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Waldemar Wołyński
Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Poland

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Cluster Analysis
functional data analysis
Discriminant analysis

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Research article
Published: 29 July 2021 in International Journal of Food Science
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The study tested how the cooking process can change the dimensions of rice grains. The impact of set times of cooking or steaming process on the characteristics such as length, width, and height of two varieties of rice, namely, long-grain white and parboiled, was investigated. The measurements of the dimension characteristics obtained at different times of the cooking process were converted to functional data. Different methods of multivariate functional data analysis, namely, functional multivariate analysis of variance, functional discriminant coordinates, and cluster analysis, were applied to discover the differences between the two varieties and the two heat treatment methods.

ACS Style

Mirosław Krzyśko; Waldemar Wołyński; Marek Domin; Zofia Hanusz; Leszek Rydzak; Łukasz Smaga; Andrzej Wojtyła. Functional Analysis of the Differences in the Dimensions of Two Types of Boiled and Steamed Rice Grains. International Journal of Food Science 2021, 2021, 1 -12.

AMA Style

Mirosław Krzyśko, Waldemar Wołyński, Marek Domin, Zofia Hanusz, Leszek Rydzak, Łukasz Smaga, Andrzej Wojtyła. Functional Analysis of the Differences in the Dimensions of Two Types of Boiled and Steamed Rice Grains. International Journal of Food Science. 2021; 2021 ():1-12.

Chicago/Turabian Style

Mirosław Krzyśko; Waldemar Wołyński; Marek Domin; Zofia Hanusz; Leszek Rydzak; Łukasz Smaga; Andrzej Wojtyła. 2021. "Functional Analysis of the Differences in the Dimensions of Two Types of Boiled and Steamed Rice Grains." International Journal of Food Science 2021, no. : 1-12.

Journal article
Published: 27 January 2021 in International Journal of Environmental Research and Public Health
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The aim of this study was to investigate if the provinces of Poland are homogeneous in terms of the observed spatio-temporal data characterizing the health situation of their inhabitants. The health situation is understood as a set of selected factors influencing inhabitants’ health and the healthcare system in their area of residence. So far, studies concerning the health situation of selected territorial units have been based on data relating to a specific year rather than longer periods. The task of assessing province homogeneity was carried out in two stages. In stage one, the original spatio-temporal data space (space of multivariate time series) was transformed into a functional discriminant coordinates space. The resulting functional discriminant coordinates are synthetic measures of the health situation of inhabitants of particular provinces. These measures contain complete information regarding 8 diagnostic variables examined over a period of 6 years. In the second stage, the Ward method, commonly used in cluster analysis, was applied in order to identify groups of homogeneous provinces in the space of functional discriminant coordinates. Sixteen provinces were divided into four clusters. The homogeneity of the clusters was confirmed by the multivariate functional coefficient of variation.

ACS Style

Mirosław Krzyśko; Waldemar Wołyńki; Marcin Szymkowiak; Andrzej Wojtyła. A Spatio-Temporal Analysis of the Health Situation in Poland Based on Functional Discriminant Coordinates. International Journal of Environmental Research and Public Health 2021, 18, 1109 .

AMA Style

Mirosław Krzyśko, Waldemar Wołyńki, Marcin Szymkowiak, Andrzej Wojtyła. A Spatio-Temporal Analysis of the Health Situation in Poland Based on Functional Discriminant Coordinates. International Journal of Environmental Research and Public Health. 2021; 18 (3):1109.

Chicago/Turabian Style

Mirosław Krzyśko; Waldemar Wołyńki; Marcin Szymkowiak; Andrzej Wojtyła. 2021. "A Spatio-Temporal Analysis of the Health Situation in Poland Based on Functional Discriminant Coordinates." International Journal of Environmental Research and Public Health 18, no. 3: 1109.

