This Article 
   
 Share 
   
 Bibliographic References 
   
 Add to: 
 
Digg
Furl
Spurl
Blink
Simpy
Google
Del.icio.us
Y!MyWeb
 
 Search 
   
Constructing Quantitative Models Using Monotone Relationships
April 1995 (vol. 7 no. 2)
pp. 294-304

Abstract—Constructing quantitative models typically requires characterizing a system in terms of algebraic relationships and then using these relationships to compute quantitative values from numerical data. For real-life systems, such as computer operating systems, an algebraic characterization is often difficult, if not intractable. This paper proposes a statistical approach to constructing quantitative models using monotone relationships. Referred to as nonparametric interpolative-estimation for monotone functions (NIMF), our approach uses monotone relationships to search historical data for bounds that provide a desired level of statistical confidence. NIMF makes no assumption about the algebraic form of the monotone relationship, not even continuity.

We present two examples of applying NIMF to computer measurements, and compare NIMF’s confidence intervals with those of least-squares regression, a traditional technique that requires specifying an algebraic relationship. Our results suggest that when an algebraic characterization is not known with precision, using NIMF with an accurate monotone relationship can produce more accurate confidence intervals than employing least-squares regression with a polynomial approximation to the unknown algebraic relationship.

[1] C. Apte and S.J. Hong,“Using qualitative reasoning to understand financial arithmetic,” Proc. Fifth Nat’l Conf. Artificial Intelligence, pp. 942-948, 1986.
[2] Y. Bard,“An analytic model of CP-67 and VM/370,” Computer Architecture and Networks,” pp. 419-460, 1974.
[3] R.E. Barlow,D.J. Bartholomew,, and H.D. Brunk,Statistical inference under order restriction, John Wiley&Sons, 1972.
[4] J. Bradley,Distribution-free statistical tests, Prentice-Hall, 1968.
[5] C.F. Christ,Econometric models and methods, John Wiley&Sons, 1967.
[6] J.P. Cunningham,“Multiple monotone regression,” Psychological Bull., vol. 3, pp. 791-800, 1982.
[7] S. Dalal,“Simultaneous confidence bands for regression with unknown, unequal variances,” Technometrics, vol. 2, pp. 173-186, 1990.
[8] C. Daniel and F.S. Wood,Fitting equations to data, John Wiley&Sons, 1980.
[9] D. DeCoste,“Dynamic across-time measurement interpretation,” Artificial Intelligence, vol. 51, pp. 273-341, 1991.
[10] J. De Kleer and J.S. Brown,“A qualitative physics based on confluences,” Artificial Intelligence, vol. 24, pp. 7-83, 1984.
[11] N.R. Draper and H. Smith,Applied regression analysis, John Wiley&Sons, 1968.
[12] D. Dvorak and E.P. Sacks,“Stochastic analysis of qualitative dynamics,” Proc. 11th Int’l Joint Conf. Artificial Intelligence, pp. 1187-1192, 1989.
[13] B. Efron,“Bootstrap methods: Another look at the jacknife,” Annals of Statistics, vol. 1, pp. 1-26, 1979.
[14] D.A. Freedman,“Bootstrapping regression models,” Annals of Statistics, vol. 6, pp. 1218-1226, 1981.
[15] J.H. Friedman and W. Stuetzle,“Projection pursuit regression,” J. Am. Statistical Assoc., vol. 376, pp. 817-823, 1981.
[16] K. Forbus, "Qualitative Process Theory," Artificial Intelligence, Vol. 24, No. 3, Dec. 1984, pp. 85-168.
[17] E. Gelenbe and I. Mitrani,Analysis and synthesis of computer systems, Academic Press, 1980.
[18] J.L. Hellerstein,“Constructing quantitative models using monotone relationships,” IBM Research Report no. RC 17664, 1992.
[19] IBM, Virtual machine monitor analysis program: User’s guide and reference, IBM Corporation, SC34-2166, 1985.
[20] R.A. Johnson and D.W. Wichern,Applied multivariate statistical analysis, Prentice Hall, 1988.
[21] L. Kleinrock,Queueing systems, vol. 1, John Wiley&Sons, 1975.
[22] B. Kuipers,“Qualitative simulation,” Artificial Intelligence, vol. 29, pp. 289-338, 1986.
[23] M.D. Morris,“Small-sample confidence limits for parameters under inequality constraints with application to quantal bioassay,” Biometrics, vol. 44, pp. 1083-1092, 1988.
[24] T. Nishida,K. Mizutani,A. Kubota,, and S. Doshita,“Automated phase portrait analysis by integrating qualitative and quantitative analysis,” Proc. Ninth Nat’l Conf. Artificial Intelligence, pp. 811-816, 1991.
[25] P. Schaefer,“Analytic solution of qualitative differential equations,” Proc. Ninth Nat’l Conf. Artificial Intelligence, pp. 830-835, 1991.
[26] C.J. Stone,“Consistent nonparametric regression,” Annals of Statistics, vol. 4, pp. 595-645, 1977.
[27] G. Wahba and S. Wold,“A completely automatic French curve: Fitting spline functions by cross-validation,” Comm. in Statistics, vol. 17, pp. 1-17, 1975.
[28] M.P. Wellman,“Probabilistic semantics for qualitative influences,” Proc. Sixth Nat’l Conf. Artificial Intelligence, pp. 660-664, 1987.

Index Terms:
Monotone function, confidence intervals, estimation, statistics, qualitative reasoning.
Citation:
Joseph L. Hellerstein, "Constructing Quantitative Models Using Monotone Relationships," IEEE Transactions on Knowledge and Data Engineering, vol. 7, no. 2, pp. 294-304, April 1995, doi:10.1109/69.382298
Usage of this product signifies your acceptance of the Terms of Use.