An uncertain number is simply written (x+-s), where the brackets () sre explicitly used. Bias shifts the answer up or down and is not both added and subtracted, it's just simply added. The standard deviation, s, is added and subtracted. Do note, this is not a confidence interval, it is just an uncertain number, if you wish to place confidence in the number reported, then use statistics to calculate the x% confidence interval.

this is well understood, I would say obvious. My question addressed the possibility to get instead a formalism for treating the uncertain number as a multidimensional number (like, for example the complex numbers are), and asking if any mathematician would be available to tackle the issue.

Well, well, well... it's obvious you say! Well, if you know how to write an uncertain number in one dimension, then simply extend to any dimension you like! It's that simple, really, that simple... elementary...

Remember that there are many different notions of uncertainty, and probability is only one of them - there are fuzzy sets,possibility theory, rough sets, interval arithmetic, non-deterministic distributions, and many others in use in different application areas - in most cases, the extension to multiple dimensions is pretty easy - the hard part is discovering and justifying any operations that combine these values into others

combining these different notions of uncertainty into one formalism seems difficult and not completely useful, unless we get a surprise (which of course is quite possible, as the history of mathematics is filled with elegant surprises)

you can consider to use stochastic arithmetic that use uncertainty to estimate numerical precision of computation. Stochastic arithmetic formalism can also manage uncertainty in inputs. CADNA library is a framework to perform computation using stochastic arithmetic in easily in C++ or Fortran.

You can check thislink for an introduction. to stochastic arithmetic :

http://calcul.math.cnrs.fr/IMG/pdf/2013_Jezequel_PRCN.pdf

and this one to acces CADNA ressources

http://www-pequan.lip6.fr/cadna/Cadna_Dir/What_is_CADNA.php

Franco, you ask

"The question is: would it be possible to formalise a definition and notation which does not depend on the choice of the method (e.g. probabilistic or not) for the category 'uncertain number', and build an arithmetic out of that?"

What do you mean by "build an arithmetic"?

An uncertain number represents an estimate of some quantity of interest, together with information about that how that estimate has been obtained. The additional information is needed to construct a statement about the quantity of interest, such as an interval at some level of confidence.

Different theories need different auxiliary information to construct a statement (e.g., degrees of freedom is not used in the Propagation of Distributions methods advocated by the recent BIPM WGGM supplements, but it is needed in the more widely adopted GUM approach).

So, I don't think you could generalise the 'arithmetic' involved. However, if you stick to one method of uncertainty calculation it is possible to automate the underlying work of doing arithmetic with estimates. There is a fair amount of software already written to do this.

hihi, Blair - using the term ``estimate'' makes it sound to me like there is supposed to be one actual correct value. and all of these other uncertain values are intended to be approximations of it

but what if there is no single correct value, rather some kind of non-stochastic variation or even a fundamental irreducible variability, such as is found in fuzzy sets or vague sets?

admittedly, the terminology is imprecise, and you may not have meant ``estimate'' the way i have just described it - if so, then i apologize, but it does bring up another important point

i agree that with one definition, you can likely define an appropriate arithmetic, but most such theories have to handle the difficulty that combining uncertain values usually increases the uncertainty, and you need some alternative processes to reduce the uncertainty, such as converging evidence, or even alternate theories

Unfortunately, as Blair states that it will be difficult to generalise the arithmetic involved. Consider a couple of examples:

a = b + c

This can be handled very simply, though the answer depends on the correlation between the uncertainties in b and c.

(u_a)^2 = (u_b)^2 + (u_c)^2 + 2 cov(u_b, u_v)

a= b times c

is a little more difficult as higher order terms in the Taylor series may need to be taken into account - meaning that you need more information about the shape of the distribution (higher order moments of the uncertainty distributions for b and c are needed to calculate the variance of the uncertainty distribution for a). An easy way to see this problem is to consider

a=b^2

if b=0, then the linear term in the Taylor series expansion gives the uncertainty in a = 0, which is incorrect.

In the general case:

a=f(b,c)

Basically depends on whether the function is sufficiently linear over the range of uncertainty in the inputs to allow the linear terms in the Taylor series expansion to be used to estimate the uncertainty in the output. If higher order terms are needed, then need more dimensions. If the assumption of linearity holds, then it is feasible, but you need to handle things differently along each axis. The advantage of such an may be further limited as there isn't the direct interaction between axes as in the case of the complex plane:

(a+ib)*(c+id) = (ac-bd) + i (ad+bc)

Hi Christopher,

You are right, I used the term 'estimate' to suggest that there is one 'correct' value.

