Skip to main content
Log in

Methods for the analysis of asymmetric pairwise relationships

  • Regular Article
  • Published:
Advances in Data Analysis and Classification Aims and scope Submit manuscript

Abstract

Asymmetric pairwise relationships are frequently observed in experimental and non-experimental studies. They can be analysed with different aims and approaches. A brief review of models and methods of multidimensional scaling and cluster analysis able to deal with asymmetric proximities is provided taking a ‘data-analytic’ approach and emphasizing data visualization.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  • Arabie P, Carroll JD (1980) MAPCLUS: A mathematical programming approach to fitting the ADCLUS model. Psychometrika 40:211–235

    Article  MATH  Google Scholar 

  • Arabie P, Carroll JD, DeSarbo WS (1987) Three-way scaling and clustering. Sage Publications, Newbury Park

    Book  Google Scholar 

  • Baier D, Frost S (2017) Relating brand confusion to ad similarities and brand strengths through image data analysis and classification. Adv Data Anal Classif. https://doi.org/10.1007/s11634-017-0282-1

    Google Scholar 

  • Bloxom B (1968) Individual differences in multidimensional scaling (Research Bulletin 68–45). Educational Testing Service, Princeton, p 68

    Google Scholar 

  • Borg I, Groenen PJF (2005) Modern multidimensional scaling. Theory and applications, 2nd edn. Springer, New York

    MATH  Google Scholar 

  • Bove G (1992) Asymmetric multidimensional scaling and correspondence analysis for square tables. Stat Appl Ital J Appl Stat 4:587–598

    Google Scholar 

  • Bove G (2006) Approaches to asymmetric multidimensional scaling with external information. In: Zani S, Cerioli A et al (eds) Data analysis, classification and the forward search. Springer, Berlin, pp 69–76

    Chapter  Google Scholar 

  • Bove G (2012) Exploratory approaches to seriation by asymmetric multidimensional scaling. Behaviormetrika 39:63–73

    Article  Google Scholar 

  • Bove G, Critchley F (1993) Metric multidimensional scaling for asymmetric proximities when the asymmetry is one-dimensional. In: Steyer R et al (eds) Psychometric methodology. Gustav Fischer, Stuttgart and New York, pp 55–60

    Google Scholar 

  • Bove G, Rocci R (1999) Methods for asymmetric three-way scaling. In: Vichi M, Opitz O (eds) Classification and data analysis. Theory and application. Springer, Berlin, pp 131–138

    Chapter  Google Scholar 

  • Bove G, Rocci R (2004) A method of asymmetric multidimensional scaling with external information. Atti della XLII Riunione Scientifica della S.I.S, Bari

    Google Scholar 

  • Bove G, Critchley F (1989) On the representation of asymmetric proximities. In: Atti delle Giornate di studio del Gruppo Italiano aderenti IFCS

  • Brossier G (1982) Classification hiérarchique à partir de matrices carrées non symétriques. Stat Anal des Données 7:22–40

    MATH  Google Scholar 

  • Busing FMTA, Groenen PJF, Heiser WJ (2005) Avoiding degeneracy in multidimensional unfolding by penalizing on the coefficient of variation. Psychometrika 70:71–98

    Article  MathSciNet  MATH  Google Scholar 

  • Carroll JD, Arabie P (1980) Multidimensional scaling. Annu Rev Psychol 31:607–649

    Article  Google Scholar 

  • Carroll JD, Arabie P (1998) Multidimensional scaling. In: Birnbaum MH (ed) Measurement, judgment and decision making. Academic, San Diego, pp 179–250

    Chapter  Google Scholar 

  • Carroll JD, Chang JJ (1970) Analysis of individual differences in multidimensional scaling via an N-way generalization of Eckart–Young decomposition. Psychometrika 35:283–319

    Article  MATH  Google Scholar 

  • Carroll JD, Chang JJ (1972) IDIOSCAL (Individual differences in Orientation SCALing): a generalization of INDSCAL allowing IDIOsyncratic reference systems as well as an analytic approximation to INDSCAL. Paper presented at meeting of the psychometric society, Princeton

