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

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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.