Journal article
Published: 25 September 2020 in International Journal of Environmental Research and Public Health
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The aim of this study was to investigate if the macroregions of Poland are homogeneous in terms of the observed spatio-temporal data characterizing their sustainable development. So far, works related to the sustainable development of selected territorial units have been based on data relating to a specific year rather than many years. The solution to the problem of macroregion homogeneity goes through two stages. In step one, the original spatio-temporal data space (matrix space) was transformed into a kernel discriminant coordinates space. The obtained kernel discriminant coordinates function as synthetic measures of the level of sustainable development of Polish macroregions. These measures contain complete information on the values of 27 diagnostic features examined over 15 years. In the second step, cluster analysis was used in order to identify groups of homogeneous macroregions in the space of kernel discriminant coordinates. The agglomeration method and the Ward method were chosen as commonly used methods. By means of both methods, three super macroregions composed of homogeneous macroregions were identified. Within the kernel discriminant coordinates, the differentiating power of a selected set of 27 features characterizing the sustainable development of macroregions was also assessed. To this end, five different and most commonly used methods of discriminant analysis were used to test the correctness of the classification. Depending on the method, the classification errors amounted to zero or were close to zero, which proves a well-chosen set of diagnostic features. Although the data relate only to a specific country (Poland), the presented statistical methodology is universal and can be applied to any territorial unit and spatial-temporal dynamic data.

ACS Style

Mirosław Krzyśko; Waldemar Wołyński; Waldemar Ratajczak; Anna Kierczyńska; Beata Wenerska. Sustainable Development of Polish Macroregions—Study by Means of the Kernel Discriminant Coordinates Method. International Journal of Environmental Research and Public Health 2020, 17, 7021 .

AMA Style

Mirosław Krzyśko, Waldemar Wołyński, Waldemar Ratajczak, Anna Kierczyńska, Beata Wenerska. Sustainable Development of Polish Macroregions—Study by Means of the Kernel Discriminant Coordinates Method. International Journal of Environmental Research and Public Health. 2020; 17 (19):7021.

Chicago/Turabian Style

Mirosław Krzyśko; Waldemar Wołyński; Waldemar Ratajczak; Anna Kierczyńska; Beata Wenerska. 2020. "Sustainable Development of Polish Macroregions—Study by Means of the Kernel Discriminant Coordinates Method." International Journal of Environmental Research and Public Health 17, no. 19: 7021.

Journal article
Published: 01 June 2020 in Biometrical Letters
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Summary There is a growing need to analyze data sets characterized by several sets of variables observed on the same set of individuals. Such complex data structures are known as multiblock (or multiple-set) data sets. Multi-block data sets are encountered in diverse fields including bioinformatics, chemometrics, food analysis, etc. Generalized Canonical Correlation Analysis (GCCA) is a very powerful method to study this kind of relationships between blocks. It can also be viewed as a method for the integration of information from K > 2 distinct sources (Takane and Oshima-Takane 2002). In this paper, GCCA is considered in the context of multivariate functional data. Such data are treated as realizations of multivariate random processes. GCCA is a technique that allows the joint analysis of several sets of data through dimensionality reduction. The central problem of GCCA is to construct a series of components aiming to maximize the association among the multiple variable sets. This method will be presented for multivariate functional data. Finally, a practical example will be discussed.

ACS Style

Tomasz Górecki; Mirosław Krzyśko; Waldemar Wołyński. Generalized canonical correlation analysis for functional data. Biometrical Letters 2020, 57, 1 -12.

AMA Style

Tomasz Górecki, Mirosław Krzyśko, Waldemar Wołyński. Generalized canonical correlation analysis for functional data. Biometrical Letters. 2020; 57 (1):1-12.

Chicago/Turabian Style

Tomasz Górecki; Mirosław Krzyśko; Waldemar Wołyński. 2020. "Generalized canonical correlation analysis for functional data." Biometrical Letters 57, no. 1: 1-12.

Article
Published: 10 November 2018 in Artificial Intelligence Review
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In the case of vector data, Gretton et al. (Algorithmic learning theory. Springer, Berlin, pp 63–77, 2005) defined Hilbert–Schmidt independence criterion, and next Cortes et al. (J Mach Learn Res 13:795–828, 2012) introduced concept of the centered kernel target alignment (KTA). In this paper we generalize these measures of dependence to the case of multivariate functional data. In addition, based on these measures between two kernel matrices (we use the Gaussian kernel), we constructed independence test and nonlinear canonical variables for multivariate functional data. We show that it is enough to work only on the coefficients of a series expansion of the underlying processes. In order to provide a comprehensive comparison, we conducted a set of experiments, testing effectiveness on two real examples and artificial data. Our experiments show that using functional variants of the proposed measures, we obtain much better results in recognizing nonlinear dependence.

