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，即，在过去的25年里，我在统计学（方法和应用）方面的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（频率论方法）的统计推断的五个问题，包括多维统计，当框架是多元的，只有空间或时间。应用环境（植物土壤，地震[1]）。.问题一：多个时间序列的M-vine分解的比较测试[2-4]之后，以及正在进行的关于不同问题的硕士论文项目，将开发一个基于可能性的测试，比较相同多个时间序列的两个M-vine分解。.问题二：点模式自相关和互相关分析的一种新的检验方法由于统计量表达式中一阶强度函数的假定恒定性，使用Ripley K和L函数进行点模式相关分析时，需要考虑均值的异方差。5]。.问题三：一种新的多元正态性检验[6，7]发展了Rao评分检验，用于pq x pq方差-协方差矩阵的可分性，该检验不需要样本量大于pq。相反，样本方差-协方差矩阵可能是奇异的，并且在检验可分性之前要评估的多元正态性的经典检验不能用于这种情况[8]。.问题四：（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标图轴的显著性检验（4.1）如何用参数检验评估区域化RDA中结构R2的显著性？这个问题结合了[9]和[10]所解决的问题的“成分”。(4.2)多尺度多变量分析中双标图轴的显著性也存在同样的问题（例如，[11]空间数据。.问题5：二元谱分析中相干性的新检验两个时间序列之间平方相干性的F检验[例如，12章4]的二元有限傅立叶变换和派生统计的渐近性质是合理的。能否找到一种基于Dutilleul的双变量多频率周期图[13]的替代相干谱估计器，并以这种方式开发一种有效的新测试？参考文献：[1] Dutilleul et al. 2015. JGR-SolidEarth 120：8494。[2]贝金2019. Usherbrooke大学数学系硕士论文。[3]Bégin等人SPL 158：108660。[4]贝杰隆2020. Usherbrooke大学数学系硕士论文。[5]Dutilleul。2011. Book，CambridgeUPress. [6]Filipiak et al. 2016. JMVA 150：105. [7]2017年。《生物医学杂志》59：192。[8]Dutilleul。2018. WIREs-ComputStat 10：4. [9]Dutilleul-Pelletier。2011.马太福音43：819. [10]Dutilleul-Pelletier。2017. SpatStat 19：21. [11]Pelletier等人，2009年。东16：467。[12]沙姆韦-斯托弗2017.电子书，SpringerNature. [13]Dutilleul。2001. CiS-TaM 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.