Uncertainty Balance Principle

The objective of this paper is to present new and simple mathematical approach to deal with uncertainty transformation for fuzzy to random or random to fuzzy data. In particular we present a method to describe fuzzy (possibilistic) distribution in terms of a pair (or more) of related random (probabilistic) events, both fixed and variable. Our approach uses basic properties of both fuzzy and random distributions, and it assumes data is both possibilistic and probabilistic. We show that the data fuzziness can be viewed as a non uniqueness of related random events, and prove our Uncertainty Balance Principle. We also show how Zadeh’s fuzzy-random Consistency Principle can be given precise mathematical meaning. Various types of fuzzy distributions are examined and several numerical examples presented.


Introduction
In studies of uncertain phenomena, several methods are employed.Two most widely used are random and fuzzy data approaches.They are typically described in terms of random and fuzzy distributions [7].These two methods look at the uncertainty from different points of view.In literature one can find various terms for fuzzy data, such as possibilistic, soft and subjective [3], as opposed to random called probabilistic, hard and objective [21].These terms are somewhat arbitrary and there are authors who used probability distributions to represent subjective information [42,43,44].Similarly, other authors used fuzzy sets and possibility distributions to describe objective imprecise information either about constants or about probability distributions [41].Historically, probability is defined in the context of some physical measurement and mathematically in terms of probability axioms by Kolmogorov [34] where probability space, events and associated probabilities are defined.Related notion of random variables are defined in terms of mappings from probabilistic space of events to real line, carrying an underlying probabilities from the original event space.Indexing with some independent variable, such as time, one can define random processes as dynamic versions of random variables.On the other hand fuzzy, possibilistic approach relates to some intuitive uncertain notion (often of human nature) of an underlying uncertain event with some confidence (presumption) levels defined.Often in fuzzy data there is no reference, at least not directly, to any experiment or hard measurement.It is more representation of our confidence level in an uncertain phenomenon.If a need arises to combine fuzzy and random data, such as in soft/hard data fusion, [7], [21], each distribution is typically handled separately for a specific problem at hand, and to our knowledge no rigorous mathematical methodologies exist for a practical uncertainty alignment between two types of data.In a fundamental paper by Zadeh [2], a concept of possibilistic fuzzy distributions was introduced as opposed to random and probabilistic distributions.The possibilistic distribution is shown to be equivalent numerically to fuzzy membership function.In classic fuzzy references [7], [8], various algebraic operations on fuzzy data are dscribed, as well as the methods as how to combine fuzzy and random data in meaningful ways.One obvious method is to normalize random data distribution to unity and combine it with the fuzzy data.Mathematically correct in principle, this method can be considered as a sort of uncertainty alignment from random to fuzzy data.Unfortunately the method is not practical because of loss of information in the process [7].Also, in our opinion this method does not have any strong conceptual ground.Another approach is to define hybrid data which retains both fuzzy and random properties of original data.One can define random fuzzy data where fuzzy distribution argument is "randomized" according to a probabilistic distribution density.Or, one can consider fuzzy random data where the value of random distribution density is fuzzified according to fuzzy distribution.From these original ideas, there was very extensive development last two decades, [9]- [19] in the area of "random fuzzy sets" and "fuzzy random variables".Neither is the focus of our paper.The subject of our paper is to consider fuzzy to random uncertainty alignment (i.e. starting with fuzzy and generating random data, or vice versa) using very basic properties of fuzzy and random distributions.Our motivation is to produce a common fuzzy or random data base to process the data further for either decision making process for a given application or a possible data filtering.In our approach we employ three step methodology, i.e.: (I) Decompose any fuzzy distribution via cumulative (probabilistic) distribution functions (CDF).We do not use probabilistic density functions (PDF), a derivative of CDF, for two reasons.First, it may not always exist [34], and second is that CDF is normalized to unity by the definition, similar to fuzzy distributions.
(II) Use basic probabilistic axioms whereas the CDFs are defined in terms of random event probabilities of the form P(X≤x), [34], [35], and combined with (I) above resulting in probability differences ∆P(Ai) for some TBA events Ai.
(III) Use Big Data or some other statistical methodologies to produce best ∆P(Ai) choices in (II).a The result of our approach is that for any unimodal fuzzy data, fuzzy distribution can be thought of as a combination of fixed and variable probability events.In the case of multimodal fuzzy data, this representation consists of a number of fixed and variable random events.We believe our approach can bring about new avenues in aligning fuzzy and random data, in particular in very important area of soft-hard (humanmachine) data fusion [21].In our previous introductory paper [40] we presented the basics of our uncertainty alignment methodology.This paper extends these results with additional fuzzy to random alignment methods, and it presents a unifying Uncertainty Balance Principle of the general form Π(x) + ∑∆P(Ai,Aj) = 1 and Π(x) = ∑∆P(Ai,Aj) for all alignment cases.The paper is organized as follows.Section 2 summarize, very briefly, basic probabilistic and possibilistic results employed in this paper.We do not aim to be complete with these summary, just what is of interest for the current paper.In Section 4 we introduce our main fuzzy to random uncertainty alignment arguments using standard triangular fuzzy distribution (TFN).The approach applies to any fuzzy distribution.We describe steps which result in fuzzy distribution as a combination of fixed and variable probabilities.In Section 4 we formalize and prove key results: (i) Probabilistic decomposition of an n-modal fuzzy distribution (ii) Universal Uncertainty Balance Principle which is presumption and x-invariant, and related (iii) Uncertainty Change Law.In Section 4.7 we point to a potential use of our methodology in Data Fusion and Decision Making situations.Section 5 presents numerical examples showing fuzzy distributions in terms of fixed (unique) and variable (non unique) random events and related probabilities.Symmetric and non symmetric TFN, convex and non convex distributions are illustrated.
The numerical examples confirm results of Section 4. Section 6 has the pseudo code for the uncertainty alignment algorithms, Conclusion is in Section 7, and key references are included in Section 8.

