Empiryczne testy istotności w taksonomii

• Autorzy: Andrzej Sokołowski

Warianty tytułu

Empirical Significance Tests in Taxonomy

• Języki publikacji: PL

Abstrakty

W niniejszej pracy zaproponowano pewne nowe taksonomiczne procedury statystyczne jak również podano propozycje uzupełnienia znanych metod o elementy weryfikacji hipotez statystycznych. (fragment tekstu)

EN

In the paper there have been proposed new taxonomic statistical procedures. The author has also suggested the expansion of some already popular methods in such a way as to incorporate the elements of verification of statistical hypotheses. The distributions of test statistics has been studied. First initial estimates of critical values have been generated and next, an adequate approximating analitical function has been searched for them. The analysis is designed around the three principal hypotheses that may be formulated while analyzing a set of operational taxonomic units. They are hypotheses of homogeneity, uniformity and unimodality. The hypotheses of homogeneity and unimodality are closely related. In order to verify the hypotheses of multidimensional uniformity five test statistics have been proposed. Their critical values have been determined and their ability to reject the false null hypothesis has been estimated. The author has put forward a method for mode estimation in the multi-dimensional distribution for any sample size and any (even multi-modal) distribution. The properties of the estimator have been analyzed, and its approximate mean square error has been determined. It is possible to demonstrate, while making some assumptions, that the distribution of the distance from the mode may be approximated by the chi-square distribution - which has led to the proposition of the test for unimodality. Taxonomic methods are mainly applied in the socio-economic sphere. In the final part of each chapter the author has presented examples of how the proposed tests may be applied for the solution of actual problems in the socio -economic field. (original abstract)

Słowa kluczowe

PL

Taksonomia; Taksonomia statystyczna

EN

Taxonomy; Statistical taxonomy

• Czasopismo: Zeszyty Naukowe / Akademia Ekonomiczna w Krakowie. Seria Specjalna, Monografie
• Rocznik: 1992
• Numer: nr 108
• Strony: 105