The usual interpretation of 'measurement' in physical metrology is as a procedure to estimate a measurand, which is unknown but for all practical purposes well-defined and fixed.

I am not familiar with problems that require a measurand to be defined in terms of vague or fuzzy sets. No doubt to include such cases in the mix would make it even more difficult to answer Franco's question.

In physical metrology, measurements are generally very well designed, so the influence of errors on a result is approximately linear. That allows us to use algorithmic methods to carry out uncertainty calculations (e.g., automatic differentiation, Monte Carlo, propagation of moments). For each the 'arithmetic' is different and all methods are approximate to some extent.

I fail to see the conundrum. A measurement is simply what you think it might be and the uncertainty, either speculative or experimental, defines the radius of convergence (analytic convergence). So, for multi-dimensional spaces, take a basis set, to each element of the basis set, either vector or scalar, one assigns to each a value for the measurement. To some basis set, of equal dimensions as the first, one assigns to each element the level of uncertainty for that particular vector or scalar measurement. Finally, one defines the sphere of uncertainty in this multi-dimensional space by adding or subtracting from the first basis set, the second basis. This would define the multi-dimensional radius of convergence or uncertainty.

P.S. complex numbers are the complex addition of two real numbers. The imaginary number simply assings the property of orthogonalism to the system under question. In like manner, one may consider a complex field to which one either adds or subtracts another complex field of numbers, where the latter are meant to represent uncertainty of some sort.

There are more kinds of uncertainties than measurement uncertainties, but blair is right about the assumption that measurement expects that there is one right answer

also, the radius of convergence is not even well-defined unless the errors are stochastically isotropic (distributed the same way in each direction) - radar, for example, is notorious for having excellent radial measurement accuracy and very poor angular ones

in any case, summarizing an uncertainty with a single number presumes a known simple shape for that uncertainty distribution

I agree Christopher, generally people assume the uncertainties are normally distributed, even though there is no valid reason for this assumption. Hence the need for more information about the distribution for some calculations.

Luisiana, while the analogy with complex numbers is useful it is limited. For addition of 2 complex numbers the same operation (addition) is applied to both the real and imaginary part. When dealing with uncertainties, this is not the case (unless the uncertainties are highly correlated). If the uncertainties are independent, then they add in a Euclidean sense. If the uncertainties are independent, then this is possible to represent if you represent the uncertainty by the variance of the distribution. However, you do need to consider the question of dependence. Take for example the simple case:

a=b+b

If we use the variance and ignore the question of dependence, then we would end up underestimating the uncertainty in a by ~30%. So the calculation of the uncertainty needs to not only take into account the magnitudes of the uncertainty, but also the dependence between the 2 quantities.

Further, multiplication of 2 numbers becomes even more difficult. In the complex plane, multiplication leads to the real and imaginary parts of the input numbers contributing to both the real and imaginary part of the result in a fairly straight forward way. This is not the case when dealing with uncertainties. As I said above, the calculations can become very complex.

I'll agree the there is more than just what has been spoken of so far, but the cross-dependencies can be handled by suitable probability theory techniques. In other words, you may make it as complicated as you like...

Probability theory has been remarkably successful in analyzing engineering measurements and other real-world phenomena, but it is not the only kind of uncertainty, and it cannot handle any kinds of cross-dependencies among uncertain non-probabilistic quantities

a common failing of some early expert systems was to take various ``subjective'' estimates and treat them as probabilities - just because a value is between 0 and 1 does not mean that it is a probability - for some number to be a probability, it must also satisfy several other conditions on long-term behavior, and there must be a hypothesis that some kind of randomness is involved

in fact, it is essentially impossible to prove that any number is a probability without postulating that it is so (of course, any number between 0 and 1 inclusive can be interpreted as a probability, but that is exactly the unjustified postulation i mentioned)

one part of the difficulty is that probability depends on the notion of randomness, which is extremely difficult to define mathematically - it must specifically be noted that random values are very constrained in the long run, most of the time

the simplest example is the difference between random and non-deterministic behavior - if i have a process that is guaranteed to produce either a 1 or a 0, then i have several possibilities for estimating how often a 1 occurs - the process could be deterministic, always producing 0 or always 1, which makes the estimation easy (if i know that in advance) - the output could be random with some frequency parameter, in which case there are long-term guarantees about the behavior and many statistical processes that allow me to estimate that parameter

but if the process is non-deterministic, then no amount of observation will produce a stable estimate of how often a 1 occurs (and there are examples from chaotic dynamics for which it is provable that no such estimate can exist) - the output is therefore not random, though it is unpredictable

there are other kinds of uncertainty that are even more difficult, but this one will suffice - uncertainty is not probability

I respectfully disagree that there are other kinds of uncertainties. As far as nonstationarity versus ergodic random dynamic systems goes... if your system is nonstationary, then it admits no mathematical description, hence, don't bother trying to mathematically represent it. In the case of ergodic systems, these systems are random, in every sense of the word, also, the only effective manner in which one may describe such a system is through statistics.

