Dependent arithmetic with fuzzy numbers

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Page abstract: 

A fuzzy number is a consonant DSS. By converting a pair of fuzzy numbers and a copula into a joint DSS, propagating the DSS and reconstructing the fuzzy number from the focal elements, dependent arithmetic between fuzzy numbers may be performed. The mass assignment on the joint DSS will come from the copula, defining the stochastic dependence between the fuzzy numbers. This will allow us to perform an arithmetic operation between two fuzzy numbers with a specific dependence. In probability bounds analysis, it is also possible to perform operations with partially known dependencies, for example when only the lower bound of the dependence structure is known. Frechet is special case of this where the lower bound is W (lowest possible bound). However others exists, for example Positive dependence is when the copula is known to be above independence. These operations generally add imprecision, and contain all dependencies above the specified copula. It may be possible to perform similar operations with fuzzy numbers my propagating the bare joint beliefs.

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In his paper "Joint distributions of random sets and their relation to copulas", B. Schmelzer explores the possibility of making a joint random set from two univariate random sets and a copula. He wishes to express a bivariate containment (belief/lower probabilities) functional in terms of two marginal containment functionals and a copula.

His main result is that in the general case this is not possible with a single copula, expect for the case of independence. You are required to split the random sets into families of increasing subsets and apply a family of sub-copula (a copula in a subset of [0, 1]^2) to each of the subsets. He does not go into much more detail about how to best select this family of copulas, and how to do this practically.

The exception is when the focal elements are already organised into a family of increasing intervals, like in a pbox (in which case a bivariate pbox can be expressed by 2 pboxes and a copula), and if the focal elements are consonant, like in a fuzzy number (in which case the bivariate membership function can be expressed by 2 fuzzy number and a copula).

He explores the consonant case further in his paper "Sklar’s theorem for minitive belief functions". He states that the joint containment (belief) can be expressed in terms of two marginal containments and a copula by:

 (eq. 1)

where psi(x, y) is the joint belief, psi1 / psi2 are the marginal beliefs and C is a copula.

EXAMPLE: finding the joint belief from 2 fuzzy numbers and a copula in some rectangle

Take:

A = Fuzzy(1,2,3) 

B = Fuzzy(1,2,3)

C(x, y) = M(x, y) = min(x, y)

L1 = [1.5,2.5]  (alpha cut = 0.5)

L2 = [1.2, 2.8] (alpha cut = 0.3) 

Since the belief is 1 - alpha (for a cut), psi1(L1) = 0.5, ps2(L2) = 0.7

The joint belief of that set = C(0.5,0.7) = min(0.5,0.7) = 0.5

For opposite: C(0.5, 0.7) = W(0.5, 0.7) = max(0.5 + 0.7 - 1, 0 ) = 0.2

For independence: C(0.7,0.5) = π(0.5,0.7) 0.5 * 0.7 = 0.35

For a gaussian copula with r = 0.5: GauCopula(0.5, 0.7; r =0.5) = 0.4216163...

etc

We can therefore construct the joint belief structure, or similarly the joint membership function. The following are examples of joint membership functions with marginals x = y = Fuzzy(1,2,3) and different copulas. 

***Note that the reverse may not be true, from a joint membership function we may not be able to get back marginal fuzzy numbers and a single copula. This is the case with bivariate pboxes***

From the joint belief, it is possible to find the basic mass assignment using the Möbius inversion. The 2 dimensional formula is: 

 (eq. 2)

Where the mass assignment is performed to the focal elements of the joint DSS. Once the mass is assigned, the operation may be performed on the focal elements (with interval arithmetic), and the belief of the resulting resulting intervals can be found by

 (eq. 3)

The alpha level of the interval is then just 1 - Belief(B).

EXAMPLE:

A = Fuzzy(1,2,3) 

B = Fuzzy(1,2,3)

Z = A + B under C = W (opp), Gau(r = -0.8), Gau(r = -0.3) , π (indp), Gau(r = 0.3), Gau(r = 0.8), M (perf)

The purple (perfect) matches levelwise arithmetic. The sum under opposite dependence is an interval in this case, and matches levelwise arithmetic but with alphaX = 1- alphaY

The above calculation can be validated with random sampling. Generally random sets may only be sampled independently, with the exception of pboxes and fuzzy numbers (ref needed for fuzzy numbers). 

To perform propagation with randomly sampled intervals from 2 fuzzy numbers x and y: 

• 1. us = rand(Nsamples, 2)

  2. xSamples = alphaCut(x, 1- us[: ,1])

  3. ySamples = alphaCut(y, 1- us[: ,2])

  4. zSamples = ySamples + zSamples

The above is with independent sampling. If you wish to perform correlated sampling, then the us are samples of a copula (eg samples of a gaussian copula). The following is the same calculation but performed with sampling.

Examples of other binary operations (some are abit rough)

                                                Product                                                                                           Subtraction                                                                                                Divide

                                                        max                                                                                            min

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Arithmetic with partially known dependence

It is possible to also perform arithmetic with partially known dependencies, where only the lower bound of the copula is known. The resulting convolutions contains all of the dependency structures above the specified copula. Frechet is an example of this, where the copula = W, the lower bound to all copulas. The resulting pbox from the operation contains all distributions from any copula combination between the inputs. A copula other than W can be used. For example if π (indp) is used, then the resulting pbox only contains positive corrections, and is tighter than Frechet. Since the lower bound of the copula is only used, if C = M (the upper bound), the convolution results in a precise operation and is equal to that of a perfect correlation convolution.

A similar operation can be performed on fuzzy numbers. In the above we calculated the basic mass assignment from the joint belief (eq. 2), and propagated the intervals with the masses. However, if we retain just the belief assigned to the joint focal elements (step before möbius), propagate the focal element and that to give it an output alpha level (1 - belief), then this corresponds to a convolution with a unknown dependence. **Not sure yet why this works**

Like in probability bounds analysis, this operation is cheaper than with a known dependence, where an integral must be performed. Finding the belief from the mass assignment (eq. 3), is a similar and costly operation. For convolution under partial dependence, only the joint belief (eq. 1) is needed.

It is also sufficient to evaluate the joint belief on nested bivariate subsets of focal elements (the diagonals in the Cartesian product).

EXAMPLE:

A = Fuzzy(1,2,3) 

B = Fuzzy(1,2,3)

Z = A op B under C_{lower} = W (opp), Gau(r = -0.8), Gau(r = -0.3) , π (indp), Gau(r = 0.3), Gau(r = 0.8), M (perf)

                                                Sum                                                                                                            Product

                                                    Subtraction                                                                                            Divide

Red corresponds to Frechet (unknown dependence), yellow is Positive dependence, and purple is perfect. The imprecision gets smaller as the lower bound of the copula approaches the perfect case.