Random (Probabilistic) Distributions
And Fuzzy (Possibilistic) Distributions For the purposes of this paper we recall few basic classic probability and cumulative distribution facts as well as elementary fuzzy distributions results used in this paper.

Probability and Cumulative Distribution
The random events A, A1, A2, etc. are the subsets of a certain event S and they are assigned probabilities P. From classic references [34], [35] we have: 0 ≤ P(A) ≤ 1, P(S) = 1, P(O) = 0 (1) P(A1UA2) = P(A1) + P(A2) -P(A1∩A2) (2) where O is an impossible event.If A1UA2 = S and two events are mutually exclusive, then P(A1UA2) = P(A1)+P(A2)=1, hence A1 and A2 are complementary with P(A1) = 1 -P(A2) = P(A2*) and A2* indicates complementary event to A2.For any event A, the following holds: If the events are independent, then we have: Mutual exclusivity and independency do not imply each other.A random variable X(ξ) is a function that assigns a real number to each outcome ξ in the sample space S of a random experiment [34], [35].If an event A is given in S such that A={ξ: X(ξ)ϵB}, where B is a subset of real line R, then A and B are equivalent events with the same probability: Cumulative distribution function (CDF) of X is defined as: which is a probability that X has a value in (-∞, x], and hence it is a function of x. Figure 1 shows uniform CDF and related PDF, which is a derivative of CDF.The properties of CDF and PDF can be found in any classic probability theory reference, such as [34] and [35].

Figure 1. Uniform CDF and PDF
In this paper we deal with the cumulative rather than density functions (which may not exist in some cases), for mathematical as well as conceptual and practical reasons.

Possibility and Fuzzy Distributions
Possibility theory was developed early on by Zadeh [2]- [5] as an extension of fuzzy sets theory, in the context of information meaning, in particular in the context of semantic variables and human soft (fuzzy) data.
Possibility was associated with fuzziness, either due to lack of knowledge or related to the subset for which possibility is defined.Since its inception possibility theory was developed in an axiomatic framework.We do not aim to discuss recent theoretical developments in possibility theory which there are many [9]- [19].We simply recall possibility distribution ΠX(ξ) as a fuzzy restriction on the values assigned to an uncertain variable X and numerically equivalent to fuzzy membership function µA(ξ), i.e.ΠX(ξ) = µA(ξ).For simplicity of the notation we use Π(x), x representing specific choice of fuzzy variable X.Here we recall just a few fuzzy distribution properties which are used in this paper.
where O is an empty set and S is a universe of discourse, with all subsets to which we can assign possibilities.For any A we also have: where A* indicates complementary event to A. The full axiomatic description can be found in [18].