ACS Style

Tomasz Górecki; Mirosław Krzyśko; Waldemar Wołyński. Independence test and canonical correlation analysis based on the alignment between kernel matrices for multivariate functional data. Artificial Intelligence Review 2018, 53, 475 -499.

AMA Style

Tomasz Górecki, Mirosław Krzyśko, Waldemar Wołyński. Independence test and canonical correlation analysis based on the alignment between kernel matrices for multivariate functional data. Artificial Intelligence Review. 2018; 53 (1):475-499.

Chicago/Turabian Style

Tomasz Górecki; Mirosław Krzyśko; Waldemar Wołyński. 2018. "Independence test and canonical correlation analysis based on the alignment between kernel matrices for multivariate functional data." Artificial Intelligence Review 53, no. 1: 475-499.

Journal article
Published: 20 September 2018 in Acta Universitatis Lodziensis. Folia Oeconomica
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Schölkopf, Smola i Müller (1998) zaproponowali analizę nieliniowych składowych głównych (NPCA) dla ustalonych danych wektorowych. Niniejszy artykuł zawiera rozszerzenie tej metody na dane czasowo‑przestrzenne oraz czasowo‑przestrzenne geograficznie ważone. Każdy obiekt jest scharakteryzowany za pomocą macierzy Xi, rozmiaru T × p, zawierającej wartości p cech zaobserwowanych w T momentach czasowych, i = 1, …, n. Macierze te są przekształcane nieliniowo do przestrzeni Hilberta i budowana jest scentrowana macierz jądrowa. Ostatecznie macierz ta jest podstawą konstrukcji nieliniowych składowych głównych. W przypadku danych geograficznie ważonych macierz Xizostaje zastąpiona macierzą wiXi, gdzie wijest dodatnią wagą geograficzną związaną z i‑tym miejscem obserwacji, i = 1, …, n. Teoria zilustrowana jest przykładem dotyczącym stanu szkolnictwa wyższego w 16 polskich województwach, notowanego w latach 2002–2016.

ACS Style

Mirosław Krzyśko; Wojciech Łukaszonek; Waldemar Ratajczak; Waldemar Wołyński. Analiza nieliniowych składowych głównych dla danych czasowo‑przestrzennych geograficznie ważonych. Acta Universitatis Lodziensis. Folia Oeconomica 2018, 4, 169 -181.

AMA Style

Mirosław Krzyśko, Wojciech Łukaszonek, Waldemar Ratajczak, Waldemar Wołyński. Analiza nieliniowych składowych głównych dla danych czasowo‑przestrzennych geograficznie ważonych. Acta Universitatis Lodziensis. Folia Oeconomica. 2018; 4 (337):169-181.

Chicago/Turabian Style

Mirosław Krzyśko; Wojciech Łukaszonek; Waldemar Ratajczak; Waldemar Wołyński. 2018. "Analiza nieliniowych składowych głównych dla danych czasowo‑przestrzennych geograficznie ważonych." Acta Universitatis Lodziensis. Folia Oeconomica 4, no. 337: 169-181.

Journal article
Published: 27 May 2018 in Statistics in Transition New Series
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ACS Style

Mirosław Krzyśko; Wojciech Łukaszonek; Waldemar Wołyński. CANONICAL CORRELATION ANALYSIS IN THE CASE OF MULTIVARIATE REPEATED MEASURES DATA. Statistics in Transition New Series 2018, 19, 75 -85.

AMA Style

Mirosław Krzyśko, Wojciech Łukaszonek, Waldemar Wołyński. CANONICAL CORRELATION ANALYSIS IN THE CASE OF MULTIVARIATE REPEATED MEASURES DATA. Statistics in Transition New Series. 2018; 19 (1):75-85.

Chicago/Turabian Style

Mirosław Krzyśko; Wojciech Łukaszonek; Waldemar Wołyński. 2018. "CANONICAL CORRELATION ANALYSIS IN THE CASE OF MULTIVARIATE REPEATED MEASURES DATA." Statistics in Transition New Series 19, no. 1: 75-85.