Twórcy

autor

Andrzej Sokołowski

Bibliografia

• Aho A.V., Hoperoft J.E., Ullman J.D., The Design and Analysis of Computer Algorithms, Adison Weseley, Reading, Massachussets 1974.
• Ajwazjan S.A., Bieiajewa Z.I., Starowierow O.W., Klassyfikacyja mnogomiernych nabludienij, Statistika, Moskwa. 1973.
• Anderberg M.R., Cluster Analysis for Application, Academic Press, New York 1973.
• Anderson T.W., The Integral of Symmetric Unimodal Function Over a Symmetric Convex Set and Some Probability Inequalities, "Proceedings of American Mathematical Society" 1955, nr 6.
• Aprauszewa N.N., Opriedielienije czsła klassow w zadaczach klassifikacii I, "Izwiestia AN SSSR, Tiechnika i Kibiernietika" 1981, nr 3.
• Aprauszewa N.N., Opriedielienije czsła klassow w zadaczach klassifikacii II, "Izwiestia AN SSSR, Tiechnika i Kibiernietika" 1981, nr 3.
• Arnold S.J., A Test for Ousters, "Journal of Marketing Research" 1979, nr 19.
• Baker F.B., Hubert L.J., Measuring the Power of Hierarchical Cluster Analysis, "Journal of the American Statistical Association". 1972, nr 70.
• Baker F.B., Stability of Two Hierarchical Grouping Techniques, Case I: Sensitivity to Data Errors, "Journal of the American Statistical Association" - 1974, t. 69.
• Ball G.H., Hall D.J., ISODATA, A Novel Method of Data Analysis and Pattern Classification, Menlo Park, California. Stanford Research Institute, April 1965.
• Bayne C.K., Beauchamp J.J., Begovich C.L., Kane V.E, Monte Carlo Comparison of Selected Clustering Procedures, "Pattern Recognition" 12,1980.
• Beale E.M.L., Cluster Analysis, Scientific Control Systems, London 1969.
• Binder D.A., Bayesian Ouster Analysis, "Biometrika" 1978, nr 65.
• Binder D.A., Comment on "Estimating Mixtures of Normal Distributions and Switching Regressions", "Journal of the American Statistical Association" 1978, t. 73.
• Blackwelder R.A., Taxonomy: A Text and Reference Book, Wiley, New York 1966.
• Blashfield R.K., Aldenderfer M.S., The Literature of Cluster Analysis, "Multivariate Behavioural Research" 1973, nr 13.
• Blashfield R.K., Mixture Model Tests of Cluster Analysis: Accuracy of Four Agglomerative Hierarchical Methods, "Psychological Bulletin" 1976, 83.
• Blashfield R.K., The Growth of Cluster Analysis: Tryon, Ward and Johnson, "Multivariate Behavioural Research" 1980, nr 15.
• Bock H.H., On Some Significance Teste in Cluster Analysis, "Journal of Classification" 1985, nr 2.
• Bock H.H., On Tests Concerning the Existence of a Classification, w: First International Symposium on Data Analysis and Informatics (Vol.2), IRIA, Rocquencourt 1977.
• Borell C., Convex Set Functions in d-space, "Period Mathematical in Hungary" 1975, nr 6.
• Borys T., Elementy teorii jakości, PWN, Warszawa 1980.
• Borys T., Kategoria jakości w statystycznej analizie porównawczej, "Prace Naukowe Akademii Ekonomicznej we Wrocławiu" 1984, nr 23.
• Borys T., Strahl D., Walesiak M., Wkład ośrodka wrocławskiego w rozwój teorii i zastosowań metod taksonomicznych, w: Taksonomia - teoria i jej zastosowania, Akademia Ekonomiczna w Krakowie, Kraków. 1990.
• Cacoullos T., Kontras M., Minimum-distance Discrimination of Spherical Distributions, w: Statistical Theory and Data Analysis, Proceedings of 1st Pacific Area Statistical Conference - Tokyo -1982, North Holland, Amsterdam 1985.
• Calinski R.B., Harabasz I.A., A Dendrite Method for Cluster Analysis, 'Communication in Statistics" 1974, nr 3.
• Catel R.B., Rp and Other Coefficients of Pattern Similarity, "Psychometrika" 1949, nr 14.
• Chernoff H., Estimation of the Mode, "Annals of Institute of Statistics and Mathematics" 1964, nr 16.
• Chinczyn A., On Unimodal Distributions, "Izwiestia Nauczno-Issliedowatielnogo Instituta Matiematiczeskoj Miechaniki Tomskogo Gosudarstwiennogo Uniwiersitieta" 1938, nr 2.
• Chomątowski S., Sokołowski A., Taksonomia struktur, "Przegląd Statystyczny" 1978, nr 2.
