The traditional approach to standardizing tree-ring time series is to divide raw ring widths by a fitted curve. Although the derived ratios are conceptually elegant and have a more homogenous variance through time than simple differences, residual heteroscedasticity associated with variance dependence on local mean ring width may remain. Incorrect inferences about climate forcing may result if this heteroscedasticity is not corrected for, or at least recognized (with appropriate caveats). A new variance stabilization method is proposed that specifically targets this source of heteroscedasticity. It is based on stabilizing the magnitude of differences from standardization curves to a common reference local mean ring width and uses data pooled from multiple radii. Application of the method to a multi-site kauri (Agathis australis (D. Don) Lindley) data set shows that (a) the heteroscedasticity issue addressed may be generic rather than radius-specific, at least for some species, (b) variance stabilization using pooled data works well for standardization curves of variable flexibility, (c) in the case of kauri, simple ratios do not appear to be significantly affected by this cause of heteroscedasticity, and (d) centennial-scale variance trends are highly sensitive to the analytical methods used to build tree-ring chronologies.

Fluorescence microscopy is a pivotal imaging technique to visualize biological processes and has been extensively utilized in live-cell morphology analysis. Despite its utility, related object detection and tracking tasks still face challenges due to large data scales, inferior data quality, and insufficient annotations, leading to reliance on adaptive thresholding. Current adaptive thresholding approaches have two significant limitations: Firstly, they cannot handle the heteroscedasticity of image data well and result in biased outputs. Secondly, they deal with frames of time-series imaging data independently and result in inconsistent detections over time. We introduce two novel optimization techniques to address these limitations and enhance detection and tracking results in live-cell imaging. The first one, ConvexVST, is a convex optimization approach to transform heteroscedastic data into homoscedastic data, making them more tractable for subsequent analysis. The second one, Joint Thresholding, is a graph-based approach to get the optimal adaptive thresholds while maintaining temporal consistency. Our methods demonstrate superior performance across various object detection and tracking tasks. Specifically, when applied to microglia imaging data, our techniques enable the acquisition of more complete cell morphology and more accurate detection of microglia tips. Furthermore, by integrating these techniques with existing frameworks, we propose an advanced pipeline for embryonic cell detection and tracking in light-sheet microscopy images, which is streets ahead of state-of-the-art peer methods and sets a new benchmark in the field. / Doctor of Philosophy / Fluorescence microscopy is an important imaging tool for observing biological processes and is widely used to study live-cell structures and activities. However, detecting and tracking objects in these images can be difficult because of the large amount of data, poor image quality, and lack of accurate annotations. It leads to the reliance on basic image segmentation approaches, which try to distinguish foreground from background by setting intensity thresholds. These methods have two main problems: they don't handle varying noise in image data well, resulting in inaccurate outputs, and they analyze each frame in a sequence of images independently, causing inconsistencies over time. To solve these issues, we developed two new techniques to improve detection performance in live-cell imaging. The first one, ConvexVST, makes the noise levels in image data more uniform, simplifying the following analysis. The second one, Joint Thresholding, can find the best intensity thresholds while maintaining consistency across frames over time. Our methods have shown significant improvements in detecting and tracking objects. For example, when applied to images of microglia (a type of brain cell), they provide more complete cell shapes and more accurate detection of cell structures. Additionally, by combining these techniques with existing frameworks, we create an advanced pipeline for detecting and tracking embryonic cells that outperforms current leading methods.

This thesis deals with the bootstrap method. Three decades after the seminal paper by Bradly Efron, still the horizons of this method need more exploration. The research presented herein has stepped into different fields of statistics where the bootstrap method can be utilized as a fundamental statistical tool in almost any application. The thesis considers various statistical problems, which is explained briefly below. Bootstrap method: A comparison of the parametric and the nonparametric bootstrap of variance is presented. The bootstrap of ranked set sampling is dealt with, as well as the wealth of theories and applications on the RSS bootstrap that exist nowadays. Moreover, the performance of RSS in resampling is explored. Furthermore, the application of the bootstrap method in the inference of contingency table test is studied. Coefficient of variation: This part shows the capacity of the bootstrap for inferring the coefficient of variation, a task which the asymptotic method does not perform very well. Information theory: There are few works on the study of information theory, especially on the inference of entropy. The papers included in this thesis try to achieve the inference of entropy using the bootstrap method.