferson, Scott <ferson@liverpool.ac.uk>; Corrado Mencar <corrado.mencar@uniba.it>; Alex <alexanderpwimbush@gmail.com>; De Angelis, Marco <mda@liverpool.ac.uk>; dominik.hose@itm.uni-stuttgart.de
Subject: Re: backcalculation in engineering design
Dear Ander, dear Scott,
thank you very much for inviting me to join the group: I'll do my best to participate. I am quite new to uncertainty modeling, so I am eager to learn more but I am afraid I cannot offer too much. As a continuation of my previous note, I wrote another where I show my thoughts on how to manage the constraint A+B=C in Possibility Theory and how, from this constraint, the necessity that A+B-C=0 is derived. Again, I am in a "learning" phase so my note may look naive (sorry for that).
Best regards,
Corrado
From: Hose, Dominik <dominik.hose@itm.uni-stuttgart.de>
Sent: 16 March 2021 13:51:48
To: Corrado Mencar; Gray, Ander
Cc: Ferson, Scott; Alex; De Angelis, Marco
Subject: AW: backcalculation in engineering design
Hi all,
I am not sure what you actually mean by backcalculations but I have a feeling that you guys are confusing yourselves.
I will try to summarize what, I gather, are your assumptions, and what you are trying to infer.
Regarding your first question, you know that A and C have marginal possibility distributions π_A and π_C, and you know that they are connected to some unknown quantity B via the precise expression A+B=C which you can equivalently write as A+B-C=0, but also as A-C = -B, B-C = -A, etc… How you write it, is irrelevant. You know the relationship. Now, you want to infer what π_B looks like under these assumptions. At least, I think that is what Scott was computing.
If you know a joint distribution of A and C, i.e. π_{A,C} then this question is easily answered in possibility theory. It is simply the natural extension under these assumptions (the relationship A+B=C and the joint dependence)
π_B(b) = sup_{ a,c : a+b=c} π_{A,C}(a,c) or
π_B(b) = sup_{ a,c : a+b-c=0} π_{A,C}(a,c) or
π_B(b) = sup_{ a,c : a+-c=-b} π_{A,C}(a,c), or …
Again, which way you write it does not matter!
But if you don’t know the joint distribution, you have to make some assumptions about the dependency between A and C first. Corrado, you implicitly assume that
π_{A,C} (a,c) = min( π_A(a), π_C(c) )
which corresponds to what Zadeh proposes in his extension principle. But if you are modeling IP distributions with π_A and π_C this is a very strong assumption (perfect dependence)! In fact, if you do not have a very good reason for doing so, I would strongly discourage you from using it. I have attached a paper where I talk about two different types of dependence assumptions, (strong) independence and the general case of unknown interaction (also called Frechet), one can make in possibility theory, and how the corresponding joint distributions look are constructed. Naturally, if you don’t know anything about the dependence between A and C, you would have to use the Frechet aggregation which would lead to
π_B(b) = \sup_{ a,c : a+b-c=0} min( 1, 2 * π_A(a), 2 * π_C(c) )
Regarding your second question, this is something entirely different. You assume that A, B and C have marginal possibility distributions π_A, π_B and π_C, and you want to infer the possibility and necessity of the event that A+B-C=0. The second half of Corrado’s note is correct, you compute the possibility and the necessity as
Pos ( A+B-C=0 ) = sup_{a,b,c : a+b-c=0} π_{A,B,C}(a,b,c) and
Nec ( A+B-C=0 ) = 1 - sup_{a,b,c : a+b-c \neq 0} π_{A,B,C}(a,b,c).
But, again, you first need a joint distribution and I would advise you not to imply perfect dependence. If you are unsure, always go with Frechet which gives you, e.g.
Pos ( A+B-C=0 ) = sup_{a,b,c : a+b-c=0} min( 1, 3 * π_A(a), 3 * π_B(b), 3* π_C(c) )
I guess, what I am trying to say is that Ander had the right idea 😉
Hope this helps,
Dominik
From: Gray, Ander
Sent: 16 March 2021 7:26 PM
To: Hose, Dominik <dominik.hose@itm.uni-stuttgart.de>; Corrado Mencar <corrado.mencar@uniba.it>
Cc: Ferson, Scott <ferson@liverpool.ac.uk>; Alex <alexanderpwimbush@gmail.com>; De Angelis, Marco <mda@liverpool.ac.uk>
Subject: Re: backcalculation in engineering design
Hi all,
I think a backcalculation is what Scott calls a "deconvolution". It's something like where you know X + Y = Z , but you only have X and Z, and you need a Y that can fit.