• Chow T.R., Hsia T.C, Shaver H.N., Distibution of City-block Distances in Pattern Recognition Problems, w: Proceedings of 4th International Joint Conference of Pattern Recognition, Kyoto 1978, New York 1979.
• Clark P.J., Evans F.G., Distance to Nearest Neighbour as a Measure of Spatial Relationship in Populations, "Ecology" 1954, nr 35.
• Clifford D.H.T., Stephenson W., An Introduction to Numerical Classification, Academic Press, New York 1975.
• Clifford P., Green N.J.B., Distances in Gaussian Point Sets, "Mathematical Proceedings of Cambridge Philosophical Society" 1985, t. 97, nr 3.
• Constanza C.M., Afifi A.A., Comparison of Stopping Rules in Forward Stepwise Discriminant Analysis, "Journal of the American Statistical Association" 1979, t. 74.
• Cormack R.M., A Review of Classification, "Journal of the Royal Statistical Society ", A, 1974, nr 134.
• Cunningham K.M., Ogilvie J.C., Evaluation of Hierarchical Grouping Techniques: A Preliminary Study, "Computer Journal" 1972, nr 15.
• Dalenius T., The Mode - a Neglected Statistical Parameter, "Journal of the Royal Statistical Society", A, 1965, nr 128.
• Davies D.L., Bouldin D.W., A Quster Separation Measure. "IEEE Transactions on Pattern Analysis and Machine Intelligence" 1979, nr 1.
• Day N.E., Estimating the Components of a Mixture of Normal Distributions, "Biometrika" 1969, nr 56.
• Day W.H.E., Edelsbrunner H., Investigation of Proportional Link linkage Clustering Methods, "Journal of Classification" 1985, Vol. 2, No. 2/3.
• Degens P.O., Federkiel H., A Monte Carlo Study on a Agglomerative Large Sample Clustering, COMPSTAT 3, Physica Verlag, Wien 1978.
• Dharmadhikari S., Joag-dev K., Unimodality, Coverity and Applications, Academic Press, London 1988.
• Dick N.P., Bowden D.G., Maximum Likelihood Estimation for Mixtures of Two Normal Distributions, "Biometrics" 1973, nr 29.
• Diday E., Some Recent Advances in Clustering, w: Hayasschi C. i in. (red.), Recend Developments in Clustering and Data Analysis, Proceedings of the Japanese-French Scientific Seminar, March 24-26,1987, Academic Press. San Diego 1988.
• Du Bien J.L., Werde W.D., A Mathematical Comparison of the Members of an Infinite Family of Agglomerative Clustering Algorithm?, "Canadian Journal of Statistics" 1979, nr 7.
• Duda R.O., Hart P.E., Pattern Classification and Scene Analysis, New York, Wiley 1973.
• Duran D.S., Odell P.L., Cluster Analysis, Springer Verlag, Berlin 1974.
• Dziubdziela W., A Note on k-th Distance Random Variables, "Zastosowania Matematyki" 1976, nr 15.
• Dziubdziela W., Kopociński B., Własności graniczne zmiennych losowych dystansowych, "Przegląd Statystyczny 1976, nr 4.
• Edmonston B., MICRO-CLUSTER: Ouster Analysis Software for Microcomputers, Ithaka, New York 1984.
• Edwards A.W.F., Cavalli-Sforza L.: A Method for Cluster Analvsis, "Biometrics" 1965, nr 21.
• Encyclopedia of the Statistical Sciences, Kotz S., Johnson N., (red.), Wiley, New York 1982.
• Engelman I., Hartigan J.A., Percentage Points of a Test for Ousters, "Journal of the American Statistical Association" 1969, t. 64.
• Everitt B.S., Ouster Analysis, Heinemann Educational Books, London 1974.
• Everitt B.S., A Monte Carlo Investigation of the Likelihood Ratio Test for the Number of Components in a Mixture of Normal Distributions, "Multivariate Behavioural Research" 1981, nr 16.
• Everitt B.S., A Test of Multivariate Normality Against the Alternative That the Distribution Is a Mixture, "Multivariate Behavioural Research" 1981, t.16.
• Everitt B.S., Hand D.J., Finite Mixture Distributions, Chapman and Hall, London 1981.
• Fisher L., Van Ness J.W., Admissible Clustering Procedures, "Biometrika" 1971, Vol. 58, No. 1.
• Fisher W.D., Clustering and Aggregation in Economics, John Hopkins, Baltimore 1969.
• Fitch W.M., Margoliash E., Construction of Phylogenetic Trees, "Science" 1967, nr 155.
• Florek K., i in., Sur le liaison et la decision des points d'un ensemble fini, "Colloquim Mathematicum" 1951, t. II, nr 3-4.