As for the 0 to 1 range... one can always map any field onto the unit interval, always! Now, thinking in terms of Shannon's information theory, if your system has equal probability for either a 1 or 0, thinking of a binary system here - of course, then you have maximum entropy for the system and the system is purely statistical in nature.

If your system has some tendency toward some number, say 1, then one could say it is skewed; but, it is better to say that the system possesses less entropy and the systems tends to some mean, which one could identify and determine the probability surrounding that mean.

Cross-dependencies can certainly be handled in probability theory... the means by which one does this are through conditional probability definitions for a random variable. Now, admittedly, one rarely ever knows the way in which random variables are conditionally dependent; nevertheless, these rules are applicable and suffice for a mathematical description.

As for the “[sic.] ...and it cannot handle any kinds of cross-dependencies among uncertain non-probabilistic quantities”. If you have a non-probabilistic measure, then why are discussing uncertainties? Obviously a non-probabilistic value would be certain, by nature and definition.

I restate: What's the conundrum?

Did you ever read de Finetti's monograph on Probability Theory.

My humble suggestion is: give it a glance!

Hm... have you ever read Papoulis' “Probability, Random Variables and Stochastic Processes“; His book is generally considered the definitive one on the subject. Anyway, tautologies aside, no one has yet convinced me there is any conundrum regarding uncertainty and the ability of mathematics to describe it! Solomon: “Nothing new under the Sun.”

"a non-probabilistic value would be certain, by nature and definition"

this statement is simply false

if i know that a real number is in the unit interval, but i don't know where, then i have a non-probabilistic uncertain number - it is common in this case to impose a uniform distribution to make it a probabilistic number and gain the strength of probability theory, but that is adding an unjustified assumption, and those baseless assumptions often render a problem unsolvable by overly constraining the possibilities - if in addition i have a second number that i know is in the unit interval, and i further know that the product of the two numbers is 0.5, then i have a dependency that is not handled by any probability theory because the numbers are not probabilistic in the first place; they are merely uncertain

we have already explained that probability is not the only viable mathematically formal model of uncertainty (it is only the most popular and the most effective where it applies), so, while papoulis' book is excellent for probability theory, it intentionally and explicitly does not address the other issues that have been raised (the table of contents makes it clear that he is treating only probability theory and not any of the other approaches that have been mentioned), for which probability theory either does not work at all, or leads to much more clumsy computation than other methods - di finetti's book is excellent in his discussion of the very notion of probability, and it hints at some of the alternative methods

on the other hand, of course, the original question was about measurement uncertainty, for which probability theory has an excellent track record and centuries of successes

if this discussion seems confusing or contentious to any of the readers, just remember that probability theory was only invented in the late 1600's, roughly the same time as calculus and certainly earlier than any of the other models - in addition, it has a much more checkered and contentious development history - the problem is that it captures some common intuitions about uncertainty, but not others - and even the basic theory remains confusing to many people to this day (check out the Monty Hall paradox for an excellent example)

i hope it is clear that i am not disparaging probability theory; i am only saying that it is not the only possibility with useful applications and methods for dealing with uncertainty

Finetti discusses the subjectivity of what we consider to be probabilistic. Well, there is very little of concern to a human outside of what is human, that is to say, we are all committed to this human condition. It is a nice philosophical discussion; but, pragmatically, if I approach any measure and haven't any certainty as to the accuracy of that measure, then I must resort to a probabilistic description.

If I have an unknown, x, within some unit interval, but do not know where, then any number along the unit interval would suffice, i.e. one would use a test function to test the validity of the initial guess. From there, one would iterate, testing over and over again, until the answer fell upon that certain number within the interval, whose whereabouts is uncertain. If you reply, "But, the unknown has no testable criteria with which to identify." Then it is quite obvious the entire unit interval will have to stand for the range of possible values.

If any relation existed, such as the product of two unknown numbers equals one half, then I am really building a polynomial relationship, hence, algebra would suffice in this case in identifying the unknown number.

