We develop a qualitative method for understanding and representing phase space structures of complex systems. To demonstrate this method, a program called MAPS has been constructed that understands qualitatively different regions of a phase space and represents and extracts geometric shape information about these regions, using deep domain knowledge of dynamical system theory. Given a dynamical system specified as a system of governing equations, MAPS applies a successive sequence of operations to incrementally extract the qualitative information and generates a complete, high level symbolic description of the phase space structure, through a combination of numerical, combinatorial, and geometric computations and spatial reasoning techniques. The high level description is sensible to human beings and manipulable by other programs. We are currently applying the method to a difficult engineering design domain in which controllers for complex systems are to be automatically synthesized to achieve desired properties, based on the knowledge of the phase space "shapes" of the systems.

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Extracting and representing qualitative behaviors of complex systems in phase spaces

Extracting and representing qualitative behaviors of complex systems in phase space

This paper describes a solution to the following problem: Given a set of weighted data points, find the cluster center points, which minimize the least squared errors. The k-means-type methods produce good results, but usually the quality of the representation depends on an initial cluster configuration. Also this does not allow a variable number of clusters for a given error tolerance.

A Voronoi Diagram Based Adaptive K-Means-Type Clustering Algorithm for Multidimenstional Weighted Data

SHIFT: a distributed runoff model using irregular triangular facets

Presents three extensions to D.G. Kirkpatrick's (1983) point location algorithm. The extensions revolve around major focuses in the point location problem, namely the location of close successive query points and the dynamic and persistent implementations. The solutions described in this paper include (a) an algorithm that requires O(1) query time for reasonably close successive query points, with O(N) space bound and O(N) time to construct the search structure; (b) a dynamic implementation that require O(N) space, O(log N) query time and O(log N) update time; and (c) a persistent implementation of the same complexity as the dynamic interpretation. Two methods for modifying the search structure (the /spl Kscr/-structure) dynamically and persistently are also discussed. The space requirements and query times of the dynamic and persistent methods are identical to their static counterparts. To affirm the practicality, conceptual simplicity and efficiency of the new methods, we implemented all these extensions, performed computational tests and compared them with other alternative schemes. The results reflect the expected efficiencies of the underlying design.

Three major extensions to Kirkpatrick's point location algorithm

A new, dynamic, hierarchical subdivision and recursive algorithm for computing Delaunay triangulations is presented. The algorithm has four main steps: (1) location of the already formed triangle that contains the point (2) identification of other adjoining triangles whose circumcircle contains the point (3) formation of the new triangles, and (4) database update. Different search procedures are analyzed. It is shown that the “oriented walk” search, when the total number of points is less than 417 or when the points are presorted by distance or coordinates. The algorithm has point-deletion capabilities which are discussed in detail.

Oscar Palacios-Velez

Papers

A dynamic hierarchical subdivision algorithm for computing Delaunay triangulations and other closest-point problems