Consistency Principle
We find it useful for our paper to recall Consistency Principle between fuzzy and random variable X defined in Zadeh's classic paper [2] as: where i=1,…,n and variable X can be interpreted both as probabilistic and possibilistic, with the corresponding distributions consisting of the same number of choices in the interval of interest.Consistency Principle carries an intuitive observation that reducing the possibility of an event tends to reduce its probability.The opposite may not hold.If there is a precise (point wise) match between possibilistic and probabilistic distributions, Pi = Πi, then: Consistency Principle as given in (9) may be useful when possibility is known about uncertain event X but not the probability.Our paper expands this idea via Uncertainty Balance Principle which produces variable probability from given possibility.In that sense our approach is similar to (10) rather than (9), as shown in Section 4. Our approach also lends itself to a precise mathematical and quantitative treatment.

Fuzzy To Random Alignment
Methodology described in this paper can be of good use in decision making process where data is inherently mixed, both soft (fuzzy) and hard (random).Often in literature one finds terms such as objective, sensor based, or machine for hard data, and subjective, human based for soft data.One particular area of interest is human-machine (soft-hard) data fusion [21].The main contribution of our work can be understood as two fold: (i) If fuzzy data is available, we can produce variable random data with variable probabilities reflecting original fuzziness.
(ii) On the other hand, if a variable random data is available we can produce corresponding fuzzy distribution, both contributions per our Uncertainty Balance Principle.

General Considerations
We proceed by considering a typical triangular fuzzy distribution number (TFN) given in Figure 2 with the interval of interest {a,b,c} for any a, b and c, and the corresponding fuzzy distribution Π(x) numerically equivalent to the fuzzy membership function µ(x), [2].At this point we will not write equations for the segments of Π(x).This is done in Section 6 with numerical examples.Note that our approach can be applied to any other fuzzy variable, unimodal or multimodal, symmetric or not, normal or non normal, convex or non convex, trapezoidal or arbitrary shaped fuzzy distribution.Section 5.5 shows additional examples of fuzzy distributions.The key is that the CDF properties [35] are satisfied.Next step is to define a pair of CDF's such as in Figure 1 to "decompose" TFN distribution Π(x): where F1(x) and F2(x) are shown in Figure 3.They are both uniform probabilistic distributions.The purpose of the decomposition ( 11) is a first step in relating fuzzy to random variables.Our first idea to define (11) came from an obvious fact that CDF is maximum at 1 similar to Π(x).Next step is to find a way how to describe both rising and falling part of Π(x), and hence (11) came as a natural solution.Recall that a CDF is a probability of an event A = "X ≤ x" as given in (6).It is critical we assume the uncertain variable X is both possibilistic and probabilistic.
Next we take another "probabilistic" step to refine (11) using basic probability relation in (2).The key is that equation (1) has negative term in it which we can associate with F2(x) in (11).This negative term can take different form depending how we move different terms around in (2).Three methods are described.