• Florek K., i in., Taksonomia wrocławska, "Przegląd Antropologiczny" 1951, t. XVIII.
• Fowlkes E.B., Mallows C.L., A Method for Comparing Two Hierarchical Clusterings, "Journal of the American Statistical Association" 1983, t. 78.
• Friedman H.P., Rubin J., On Some Invariant Criteria for Grouping Data, "Journal of the American Statistical Association" 1967, No. 62.
• Fry T., Van Groenewoud H., A Cluster Analysis of the D-squared Matrix of White Spruce Stands in Saskatchewan Based on the Maximum-minimum Principle, "Journal of Ecology" 1972, nr 60.
• Fukunaga K., Koontz W.L.G., A Criterion and an Algorithm for Grouping Data, "IEEE Transactions on Computers" 1970, C-19.
• Ghosh P., On Generalized Unimodality, "Communication Statistics" 1974, nr 3.
• Gnanadesikan R., Kettenring J.R., Landwehr J.M., Interpreting and Assesing the Results of Cluster Analysis, "Bulletin of the International Statistical Institute" 1977, nr 47.
• Good I.J., An Index of Separateness of Clusters and a Permutation Test for its Statistical Significance, "Journal of Statistical Computing and Simulation" 1982, nr 15.
• Goodman L.A., Kruskal W.H., Measures of Association for Cross Classifications, "Journal of the American Statistical Association" 1954, 149.
• Goower J.C., A Comparison of Some Methods of Cluster Analysis, "Biometrics" 1967, t. 23.
• Gordon A.D., Classification: Methods for the Exploratory Analysis of Multivariate Data, Chapman and Hall, London 1981.
• Grabiński T., Dynamiczne modele analizy taksonomicznej, Akademia Ekonomiczna w Krakowie, 1975, praca doktorska, maszynopis.
• Grabiński T., Metody określania charakteru zmiennych w wielowymiarowej analizie porównawczej, "Zeszyty Naukowe Akademii Ekonomicznej w Krakowie" 1985, nr 213.
• Grabiński T., Sokołowski A., The Effectiveness of Some Signal Identification Procedures, w: Signal Processing: Theories and Applications, Kunt M. and de Coulon F. (ed.), North-Holland Publishing Company, EURASIP, Amsterdam 1980.
• Grabiński T., Sokołowski A., Z badań nad efektywnością wybranych procedur taksonomicznych, "Zeszyty Naukowe Akademii Ekonomicznej w Krakowie" 1984, nr 181.
• Grabiński T., Wielowymiarowa analiza porównawcza w badaniach dynamiki zjawisk ekonomicznych, Akademia Ekonomiczna w Krakowie, seria specjalna; "Monografie", Kraków 1984.
• Grabiński T., Wydymus S., Zeliaś A., Metody taksonomii numerycznej w modelowaniu zjawisk społeczno-gospodarczych, PWN, Warszawa 1989.
• Grenander U., Some Direct Estimates of the Mode, "Annals of Mathematical Statistics" 1965, nr 36.
• Hamann V., Merkmalbestand raid Verwandtdchaftsbeziehungen der Farinosae. Ein Beitragzum System der Monokotyledonen, "Willdenowia" 1961, nr 2.
• Hand D.J., Discrimination and Classification, John Wiley & Sons, Chichester 1986.
• Hartigan J.A., Asymptotic Distributions for Clustering Criteria, "The Annals of Statistics" 1978, nr 6.
• Hartigan J.A., Clustering Algorithms, John Wiley & Sons, New York 1975.
• Hartigan J.A., Distribution Problems in Clustering, w: Van Ryzm J. (ed.), Classification and Clustering, Academic Press, New York 1977.
• Hartigan J.A., Hartigan P.M., The Dip Test for Unimodality, "The Annals of Statistics" 1981, nr 13.
• Hartigan J.A., Statistical Theory of Clustering, "Journal of Classification" 1985, nr 2.
• Hartigan J.A., The Span Test for Unimodality, w: Classifacation and Related Methods of Data Analysis, Bock H.H. (red.), Elsevier Science Publishing B.V. (North Holland), Amsterdam 1988.
• Hartigan P.M., Computation of the Dip Statistic to Test for Unimodality, "Applied Statistics" 1985, t. 34, nr 3.
• Hellwig Z., Aproksymacja stochastyczna, PWE, Warszawa 1965.
• Hellwig Z., On the Testing a Hypothesis That an n-dimensional Random Variable is Normal, "Prace Naukowe Akademii Ekonomicznej we Wrocławiu" 1976, nr 70.
