Mirosław Krzyśko

Canonical correlation analysis for functional data

Summary Classical canonical correlation analysis seeks the associations between two data sets, i.e. it searches for linear combinations of the original variables having maximal correlation. Our task is to maximize this correlation, and is equivalent to solving a generalized eigenvalue problem. The maximal correlation coefficient (being a solution of this problem) is the first canonical correlation coefficient. In this paper we propose a new method of constructing canonical correlations and canonical variables for a pair of stochastic processes represented by a finite number of orthonormal basis functions.

A comparison of some tests for determining the number of nonzero canonical correlations

When considering the relationships between two sets of variates, the number of nonzero population canonical correlations may be called the dimensionality. In the literature, several tests for dimensionality in the canonical correlation analysis are known. A comparison of seven sequential test procedures is presented, using results from some simulation study. The tests are compared with regard to the relative frequencies of underestimation, correct estimation, and overestimation of the true dimensionality. Some conclusions from the simulation results are drawn.

Functional Discriminant Coordinates

In this article we propose a new method of construction of discriminant coordinates and their kernel variant based on the regularization (ridge regression). Moreover, we compare the case of discriminant coordinates, functional discriminant coordinates and the kernel version of functional discriminant coordinates on 20 data sets from a wide variety of application domains using values of the criterion of goodness and statistical tests. Our experiments show that the kernel variant of discriminant coordinates provides significantly more accurate results on the examined data sets.

A Closed Testing Procedure for Canonical Correlations

ABSTRACT The problem of deciding on dimensionality in the canonical correlation model is considered. It is related to the investigation of significant empirical departures from the overall hypothesis on the nullity of the population canonical correlations. A closed testing procedure for a sequence of relevant dimensionality hypotheses is proposed, following an earlier suggestion made for testing dimensionality in a MANOVA model. Unlike the classical procedures based on asymptotic distributions, the proposed method ensures that the Type I familywise error rate does not exceed the nominal α-level. In the derivation of the procedure, the problem is considered in the framework of a multivariate linear regression model, which allows the hypotheses on canonical correlations to be treated as linear hypotheses under such a model. Two examples are given to illustrate application of the proposed procedure and to compare it with some other methods.

An extension of Bar-Hen's preliminary test procedure

An asymptotic test procedure, proposed by Bar-Hen (J. Multivariate Anal. 57 (1996) 266), for deciding if a given set of data represents a new population or one of k a priori known populations, is extended to the case when the new population is described by more than one parameter.

Nonlinear Principal Component Analysis for Geographically Weighted Temporal‑spatial Data

Scholkopf, Smola and Muller (1998) have proposed a nonlinear principal component analysis (NPCA) for fixed vector data. In this paper, we propose an extension of the aforementioned analysis to temporal ‑ spatial data and weighted temporal ‑ spatial data. To illustrate the proposed theory, data describing the condition of state of higher education in 16 Polish voivodships in the years 2002–2016 are used.

Correlation Analysis for Multivariate Functional Data

The relationship between two sets of real variables defined for the same individuals can be evaluated by few different correlation coefficients. For the functional data we have only one important tool: the canonical correlations. It is not immediately straightforward to extend other similar measures to the context of functional data analysis. In this work we show how to use commonly known measures of correlation for two sets of variables: \(\mathop{\mathrm{rV}}\nolimits\) coefficient and distance correlation coefficient for multivariate functional case. Finally, these three different coefficients are compared and their use is demonstrated on two real examples.

Multivariate Functional Regression Analysis with Application to Classification Problems

Multivariate functional data analysis is an effective approach to dealing with multivariate and complex data. These data are treated as realizations of multivariate random processes; the objects are represented by functions. In this paper we discuss different types of regression model: linear and logistic. Various methods of representing functional data are also examined. The approaches discussed are illustrated with an application to two real data sets.

The mineralization effect of wheat straw on soil properties described by MFPC analysis and other methods

Abstract Recycling of crop residues is essential to sustain soil fertility and crop production. Despite the positive effect of straw incorporation, the slow decomposition of that organic substance is a serious issue. The aim of the study was to assess the influence of winter wheat straws with different degrees of stem solidness on the rate of decomposition and soil properties. An incubation experiment lasting 425 days was carried out in controlled conditions. To perform analyses, soil samples were collected after 7, 14, 21, 28, 35, 49, 63, 77, 91, 119, 147, 175, 203, 231, 259, 313, 341, 369, 397 and 425 days of incubation. The addition of two types of winter wheat straw with different degree of stem solidness into the sandy soil differentiated the experimental treatments. The results demonstrate that straw mineralization was a relatively slow process and did not depend on the degree of filling of the stem by pith. Multivariate functional principal component analysis (MFPC) gave proof of significant variation between the control soil and the soil incubated with the straws. The first functional principal component describes 48.53% and the second 18.55%, of the variability of soil properties. Organic carbon, mineral nitrogen and sum of bases impact on the first functional principal component, whereas, magnesium, sum of bases and total nitrogen impact on the second functional principal component.

Kernel functional canonical correlation analysis

Canonical correlation methods for data representing functions or curves have received much attention in recent years. Such data, known in the literature as functional data (Ramsay and Silverman, 2005), has been the subject of much recent research interest. Examples of functional data can be found in several application domains, such as medicine, economics, meteorology and many others. Unfortunately, the multivariate data canonical correlation methods cannot be used directly for functional data, because of the problem of dimensionality and difficulty in taking into account the correlation and order of functional data. The problem of constructing canonical correlations and canonical variables for functional data was addressed by Leurgans et al. (1993), and further developments were made by Ramsay and Silverman (2005). In this paper we propose a new method of constructing canonical correlations and canonical variables for functional data.