Method 1
This method is described in full details in our earlier paper [40].It is summarized here.We rewrite (2) as: P(A1) -P(A1∩A2) = P(A1UA2) -P(A2) Each side of Equation ( 12) is a probability and it satisfies basic probabilistic axioms in (1) and ( 2).Nice property of ( 12) is that both sides have negative terms, as does Equation (11).See also Figure 4. We proceed in two Π(x) parts, rising and falling.
Rising Part.We equate left side of ( 12) with (11): from where F1(x) and F2(x) could be uniquely associated with the corresponding probabilities in (13).The aim of this step is to formally align F1(x) with P(A1) and F2(x) with P(A1∩A2), with an idea that an intersection of two events A1∩A2 can produce probability non uniqueness, whereas A1 would be fixed.We again recall the fact that CDF is a probability of an event per (6).Next, the events A2 and A1∩A2 (their probabilities P(A2) and P(A1∩A2) given via F1(x) and F2(x)) are to be determined.What is not uniquely determined is A2 because different A2 can produce the same intersection A1∩A2.A little reflection on set theory brings us to: producing the same P(A1∩A2).The right side 1 -Π(x) ≤ Π*(x) represents complementary fuzzy distribution to Π(x), which upholds the condition in (10).Note that non unique A2 corresponds to x≤b, while A2 is unique for b ≤ x, due to a simultaneous action of conditions in (14).One can consider that the interplay of unique P(A1) and non unique P(A2) in P(A1∩A2), produces "fuzziness" on the left hand side of Π(x).See also Figures 5.
Falling Part.We equate right side of ( 12) with (11): where P(A2) is uniquely defined in (15).The events A1UA2 and A1 are to be determined.

Figures 4. Method 1: CDF's F1(x) and F2(x) Figures 5. Rising part bounds on P(A2) Figures 6. Falling part bounds on P(A1)
Note that the forms of F1 and F2 are same as before but we interpret them differently, i.e. as P(A1UA2) and P(A2).To further clarify A1, we consider union A1UA2, which produces the following condition on P(A1): Figures 6 summarize P(A1) and bounds given in (16) for Π(x) using P(A1UA2).The gray area in P(A1) indicates its non unique choices.They will all generate the same F1=P(A1UA2).Note that the grey area corresponds to b ≤ x, while P(A1) is uniquely defined for x ≤ b, due to a simultaneous action of conditions (16).As in Rising Part, one can consider that the interplay of unique A2 and P(A2) and non unique A1 and P(A1UA2) is equivalent to "fuzziness" of the right hand side of Π(x), when b ≤ x.By combining two parts, we conclude that non unique choices for A1 and A2 and their corresponding probabilities P(A1) and P(A2), correspond to the non zero part of the distribution Π(x).Outside of that, when Π(x)=0, they can be considered independent for the trivial cases of probabilities 0 or 1, per Table 1, where we used the notation P1=P(A1) and P2=P(A2).Note that the non zero Π(x) corresponds to the gray shaded areas in Table 1.
Table 1.Π(x), Method 1. Equations ( 16) and ( 19) Final note is that if we choose (13) for the Falling Part instead of (15) we end up with P(A1) and P(A2) as constant probabilities, given Π(x), and there will be no "fuzziness" induced by variable probabilities.

Other Methods
Before we state the main results in the next Section 4, few comments are in order.By considering Section 3.1 and Equation (11) we can attempt to decompose Π(x) in other ways.For example an "obvious" choice is Π(x) = F1(x) -F2(x) = P(A1) -P(A2).The problem with this choice is that it does not offer any probability variations (non uniqueness).Similarly if we choose Π(x) = F1(x) -F2(x) = [P(A1) + P(A2)] -[P(A3) + P(A4)], the same comment applies, i.e. once we choose say P(A1), then P(A2) follows uniquely, and the same for P(A3) and P(A4).We comment on this further in Section 4.5.

Uncertainty Balance Principle
This section advances Section 4 results and states several general results.The main goal is to produce a usable and practical result to relate fuzzy and variable random data.

General Considerations
For simplicity we assume TFN within the intervals {a,b} and {b.c} non zero Π(x).The results are general for any fuzzy distribution Π(x) which can be represented by repeated procedure (11) for increasing x values.These distributions can be non convex, non normalized, and of other shapes, symmetric, non symmetric, unimodal and multimodal.Figure 10 shows a bimodal fuzzy distribution consisting of two nonoverlapping TFNs.They can also overlap.

