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

我的提案与“时空异质性分析”有关，这是我过去25多年来在统计学（方法和应用）研究工作的一个主要轴。在接下来的五年里，我计划研究时空数据统计推断的五个问题（频率主义方法），包括多维统计，当框架是多元的，只有空间或时间。应用于环境（植物-土壤，地震bbb）。问题1:M-vine分解对多个时间序列的比较检验以下[2-4]，以及一个正在进行的关于不同问题的硕士论文项目，将开发一个基于似然的检验，比较相同多个时间序列的两个M-vine分解。均值的异质性是Ripley的K和L函数与点模式相关分析的用户关心的问题，因为在统计量的表达中假设一阶强度函数是恒定的[5，第5章]。问题3：一种新的多元正态性检验[6,7]发展了pq x pq方差-协方差矩阵可分性的Rao分数检验，该检验不需要大于pq的样本量。相反，样本方差-协方差矩阵可能是奇异的，而在检验可分性之前进行评估的多元正态性的经典检验不能在这种情况下使用[b]。问题4:（4.1）共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R2s的显著性？这个问题结合了[9]和[10]解决的问题的“成分”。（4.2）对于空间数据bbb的多尺度多变量分析（如RDA），双标图轴的意义也是同样的问题。用二元有限傅里叶变换的渐近性质和导出的统计量证明了两个时间序列之间的平方相干性的f检验[例如，12，第4章]是正确的。是否可以找到一种基于Dutilleul的二元多频周期图[13]的替代相干谱估计器，并以这种方式开发一个有效的新测试？参考文献：[1]Dutilleul et al. 2015。JGR-SolidEarth 120:8494。[2]。2019. ms . thesis, dsampath - sammatiques, usher - brooke。[3] bassimin等。SPL 158:108660。2020年，Bergeron。ms . thesis, dsampath - sammatiques, usher - brooke。Dutilleul, 2011。书,CambridgeUPress。[10] Filipiak et al. 2016。JMVA 150:105。2017年[7]。BiomJ 59:192。Dutilleul, 2018。WIREs-ComputStat 10:4。[9] Dutilleul-Pelletier。2011. MatGeosci 43:819。[10] Dutilleul-Pelletier。2017. SpatStat十九21。[b] Pelletier等。2009。e 16:467。[12] Shumway-Stoffer。2017. 电子书,SpringerNature。[qh] [qh] 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.