Vladik was talking about it for intervals in his vice presentation (even though he didn't say backcalculation).
Thanks for the responses, I haven't had a chance to look through the maths thoroughly yet. I think Doms and Corrado are on the right track though.
I think we need to state what the problem is specifically. What is it that we know? Is it only Pi_x and Pi_z, or is there some other information that we have, like copulas?
If it's only, Pi_x and Pi_z, then I think Dominik is right about the Frechet. But if I remember from the interval version of problem, there is some issues about using Frechet. Like some of the solutions don't fit or something. But it should bound the correct Y at least?
Solutions might be tightened by knowing C_xy for example.
Ander
From: Hose, Dominik <dominik.hose@itm.uni-stuttgart.de>
Sent: 17 March 2021 5:49 AM
To: Gray, Ander <Ander.Gray@liverpool.ac.uk>; Corrado Mencar <corrado.mencar@uniba.it>
Cc: Ferson, Scott <ferson@liverpool.ac.uk>; Alex <alexanderpwimbush@gmail.com>; De Angelis, Marco <mda@liverpool.ac.uk>
Subject: AW: backcalculation in engineering design
Perhaps this paper will interest you:
>> https://core.ac.uk/download/pdf/50531098.pdf <<
From: Ferson, Scott
Sent: 17 March 2021 9:32 AM
To: Gray, Ander <Ander.Gray@liverpool.ac.uk>
Cc: 'Corrado Mencar' <corrado.mencar@uniba.it>; Alex <alexanderpwimbush@gmail.com>; De Angelis, Marco <mda@liverpool.ac.uk>; 'dominik.hose@itm.uni-stuttgart.de' <dominik.hose@itm.uni-stuttgart.de>
Subject: RE: backcalculation in engineering design
Yes, it is a dependence problem exactly. Ander has stated the problem: If Z = X + Y, [and] if you only have Fz, Fx and Cxy, then you would have to do some sort of calculation to get Czx. This is the deconvolution problem, which is to find a solution to the equation. Of course, deconvolutions rarely actually exist mathematically, especially if the margins and dependence are specified precisely. Backcalculation is actually a somewhat different problem. Backcalculation also has the issue of ensuring that the solution Y is such that the sum X+Y is inside the constraint Z. Here, solutions commonly exist (although not always...in which case, see "control" solutions).
By the way, I told Corrado that you guys have said that Kaufman and Gupta’s fuzzy arithmetic (which is level-wise interval arithmetic) is not the correct calculus for arithmetic operations on possibility distributions, even though almost everyone uses it. I wasn’t confused about this, was I? Is there a paper or a draft of a paper, or even just some examples on the website, that we can point to about this? The ESREL paper? What is the link?
I am pretty sure that what we worked out for backcalculation for p-boxes is correct. It will be very interesting to understand how backcalculation should work with possibility distributions in view of the ‘modality’ idea [hmm, maybe we need a name for that idea].
Backcalculations are not unique in the case of p-boxes. You can get different answers that extremely satisfy the constraint, which of course means that you can choose among them to best address another issue such as cost or tail weight. Backcalculations are definitely inverse operations; indeed they are kind of a bizarro world of uncertainty. For instance, when solving for B such that A+B=C, assuming independence between A and B makes the backcalculation result wider than assuming nothing about their dependence.
Dominik described two situations for which he has solutions:
If you know a joint distribution of A and C…
But if you don’t know the joint distribution…
But of course neither of these is the situation we have in backcalculation. What we have is a copula (or maybe Fréchet) for A and B, but not directly for A and C. It can be hard to find the solution, but it is easy to check whether a proposed solution works. In practice, we can use an obvious trial-and-error approach to figure out whether a suggested backcalculation solution is correct.
Finally, and maybe this is a terribly pedantic point, but I notice something in your emails. Ander mentioned at one point If Z = X + Y, then sure Y = Z − X, and Dominik went on to introduce the problem this way:
you know that A and C have marginal possibility distributions πA and πC, and you know that they are connected to some unknown quantity B via the precise expression A+B=C which you can equivalently write as A+B−C=0, but also as A−C = −B, B−C = −A, etc… How you write it, is irrelevant. You know the relationship.
