Modern Directional Statistics

Content: Introduction OverviewA brief introduction to directional statisticsA brief outline of the theoretical advances presented in this bookDirectional datasets PaleomagnetismPolitical sciencesText miningWildfire orientationLife sciences and bioinformaticsBasics and notations Plan of the bookAdvances in flexible parametric distribution theory IntroductionFlexible parametric modeling: an active research area on RPOrganization of the remainder of the chapterFlexible circular distributionsFour ways to construct circular densitiesThe classics: von Mises, cardioid and wrapped Cauchy distributionsBeyond the classics modern flexible circular modelingFlexible modeling of symmetric data: the Jones-Pewsey distributionSine-skewing: a simple tool to skew any symmetric distributionSkewness combined with unimodality: the scale-transforming approachA general device for building symmetric bipolar distributionsA brief description of three other flexible modelsFlexible spherical distributionsClassical spherical distributionsRotationally symmetric distributionsA general method to skew-rotationally symmetric distributionsFlexible toroidal and cylindrical distributionsSome history, motivations and goalsThe bivariate von Mises distribution and its variantsMardia-Sutton type cylindrical distributionsJohnson-Wehrly type cylindrical distributionsThe copula approachFurther readingAdvances in kernel density estimation on directional supports IntroductionKernel density estimation on the real lineOrganization of the remainder of the chapterDefinitions and main propertiesSpherical kernel density estimationCylindrical kernel density estimationA delicate yet crucial issue: bandwidth choiceSpherical AMISB and bandwidth selectionRule of thumb based on the FvML distributionA gain in generality: AMISE via mixtures of FvML densitiesThree further proposalsBandwidth selection in the cylindrical settingInferential proceduresNon-parametric goodness-of-fit test for directional dataNon-parametric independence test for cylindrical dataAn overview of non-parametric regressionFurther readingComputational and graphical methods Ordering data on the sphere: quantiles and depth functionsOrdering on R and RP, and organization of the remainder of the sectionClassical depth functions on the sphereProjected quantiles and related asymptotic resultsThe angular Mahalanobis depthStatistical procedures based on projected quantities and the angular Mahalanobis depthStatistical inference under order restrictions on the circle Isotomic regression estimation and organization of the remainder of the sectionOrder restrictions on the circleCircular isotonic regressionExploring data features with the CircSiZerThe SiZer, scale space theory and organization of the remainder of the sectionThe CircSiZerKernel choice based on causality: the special role of the wrapped normalComputationally fast estimation for high-dimensional FvML distributionsMaximum likelihood expressions for the parameters of FvML distributions and organization of the sectionApproximations for the concentration parameter from Mardia & Jupp (200) and their limitations in high dimensionsNew (high-dimensional) approximations for the concentration parameterFurther readingLocal asymptotic normality for directional data IntroductionThe LAN property on R and its deep impact on asymptotic statisticsOrganization of the remainder of the chapterLocal asymptotic normality and optimal testingContiguityLocal asymptotic normalityOptimal testing in LAN experimentsLAN, semiparametric efficiency and invarianceLAN for directional dataThe Le Cam methodology for curved experiments and associated efficient testsLAN property for rotationally symmetric distributionsApplication 1: Optimal inference based on signed-ranksApplication 2: ANOVA on spheresApplication 3: Asymptotic power of tests of concentration Further readingRecent results for tests of uniformity and symmetryIntroduction Organization of the remainder of the chapterRecent advances concerning the Rayleigh test of uniformity Sobolev tests of uniformity Uniformity tests based on random projections Testing for uniformity with noisy data Tests of reflective symmetry on the circle Tests of rotational symmetry on hyperspheres Testing for spherical location in the vicinity of the uniform distribution Further reading High-dimensional directional statistics IntroductionHigh-dimensional techniques in RP Organization of the remainder of the chapterDistributions on high-dimensional spheres Testing uniformity in the high-dimensional case Location tests in the high-dimensional case Concentration tests in the high-dimensional case Principal nested spheres Further reading