Checking If There Exists a Monotonic Function That Is Consistent with the Measurements: An Efficient Algorithm

In many problems in science and engineering ranging from astrophysics to geosciences to financial analysis, we know that a physical quantity y depends on the physical quantity x, i.e., y = f(x) for some function f(x), and we want to check whether this dependence is monotonic. Specifically, finitely many measurements of xi and y = f(x) have been made, and we want to check whether the results of these measurements are consistent with the monotonicity of f(x). An efficient parallelizable algorithm is known for solving this problem when the values xi are known precisely, while the values yi are known with interval uncertainty. In this paper, we extend this algorithm to a more general (and more realistic) situation when both xi and yi are known with interval uncertainty.

[1]  Guy E. Blelloch,et al.  Prefix sums and their applications , 1990 .

[2]  D. Thomson,et al.  WT4 millimeter waveguide system: Spectrum estimation techniques for characterization and development of WT4 waveguide — II , 1977, The Bell System Technical Journal.

[3]  Scott A. Starks,et al.  Towards reliable sub-division of geological areas: interval approach , 2000, PeachFuzz 2000. 19th International Conference of the North American Fuzzy Information Processing Society - NAFIPS (Cat. No.00TH8500).

[4]  Vladik Kreinovich,et al.  From interval computations to modal mathematics: applications and computational complexity , 1998, SIGS.

[5]  Joseph JáJá,et al.  An Introduction to Parallel Algorithms , 1992 .

[6]  Karl Nickel Interval acceleration of convergence , 1988 .

[7]  V. Kreinovich Computational Complexity and Feasibility of Data Processing and Interval Computations , 1997 .

[8]  Vladik Kreinovich,et al.  HPC-ICTM: The Interval Categorizer Tessellation-Based Model for High Performance Computing , 2004, PARA.

[9]  D. Thomson Spectrum estimation techniques for characterization and development of WT4 waveguide–I , 1977, The Bell System Technical Journal.

[10]  Ernest Gardeñes,et al.  Modal Intervals: Reason and Ground Semantics , 1985, Interval Mathematics.

[11]  Eric J. Pauwels,et al.  Image segmentation based on statistically principled clustering , 2001, Proceedings 2001 International Conference on Image Processing (Cat. No.01CH37205).

[12]  Graçaliz Pereira Dimuro,et al.  ICTM: An Interval Tessellation-Based Model for Reliable Topographic Segmentation , 2004, Numerical Algorithms.

[13]  Vladik Kreinovich,et al.  Interval Computations No 4 , 1993 A Linear-Time Algorithm That Locates Local Extrema of a Function of One Variable From Interval Measurement Results , 1993 .

[14]  John E. Dennis,et al.  Numerical methods for unconstrained optimization and nonlinear equations , 1983, Prentice Hall series in computational mathematics.

[15]  Vladik Kreinovich,et al.  A Feasible Algorithm for Locating Concave and Convex Zones of Interval Data and Its Use in Statistics-Based Clustering , 2004, Numerical Algorithms.

[16]  Ramon E. Moore Methods and applications of interval analysis , 1979, SIAM studies in applied mathematics.

[17]  M. Stadtherr,et al.  Reliable nonlinear parameter estimation using interval analysis: error-in-variable approach , 2000 .

[18]  Ronald R. Willis,et al.  New Computer Methods for Global Optimization , 1990 .

[19]  Guido Deboeck,et al.  Financial Applications of Neural Networks and Fuzzy Logic , 1994 .

[20]  R. B. Kearfott Rigorous Global Search: Continuous Problems , 1996 .

[21]  D. Thomson,et al.  Spectrum estimation and harmonic analysis , 1982, Proceedings of the IEEE.

[22]  D. Thomson,et al.  Robust-resistant spectrum estimation , 1982, Proceedings of the IEEE.

[23]  M. Stadtherr,et al.  Reliable Nonlinear Parameter Estimation in VLE Modeling , 2000 .

[24]  I. Good,et al.  Density Estimation and Bump-Hunting by the Penalized Likelihood Method Exemplified by Scattering and Meteorite Data , 1980 .

[25]  Eric J. Pauwels,et al.  Image Segmentation by Nonparametric Clustering Based on the Kolmogorov-Smirnov Distance , 2000, ECCV.

[26]  B W Silverman On a Test for Multimodality Based on Kernel Density Estimates. , 1981 .

[27]  Vladik Kreinovich,et al.  A New Differential Formalism for Interval-Valued Functions and its Potential Use in Detecting 1-D Landscape Features , 2003 .

[28]  Ramon E. Moore Global optimization to prescribed accuracy , 1991 .

[29]  Mark A. Stadtherr,et al.  NONLINEAR PARAMETER ESTIMATION USING INTERVAL ANALYSIS , 1998 .

[30]  Vladik Kreinovich,et al.  If we measure a number, we get an interval. What if we measure a function or an operator? , 1996, Reliab. Comput..