In the context of uncertain numbers—under an algebra of uncertain numbers—I'm actually not sure it is right that how you write it is irrelevant. The 'relationship' among the variables surely has to include not merely the fact that one is a function of the others, but also their dependency. You see, real-number algebra doesn't have dependency to worry about. Dependencies don't matter. Of course it wouldn't matter how you write the relationship if you are talking about real numbers, but we're not talking about real numbers here. If X, Y and Z are uncertain numbers, then Z = X + Y => Y = Z − X is certainly false, right? In the same vein, Ander's expression Y = Z − X is dangerously ambiguous because it suggests a subtraction convolution, but that of course would give the wrong answer for sure. So I guess I'd like to suggest that it might be helpful always add a caveat that you're only talking about the real-valued realisations (maybe emphasised by using a different font or case like the probabilists do) or, maybe better, to stick to one expression for what the relationship is. It would usually be the one where the independence is expressed, wouldn't you think?
Scott
From: Gray, Ander
Sent: 17 March 2021 12:16 PM
To: Ferson, Scott <ferson@liverpool.ac.uk>
Cc: 'Corrado Mencar' <corrado.mencar@uniba.it>; Alex <alexanderpwimbush@gmail.com>; De Angelis, Marco <mda@liverpool.ac.uk>; 'dominik.hose@itm.uni-stuttgart.de' <dominik.hose@itm.uni-stuttgart.de>
Subject: Re: backcalculation in engineering design
We have a conference paper for ISIPTA submitted on copulas and possibilities. The only example there is sum however, but you can see it for a range of dependencies.
assuming independence between A and B makes the backcalculation result wider than assuming nothing
Wow, that's quite counterintuitive, but it must be true if you're doing the inverse.
Ander
From: Hose, Dominik <dominik.hose@itm.uni-stuttgart.de>
Sent: 18 March 2021 3:22 PM
To: Ferson, Scott <ferson@liverpool.ac.uk>; Gray, Ander <Ander.Gray@liverpool.ac.uk>; Corrado Mencar <corrado.mencar@uniba.it>
Cc: Alex <alexanderpwimbush@gmail.com>; De Angelis, Marco <mda@liverpool.ac.uk>
Subject: AW: backcalculation in engineering design
Ok, I think I better understand the problem now. As a preface, I agree that it is a dependence problem… a quite exciting one.
But first, let me elaborate on how I view the irrelevance of how you state A+B=C. We should remember that this is an abuse of notation because, really, random variables/uncertain numbers/... are functions from the universe of discourse Ω to the reals. That is, what we really express by A+B=C is that we know that, for all ω ∈ Ω, we know that A(ω) + B(ω) = C(ω). This is a clear relationship and it has to be fulfilled for all ω. Yes, it tells us a lot about the dependencies between A,B and C because if we know the values of two of them, we know the third. But how we write it does not matter. I will keep the notation A(ω), B(ω), C(ω) just to make this clear... and maybe to annoy Scott ;)
What I did not understand at first was how the problem is actually stated. I will try and do this as precisely as possible now and you tell me if I missed something.
We know that A(ω) + B(ω) = C(ω) for all ω ∈ Ω and we know the marginal possibility distributions π_A and π_C of A and C, respectively. Furthermore, we assume some type of dependence between A and B which could e.g. be given by a copula. I wrongly assumed that you knew some dependence between A and C but I think this is not correct, right? You actually have two different types of information about dependencies, i.e. precise dependencies (A+B=C) and stochastic dependencies (Copula C_{A,B}).
Now, we want to find an expression for π_B such that consistency is guaranteed for all probabilities P that fulfill these relationships, i.e.
P( ω : π_B( B(ω) ) ≤ α ) ≤ α for all α ∈ [0,1]
Unfortunately, I don’t have an answer. But I have some ideas we can discuss. How did you solve this?
What I would also be interested in is how these problems arise. Or are they just intellectual pastimes? I can see that how you would know a distribution of A, a measurement C, and you want to infer an unknown parameter B but then the problem is very different because B is a parameter and not a random variable.
From: Ferson, Scott
Sent: 19 March 2021 12:16 PM
To: 'Hose, Dominik' <dominik.hose@itm.uni-stuttgart.de>; Gray, Ander <Ander.Gray@liverpool.ac.uk>; Corrado Mencar <corrado.mencar@uniba.it>
Cc: Alex <alexanderpwimbush@gmail.com>; De Angelis, Marco <mda@liverpool.ac.uk>
Subject: RE: backcalculation in engineering design
We’re probably annoying the long list of email recipients. Would it be better to move the discussion off email and onto the website? It would be automatically archiving, and it allows easier editing.
