Problems and Techniques

Abstract

This edition of Problems and Techniques features articles on orthogonal rational functions with applications in identification and control, and methods for cancer classification. Our lead-off article, by Wahlberg, is motivated by applications in control theory and signal processing where finite impulse response (FIR) and infinite impulse response (IIR) models are often invoked. The former are closely related to orthogonal polynomials and have nice computational properties, while the latter often give more compact representations and improved accuracy. Our author develops a fractional transformation framework to map the original problem to a new domain where FIR descriptions have rapid convergence. When mapped back to the original domain, the FIR models generate an IIR description with an orthogonal rational basis and extends the theory available for FIR models to the IIR case. Our second article, by Rifkin, Mukherjee, Tamayo, Ramaswamy, Yeang, Angelo, Reich, Poggio, Lander, Golub, and Mesirov, an interdisciplinary team centered at the Whitehead Institute/MIT Center for Genome Research, describes a molecular approach to the classification of cancers. This technology has important ramifications when developing correct diagnoses and appropriate cancer treatment strategies. The technique proposed by the authors uses DNA microarrays to assemble a large database of expressed genes in the tumor samples. Gene expression profiles are analyzed by combining multiple binary support vector machine (SVM) classifiers trained to predict class membership. The cover of this issue of SIAM Review depicts six different types of cancer of the fourteen studied in this article.