Bootstrap resampling in wavelet analysis and statistical methodologies in ecological research.

摘要

Part I. Bootstrap resampling in wavelet analysis . Wavelet smoothing is often used to estimate an unknown signal function from observations of "function + noise" at n equi-spaced times. When n is very large, a big proportion of the discrete wavelet transform (DWT) coefficients are often very small compared to the others. In this scenario, it is reasonable to replace most small empirical DWT coefficients with zeros and leave the rest intact for the estimation of the DWT coefficients, which forms a variable selection problem. To implement variable selection, estimation of the noise standard deviation sigma is inevitable. The conventional estimator of sigma is the adjusted median absolute deviation of the finest empirical DWT coefficients, which is useful mostly for low frequency signals. To account for all types of signals, especially for high frequency signals, we developed the moving block estimator. A very popular variable selection method in literature is thresholding, and the common threshold level is 2logn , which is mostly based on the low frequency signals. We found a better threshold level, 2logn , based on all types of signals as well as the resampling behavior. We also developed another variable selection method: the optimal choice method, based on an estimate of expected loss. Beyond the point estimator of the DWT coefficients with variable selection, confidence sets based on two approximately pivotal quantities are constructed. Beran's key idea of shrink bootstrap is used in constructing the confidence sets.; Part II. Statistical methodologies in ecological research. Many statistical methodologies have been used by the author in his research work with ecological data. In this dissertation, three types of methodologies are discussed: (1) explorative (ordination) methodologies, including correspondence analysis and nonmetric scaling, (2) inferential (regression) methods, including analysis of covariance, mixed-effect models and partial least squares regression, (3) hybrid (combination of ordination and regression) methodologies, including redundancy analysis, canonical correspondence analysis and co-correspondence analysis.