This paper is concerned with regression under a "sum" of partial order constraints. Examples include locally monotonic, piecewise monotonic, runlength constrained, and unimodal and oligomodal regression. These are of interest not only in nonlinear filtering but also in density estimation and chromatographic analysis. It is shown that under a least absolute error criterion, these problems can be transformed into appropriate finite problems, which can then be efficiently solved via dynamic programming techniques. Although the result does not carry over to least squares regression, hybrid programming algorithms can be developed to solve least squares counterparts of certain problems in the class. Index Terms--- Dynamic programming, locally monotonic, monotone regression, nonlinear filtering, oligomodal, piecewise monotonic, regression under order constraints, runlength constrained, unimodal. I.

Locally monotonic regression is the optimal counterpart of iterated median filtering. In [1], Restrepo and Bovik developed an elegant mathematical framework in which they studied locally monotonic regressions in R N . The drawback is that the complexity of their algorithms is exponential in N . In this paper, we consider digital locally monotonic regressions, in which the output symbols are drawn from a finite alphabet, and, by making a connection to Viterbi decoding, provide a fast O(jAj 2 ffN) algorithm that computes any such regression, where jAj is the size of the digital output alphabet, ff stands for lomo-degree, and N is sample size. This is linear in N , and it renders the technique applicable in practice. I. Introduction Local monotonicity is a property that appears in the study of the set of root signals of the median filter [2], [3], [4], [5], [6], [7], [8]; it constraints the roughness of a signal by limiting the rate at which the signal undergoes changes of trend (inc...

Many clustering algorithms have been proposed to partition a set of static data points into groups. In this paper, we consider an evolutionary clustering problem where the input data points may move, disappeare, and emerge. Generally, these changes should result in a smooth evolution of the clusters. Mining this naturally smooth evolution is valuable for providing an aggregated view of the numerous individual behaviors. We solve this novel and generalized form of clustering problem by converting it into a Bayesian learning problem. Analogous to that the EM clustering algorithm clusters static data points by learning a Gaussian mixture model, our method mines the evolution of clusters from dynamic data points by learning a hidden semi-Markov model (HSMM). By utilizing characteristics

Nonlinear regression and nonlinear regularization are two powerful approaches to segmentation and nonlinear filtering. In this correspondence, we propose a hybrid approach that effectively combines the best of both worlds and can be efficiently implemented via the Viterbi algorithm. I.

) C. Faloutsos Univ. of Maryland christos@cs.umd.edu H. V. Jagadish AT&T Labs jag@research.att.com A. O. Mendelzon Univ. of Toronto mendel@db.toronto.edu T. Milo Tel Aviv Univ. milo@math.tau.ac.il 1 Introduction Sequences of real-valued data arise in many applications ranging from the stock market to electro-cardiograms. Often, it is of interest to locate sequences that are similar to a specified query sequence. The notion of similarity is application dependent, and even within a single application, may vary from one query to the next. Work in this area is usually specific to one particular domain and uses one specific notion of similarity. For example, Faloutsos et al [6, 1] studied the problem of searching a database of time sequences for sequences similar to one given. They reduced sequences to points in a low-dimensional space by using Fourier transforms and used the Euclidean distance in this space to measure similarity. This notion of similarity is extended in [21] by...