Figure 10. Bimodal TFN
The Π(x) decomposition consists of two pairs of cumulative probabilistic distributions: By an induction extension of (2), for "n" modal TFN we have the following general result: Theorem 1. Fuzzy n-modal distribution function Π(x) can be decomposed as a difference of sums of probabilistic cumulative distributions: with i = 2k-1, j = 2k, k = 1,2,…,n, for any x, where the odd functions amount for rising portion of fuzzy distribution and even for the falling portion.

Method 1
Using (26) and Section 3.2 results for bimodal case, we have: for two rising parts of Π(x) and: for two falling parts of Π(x).With ( 27) and (28a,b) we have the following: Corollary 1.1.
Fuzzy n-modal distribution function given in Theorem 1 can be further expressed as a difference of sums of probabilities: for two rising parts of Π(x), and: for two falling parts of Π(x), with i = 2k-1 and j = 2k, k = 1,2,…,n.
When n=1, for unimodal fuzzy distribution, (29a,b) reduce to (13) and (15), and for n=2, bimodal fuzzy distribution, (29a,b) reduce to (28a,b), respectively.Next, we define: as total probability change, with: as probability range for event Ak, where "M" stands for maximum value, and "m" is for minimum value.We now state the following general result which relates a fuzzy distribution and a set of changes in related random event probabilities.
Note that practical implication of this Theorem is to be able to express fuzzy data distribution as a combination of a number of variable random events and corresponding probabilities.We illustrate this notion in Section 5 with numerical examples, and in particular in Example of Section 5.5 which discusses a specific fuzzy distribution and specific resulting variable probabilities.To continue, note that for simplicity, we did not burden the notation in Theorem 2 with stating dependency of ∆P's on x.The key feature of Theorem 2 is that it holds for any x and any presumption level of Π(x).We prove the unimodal case when n=1, for TFN in Figure 2. The proof for any n and Π(x) is straightforward, by repeating the procedure n times.From ( 14) and ( 16) we obtain (see also Section  Note that the point "b" is maximum Π(x) point of a TFN, or any other unimodal fuzzy distribution.Replacing (33) and ( 34) into (32), for n = 1, we obtain: Π(x) + ∆P(A1) + ∆P(A2) = 1 (35) or: holding across the full range of argument x and Π(x), and (36b) follows from (10).One can interpret Theorem 2 result as "randomness" pool left to form fuzzy distribution to a random certainty.This also means that for higher "presumption" levels, near 1, corresponding randomness pool is smaller (less uncertainty to adjust) and for lower "presumption" levels it is larger (more uncertainty to adjust).Examples in Section 7 and for any Fi(x).This is also illustrated in Section 6 with various distributions Π(x).

Method 2
In this case Theorem 1 still holds as stated above.Instead of Corollary 1.1 and using (17ab), we have: Fuzzy n-modal distribution function given in Theorem 1 can be expressed as a difference of sums of probabilities: with i = 2k-1 and j = 2k, k = 1,2,…,n.
Note that the results of this section can be also used in hard-soft data fusion where the staring point are probabilistic rather than possibilistic data.This gives our results an universal applicability in either fuzzyrandom or random-fuzzy uncertainty data alignment.

Other Methods
As described in Section 4.5 we may consider to use other probability decomposition methods in ( 14) such as Π(x) = F1(x) -F2(x) = [P(A1) + P(A2)] -[P(A3) + P(A4)].These choices do not offer any variability of probabilities, once one is chosen, the other follow uniquely.Hence there is no "fuzziness" involved, i.e. all ∆P(Ai,Ai+1) are zero.For such cases Uncertainty Balance Principle reduces to "Certainty Principle" of the form: Π(x) = P(A1) + P(A2) (69a) Π(x) + P(A3) + P(A4) = 1 (69b) for x ≤ b and b ≤ x respectively and it looks to be of less practical use.Note that for probability variability we need presence of non zero intersection of the random events and non zero probability P(Ai∩Aj).This is because specific P(Ai∩Aj) can be generated by a variety of random events Ai and Aj and their corresponding probabilities.This is illustrated in Section 5.5.

