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 Myllymäki 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.
A method of estimation of intrinsic volume densities for stationary random closed sets in Rd based on estimating volumes of tiny collars has been introduced in T. Mrkvička and J. Rataj, On estimation of intrinsic volume densities of stationary random closed sets, Stoch. Proc. Appl. 118 (2008), 2, 213-231. In this note, a stronger asymptotic consistency is proved in dimension 2. The implementation of the method is discussed in detail. An important step is the determination of dilation radii in the discrete approximation, which differs from the standard techniques used for measuring parallel sets in image analysis. A method of reducing the bias is proposed and tested on simulated data.
A new method of testing the random closed set model hypothesis (for example: the Boolean model hypothesis) for a stationary random closed set Ξ⊆\rd with values in the extended convex ring is introduced. The method is based on the summary statistics - normalized intrinsic volumes densities of the \ep-parallel sets to Ξ. The estimated summary statistics are compared with theirs envelopes produced from simulations of the model given by the tested hypothesis. The p-level of the test is then computed via approximation of the summary statistics by multinormal distribution which mean and the correlation matrix is computed via given simulations. A new estimator of the intrinsic volumes densities from \cite{MR06} is used, which is especially suitable for estimation of the intrinsic volumes densities of \ep-parallel sets. The power of this test is estimated for planar Boolean model hypothesis and two different alternatives and the resulted powers are compared to the powers of known Boolean model tests. The method is applied on the real data set of a heather incidence.
This study investigated the post-spawning dispersal of seven species occurring in a tributary of the Římov Reservoir during
the years 2000-2004. Fish were captured during spawning migration to the tributary, marked and released. The subsequent distribution
of marked fish was followed in the reservoir and tributary during three successive periods 1) early summer, 2) late summer and 3)
the next spawning season. Species were divided into two groups – obligatory tributary spawners (white bream
Blicca bjoerkna
, chub
Squalius cephalus
, bleak
Alburnus alburnus
and asp
Aspius aspius
) that did so predominantly in the tributary of the reservoir and
generalists (bream
Abramis brama
, perch
Perca fluviatilis
and roach
Rutilus rutilus
) that usually spawned in the tributary as well as at
different sites within the reservoir main body. We hypothesized that obligatory tributary spawners would distribute across the reservoir
after spawning according to their species-specific preferences for certain feeding grounds. We expected a relatively low or erratic post-
spawning dispersal for spawning generalists. The results of the study revealed that the post-spawning dispersal of obligatory tributary
spawners is consistent with our hypothesis and they most likely dispersed according to their feeding ground requirements. The post-
spawning dispersal of generalists revealed that the assumed low dispersal was relevant for bream and perch while erratic dispersal was
observed in roach.
We discuss the prediction of a spatial variable of a multivariate mark composed of both dependent and explanatory variables. The marks are location-dependent and they are attached to a point process. We assume that the marks are assigned independently, conditionally on an unknown underlying parametric field. We compare (i) the classical non-parametric Nadaraya-Watson kernel estimator based on the dependent variable (ii) estimators obtained under an assumption of local parametric model where explanatory variables of the local model are estimated through kernel estimation and (iii) a kernel estimator of the result of the parametric model, supposed here to be a Uniformly Minimum Variance Unbiased Estimator derived under the local parametric model when complete and sufficient statistics are available. The comparison is done asymptotically and by simulations in special cases. The procedure for better estimator selection is then illustrated on a real-life data set.