Measurement . . . Expert estimates and . . . Reliability of interval data Need to gauge the . . . Quantum Computation Need to gauge . . . How to gauge . . . Techniques for Gauging Resulting . . . Results Reliability of Interval and Algorithm: description . . . Fuzzy Data Use of quantum . . . Conclusion Luc Longpr´ e and Christian Servin Conclusions (cont-d) Department of Computer Science Acknowledgments University of Texas at El Paso Title Page El Paso, TX 79968 Contact email christians@miners.utep.edu ◭◭ ◮◮ ◭ ◮ Page 1 of 16 Go Back Full Screen
Measurement . . . Expert estimates and . . . 1. Outline Reliability of interval data Need to gauge the . . . • Traditionally, we assume that the interval bounds are Need to gauge . . . correct, and that (fuzzy) expert estimates are correct. How to gauge . . . • In practice, measuring instruments and experts are not Resulting . . . 100% reliable. Results • Usually, we know the percentage of such outlier un- Algorithm: description . . . reliable measurements. Use of quantum . . . Conclusion • However, it is desirable to check that the reliability of Conclusions (cont-d) the actual data is indeed within the given percentage. Acknowledgments • The problem of checking (gauging) this reliability is, Title Page in general, NP-hard. ◭◭ ◮◮ • In reasonable cases, there exist feasible algorithms for ◭ ◮ solving this problem. Page 2 of 16 • We show that quantum computing can speed up the computation of reliability of given data. Go Back Full Screen
Measurement . . . Expert estimates and . . . 2. Two main sources of information Reliability of interval data Need to gauge the . . . • In practice: we want to know the state of objects. Need to gauge . . . • In science: we are simply interested in this state. How to gauge . . . • Example: we want to know the river’s water level. Resulting . . . Results • In engineering: we need the information about the Algorithm: description . . . state of the world to change the situation. Use of quantum . . . • Example: how to build a dam to prevent flooding. Conclusion • Most accurate reliable estimate of each quantity: mea- Conclusions (cont-d) surement. Acknowledgments Title Page • In many cases, it is too difficult or too expensive to measure all the quantities. ◭◭ ◮◮ • In such situations, we can ask the experts to estimate ◭ ◮ the values of these quantities. Page 3 of 16 • Measurements and expert estimates are thus the two Go Back main sources of information about the real world. Full Screen
Measurement . . . Expert estimates and . . . 3. Measurement uncertainty and interval data Reliability of interval data Need to gauge the . . . • The result � x of a measurement is usually somewhat Need to gauge . . . different from the actual (unknown) value x . How to gauge . . . • Usually, the manufacturer of the measuring instrument Resulting . . . (MI) gives us a bound ∆ on the measurement error: Results def | ∆ x | ≤ ∆ , where ∆ x x − x Algorithm: description . . . = � Use of quantum . . . • Once we know the measurement result � x , we can con- Conclusion x − ∆ , � clude that the actual value x is in [ � x + ∆]. Conclusions (cont-d) • In some situations, we also know the probabilities of Acknowledgments different values ∆ x ∈ [ − ∆ , ∆]. Title Page • In this case, we can use statistical techniques. ◭◭ ◮◮ • However, often, we do not know these probabilities; we ◭ ◮ def only know that x is in the interval x = [ � x − ∆ , � x + ∆]. Page 4 of 16 • So, we need to process this interval data. Go Back Full Screen
Measurement . . . Expert estimates and . . . 4. Expert estimates and fuzzy data Reliability of interval data Need to gauge the . . . • There is no guarantee of expert’s accuracy. Need to gauge . . . • We can only provide bounds which are valid with some How to gauge . . . degree of certainty. Resulting . . . • This degree of certainty is usually described by a num- Results ber from the interval [0 , 1]. Algorithm: description . . . Use of quantum . . . • So, for each β ∈ [0 , 1], we have an interval x ( α ) con- Conclusion taining the actual value x with certainty α = 1 − β . Conclusions (cont-d) • The larger certainty we want, the broader should the Acknowledgments corresponding interval be. Title Page • So, we get a nested family of intervals corresponding ◭◭ ◮◮ to different values α . ◭ ◮ • Alternative: for each x , describe the largest α for which Page 5 of 16 x is in x ( α ); this α largest is a membership function µ ( x ). Go Back Full Screen