I wrongly assumed that you knew some dependence between A and C but I think this is not correct, right?
Um, yes, that is not correct. What we know—or more typically are making assumptions about—is the dependence between A and B. Ander will tell us that, if we know the function and knew the marginal for B, knowing the dependence between A and B implies something about the dependence between A and C that he could theoretically deduce, but we don’t start there.
You actually have two different types of information about dependencies, i.e. precise dependencies (A+B=C) and stochastic dependencies (Copula C_{A,B}).
No, I don’t think so. The statement A+B=C applies without ambiguity only to the realisations. The dependence between uncertain numbers is always described by a copula, or more generally a family of copulas.
I can see that how you would know a distribution of A, a measurement C, and you want to infer an unknown parameter B but then the problem is very different because B is a parameter and not a random variable.
Your example of dispatching measurement error is actually a deconvolution problem, not usually a backcalculation problem. See the slides in "untangling uncertain equations.ppt" below for the distinction.
What I would also be interested in is how these problems arise. Or are they just intellectual pastimes?
No, not at all. Backcalculation problems absolutely arise all the time in engineering. In fact, engineering design can be thought of as this backcaculation problem. See the “Backcalc for NASA example” [at the bottom of the webpage] for the space station shielding problem for an example. How much heavy shielding do I need to bring up into orbit to protect astronauts during a mission when we are uncertain about the space weather they will experience? Another example is how clean does the water in San Francisco Bay need to be in order to ensure that mussels that grow there will surely be fit for human consumption. That’s a backcalulation across a foodweb and multiple chained equations.
Unfortunately, I don’t have an answer. But I have some ideas we can discuss. How did you solve this?
It's indulgent, but I’ll add some more papers and slide sets to the webpage to explain the issues. See “ewri backcalc.pdf” for the actual algorithm, but it’s be less useful to you than you might expect.
I will keep the notation A(ω), B(ω), C(ω) just to make this clear... and maybe to annoy Scott ;)
So you’re saying you’ll use A(ω) to denote a realization of the uncertain number A (i.e., interval, distribution or p-box), is that right? That’s fine, but I think we should perhaps acknowledge the convention in statistics. Googling “capitals and lower case in statistics” yields:
Introduction to Discrete Random Variables - Lumen Learning
When to use lowercase or capital letters for variables ... - reddit
25 Aug 2015 — Usually, the uppercase is for the random variable, and lowercase is for a particular value of the variable. A sample from X can take values x1, x2, etc.
When exactly does one use capital and lowercase letters for ...
What is the difference between "X" and "x" in regards to ...
What's the difference between upper case and lower case ...
www.quora.com › Whats-the-difference-between-upper...
Upper case letters are usually used to refer to random variables and lower case are usually used to refer simply to variables.
I think the statistics convention got it backwards. I want to make the complex object, the uncertain number, seem less complex, so it should be indicated by the lowly lower case letter. That leaves uppercase letter to denote the real-valued realisations. I also want people feel familiarity towards uncertain numbers, which I think giving them lower case letters encourages. (Simultaneously, I want to encourage “x-think” rather than “p-think”. So they should think in terms of the magnitudes of the x-values rather than focusing on the fact that those values are associated with some measure function. Maybe if we used complementary cumulative distributions as the primary characterisations for a distribution rather than its CDF, this ass-backwardism could have been avoided.) All this is to say we should start treating uncertain numbers like numbers that get operated on with arithmetic, and compared in terms of magnitude, so writing x + y denotes a convolution, but can be understood abstractly as a simple addition.
Directory of attachments below
Corrado Mencar's discussion of backcalculations with possibility distributions ESERCIZIO.PDF and ESERCIZIO2.PDF
Engineering design under uncertainty, inward-directed rounding BACKCALCULATION.PPTX
Backcalculation v. deconvolution v. updating UNTANGLING UNCERTAIN EQUATIONS.PPT
NASA shielding example BACKCALC FOR NASA EXAMPLE.DOC
Algorithm for backcalculation with p-boxes EWRI BACKCALC.PDF
Backcalculation through multiple equations, "use this formula" tables for backcalculation/control solutions 9 CLEANUP.PPT
Multitoxicant cleanup TUCKERFERSON06A.PDF
How clean does soil have to be to protect wildlife? ECOSSL_SETAC.PDF
Approximate backcalculation/deconvolution SOIL.PDF
Deconvolution to reduce uncertainty DECON.PDF
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