• Hellwig Z., Taksonometria ekonomiczna, jej osiągnięcia, zadania i cele, w: Taksonomia - teoria i jej zastosowania, Akademia Ekonomiczna w Krakowie, Kraków 1990.
• Hill R.S., A Stopping Rule for Partitioning Dendrograms, "Botanical Gazette" 1980, nr 141.
• Hosmer D.W., A Comparison of Iterative Maximum Likelihood Estimates of the Parameters of a Mixture of Two Normal Distributions under Three Different 'types of Sample, "Biometrics" 1973, nr 29.
• Hubert L.J., Arabie P., Comparing Partitions, "Journal of Classification" 1985, t. 2, nr 2-3.
• Hubert L.J., Levin J.R., A General Statistical Framework forAssesing Categorical Clustering in Free Recall, "Psychological Bulletin" 1976, No. 83.
• Hussain M., Taksonomiczne metody podziału zbiorów skończonych, Akademia Ekonomiczna w Krakowie, 1982, praca doktorska, maszynopis.
• Ibragimow J.A., On the Composition of Unimodal Distributions,"Theory of Probability Applications" 1956, nr 1.
• Jaccard P., Nouvelles recherches sur la distribution florale, "Bull Soc. Vaud. Sci Nat." 1908, t. 44.
• Jain A.K., Dubes R.C., Algorithms for Clustering Data, Prentice Hall, New York 1988.
• Jajuga K., Statystyczna teoria rozpoznawania obrazów, Państwowe Wydawnictwo Naukowe, 1990.
• Jardine N., Sibson R., Mathematical Taxonomy, Wiley, New York 1971.
• Jevons W.S., Zasady nauki, PWN, Warszawa 1980.
• Johnson S.C., Hierarchical Clustering Schemes, "Psychometrika" 1967, t. 32.
• Kaufmann H.L., Rousseew P.J., Finding Groups in Data. An Introduction to Cluster Analysis, John Wiley & Sons, New York 1990.
• Kaufmann H.L., Engelhardt K., Zur Klassenzahlbestimmung beimMischver- tailungsverfahren, Numer, und Nichtnumer. Klasif. zwischen Theor. und. Prax., Proc. 5 fachtag Ges. Klassif., Hofgeismar 1981, Frankfurt 1982.
• King B.F., Step-wise Clustering Procedures, "Journal of the American Statistical Association" 1967, t. 62.
• Kopociński B., Rozkład zmiennej losowej dystansowej, "Prace Naukowe WSE we Wrocławiu" 1967, nr 6.
• Korhonen P., A Stepwise Procedure for Multivariate Clustering, University of Helsinki, Research Report, No. 7, Helsinki 1979.
• Kruskal J.B., Landwehr J.M., McRae I.E., ICICLE Plot Package for Hierarchical Clustering, AT and Bell Laboratories, Murray Hill, New York 1984.
• Krzanowski W.J., Lai Y.T., A Criterion for Determining the Number of Groups in a Data Set Using Sum-of-squares Clustering, "Biometrics" 1988, nr 44.
• Krzyśko M., Stolarski P., Probabilistyczna metoda analizy skupień, "Roczniki Akademii Rolniczej w Poznaniu(r) 1979, nr 116.
• Kucharczyk J., Algorytmy analizy skupień w języku ALGOL 60, PWN, Warszawa 1982.
• Kuiper F.K., Fisher L., A Monte Carlo Comparison of Six Clustering Procedures, "Biometrics" 1975, nr 31.
• Kulp R.W., Nagarscnker B.N., An Asymptotic Expansion of the Nonnull Distribution of Wilks Criterion for Testing the Multivariate Linear Hypothesis, "The Annals of Statistics" 1984, L 12, nr 4.
• Lance G.H., Williams W.T., A General Theory of classificatory Sorting Strategies, 1. Hierarchical Systems, "Computer Journal" 1967, t. 9.
• Lance G.H., Williams W.T., A Generalized Sorting Strategy for Computer Classifications, "Computer Journal" 1966, nr 9.
• Lance G.H., Williams W.T., Computer Programs for Hierarchical Polythelia Classification, "Computer Journal" 1966, nr 9.
• Lance G.H., Williams W.T., Generalized Sorting Strategy for Computer Classifications, "Nature" 1966, nr 212.
• Lee K.L., Multivariate Tests for Clusters, "Journal of the American Statistical Association" 1979, nr 367.
• Lefkovitch L.P., Further Nonparametric Tests for Comparing Dissimilarity Matrices Based on the Relative Neighbour Graphs, "Mathematical Bioscience" 1985, t.73, nr 1.
• Lerich W., On the Distance Random Variables, "Prace Naukowe Akademii Ekonomicznej we Wrocławiu" 1984, nr 285.
• Lerman I.C., Comparing partitions (Mathematical and Statistical aspects), w: Classification and Related Methods of Data Analysis, Bock H.H. (red.), Elsevier Science Publishers B.V. (North Holland), Amsterdam 1988.