Consistency Principle as Uncertainty Balance Principle
Referring back to Zadeh's Consistency Principle [2], as given in (7a) for an unimodal fuzzy distribution, one can re interpret it in the light of our Theorems 2, 3 and 4 which hold for any multimodal fuzzy distribution, any presumption level and any argument x.In this reinterpretation the Principle has a clear conceptual and numeric meaning, as well as an intuitive rationale.We can consider it as a "Fuzzy-Random Uncertainty Balance Principle" and it can be an alternative to Consistancy Principle given in (9).Instead of multiplying Pi and Πi, we can use the sum of ∆Pi's and Πi in the spirit of Theorems 2, 3 and 4 where "i" now points to a different x, that is xi.Recall that all Theorems hold for any presumption level Π(x) as well as any x.For example, from Theorem 2 we can redefine Consistence Principle using our Uncertainty Balance Principle general form Πi + ∆Pi = 1, equivalent to Πi = 1 -∆Pi = ∆P*i or Π*i ≥ ∆Pi which hold for any xi, (i.e. on both sides of Π(x)) as: where i=1,…,n and ∆Pi is the corresponding total probability change in the x range where Πi's are non zero.Complementary values are indicated with "*".Theorems 3 and 4 produce similar result, with slight modification: where i and j can go from 1 to n/2 or some other ratio.
We see how Consistency Principle as defined above is a reflection of our Uncertainty Balance Principle, and besides an intuitiveness it has a definite numerical meaning as well.For example we can agree that ΓX = 10 is better consistency than ΓX = 5, if 10 and 5 are number of arguments xi for which we have the agreement (or knowledge) that Πi = ∆Pi.This can be used in decision making situations when we need to combine soft (fuzzy) with hard (random) data, starting from either one.The assumption is, as stated earlier, that the uncertain variable X is both possibilistic as well as probabilistic.See also Table 4 bellow.We will elaborate on various applications of Uncertainty Balance Principle in our currently going research work on soft-hard or hard-soft data fusion.

Note On Soft-Hard Data Fusion
One of the practical motivations for this work, is to have a methodology to transform fuzzy data to random, and vice versa, so we can apply unique approach and available tools to both.In the first case it is probabilistic methodology, once fuzzy data are described in terms of certain variable probabilities.At the same time we can reverse the process, and given probabilistic description of some phenomenon, where there is a natural variability of probabilities, we can transform random to fuzzy and use all the fuzzy tools available.In either case our approach can enhance decision making process where both soft and hard data are present, and both are to be used to make some decision.Section 3.6 points to one way to judge level of alignment of fuzzy and random data, in the situations where both are generated for a phenomenon which can be treated both as a fuzzy and as random.This phenomenon may come from a system of sensors or soft valuations such as coming from a human operator [21].

Other Fuzzy Distributions
Figures 14 show other types of fuzzy distributions which can be handled by our approach.The first one is a trapezoidal distribution which can be decomposed using a pair of CDFs.The second one is a bimodal and a combination of two distributions put together (gray area can belong to either).It can be decomposed by using two pairs of CDFs.The next one is a convex distribution with the maximum at "b".It can be decomposed by a pair of CDFs, with the break at "b".The last one is a concave distribution.First two fuzzy distributions consist of uniform random distributions, and the last two are not uniform.Any combination of the above distributions is possible too.Uncertainty Balance Principle and Uncertainty Change Law hold in any case, for uniform or non uniform distributions.In Section 6 we show four numerical examples, two uniform, two non uniform fuzzy distributions.Note that in every case the continuity conditions for CDFs are observed when Π(x) is expressed in terms of Theorem 1.

Examples of Π(x) Distributions
In this section we consider four numerical examples which illustrate the main results of the paper.Figure 2 and Equation (11) give a simple TFN decomposition with the CDFs as: Recall that F1 and F2 are equal to various probabilities as described in Methods 1, 2 and 3.

