Estimation and testing for spatially indexed curves with application to ionospheric and magnetic field trends

ISRN Probability and Statistics

2012

Self Cite

This paper reviews recent research on dependent functional data. After providing an introduction to functional data analysis, we focus on two types of dependent functional data structures: time series of curves and spatially distributed curves. We review statistical models, inferential methodology, and possible extensions. The paper is intended to provide a concise introduction to the subject with plentiful references.

Section: Estimation Of the Mean Function And Of The Fpcsmentioning

confidence: 99%

“…Estimation of the v j in the spatial setting involves issues similar to those related in the estimation of the mean function μ, but the notation and approaches are more complex. They are discussed in Gromenko et al 97 and Hörmann and Kokoszka 98 .…”

Section: Estimation Of the Fpcsmentioning

confidence: 99%

Dependent Functional Data

ISRN Probability and Statistics

2012

Self Cite

“…In this section, we propose an extension of the methodology for the estimation of the mean function and the FPCs that are introduced by Gromenko et al. (2012).…”

Section: Estimation Of the Mean And Covariance Functionsmentioning

confidence: 99%

Testing the Equality of Mean Functions of Ionospheric Critical Frequency Curves

Gromenko

Journal of the Royal Statistical Society Series C: Applied Statistics

2012

Self Cite

Summary. The paper develops a significance test for evaluating the equality of the mean functions in two samples of spatially indexed functional data. The problem is motivated by an important question in space physics research which is related to the hypothesis of ionospheric global cooling (as opposed to the conjectured global warming of near surface atmosphere). The critical electron frequency of the ionosphere's F2 region, foF2, can be used to test conjectures about the trends in the ionosphere empirically. We apply the proposed test to the foF2‐records over eastern and western Europe to verify whether there is a conjectured difference between first‐order behaviour of these records over regions with different magnetic inclinations. It is found that the difference between the means is statistically significant for night‐time records. The implications of this result are discussed. Finite sample performance of the test proposed is validated via numerical simulations.

“…If the conclusion does not depend on d, we can be confident that it is correct. This approach has been used in applied research, see Gromenko et al (2012) for a recent analysis of this type. The second approach, would be to let d increase with the sample size N, and derive a test statistic based on the limit.…”

Section: Introductionmentioning

confidence: 99%

Functional data analysis with increasing number of projections

Fremdt

Horváth

Journal of Multivariate Analysis

et al. 2014

Self Cite

Abstract. Functional principal components (FPC's) provide the most important and most extensively used tool for dimension reduction and inference for functional data. The selection of the number, d, of the FPC's to be used in a specific procedure has attracted a fair amount of attention, and a number of reasonably effective approaches exist. Intuitively, they assume that the functional data can be sufficiently well approximated by a projection onto a finite-dimensional subspace, and the error resulting from such an approximation does not impact the conclusions. This has been shown to be a very effective approach, but it is desirable to understand the behavior of many inferential procedures by considering the projections on subspaces spanned by an increasing number of the FPC's. Such an approach reflects more fully the infinite-dimensional nature of functional data, and allows to derive procedures which are fairly insensitive to the selection of d. This is accomplished by considering limits as d → ∞ with the sample size.We propose a specific framework in which we let d → ∞ by deriving a normal approximation for the partial sum processwhere N is the sample size and ξ i,j is the score of the ith function with respect to the jth FPC. Our approximation can be used to derive statistics that use segments of observations and segments of the FPC's. We apply our general results to derive two inferential procedures for the mean function: a change-point test and a two-sample test. In addition to the asymptotic theory, the tests are assessed through a small simulation study and a data example.