• Ling R.F., A Probability Theory of Cluster Analysis, "Journal of the American Statistical Association" 1973, t. 68.
• Ling R.F., On the Theory and Construction of k-cluster, "Computer Journal" 1972, t.15.
• Łybacka K., Random Partitions of an Interval Under Some Distribution Points, "Fasc. math." 1985, nr 14.
• MacQueen J.B., Some Methods for Classifications and Analysis of Multivariate Observations, w: Le Cam L.M., Neyman (red.) Proceedings of the 5th Symposium on Mathematical Statistics and Probability, t.l, University of California Press, Berkeley 1967.
• Mahalanobis P.C., On the Generalized Distance in Statistics, "Proceedings of Indian National Institute of Science" 1936, nr 2.
• Marcotorchino F., Utilization des Comparaisons par paires en statistique des contingences (Partie I), Etude F-069, Centre Scientifique IBM-France, 1984.
• Marek T., Analiza skupień w badaniach empirycznych. Metody SAHN, Państwowe Wydawnictwo Naukowe, Warszawa 1989.
• Marriot F.H.C., Practical Problems in a Method of Ouster Analysis, "Biometrics" 1971, nr 27.
• McQain J.O., Rao V.R., CLUSTIZ: A Program to Test for the Quality of Clustering of a Set of Objects, "Journal of Marketing Research" 1975, nr 12.
• McLachlan G.J., Basford K.E., Mixture Models: Inference and Applications to Clustering, Dekker, New York 1988.
• McLachlan G.J., The Classification and Mixture Maximum Likelihood Approaches to Cluster Analysis, w: Handbook of Statistics, t. 2, Krishnaiah P.R., Kanal L.N. (red.), North Holland, Amsterdam 1982.
• Milligan G.W., An Algorithm for Generating Artificial Test Ousters, "Psychometrika" 1985, t. 50, nr 1.
• Milligan G.W., An Examination of the Effect of Six Types of Error Perturbation on Fifteen Clustering Algorithms, "Psychometrika" 1980, nr 45.
• Milligan G.W., Cooper M.C, A Study of Standarization of Variables in Cluster Analysis, "Journal of Classification" 1988, nr 5.
• Milligan G.W., Cooper M.C., An Examination of Procedures for Determining the Number of Clusters in a Data Set, "Psychometrika" 1985, t. 50, nr 2.
• Mojena R., Hierarchical Grouping Method and Stopping Rules: An Evaluation, "Computer Journal" 1977, nr 20.
• Mountford M.D., A Test for the Difference Between Clusters, w: Patii G.P., Pielou EC, Waters W.E., (red.), Statistical Ecology, t. 3, University Park. Pa., Pennsylvania State University Press, 1970.
• Naylor T.H., Computer Simulation Experiments with Models of Economic Systems, John Wiley and Sons, New York 1971.
• Nelson L.S., Critical Values for a Cluster Analysis Approach to Multiple Comparisons, "Techmometrics" 1990, nr 1.
• Noreen E.W., Computer-Intensive Methods for Testing Hypothesis. An Introduction, John Wiley & Sons, New York 1989.
• Nowak E., Metody taksonomiczne w klasyfikacji obiektów społeczno-ekonomicznych, PWE, Warszawa 1990.
• Nowak E., Wskaźnik podobieństwa wyników podziałów, "Przegląd Statystyczny" 1985, nr 1.
• Ochiai A., Zoogeographie Studies on the Soleoid Fishes Found in Japan and its Neighboring Regions, "Bull. Jap. Soc. Sci. Fish" 1957, t. 22.
• Orloci L., An Agglomerative Method for Classification of Plant Communities, "Journal of Ecology" 1967, nr 55.
• Pamuła J., Sokołowski A., Propozycja wyznaczania podziału wynikowego aglomeracyjnych algorytmów taksonomicznych, "Zeszyty Naukowe Akademii Ekonomicznej w Krakowie" 1980, nr 127.
• Parzen E., On Estimation of a Probability Density Function and Mode, "Annals of Mathematical Statistics" 1962, nr 33.
• Pawlenko T., G-ocienka rasstojanija Mahalanobisa dla sluczaja proizwolno nieproerywnogo raspriedielienija nabludajemych wiektorow, "Trudy Wsiesojuznogo Gosudarstwiennogo Uniwiersitieta" 1988, nr 56.
• Pearson K., On the Coefficient of Racial Likeness, "Biometrika" 1926, t. 18, nr 105.
• Pearson K.P., Contributions to the Mathematical Theory of Evolution - II: Skew Variation in Homogenous Material, "Philosophical Transactions of Royal Society in London Academy" 1895, nr 186.
