Autoregressive model for a finite random sequence on the unit circle for investigating the fluctuations of residual stresses in the rims of new railroad wheels

This paper presents an autoregressive model for a finite sequence of random variables that are observed at points equally spaced on the unit circle. The proposed model is an extension of the well‐known autoregressive model of time series. We demonstrate that this model amounts to a linear transformation of a vector of independent and identically distributed random variables. The second‐order properties of the multivariate distribution were examined. The least squares estimators of the model parameters were obtained. The connection between the proposed first‐order model and a second‐order, stationary, mean‐square‐continuous, real‐valued random process on the unit circle was considered. We used the model presented to describe the fluctuations of hoop residual stresses in the rims of new railroad wheels. The stress measurement was performed using an ultrasonic method. The stress fluctuation model allowed us to determine the number of measurement points required to assess residual stress levels in the wheels. Copyright © 2014 John Wiley & Sons, Ltd.