Journal article
Published: 01 January 2018 in Prace Naukowe Uniwersytetu Ekonomicznego we Wrocławiu
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ACS Style

Mirosław Krzyśko; Waldemar Wołyński; Wojciech Łukaszonek; Waldemar Ratajczak. PRINCIPAL COMPONENT ANALYSIS FOR TEMPORAL-SPATIAL DATA. Prace Naukowe Uniwersytetu Ekonomicznego we Wrocławiu 2018, 115 -123.

AMA Style

Mirosław Krzyśko, Waldemar Wołyński, Wojciech Łukaszonek, Waldemar Ratajczak. PRINCIPAL COMPONENT ANALYSIS FOR TEMPORAL-SPATIAL DATA. Prace Naukowe Uniwersytetu Ekonomicznego we Wrocławiu. 2018; (507):115-123.

Chicago/Turabian Style

Mirosław Krzyśko; Waldemar Wołyński; Wojciech Łukaszonek; Waldemar Ratajczak. 2018. "PRINCIPAL COMPONENT ANALYSIS FOR TEMPORAL-SPATIAL DATA." Prace Naukowe Uniwersytetu Ekonomicznego we Wrocławiu , no. 507: 115-123.

Conference paper
Published: 05 July 2017 in Data Science
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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.

ACS Style

Tomasz Górecki; Mirosław Krzyśko; Waldemar Wołyński. Correlation Analysis for Multivariate Functional Data. Data Science 2017, 243 -258.

AMA Style

Tomasz Górecki, Mirosław Krzyśko, Waldemar Wołyński. Correlation Analysis for Multivariate Functional Data. Data Science. 2017; ():243-258.

Chicago/Turabian Style

Tomasz Górecki; Mirosław Krzyśko; Waldemar Wołyński. 2017. "Correlation Analysis for Multivariate Functional Data." Data Science , no. : 243-258.

Journal article
Published: 01 January 2017 in Applicationes Mathematicae
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ACS Style

Dominik Szynal; Waldemar Wołyński. On simulation powers of new normality tests. Applicationes Mathematicae 2017, 44, 57 -83.

AMA Style

Dominik Szynal, Waldemar Wołyński. On simulation powers of new normality tests. Applicationes Mathematicae. 2017; 44 (1):57-83.

Chicago/Turabian Style

Dominik Szynal; Waldemar Wołyński. 2017. "On simulation powers of new normality tests." Applicationes Mathematicae 44, no. 1: 57-83.

Journal article
Published: 01 December 2016 in Biometrical Letters
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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.

ACS Style

Monika Jakubus; Mirosław Krzyśko; Waldemar Wołyński; Małgorzata Graczyk. The mineralization effect of wheat straw on soil properties described by MFPC analysis and other methods. Biometrical Letters 2016, 53, 133 -147.

AMA Style

Monika Jakubus, Mirosław Krzyśko, Waldemar Wołyński, Małgorzata Graczyk. The mineralization effect of wheat straw on soil properties described by MFPC analysis and other methods. Biometrical Letters. 2016; 53 (2):133-147.

Chicago/Turabian Style

Monika Jakubus; Mirosław Krzyśko; Waldemar Wołyński; Małgorzata Graczyk. 2016. "The mineralization effect of wheat straw on soil properties described by MFPC analysis and other methods." Biometrical Letters 53, no. 2: 133-147.

Book chapter
Published: 04 August 2016 in Data Science
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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.

ACS Style

Tomasz Górecki; Mirosław Krzyśko; Waldemar Wołyński. Multivariate Functional Regression Analysis with Application to Classification Problems. Data Science 2016, 173 -183.

AMA Style

Tomasz Górecki, Mirosław Krzyśko, Waldemar Wołyński. Multivariate Functional Regression Analysis with Application to Classification Problems. Data Science. 2016; ():173-183.

Chicago/Turabian Style

Tomasz Górecki; Mirosław Krzyśko; Waldemar Wołyński. 2016. "Multivariate Functional Regression Analysis with Application to Classification Problems." Data Science , no. : 173-183.

