Wavelets in robust designs and longitudinal time series analysis

实验人员经常在一个过程中寻找能提供最佳产量的输入变量的值。在大多数情况下，输入变量的数量很多，系统可以用非线性模型来表示。在某些情况下，可以猜测非线性函数的结构，以便在一定程度上精确地表示系统。然而，在许多情况下，函数可能是未知的。事实证明，如果对系统中非线性的结构知之甚少或一无所知，则可以使用一类称为小波的函数来非常密切地模拟非线性，以便进行分析，预测和推理。我们打算利用小波的这种挠性来构造非线性系统的设计。我们的目标是关注有多个输入变量的系统。此外，在纵向时间序列中，该序列的方差可能因受试者而异，或者一组受试者的序列方差可能不同于其他组的方差。此外，每个序列也可能有一个趋势成分，在整个数据中可能不相同。在分析纵向数据之前，重要的是要知道(i)每个序列的方差是否相同，以及（ii）在每个序列中存在相关结构的情况下，可以安全地假设来自每个主题的序列的趋势分量相同。我们的目标是开发基于小波的程序，用于评估每个序列的平均值的非平稳性的性质，并确定它是否在纵向设置的几个时间序列中相同。我们的方法将考虑到每个序列的相关结构。这将对如何执行跨这些时间序列的方差相等检验产生影响。据我们所知，目前的方法假设每个序列的趋势是相同的。我们的目标是将这些结果扩展到趋势可能从一个序列延迟到另一个序列的情况。所提出的研究在设计、建造和医学研究、植物生理学和动物生长曲线等经常出现曲线比较问题的领域是有用的。

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.