Method 1
Example 1 The symmetric TFN triplet {a, b, c} in Figure 2 is {2,3,4}.Table 5 has the values for x and the corresponding fuzzy "presumption" Π(x) levels.The gray areas show Π(x) and ∆P(A2) and ∆P(A1).For x ≤ b, the probability P(A1) is fixed for a fixed x.On the other hand, P(A2) resides in ∆P(A2).We observe that for small Π(x) values (low fuzzy "presumption") the corresponding range of P(A2) is wider (more uncertainty), and for bigger values of Π(x) (high fuzzy "presumption" level), range of P(A2) is narrower (less uncertainty).This makes intuitive sense.We have the same situation for b ≤ x, except that the non unique probability is now P(A1) residing in ∆P(A1).
Example 2. We change the triplet {a,b,c} in Figure 5 to {10,15,30}, a non symmetric TFN with a larger spread of x.Table 6 shows numerical values.The same comments apply as in Example 1.Note that Π(x) values and probability ranges ∆P(A) are as in Example 1 (confirming x-and fuzzy presumption invariance).

Method 2
We use Example 1 from Method 1 again.Now we have more complexity due to more variable probabilities, P(A1) and P(A2) but also P(A1∩A2), which is not always zero.This gives more options to form fuzzy distribution Π(x).Conditions (19), (20) and ( 21) are used to determine values of various probabilities, as summarized in Tables 9.1 and 9.2 bellow.For simplicity we only included minimum number of values of fuzzy distribution Π(x), due to many different combinations of individual probabilities P(A1), P(A2), and P(A1∩A2).Table 9.1 corresponds to x ≤ b, rising part of Π(x) and F1(x), and Table 9.2 corresponds to b ≤ x and the falling part of Π(x) and F2(x).Table 9.1 confirms Theorem 3 and Corollary 3. Of many options in Tables 9 we can simplify things by choosing, for example, various probabilities to be 1 or 0 at the critical points [a,b,c] (boldfaced).Also, between the critical points we can reduce number of options.Double lines indicate breaks in Tables 9 where probability values in between are obvious.Examples 2, 3 and 4, are not repeated for simplicity.

Method 3
For simplicity we will not repeat all the details for Method 3. The key is for the probabilities to follow conditions ( 23) and ( 25), given in Theorem 4 proof.
Other comments given for Method 3 apply for Method 2 as well.As in previous two Methods, we can see a variety of probability choices which generate the same presumption level Π(x).Depending on the specific application we can choose specific ∆Pi's.

