By Christian Servin, Vladik Kreinovich
On a variety of examples starting from geosciences to environmental sciences, this
book explains the right way to generate an enough description of uncertainty, the way to justify
semiheuristic algorithms for processing uncertainty, and the way to make those algorithms
more computationally effective. It explains in what feel the present technique to
uncertainty as a mixture of random and systematic parts is simply an
approximation, offers a extra enough three-component version with an additional
periodic blunders part, and explains how uncertainty propagation suggestions can
be prolonged to this version. The e-book presents a justification for a virtually efficient
heuristic procedure (based on fuzzy decision-making). It explains how the computational
complexity of uncertainty processing might be decreased. The publication additionally indicates how to
take under consideration that during actual lifestyles, the data approximately uncertainty is frequently only
partially identified, and, on numerous useful examples, explains find out how to extract the missing
information approximately uncertainty from the to be had data.
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Additional info for Propagation of Interval and Probabilistic Uncertainty in Cyberinfrastructure-related Data Processing and Data Fusion
Thus, when we have a uniform distribution on an arbitrary interval [c, c], the probability p to be in a subinterval [d, d] ⊆ [c, c] is equal to the ratio of the widths of these intervals: d −d . p= c−c In our case, the width of the big interval is equal to ((a − b) + (Δa + Δb )) − ((a − b) − (Δa + Δb )) = 2 · (Δa + Δb ), and thus, the probabilities Prob(a > b) and Prob(a < b) are equal to Prob(a > b) = (a − b) + (Δa + Δb ) ; 2 · (Δa + Δb ) Prob(a < b) = (Δa + Δb ) − (a − b) . 2 · (Δa + Δb ) So, the desired difference Prob(a > b) − Prob(b > a) takes the form Prob(a > b) − Prob(b > a) = (a − b) + (Δa + Δb ) (Δa + Δb ) − (a − b) − = 2 · (Δa + Δb ) 2 · (Δa + Δb ) 2 · (a − b) a−b = .
Finally, let us consider the problem of propagating the periodic components. , to Δ y p = ∑ ∑ Ci j · A i · (cos(ω · ti j ) · cos(ϕ i ) − sin(ω · ti j ) · sin(ϕ i )). i j By combining the terms corresponding to different j, we conclude that Δ y p = ∑ A i · Kci · cos(ϕ i ) + ∑ A i · Ksi · sin(ϕ i ), i def i def where Kci = ∑ Ci j · cos(ω · ti j ) and Ksi = ∑ Ci j · sin(ω · ti j ). j j The values Kci and Ksi can be explicitly described. Namely: • One can easily see that if for some small value δ > 0, for this sensor i, we take Δ xi (ti j ) = δ · cos(ω ·ti j ) for all j, and for all other sensors i , we take Δ xi (ti j ) = 0, then the resulting increase in y will be exactly equal to δ · Kci .
In precise terms: • we have resolution estimates x1 , . . , xn of the values x1 , . . , xn within several small spatial cells; these estimates correspond to models with a higher spatial resolution n • we also have estimates X j for the weighted averages X j = ∑ w j,i · xi ; these estii=1 mates correspond to models with a lower spatial resolution. Comment. In this chapter, we assume that we know the values of the weights w j,i . This assumption makes perfect sense for geophysical problems, because in these problems, these weights are indeed known.
Propagation of Interval and Probabilistic Uncertainty in Cyberinfrastructure-related Data Processing and Data Fusion by Christian Servin, Vladik Kreinovich