Spatial, Temporal and Multidimensional Statistics: Estimation, testing, and applications in the environmental sciences

我的建议与“时空异质性分析”有关，这是我在过去25年多的统计(方法和应用)研究工作的一个主轴。在接下来的五年里，我计划研究空间和时间数据的统计推断的五个问题(频率法)，包括当框架是多变量且仅限于空间或时间时的多维统计。应用领域是环境(植物-土壤、地震学[1])。。问题1：在[2-4]之后的多个时间序列的M-Vine分解的比较检验，以及正在进行的M.Sc。论文计划在不同的问题上，将开发一种基于似然检验的测试，比较同一多个时间序列的两个M-Vine分解。。问题2：点模式自相关分析和互相关分析的一种新的检验方法均值的异质性是Ripley的K和L函数用于点模式相关分析的使用者所关注的问题，因为在统计量的表达式中假设一阶强度函数是恒定的[5，第一章。5]。。问题3：一种新的多元正态检验[6，7]发展了Rao关于PQ×PQ方差-协方差矩阵的可分性的分数检验，该检验不要求样本量大于PQ。相反，样本方差-协方差矩阵可能是奇异的，在检验可分性之前要评估的多变量正态分布的经典检验不能在这种情况下使用[8]。。问题4：(4.1)协区域化线性模型下的结构决定系数和(4.2)空间数据多变量分析中的双曲线轴的显著性检验(4.1)如何用参数检验来评估区域化RDA中结构R2的重要性？这个问题结合了[9]和[10]所解决问题的“成分”。(4.2)关于空间数据的多尺度多变量分析(例如，RDA)中双曲线轴的重要性的同样问题[11]。。问题5：双变量谱分析中一致性的新检验两个时间序列之间的平方一致性的F检验[例如，第12章。4]由二元有限傅立叶变换和导出统计量的渐近性质证明了这一结论。能否找到一种基于Dutilleul的二元多频周期图的替代相干谱估计器[13]，并以这种方式开发出一种有效的新检验？参考文献：[1]Dutilleul等人。2015年。JGR-SolidEarth 120：8494。[2]贝金。2019年。首页--期刊主要分类--期刊细介绍--期刊题录与期刊详细文摘内容[3]Bégin et al.SPL158：108660。[4]Bergeron.2020年。首页--期刊主要分类--期刊细介绍--期刊题录与期刊详细文摘内容[5]杜蒂勒尔。2011年。书，剑桥大学出版社。[6]Filipiak等人。2016年。JMVA 150：105。[7]2017年。BiomJ 59：192。[8]Dutilleul.2018年。电线-ComputStat 10：4.[9]Dutilleul-Pelletier.2011年。马吉奥西43：819。[10]杜蒂勒尔-佩尔蒂埃。2017年。斯帕特斯塔特19：21。[11]Pelletier等人。2009年。EEST 16：467。[12]Shumway-Stoffer.2017年。电子书，春天自然。[13]杜蒂勒尔。2001年。顺应-30：1063。

My proposal is related to "Spatio-Temporal Heterogeneity Analysis", i.e., a major axis of my research work in Statistics (Methods and Applications) in the last 25+ years. In the next five years, I plan to work on five problems of statistical inference with spatial and temporal data (frequentist approach), including multidimensional statistics when the framework is multivariate and only spatial or temporal. Applications are environmental (plant-soil, seismological [1]). . Problem 1: A test of comparison of M-vine decompositions for multiple time series Following [2-4], and an ongoing M.Sc. thesis project on a different problem, a likelihood-based test comparing two M-vine decompositions for the same multiple time series will be developed. . Problem 2: A new testing procedure for auto- and cross-correlation analyses with point patterns Heterogeneity of the mean is a concern to users of Ripley's K and L functions for correlation analysis with point patterns, because of the assumed constancy of the first-order intensity function in the expression of statistics [5, Chap. 5]. . Problem 3: A new test of multivariate normality [6, 7] developed Rao's score test for separability of a pq x pq variance-covariance matrix, which does not require a sample size greater than pq. In counterpart, the sample variance-covariance matrix may be singular and the classical tests for multivariate normality, to be assessed prior to testing separability, cannot be used in that case [8]. . Problem 4: Tests of significance for (4.1) structural coefficients of determination under the linear model of coregionalization and (4.2) biplot axes in multivariate analyses of spatial data (4.1) How to assess the significance of structural R2s in a regionalized RDA with a parametric test? This problem combines the `ingredients' of the problems solved by [9] and [10]. (4.2) Same question for the significance of biplot axes in multi-scale multivariate analyses (e.g., RDA) of spatial data [11]. . Problem 5: A new test for coherency in bivariate spectral analysis The F-test for squared coherence between two time series [e.g., 12, Chap. 4] is justified by asymptotic properties of the bivariate finite Fourier transform and derived statistics. Could one find an alternative coherency spectrum estimator based on Dutilleul's bivariate multi-frequential periodogram [13], and this way develop a valid new test? References: [1] Dutilleul et al. 2015. JGR-SolidEarth 120:8494. [2] Bégin. 2019. MScthesis, Déptmathématiques, USherbrooke. [3] Bégin et al. SPL 158:108660. [4] Bergeron. 2020. MScthesis, Déptmathématiques, USherbrooke. [5] Dutilleul. 2011. Book, CambridgeUPress. [6] Filipiak et al. 2016. JMVA 150:105. [7] 2017. BiomJ 59:192. [8] Dutilleul. 2018. WIREs-ComputStat 10:4. [9] Dutilleul-Pelletier. 2011. MatGeosci 43:819. [10] Dutilleul-Pelletier. 2017. SpatStat 19:21. [11] Pelletier et al. 2009. EEST 16:467. [12] Shumway-Stoffer. 2017. eBook, SpringerNature. [13] Dutilleul. 2001. CiS-TaM 30:1063.