Grant: Discovery Grants Program - Individual

Experimenters are often searching for the values of input variables that will provide the optimal yield in a process. In most cases, the number of input variables are many and the system can be represented by nonlinear models. In some situations it may be possible to guess the structure of the nonlinear function needed to represent the system to some degree of accuracy. However, in many cases the function may be unknown. It turns out that if little or nothing is known about the structure of the nonlinearity in a system, it has been found that a class of functions known as wavelets can be used to model the nonlinearity very closely for the purpose of analysis, prediction and inference. We intend to exploit this flexible property of wavelets to construct designs for nonlinear systems. Our goal is to focus on systems in which there are more than one input variable. Furthermore, in longitudinal time series, the variance of the series may vary from subject to subject or the variance of the series from a group of subjects may differ from the variance of the other groups. In addition, each of the series may also have a trend component which may not be the same across the entire data. Before analyzing the longitudinal data, it is then important to know if (i) the variance of each of the series is the same, and (ii) it is safe to assume that the trend component is the same for the series from each subject, in the presence of the correlation structure within each series. Our goal is to develop wavelet based procedures for assessing the nature of the nonstationarity in the mean of each series and determining if it is the same across several time series in a longitudinal set up. Our approach will take into account the correlation structure of each series. This will then have an impact on how tests for equality of variances across these time series will be performed. To our knowledge, current methods have assumed that the trend in each of the series is the same. Our objective is to extend these results to the case where the trend may defer from series to series. The proposed research is useful in design construction and in medical research, plant physiology and animal growth curves - areas in which the problem of comparison of curves frequently arise.