I think what is being confused is an unknown number versus that of an uncertain number.

@Luisiana Cundin

I never used 'unknown', only 'uncertain', which is different.

The use of probability is not today the only means to treat an uncertain number.

@Christopher Landauer

@Lorenzo Peccati

I am not dealing with the merits of a theory for treating uncertain numbers, like the probability theory, or others. I am asking about a totally different issue. I admit that it may appear odd or a wishful thinking.

I wrote 'non necessarily probabilistic' because possibly for one case it may be simpler than for another. As Chris said there are many. Incidentally, why 'measurement uncertainty' should restrict the variety of methods for expressing and evaluating 'uncertanty' at large?

@Barry Croke

@all

To make an example different from the complex numbers of an algorithm established for a new category of numbers: interval mathematics (I attach a public report on the subject). In this case the 'number' is defined as a pair of number representing the bounds of an interval, or as a pair of numbers, the first representing the middle of the interval and the second the semi-width of the interval. Arithmetics and statistics now exists for them.

Here is the attachment

When using interval uncertainty, all you are saying is that the true value lies somewhere between an upper and lower bound. This is a non-probabilistic measure of uncertainty. As is stated in the report an uncertain number can represented as x +- e. However, the e could be the standard deviation of the distribution of the uncertainty, the standard error, or a half range. I would add additional options like a confidence limit (e.g. 95%). If using the half range, then you are using interval uncertainty, and interval arithmetic can be used. However, this can lead to an overly pessimistic estimate of the final uncertainty as this basically ignores the question of independence, and assumes the uncertainties are dependent.

So, yes, interval arithmetic can be used, is very straightforward, and would give an upper limit on the final uncertainty. However, if you have some information about the distribution of the uncertainties (e.g. the standard deviation), then you can use the more powerful formulation I mentioned above. However, this is made more difficult if you are using functions that are suffiently non-linear over the range of uncertainty. This means that if you use the standard approach (using the first derivative), you can introduce errors in the uncertainty estimate.

The best way to describe the uncertain data according to different modalities of information is to use the theory of fuzzy measures

The best way to describe the uncertain data according to different modalities of information is to use the theory of fuzzy measures

@Barry

yes, interval statistics is not probabilistic.

In fact, my question is not intended to only address probabilistic frame: it is not the only one existing, though the historical first one and the one that requires more information to be used sensibly.

Actually, in many instances the information available is not much. For example, when in so many cases the uniform probability distribution is used, basically meaning ignorance, and the distribution is bounded, why should not interval statistics be better?

However, the issue of interval statistics is that, from a computational viewpoint, in many cases it does not provide a result: it is not so straightforward, as far as I know. This is basically due, I believe, to the fact that the interval is supposed to be always continuous (I think is called 'compact' in mathematics). From the point of view of measurement science, this is not necessary: one is only interested in the bounds of the interval resulting from an algebraic operation, even if in its inside not all values are reachable.

@Victor

Of course fuzzy theory is an alternative to probability. I do not know enough about it. How is it formalised an uncertain number?

However, it looks that my question is not aswered so far.

I think that it requires more the interest of mathematicians than of statisticians, since it is the matter of developing a formalism, a classical cross-discipline issue, always very difficult to tackle.

It is possible by new maximum likelihood algorithm. The advantage of this algorithm is that precisely calculates the function for generalized assumptions.

https://www.researchgate.net/post/What_is_the_maximum_likelihood_estimation_of_dynamic_systems?cp=re72_x_p2&ch=reg&loginT=qwDPClDRBsfw8natUEvDq6eTvtxHzDdBIIOFzGFle7M%2C&pli=1#view=52a20223cf57d7b7428b4573

To answer your question, I must divide the answer into two parts:

Stochastic Answer: In my article "Boolean Powers and stochastic spaces" you can have a formal method of introducing "random real numbers"

In the second part you can use "fuzzy reals", which you can find in any book on Fuzzy Logic.

I like to resume this question for informing you of a paper just published on Acta IMEKO vol.4 N.4 (2015) pp4-8, downloadable for free from https://acta.imeko.org/index.php/acta-imeko/issue/view/13:

Physical quantity as a pseudo-euclidean vector, Valery Mazin

tackling this issue.

Abstract: A presentation of a physical quantity by a vector of the pseudo-Euclidean plane including on a level with the quantity value it degree of uncertainty is proposed. Four new matters appearing by such presentation are discussed.

I would be interested in discussing its analytical representation.

I recall here that the interval representation was already pointed out as one possibility.