• Peck R., Fisher L., Van Ness J., Approximate Confidence Intervals for the Number of Clusters, "Journal of the American Statistical Association" 1989, t. 84.
• Perkal J., Taksonomia wrocławska, "Przegląd Antropologiczny", t. XIX, 1951.
• Pociecha J., Kryteria oceny procedur taksonomicznych, "Przegląd Statystyczny" 1982, nr 1/2.
• Pociecha J., Sokołowski A., The Multidimensional Uniformity, referat na konferencję "Classification and Related Methods of Data Analysis", Aachen 1988, maszynopis.
• Pociecha J., Zając K., Wkład ośrodka krakowskiego w rozwój teorii i zastosowań metod taksonomicznych, w: Taksonomia - teoria i jej zastosowania, Akademia Ekonomiczna w Krakowie, Kraków 1990.
• Pociecha J.,Podolec B., Sokołowski A., Zając K., Metody taksonomiczne w badaniach społeczno-ekonomicznych, PWN, Warszawa 1988.
• Polhemus N.W., STATGRAPHICS. An Interactive Statistical Graphics System in APL, Statistical Graphics Corporation, Princeton, New York 1984.
• Pollard D., A Central Limit Theorem for k-means Clustering, "Annals of Probability" 1982, nr 10.
• Rand W.M., Objective Criteria for the Evaluation of Clustering Methods, "Journal of the American Statistical Association" 1971, t. 66, nr 336.
• Rao C.R., The Utilization of Multiple Measurement in Problems of Biological Classification, "Journal of the Royal Statistical Society", B, 1948, t.10.
• Ratkowsky D.A., Lance G.N., A Criterion for Determining the Number of Groups in a Classification, "Australian Computer Journal" 1978, nr 10.
• Rinott Y., On Convexity of Measures, "Annals of Probability" 1976, nr 4.
• Robertson T., Cryer J., An Iterative Procedure for Estimating the Mode, "Journal of the American Statistical Association" 1974, nr 69.
• Rohlf F.J., Methods of Comparing Classifications, "Annual Review of Ecology and Systematics" 1974, nr 5.
• Rosemburg H.C.X, Marshall K., User's Manual for CLUSTER/CLUSTID Computer Program for Hierarchical Cluster Analysis, Lifetime Learning Publications, Belmont 1984.
• Roy S.N., Some Aspects of Multivariate Analysis, John Wiley & Sons, New York 1957.
• Sager T., An Iterative Method for Estimating a Multivariate Mode and Isopleth, "Journal of the American Statistical Association" 1979, t. 74, nr 366.
• Sager T., Consistency in Nonparametric Estimation of the Mode, "Annals of Statistics" 1975, nr 3.
• Sager T., Estimation of a Multivariate Mode, "Annals of Statistics" 1978, nr 6.
• Sager T., Thisted R., Isotonic Model Regression, "Annals of Statistics" 1982, nr 10.
• Sager T., Estimating Modes and Isophlets, "Communication Statistical Theory and Methods", 1983, t. 12, nr 5.
• Scott A.J., Symons M.J., Clustering Methods Based on Likelihood Ratio Criteria, "Biometrics" 1971, nr 27.
• Shepp L.A., Symmetric Random Walk, Transactions of American Mathematical Society" 1962, nr 104.
• Sherman S., A Theorem on Convex Sets with Applications, "Annals of Mathematical Statistics" 1955, nr 26.
• Sheuer E., Stoller D.S., On the Generation of Normal Random Vectors, "Technometrics" 1962, nr 4.
• Siedlecka U., Zastosowanie metody taksonomii stochastycznej do dyskryminacji zbiorów skończonych, "Przegląd Statystyczny" 1976, nr 6.
• Smoluk A., Uwagi o dyskryminacji w zwartych przestrzeniach metrycznych, "Przegląd Statystyczny" 1976, nr 23.
• Sneath P.H.A., Sokal R.R., Numerical Taxonomy, W.H. Freeman, San Francisco 1973.
• Sneath P.H.A., A Method for Testing the Distinctness of dusters: A Test of the Disjunction of Two Clusters in Euclidean Space as Measured by Their Overlap, "Mathematical Geology" 1977, nr 9.
• Sneath P.H.A., Some Empirical Tests for Significance of dusters, w: "Data Analysis and Information Processing, 2nd International Symposium, Versailles 1979", North Holland, Amsterdam 1980.
• Sokal R.R., Michener C.D., A Statistical Method for Evaluating Systematic Relationship, "University of Kansas Science Bulletin" 1958, nr 38.
