Joint Distributions of Weighted Medians on Overlapping Windows in Image Analysis

In this paper, we derive the joint distribution function of the output of weighted median filters with independent but not identically distributed inputs on overlapping windows which may contain an image feature such as a line or edge. These results are used to derive the probability distributions of the difference between the weighted medians calculated on two adjacent overlapping windows. As an illustration, these latter distributions are used to compute the power functions for detecting a step change in the image for a variety of range of existing weighting schemes. These comparisons demonstrate that optimal structure preserving weighting schemes are not optimal in terms of power to detect an edge in the presence of noise

Maximum Likelihood Estimation for MA(1) Processes with a Root on or near the Unit Circle

This paper considers maximum likelihood estimation for the moving average parameter θ in an MA(1) model when θ is equal to or close to 1. A derivation of the limit distribution of the estimate θ LM , defined as the largest of the local maximizers of the likelihood, is given here for the first time. The theory presented covers, in a unified way, cases where the true parameter is strictly inside the unit circle as well as the noninvertible case where it is on the unit circle. The asymptotic distribution of the maximum likelihood estimator sub MLE is also described and shown to differ, but only slightly, from that of θ LM . Of practical significance is the fact that the asymptotic distribution for either estimate is surprisingly accurate even for small sample sizes and for values of the moving average parameter considerably far from the unit circle.

Time series analysis of real-time music perception: approaches to the assessment of individual and expertise differences in perception of expressed affect

<div><p>We use time series analysis methods to detect differences between individuals and expertise groups in continuous perceptions of the arousal expressed by Wishart's electroacoustic piece <i>Red Bird</i>. The study is part of a project in which we characterise dynamic perception of the structure and affective expression of music. We find that individual series of perceptions of expressed arousal often show considerable periods of stasis. This may challenge conventional time series methodologies, so we test their validity by application of a general linear autoregressive moving average (GLARMA) approach, which supports it. Acoustic intensity is a dominant predictor of perceived arousal in this piece. We show that responses are time-variant and that animate sounds influence the conditional variance of perceived arousal. Using vector autoregression and cross-sectional time series analysis (which preserves the integrity of each individual response series), we find differences between musical expertise groups (non-musicians, musicians, and electroacoustic musicians). Individual differences within each group are greater than those between expertise groups. The companion paper applies the developed methods to all four pieces in our overall project (Dean, R.T., F. Bailes, and W.T.M. Dunsmuir. 2014. “Shared and Distinct Mechanisms of Individual and Expertise-Group Perception of Expressed Arousal in Four Works.” <i>Journal of Mathematics and Music</i> 8 (3): 207–223). An Online Supplement is available at http://dx.doi.org/10.1080/17459737.2014.928752.</p></div