Practical Example of Π(x)
Finally, we illustrate our methodology by a specific TFN fuzzy distribution describing people productivity, and other similar fuzzy applications, [23]- [29].The triplet {a,b,c} is {10,45,90} expressed in years.Assumption is that an average person is the most productive around age of 45 which corresponds to Π(x) = 1.On the opposite end, it is assumed person has zero productivity at age of 10 and 95, hence Π(x) =0.Obviously this is just an approximation but it serves our purposes here.Using (71) we obtain Table 10 where the second row will be described shortly.So how do we interpret the results ?It is assumed the variable X is both possibilistic (fuzzy) and probabilistic (random).Data produced by different sources related to peoples productivity may be soft (subjective, expert opinions) or hard (statistics, random analysis, objective).Hence we can assume that the fuzzy presumption level Π(x) is generated by an interplay of P(Ai) and P(Ai∩Ai+1) per of Corollary 1.2, i.e.: For x ≤ 45: Π(x) = P(A1) -P(A1∩A2) For 45 ≤ x: Π(x) = P(A3) -P(A3∩A4) From Theorem 3 we know that all of the probabilities vary for a given presumption level Π(x) hence producing "fuzziness" of uncertain variable X.Table 10 indicates that as well.Next we can assume that the random (probabilistic) events A1 through A2 are related to level of people's productivity.For example, given the age x and presumption level Π(x) we can assume there are other factors playing the role in productivity for the given age.One possibility may be as given in Table 11.There are other posibilities as well, per specific interest.Second row in Table 10 indicates choices from Table 11.All of the Ai's and their probabilities ∆P's can be described by some CDF, uniform or non uniform, obtained analytically or statistically.Once Ai are defined we can interpret variable probabilities accordingly.For example for x=36.25 years in Table 11, presumption level is Π(x) = 0.75.Corresponding ∆P's are all 0.25 on rising part of Π(x) and that can be interpreted as variability of education, employment and their intersection for that age.On the other hand for the falling part of Π(x), when Π(x)=0.75 we note larger variability of 0.75 of ∆P's, and this refers to health, marriage and their intersection.Methods 1 and 3 could have been used as well.Which Method do we use may depend on a specific fuzzy distribution and application, and more research needs to be done in this subject, looking into specific applications.Note also that Big Data methodology or various statistical methods can be used to have specific choices for the probabilities in Table 11, which exhibit strongest correlations with the original uncertain soft data.
Per Corollary 5, one can reverse the problem.Namely, instead of starting with the possibilistic description and produce (or interpret it via) a probabilistic one, we can start with the probabilistic and produce possibilistic description using essentially the same methodology described in this paper, just in a reverse order.This may be advantageous in some specific cases where we have the fuzzy tools or other fuzzy applications available, but the initial data is random and probabilistic in nature.
any fuzzy distribution.The main results is a universal fuzzy-random (possibilistic-probabilistic) uncertainty alignment law which we named Uncertainty Balance Principle, for its simple statement Π(x) + ∑∆P(Ai,Aj) = 1, which is a linear law, fuzzy presumption-invariant and fuzzy argument x invariant for any fuzzy distribution.Another byproduct of this Principle is also a linear law, Uncertainty Change Law which relates changes in fuzzy distribution against corresponding changes in probabilities.Our results hold for any fuzzy distributions, triangular, trapezoidal, convex, non convex, symmetric or not, normalized or not, uni modal or multi modal alike.This universal range of applicability comes as a result of employing CDF rather than PDF.The results of this paper can be employed effectively in a variety of data fusion and decision problems where both objective (hard, random, probabilistic, sensor based) data are to be fused with subjective (soft, fuzzy, possibilistic, human based) data [9], [10], [21].They can be also used to generate random from fuzzy data for other applications.Additional feature of our approach is a reverse applicability of the results, i.e. going from random to fuzzy.If the range of probabilities is given we can form corresponding fuzzy distribution satisfying Uncertainty Balance principle.Another way to interpret the results is as precise mathematical description of Consistency Principle first introduced by Zadeh in his classic "possibility" paper [2], as a loose and intuitive notion connecting fuzzy and random data.As defined in [2] this Principle relates fuzzy and random distributions in an intuitive way.In our paper, a precise mathematical definition is given for the modified Consistency Principle in terms of Uncertainty Balance Principle of this paper, as a measure of agreement between fuzzy and random data distributions.This can be effectively used to measure level of agreement between related fuzzy and random data in decision making process.In our future work we will extend the results in decision making areas such as machine-human data fusion where using both types of data is crucial for the fusion usefulness.Also, further properties of both random and fuzzy data will be analyzed in the light of the paper's main results.In particular we will consider relationship between probabilistic and possibilistic axioms in the light of this paper results, as well as special cases such as random events independence or dependence.

Figures 11 : 2 Corollary 1
Figures 11: Method 1: Theorem 2 and Corollary 2 Corollary 1.2.Fuzzy n-modal distribution function given in Theorem 1 can be expressed as a difference of sums of probabilities:

Figure 14 .
Figure 14.Various fuzzy distributions which is used in Examples 1 and 2 bellow.Last two distributions in Figures 14 are used for Examples 3 and 4. Recall that F1 and F2 are equal to various probabilities as described in Methods 1, 2 and 3.

Table 2
. Figures 8 show probability diagrams.The shaded areas indicate non unique probabilities.The probability limits are obtained for x ≤ b from Figures 7, 8, and from (17a):

Table 4 .
Table 4 has an intuitive summary of various equivalent descriptions and attributes found in literature on soft (human generated) and hard (sensor or machine) generated data.Other views on what is hard and soft and when to apply fuzzy vs. random are possible as well [41]-[44].Intuitive Soft and Hard Data Designations

Table 6
has the results.Note that the Π(x) values are not uniformly distributed.The distribution changes the fastest right from x=1 and left from x=3, as in Table7.Still linear Uncertainty Balance Principle holds.

Table 11 .
Random event categories