Measurement . . . Expert estimates and . . . 5. Reliability of interval data Reliability of interval data Need to gauge the . . . • Usual assumption: all measuring instruments (MI) func- Need to gauge . . . tioned correctly. How to gauge . . . • Conclusion: the resulting intervals [ � x − ∆ , � x + ∆] con- Resulting . . . tain the actual value x . Results • In practice: a MI can malfunction, producing way-off Algorithm: description . . . values (outliers). Use of quantum . . . • Problem: outliers can ruin data processing. Conclusion Conclusions (cont-d) • Example: average temperature in El Paso Acknowledgments – based on measurements, 95 + 100 + 105 = 100 . Title Page 3 – with outlier, 95 + 100 + 105 + 0 ◭◭ ◮◮ = 75. 4 ◭ ◮ • Natural idea: describe the probability p of outliers. Page 6 of 16 def • Solution: out of n results, dismiss k = p · n largest Go Back values and k smallest. Full Screen
Measurement . . . Expert estimates and . . . 6. Need to gauge the reliability of interval data Reliability of interval data Need to gauge the . . . • Ideal case: all measurements of the same quantity are Need to gauge . . . correct. How to gauge . . . • Fact: resulting intervals x (1) , . . . , x ( n ) contain the same Resulting . . . (actual) value x . Results � n x ( i ) � = ∅ . Algorithm: description . . . • Conclusion: i =1 Use of quantum . . . � n x ( i ) = ∅ . Conclusion • Reality: we have outliers far from x , so Conclusions (cont-d) i =1 • Expectation: out of n given intervals, ≥ n − k are cor- Acknowledgments rect – and hence have a non-empty intersection. Title Page • Conclusion: ◭◭ ◮◮ – to check whether our estimate p for reliability is ◭ ◮ correct, Page 7 of 16 – we must check whether out of n given intervals, Go Back n − k have a non-empty intersection. Full Screen
Measurement . . . Expert estimates and . . . 7. Need to gauge reliability of interval data: multi-D Reliability of interval data case Need to gauge the . . . Need to gauge . . . • In practice, a measuring instrument often measure sev- How to gauge . . . eral different quantities x 1 , . . . , x d . Resulting . . . • Due to uncertainty, after the measurement, for each Results quantity x i , we have an interval x i of possible values. Algorithm: description . . . • So, the set of all possible values of the tuple x = Use of quantum . . . ( x 1 , . . . , x d ) is a box Conclusion Conclusions (cont-d) X = x 1 × . . . × x d = { ( x 1 , . . . , x d ) : x 1 ∈ x 1 , . . . , x d ∈ x d } . Acknowledgments Title Page • Thus: ◭◭ ◮◮ – to check whether our estimate p for reliability is correct, ◭ ◮ – we must be able to check whether out of n given Page 8 of 16 boxes, n − k have a non-empty intersection. Go Back Full Screen
Measurement . . . Expert estimates and . . . 8. How to gauge reliability of fuzzy data Reliability of interval data Need to gauge the . . . • Fact: experts are sometimes wrong, so their estimates Need to gauge . . . are way off. How to gauge . . . • Idea: to gauge the reliability of the experts by the Resulting . . . probability p that an expert is wrong. Results • Example: p = 0 . 1 means that we expect 90% of the Algorithm: description . . . experts to provide us with correct bounds X (0). Use of quantum . . . Conclusion • Comment: we may have different probabilities p for Conclusions (cont-d) different certainty levels α . Acknowledgments • Conclusion: Title Page – to check whether the data fits given reliability es- ◭◭ ◮◮ timates, ◭ ◮ – we must therefore be able to check whether out of Page 9 of 16 n given boxes, n − k have a non-empty intersection. Go Back Full Screen
Measurement . . . Expert estimates and . . . 9. Resulting Computational problem: box intersec- Reliability of interval data tion problem Need to gauge the . . . Need to gauge . . . Thus, both in the interval and in the fuzzy cases, we need How to gauge . . . to solve the following computational problem: Resulting . . . • Given: Results • integers d , n , and k ; and Algorithm: description . . . Use of quantum . . . • n d -dimensional boxes Conclusion X ( j ) = [ x ( j ) 1 , x ( j ) 1 ] × . . . × [ x ( j ) n , x ( j ) n ] , Conclusions (cont-d) j = 1 , . . . , n , with rational bounds x ( j ) and x ( j ) Acknowledgments i . i Title Page • Check whether ◭◭ ◮◮ – we can select n − k of these n boxes ◭ ◮ – in such a way that the selected boxes have a non- Page 10 of 16 empty intersection. Go Back Full Screen
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