Mass comparison

Comparing the probability mass calculated from a Dempster-Shafer Structure, a p-box and a fuzzy number.

Q: When does one structure (pbox/fuzzy) perform better than the other? Can we find an example where only using both structures do we get the correct mass of the DDS?

Hypothesis pboxes are 'good' (i.e. they provide tight bounds) for the tail risks but 'bad' for internal risks; FNs are the opposite.

DSS   -> [1,3], [2,4], [2,2.5]

and the translated p-box and fuzzy numbers:

pbox  -> [1,2.5], [2,3], [2,4]                        (sort endpoints)

fuzzy -> [1,4], [1,3], [2,2.5]                         (make consonant, not the only translation, alternative: fuzzy -> [1,4], [2,4], [2,2.5])

                        DSS                                              pbox                               fuzzy number

How to calculate the mass from each structure in some interval A = [a,b] : 

Some calculations: 

Internal

interval : [1,3] (Fuzzy good)

DSS = [2/3 , 1]

pbox = [1/3, 1]

fuzzy = [2/3, 1]               

Internal

interval : [2,2.5] (Fuzzy good)

DSS = [1/3, 1]

pbox = [0, 1]

fuzzy = [1/3, 1]

Tail

interval : [1,1.5] (pbox good)

DSS = [0, 1/3]

pbox = [0, 1/3]

fuzzy = [0, 2/3]

Tail

interval : [3,4] (both good)

DSS = [0, 2/3]

pbox = [0, 2/3]

fuzzy = [0,2/3]

Internal

interval : [2,3] (Fuzzy good)

DSS = [1/3, 1]

pbox = [0, 1]

fuzzy = [1/3, 1]

Arb

interval : [2.7, 10] (both good)

DSS = [0, 2/3]

pbox = [0, 2/3]

fuzzy = [0, 2/3]

Questions:

Can we construct an example where both a pbox and fuzzy are needed to bound the mass of a DSS?

eg. lower bound from pbox, upper from fuzzy. Individually both are wrong, together they are right

Does this hold when we perform arithmetic (sum under independence)

example: DSS + DSS, pbox + pbox, fuzzy + fuzzy.

Where the variables are those defined above. Can we bound: mass, quantiles, moments

Sum under independence of uncertain numbers: 

DSS                                              pbox                                  fuzzy number

Above plots show DSS+ DSS, pbox + pbox, fuzzy + fuzzy under independence

Mass comparison of sum:

DSS

x Intervals  |           masses

[2, 3]       |        [0, 0.555556]

[2, 4]       |        [0, 1]

[4, 5]       |        [0.111111, 1]

[5, 7]       |        [0, 1]

[6, 8]       |        [0, 0.666667]

[2, 6]       |        [0.25, 1]

P-box

x Intervals  |           masses

[2, 3]       |        [0, 0.560001]

[2, 4]       |        [0, 1]

[4, 5]       |        [0, 1]

[5, 7]       |        [0, 1]

[6, 8]       |        [0, 0.665001]

[2, 6]       |        [0.219999, 1]

Fuzzy

x Intervals  |           masses

[2, 3]       |        [0, 0.890001]

[2, 4]       |        [0, 1]

[4, 5]       |        [0.109999, 1]

[5, 7]       |        [0, 1]

[6, 8]       |        [0, 0.890001]

[2, 6]       |        [0.439999, 1]

Revised fuzzy for sum under independence:

x Intervals  |           masses

[2, 3]       |        [0, 0.895001]

[2, 4]       |        [0, 1]

[4, 5]       |        [0.104999, 1]

[5, 7]       |        [0, 1]

[6, 8]       |        [0, 0.565]

[2, 6]       |        [0.435, 1]