Abstract — This paper introduces multi-rate coupled hidden Markov models (multi-rate HMMs for short) for multiscale modeling of nonstationary processes, extending traditional HMMs from single to multiple time scales with hierarchical representations of the process state and observations. Scales in the multi-rate HMMs are organized in a coarse-to-fine manner with Markov conditional independence assumptions within and across scales, allowing for a parsimonious representation of both shortand long-term context and temporal dynamics. Efficient inference and parameter estimation algorithms for the multi-rate HMMs are given, which are similar to the analogous algorithms for HMMs. The model is applied to the classification of tool wear in titanium milling, for which acoustic emissions exhibit multiscale dynamics and long-range dependence. Experimental results show that the multi-rate extension outperforms HMMs in terms of both wear prediction accuracy and confidence estimation. Index Terms — Hidden Markov model, multi-rate hidden Markov model, multiscale statistical modeling, confidence, tool

Many of the currently used tool materials will dissolve, diffuse, and/or react with titanium due to high temperatures at the tool/work-piece interface. Potential next generation tool materials that would improve the machining of titanium and eliminate the contamination of the work-piece are being developed. One such material, AlMg B14, is the subject of the research presented in this paper. Specimens of the newly developed tool, AlMg B14 were fabricated into a standard tool geometry. This tool material was compared with a standard WC-Co tool material to machine a Ti-6Al-4V work material. Thermal and force measurements were conducted using both types of tools during orthogonal cutting. Results are also compared with finite element simulations. The paper outlines that higher temperatures are obtained by AlMgB14. It is also shown that this material is more appropriate to be used with high cutting speeds. Nevertheless, a weakness of this newly developed AlMgB14 B is its toughness that needs to be improved for better performance in an industrial environment. 1 Laurent Deshayes is currently a guest researcher at the National Institute of Standards and Technology.