Generalized Levinson–Durbin sequences, binomial coefficients and autoregressive estimation

AbstractFor a discrete time second-order stationary process, the Levinson–Durbin recursion is used to determine the coefficients of the best linear predictor of the observation at time k+1, given k previous observations, best in the sense of minimizing the mean square error. The coefficients determined by the recursion define a Levinson–Durbin sequence. We also define a generalized Levinson–Durbin sequence and note that binomial coefficients form a special case of a generalized Levinson–Durbin sequence. All generalized Levinson–Durbin sequences are shown to obey summation formulas which generalize formulas satisfied by binomial coefficients. Levinson–Durbin sequences arise in the construction of several autoregressive model coefficient estimators. The least squares autoregressive estimator does not give rise to a Levinson–Durbin sequence, but least squares fixed point processes, which yield least squares estimates of the coefficients unbiased to order 1/T, where T is the sample length, can be combined to construct a Levinson–Durbin sequence. By contrast, analogous fixed point processes arising from the Yule–Walker estimator do not combine to construct a Levinson–Durbin sequence, although the Yule–Walker estimator itself does determine a Levinson–Durbin sequence. The least squares and Yule–Walker fixed point processes are further studied when the mean of the process is a polynomial time trend that is estimated by least squares

Generalized Levinson-Durbin sequences, binomial coefficients and autoregressive estimation

For a discrete time second-order stationary process, the Levinson-Durbin recursion is used to determine the coefficients of the best linear predictor of the observation at time k+1, given k previous observations, best in the sense of minimizing the mean square error. The coefficients determined by the recursion define a Levinson-Durbin sequence. We also define a generalized Levinson-Durbin sequence and note that binomial coefficients form a special case of a generalized Levinson-Durbin sequence. All generalized Levinson-Durbin sequences are shown to obey summation formulas which generalize formulas satisfied by binomial coefficients. Levinson-Durbin sequences arise in the construction of several autoregressive model coefficient estimators. The least squares autoregressive estimator does not give rise to a Levinson-Durbin sequence, but least squares fixed point processes, which yield least squares estimates of the coefficients unbiased to order 1/T, where T is the sample length, can be combined to construct a Levinson-Durbin sequence. By contrast, analogous fixed point processes arising from the Yule-Walker estimator do not combine to construct a Levinson-Durbin sequence, although the Yule-Walker estimator itself does determine a Levinson-Durbin sequence. The least squares and Yule-Walker fixed point processes are further studied when the mean of the process is a polynomial time trend that is estimated by least squares.Binomial coefficients Generalized Levinson-Durbin sequence Least squares estimator Levinson-Durbin sequence Partial correlations Yule-Walker estimator

On the Recognition of Emotion from Physiological Data

This work encompasses several objectives, but is primarily concerned with an experiment where 33 participants were shown 32 slides in order to create ‗weakly induced emotions‘. Recordings of the participants‘ physiological state were taken as well as a self report of their emotional state. We then used an assortment of classifiers to predict emotional state from the recorded physiological signals, a process known as Physiological Pattern Recognition (PPR). We investigated techniques for recording, processing and extracting features from six different physiological signals: Electrocardiogram (ECG), Blood Volume Pulse (BVP), Galvanic Skin Response (GSR), Electromyography (EMG), for the corrugator muscle, skin temperature for the finger and respiratory rate. Improvements to the state of PPR emotion detection were made by allowing for 9 different weakly induced emotional states to be detected at nearly 65% accuracy. This is an improvement in the number of states readily detectable. The work presents many investigations into numerical feature extraction from physiological signals and has a chapter dedicated to collating and trialing facial electromyography techniques. There is also a hardware device we created to collect participant self reported emotional states which showed several improvements to experimental procedure