• Sokal R.R., Sneath P.H.A., Principles of Numerical Taxonomy, Freeman, London 1963.
• Sokołowski A., Generowanie losowego podziału zbioru skończonego, Trace Naukowe Akademii Ekonomicznej we Wrocławiu" 1979, nr 182.
• Sokołowski A., HIERA - Hierarchiczne metody taksonomiczne, maszynopis.
• Sokołowski A., Metoda porównywania wyników podziału zbioru skończonego, XII Konferencja Naukowa Ekonometryków, Statystyków i Matematyków Akademii Ekonomicznych Polski Południowej, Karpacz 1976.
• Sokołowski A., Metody badania stacjonarności jednowymiarowych ciągów losowych, Akademia Ekonomiczna w Krakowie, 1977, praca doktorska, maszynopis.
• Sokołowski A., O zagadnieniach taksonomicznych, "Zeszyty Naukowe Akademii Ekonomicznej w Krakowie" 1982, nr 165.
• Sokołowski A., Wybrane zagadnienia pomiaru i ważenia cech w taksonomii, "Zeszyty Naukowe Akademii Ekonomicznej w Krakowie" 1934, nr 203.
• Spath H., Cluster Analysis Algorithms, Ellis Horwood Ltd., Chichester 1980.
• Steinhausen D., Langer K., Clusteranalyse, Walter de Gruyter, Benin-New York 1977.
• Szmigiel C., Modyfikacje wskaźnika zgodności kryteriów podziału, "Prace Naukowe Akademii Ekonomicznej we Wrocławiu" 1981, nr 190.
• Szmigiel G., Wskaźnik zgodności kryteriów podziału, "Przegląd Statystyczny" 1976, nr 4.
• Szymanowicz K., Propozycja w zakresie cech określających stopień rozwoju społeczno-ekonomicznego niektórych krajów, w: Studia z zakresu zastosowań metod ilościowych w ekonomii, demografii i socjologii, PAN, Oddział w Krakowie, Kraków 1977.
• Trybuś G., Zmienna losowa dystansowa. Teoria i zastosowania, "Prace Naukowe Akademii Ekonomicznej we Wrocławiu" 1981, nr 173, Seria Monografie i Opracowania, nr 1.
• Tryon R.C., Bailey D.C., Cluster Analysis, McGraw-Hill, New York 1970.
• Van Ness J.W., Admissible Clustering Procedures, "Biometrika" 1973, Vol. 60, No. 2.
• Venter J.H., On Estimation of the Mode, "Annals of Mathematical Statistics" 1967, nr 38.
• Walesiak M., Porównawcza analiza miar zgodności wyników klasyfikacji w świetle postulowanych własności, w: Taksonomia - teoria i jej zastosowania, Akademia Ekonomiczna w Krakowie, Kraków 1990.
• Walesiak M., Sposoby rejestracji zmian w czasie w wynikach klasyfikacji, "Prace Naukowe Akademii Ekonomicznej we Wrocławiu" 1985, nr 311.
• Ward J.H., Hierarchical Grouping to Optimize an Objective Function, "Journal of the American Statistical Association" 1963, t. 58.
• Ward J.H., Hook M.E., Application of an Hierarchical Grouping Procedure to a Problem of Grouping Profiles, "Educational Psychology Measurements" 1963, t. 23.
• Wegman E.J., A Note on the Estimation of the Mode, "Annals of Mathematical Statistics" 1971, nr 42.
• Wilks S.S., Certain Generalizations in the Analysis of Variance. "Biometrika" 1932, t. 24.
• Wilks S.S., On the Independence of k Sets of Normally Distributed Statistical Variables, "Econometrica" 1935, t. 3.
• Williams W.T. i inni, Controversy Concerning the Criterion for Taxonometric Strategies, "Computer Journal" 1971, t.14.
• Wojcik T., Zarys teorii klasyfikacji, PWN, Warszawa 1965.
• Wolfe J.H., A Monte Carlo Study of the Sampling Distribution of the Likelihood Ratio for Mixtures of Multinomial Distributions, Research Memorandum 72-2, Naval Personnel and Training Research laboratory, San Diego 1971.
• Wolfe J.H., Pattern Clustering by Multivariate Mixture Analysis, "Multivariate Behavioural Research" 1970, nr 5.
• Wong A.K.C., Konigs R.M., A Clustering Algorithm Based on Global and Local Potential Functions, w: Proceedings of International Conference on Cybernetics and Sociology, Denver, Colorado 1979, New York 1979.
• Yule G.U., An Introduction of the Theory of Statistics, Charles Griffin and Co. Ltd., London 1911.
• Yule G.U., On the Methods of Measuring the Association Between Two Attributes. "Journal of the Royal Statistical Society" 1912, t. 75.
• Zając K., Zarys metod statystycznych, PWE, Warszawa 1987.