Journal article
Published: 23 February 2016 in Statistical Papers
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Data in the form of a continuous vector function on a given interval are referred to as multivariate functional data. These data are treated as realizations of multivariate random processes. The paper is devoted to three statistical dimension reduction techniques for multivariate data. For the first one, principal components analysis, the authors present a review of a recent paper (Jacques and Preda in, Comput Stat Data Anal, 71:92–106, 2014). For two others one, canonical variables and discriminant coordinates, the authors extend existing works for univariate functional data to multivariate. These methods for multivariate functional data are presented, illustrated and discussed in the context of analyzing real data sets. Each of these techniques is applied on real data set.

ACS Style

Tomasz Górecki; Mirosław Krzyśko; Łukasz Waszak; Waldemar Wołyński. Selected statistical methods of data analysis for multivariate functional data. Statistical Papers 2016, 59, 153 -182.

AMA Style

Tomasz Górecki, Mirosław Krzyśko, Łukasz Waszak, Waldemar Wołyński. Selected statistical methods of data analysis for multivariate functional data. Statistical Papers. 2016; 59 (1):153-182.

Chicago/Turabian Style

Tomasz Górecki; Mirosław Krzyśko; Łukasz Waszak; Waldemar Wołyński. 2016. "Selected statistical methods of data analysis for multivariate functional data." Statistical Papers 59, no. 1: 153-182.

Journal article
Published: 01 January 2016 in Statistics in Transition New Series
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ACS Style

Tomasz Górecki; Mirosław Krzyśko; Waldemar Ratajczak; Waldemar Wołyński. AN EXTENSION OF THE CLASSICAL DISTANCE CORRELATION COEFFICIENT FOR MULTIVARIATE FUNCTIONAL DATA WITH APPLICATIONS. Statistics in Transition New Series 2016, 17, 449 -466.

AMA Style

Tomasz Górecki, Mirosław Krzyśko, Waldemar Ratajczak, Waldemar Wołyński. AN EXTENSION OF THE CLASSICAL DISTANCE CORRELATION COEFFICIENT FOR MULTIVARIATE FUNCTIONAL DATA WITH APPLICATIONS. Statistics in Transition New Series. 2016; 17 (3):449-466.

Chicago/Turabian Style

Tomasz Górecki; Mirosław Krzyśko; Waldemar Ratajczak; Waldemar Wołyński. 2016. "AN EXTENSION OF THE CLASSICAL DISTANCE CORRELATION COEFFICIENT FOR MULTIVARIATE FUNCTIONAL DATA WITH APPLICATIONS." Statistics in Transition New Series 17, no. 3: 449-466.

Journal article
Published: 01 January 2015 in Statistics in Transition New Series
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ACS Style

Tomasz Górecki; Mirosław Krzyśko; Waldemar Wołyński. CLASSIFICATION PROBLEMS BASED ON REGRESSION MODELS FOR MULTI-DIMENSIONAL FUNCTIONAL DATA. Statistics in Transition New Series 2015, 16, 97 -110.

AMA Style

Tomasz Górecki, Mirosław Krzyśko, Waldemar Wołyński. CLASSIFICATION PROBLEMS BASED ON REGRESSION MODELS FOR MULTI-DIMENSIONAL FUNCTIONAL DATA. Statistics in Transition New Series. 2015; 16 (1):97-110.

Chicago/Turabian Style

Tomasz Górecki; Mirosław Krzyśko; Waldemar Wołyński. 2015. "CLASSIFICATION PROBLEMS BASED ON REGRESSION MODELS FOR MULTI-DIMENSIONAL FUNCTIONAL DATA." Statistics in Transition New Series 16, no. 1: 97-110.

Journal article
Published: 01 December 2014 in Biometrical Letters
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In this paper we consider a set of T repeated measurements on p characteristics on each of n individuals. The n individuals themselves may be divided and randomly assigned to K groups. These data are analyzed using a mixed effect MANOVA model, assuming that the data on an individual have a covariance matrix which is a Kronecker product of two positive definite matrices. Results are illustrated on a data set obtained from experiments with varieties of winter rye.

ACS Style

Mirosław Krzysko; Tadeusz Smiałowski; Waldemar Wołynski. Analysis of multivariate repeated measures data using a MANOVA model and principal components. Biometrical Letters 2014, 51, 103 -114.

AMA Style

Mirosław Krzysko, Tadeusz Smiałowski, Waldemar Wołynski. Analysis of multivariate repeated measures data using a MANOVA model and principal components. Biometrical Letters. 2014; 51 (2):103-114.

