Expanding the Spectral Envelope

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Stoffer DMS-9703720 The concept of the spectral envelope was recently introduced as a general statistical method for the frequency domain analysis and scaling of qualitative-valued time series. In the process of developing the technology, many other interesting extensions became evident. This research involves the extension of the fundamental concept in various directions. The first extension involves matching two categorical sequences in an effort to discover whether they contain similar patterns and is motivated by the problem of matching of two DNA sequences. The approach builds on the ideas used in defining the spectral envelope and focuses on what could be called coherency envelopes for measuring the similarity between two categorical time series. Estimation is based on the fast Fourier transform so that the methods are computationally simple and fast, and can be applied to long sequences. In another direction, the technology is extended to the analysis of qualitative spatial data. The applications that motivate this project are automated image retrieval and pattern recognition with potential use in computer vision. Specifically, this part of the research will focus on the analysis of categorical random fields via optimal scaling and the wave number spectral envelope. Since many of the applications where the spectral envelope methodology has been an asset are situations where the assumptions of stationarity and homogeneity are not realistic, this research will explore the benefits of wavelet-based analysis over dynamic Fourier analysis. Another extension of this research is to apply the concept to the analysis of real-valued time series collected in an experimental design where the primary interest is whether any, and how many, have common cyclic components. This problem is motivated by numerous applications in the medical and behavioral sciences. A statistical concept called the spectral envelope was recently introduced as a general method to study patterns in long seq uences of letters or symbols (such as codes). The most well known application of this methodology is in biotechnology, specifically in the analysis of DNA. In the process of developing the concept, many other practical extensions became evident. This research involves the extension of the fundamental concept in various directions. The first extension involves matching two non-numeric sequences in an effort to discover whether they contain similar patterns and is motivated by the biotechnical problem of finding similarities in two otherwise different genetic codes. Another direction is to extend the technology to qualitative spatial data (such as images) with applications in automated image retrieval, automated pattern recognition, and computer vision. This research will have an impact on robotics and automation technologies that will be useful, for example, in industrial manufacturing processes. Moreover, the methodology used in matching sequences could have applications in computer vision and robotics where the problem is to automatically and quickly align two pictures of the same scene taken from different camera angles. In addition, this research will focus on ways to incorporate this new technology into current medical applications such as understanding changes in biorythms in stressful situations.