arXiv subjects

Tomas Mrkvicka

Publications and source records attributed to Tomas Mrkvicka.

5 recordsLinked to original sources

arXiv2020

New methods for multiple testing in permutation inference for the general linear model

Permutation methods are commonly used to test significance of regressors of interest in general linear models (GLMs) for functional (image) data sets, in particular for neuroimaging applications as they rely on mild assumptions. Permutation inference for GLMs typically consists of three parts: choosing a relevant test statistic, computing pointwise permutation tests and applying a multiple testing correction. We propose new multiple testing methods as an alternative to the commonly used maximum value of test statistics across the image. The new methods improve power and robustness against inhomogeneity of the test statistic across its domain. The methods rely on sorting the permuted functional test statistics based on pointwise rank measures; still they can be implemented even for large brain data. The performance of the methods is demonstrated through a designed simulation experiment, and an example of brain imaging data. We developed the R package GET which can be used for computation of the proposed procedures.

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arXiv2020

A one-way ANOVA test for functional data with graphical interpretation

A new functional ANOVA test, with a graphical interpretation of the result, is presented. The test is an extension of the global envelope test introduced by Myllymaki et al. (2017, Global envelope tests for spatial processes, J. R. Statist. Soc. B 79, 381--404, doi: 10.1111/rssb.12172). The graphical interpretation is realized by a global envelope which is drawn jointly for all samples of functions. If a mean function computed from the empirical data is out of the given envelope, the null hypothesis is rejected with the predetermined significance level $α$. The advantages of the proposed one-way functional ANOVA are that it identifies the domains of the functions which are responsible for the potential rejection. We introduce two versions of this test: the first gives a graphical interpretation of the test results in the original space of the functions and the second immediately offers a post-hoc test by identifying the significant pair-wise differences between groups. The proposed tests rely on discretization of the functions, therefore the tests are also applicable in the multidimensional ANOVA problem. In the empirical part of the article, we demonstrate the use of the method by analyzing fiscal decentralization in European countries. The aim of the empirical analysis is to capture differences between the levels of government expenditure decentralization ratio among different groups of European countries. The idea behind, based on the existing literature, is straightforward: countries with a longer European integration history are supposed to decentralize more of their government expenditure. We use the government expenditure centralization ratios of 29 European Union and EFTA countries in period from 1995 to 2016 sorted into three groups according to the presumed level of European economic and political integration.

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arXiv2020

Functional Outlier Detection and Taxonomy by Sequential Transformations

Functional data analysis can be seriously impaired by abnormal observations, which can be classified as either magnitude or shape outliers based on their way of deviating from the bulk of data. Identifying magnitude outliers is relatively easy, while detecting shape outliers is much more challenging. We propose turning the shape outliers into magnitude outliers through data transformation and detecting them using the functional boxplot. Besides easing the detection procedure, applying several transformations sequentially provides a reasonable taxonomy for the flagged outliers. A joint functional ranking, which consists of several transformations, is also defined here. Simulation studies are carried out to evaluate the performance of the proposed method using different functional depth notions. Interesting results are obtained in several practical applications.

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arXiv2019

Revisiting the random shift approach for testing in spatial statistics

We consider the problem of non-parametric testing of independence of two components of a stationary bivariate spatial process. In particular, we revisit the random shift approach that has become a standard method for testing the independent superposition hypothesis in spatial statistics, and it is widely used in a plethora of practical applications. However, this method has a problem of liberality caused by breaking the marginal spatial correlation structure due to the toroidal correction. This indeed causes that the assumption of exchangability, which is essential for the Monte Carlo test to be exact, is not fulfilled. We present a number of permutation strategies and show that the random shift with the variance correction brings a suitable improvement compared to the torus correction in the random field case. It reduces the liberality and achieves the largest power from all investigated variants. To obtain the variance for the variance correction method, several approaches were studied. The best results were achieved, for the sample covariance as the test statistics, with the correction factor $1/n$. This corresponds to the asymptotic order of the variance of the test statistics. In the point process case, the problem of deviations from exchangeability is far more complex and we propose an alternative strategy based on the mean cross nearest-neighbor distance and torus correction. It reduces the liberality but achieves slightly lower power than the usual cross $K$-function. Therefore we recommend it, when the point patterns are clustered, where the cross $K$-function achieves liberality.

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arXiv2019

A nonparametric graphical tests of significance in functional GLM

A new nonparametric graphical test of significance of a covariate in functional GLM is proposed. Our approach is especially interesting due to its functional graphical interpretation of the results. As such it is able to find not only if the factor of interest is significant but also which functional domain is responsible for the potential rejection. In the case of functional multi-way main effect ANOVA or functional main effect ANCOVA models it is able to find which groups differ (and where they differ), in the case of functional factorial ANOVA or functional factorial ANCOVA models it is able to find which combination of levels (which interactions) differ (and where they differ). The described tests are extensions of global envelope tests in the GLM models. It applies Freedman-Lane algorithm for the permutation of functions and as such it approximately achieve the desired significance level.

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