Interval Computations are Also Useful for Statistical Estimates

1. Julian Ortiz Perez Department of Computer Science University of Texas at El Paso jortizperez@gmail.com Interval Computations are Also Useful for Statistical Estimates

2. Outline • Introduction • Estimating Errors • Error Estimation in: • Traditional Engineering Approach • Interval Computations • Open Problems • Our Solution • Proof • Conclusions

3. Introduction • We are interested in: • Value of a physical quantity y • Problem • Difficult to measure directly • Solution • Find quantities xi related to y: • y = f(x1, . . . , xn) • Measure xi • Estimate Y= f(X1, . . . , Xn)

4. Estimating Errors • Ideal case • We know the exact values of x1, . . . , xn • We find y= f(x1, . . . , xn) • Practical case: • We know approximate measurement results of X1, . . . , Xn • We find Y= f(X1, . . . , Xn) • Since Xi ≠ xi, we have Δy = Y - y = 0

5. Error Estimation: Traditional Engineering Approach • Assumptions: • Measurement errors Δxi are independent • Δxi are normally distributed with 0 mean and known σi

6. Error Estimation: Interval Computations • In practice: • We are often not sure that Δxi are independent • We are not sure that Ε[Δxi] = 0 • Available information: • Upper bound Δion Δxi : |Δxi| ≤ Δi • Conclusion: xi in xi = [Xi - Δi, Xi+ Δi] • Interval computations: Estimate the range y= {f(x1, . . . , xn) | xi in xi, . . ., Xn in xn}

7. Open Problems • Previously known • Statistical approach – full Info about probabilities • Interval computations – no info about probabilities • In reality: we often have partial info about probabilities • Example: • Measurement errors are independent • (almost) normally distributed • Ε[Δxi] may be non-0

8. Our Solution • Problem (reminder): • Measurement errors are independent • (almost) normally distributed • Ε[Δxi] may be non–0 • Solution: we get the same set yof possible values of y as in interval computations • Additional comment: for each subinterval Yof y, Prob (y in Y) can take any number from [0, 1]

9. Proof • Reminder: y= f(x1, . . . , xn), and each xi is in xi • Since xiare in xi, we have y in y • Vice versa, let Y be a narrow subinterval of y, and let y be in Y • Since y is in y, there exist xis.t. y= f(x1, . . . , xn) • For normal distributions with E = xi and small σi, Prob (y in Y) is close to 1 • For y not in Y, also y= f(x1, . . . , xn) for some xifrom xi • For normal distributions, Prob (y in Y) is close to 0 • By continuity, we get probability from (0, 1) as well

10. Conclusions • Traditionally, interval computations are mainly used when we have no info about probabilities • Our idea: they can also be helpful when we have info about probabilities

11. Acknowledgments • Dr. Luc Longpré • Dr. Vladik Kreinovich • University of Texas at El Paso (UTEP) • Cyber-ShARE Center of Excellence (Cyber-ShARE) • National Science Foundation

12. Questions?