Marked regularity models

We present a generalization of the regularity model, which is a stationary point process model describing how often and how regularly a random "event" occurs. The generalization allows the amplitude of each event to be a sample from a random process. First, we developed closed-form approximations of the power spectra of data segments; then we examined the accuracy of a procedure that estimates the regularity and mark process parameters by minimizing the error between measured spectra and the approximations. We found the following. In the absence of measurement noise, joint estimation of both mark and regularity parameters is accurate only if the ratio of the square of the mean of the marks to the variance of the marks (the SMNPR) is small. Marginal estimation of the regularity process parameters can be accurate if the mark process is taken into account by minimizing overall parameters; the accuracy then depends on both measurement noise and SMNPR. Error in the marginal estimation of the regularity process parameters will be inversely proportional to the SMNPR if the marks are ignored by minimizing only with respect to the regularity parameters, so ignoring the marks can cause a substantial degradation in accuracy when the SMNPR is small. We illustrate these findings with an acoustic scattering example in which simulated ultrasound measurements of tissue samples are characterized by their description in the parameter space.