Chicago/Turabian Style

Mirosław Krzysko; Tadeusz Smiałowski; Waldemar Wołynski. 2014. "Analysis of multivariate repeated measures data using a MANOVA model and principal components." Biometrical Letters 51, no. 2: 103-114.

Original articles
Published: 08 January 2014 in Communications in Statistics - Simulation and Computation
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The construction of kernel discriminant coordinates reduces to the solution of a generalized eigenvalue problem in which both matrices are nonnegative definite. Six different algorithms for solving that problem are described, and the performance of these algorithms is tested on 26 different datasets. The percentage of misclassifications using a linear discriminant function is noted, and the algorithms’ running times are ascertained. Classification is also performed in the space of classical discriminant coordinates.

ACS Style

Mirosław Krzyśko; Łukasz Waszak; Waldemar Wołyński. Comparison of Kernel Discriminant Coordinates Algorithms. Communications in Statistics - Simulation and Computation 2014, 43, 2138 -2148.

AMA Style

Mirosław Krzyśko, Łukasz Waszak, Waldemar Wołyński. Comparison of Kernel Discriminant Coordinates Algorithms. Communications in Statistics - Simulation and Computation. 2014; 43 (9):2138-2148.

Chicago/Turabian Style

Mirosław Krzyśko; Łukasz Waszak; Waldemar Wołyński. 2014. "Comparison of Kernel Discriminant Coordinates Algorithms." Communications in Statistics - Simulation and Computation 43, no. 9: 2138-2148.

Journal article
Published: 01 January 2014 in Discussiones Mathematicae Probability and Statistics
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ACS Style

Dominik Szynal; Waldemar Wołyński. On two families of tests for normality with empirical description of their performances. Discussiones Mathematicae Probability and Statistics 2014, 34, 169 .

AMA Style

Dominik Szynal, Waldemar Wołyński. On two families of tests for normality with empirical description of their performances. Discussiones Mathematicae Probability and Statistics. 2014; 34 (1):169.

Chicago/Turabian Style

Dominik Szynal; Waldemar Wołyński. 2014. "On two families of tests for normality with empirical description of their performances." Discussiones Mathematicae Probability and Statistics 34, no. 1: 169.

Journal article
Published: 04 October 2013 in International Journal of Pure and Apllied Mathematics
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ACS Style

D. Szynal; W. Wo{\l}Y\'{N}Ski. SIMULATION SUPPLEMENT TO GOODNESS-OF-FIT TESTS DERIVED FROM CHARACTERIZATIONS OF CONTINUOUS DISTRIBUTIONS VIA RECORD VALUES. International Journal of Pure and Apllied Mathematics 2013, 87, 1 .

AMA Style

D. Szynal, W. Wo{\l}Y\'{N}Ski. SIMULATION SUPPLEMENT TO GOODNESS-OF-FIT TESTS DERIVED FROM CHARACTERIZATIONS OF CONTINUOUS DISTRIBUTIONS VIA RECORD VALUES. International Journal of Pure and Apllied Mathematics. 2013; 87 (4):1.

Chicago/Turabian Style

D. Szynal; W. Wo{\l}Y\'{N}Ski. 2013. "SIMULATION SUPPLEMENT TO GOODNESS-OF-FIT TESTS DERIVED FROM CHARACTERIZATIONS OF CONTINUOUS DISTRIBUTIONS VIA RECORD VALUES." International Journal of Pure and Apllied Mathematics 87, no. 4: 1.

Journal article
Published: 01 January 2011 in Discussiones Mathematicae Probability and Statistics
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ACS Style

Mirosław Krzyśko; Michał Skorzybut; Waldemar Wołyński. Classifiers for doubly multivariate data. Discussiones Mathematicae Probability and Statistics 2011, 31, 5 .

AMA Style

Mirosław Krzyśko, Michał Skorzybut, Waldemar Wołyński. Classifiers for doubly multivariate data. Discussiones Mathematicae Probability and Statistics. 2011; 31 (1):5.

Chicago/Turabian Style

Mirosław Krzyśko; Michał Skorzybut; Waldemar Wołyński. 2011. "Classifiers for doubly multivariate data." Discussiones Mathematicae Probability and Statistics 31, no. 1: 5.