  • Caussinus H (1966) Contribution à l’analyse statistique des tableaux de corrélation. Annales de la Faculté des Sciences de Toulouse 29:77–183

    Article  MATH  Google Scholar 

  • Caussinus H, de Falguerolles A (1987) Tableaux carrés: modélisation et methods factorielles. Revue de Stat Appl 35:35–52

    Google Scholar 

  • Chino N (1978) A graphical technique for representing the asymmetric relationships between \(N\) objects. Behaviormetrika 5:23–40

    Article  Google Scholar 

  • Chino N (1990) A generalized inner product model for the analysis of asymmetry. Behaviormetrika 17:25–46

    Article  Google Scholar 

  • Chino N (2012) A brief survey of asymmetric MDS and some open problems. Behaviormetrika 39(1):127–165

    Article  Google Scholar 

  • Chino N, Shiraiwa K (1993) Geometrical structures of some non-distance models for asymmetric MDS. Behaviormetrika 20:35–47

    Article  Google Scholar 

  • Constantine AG, Gower JC (1978) Graphical representation of asymmetric matrices. Appl Stat 3:297–304

    Article  MATH  Google Scholar 

  • Coombs CH (1964) A theory of data. Wiley, New York

    Google Scholar 

  • Cox TF, Cox MAA (2001) Multidimensional scaling. CRC/Chapman and Hall, London

    MATH  Google Scholar 

  • Critchley F (1988) On characterization of the inner products of the space of square matrices of given order which render orthogonal the symmetric and the skew-symmetric subspaces. Warwick statistics research report 171, University of Warwick, Coventry

  • De Leeuw J, Heiser WJ (1982) Theory of multidimensional scaling. In: Krishnaiah PR, Kanal LN (eds) Handbook of statistics, vol 2. North Holland, Amsterdam, pp 285–316

    Google Scholar 

  • De Rooij M (2001) Distance models for the analysis of transition frequencies. Unpublished Doctoral dissertation, Leiden University

  • De Rooij M (2009) Trend vector models for the analysis of change in continuous time for multiple groups. Comput Stat Data Anal 53:3209–3216

    Article  MathSciNet  MATH  Google Scholar 

  • De Rooij M (2015) Transitional modelling of experimental longitudinal data with missing values. Adv Data Anal Classif. https://doi.org/10.1007/s11634-015-0226-6

    Google Scholar 

  • De Rooij M, Heiser WJ (2000) Triadic distance models for the analysis of asymmetric three-way proximity data. Br J Math Stat Psychol 53:99–119

    Article  MathSciNet  Google Scholar 

  • De Rooij M, Heiser WJ (2003) A distance representation of the quasi-symmetry model and related distance models. In: Yanai H, Okada A et al (eds) New developments in psychometrics. Springer, Tokyo, pp 487–494

    Chapter  Google Scholar 

  • De Rooij M, Heiser WJ (2005) Graphical representations and odds ratios in a distance association for the analysis of cross-classified data. Psychometrika 70:99–122

    Article  MathSciNet  MATH  Google Scholar 

  • DeSarbo WS (1982) GENNCLUS: new models for general nonmetric clustering analysis. Psychometrika 47:449–475

    Article  MathSciNet  MATH  Google Scholar 

  • DeSarbo WS, John MD, Manrai AK, Manrai LA, Edwards EA (1992) TSCALE: a new multidimensional procedure based on Tvresky’s contrast model. Psychometrika 57:43–69

    Article  MATH  Google Scholar 

  • Dossou-Gbété S, Grorud A (2002) Biplots for matched two-way tables. Annales de la Faculté des Sciences de Toulouse 11:469–483

    Article  MathSciNet  MATH  Google Scholar 

  • Escofier B (1983) Analyse de la différence entre deux mesures sur le produit de deux même ensembles. Cah l’Anal des Données 3:325–329

    MATH  Google Scholar 

  • Escoufier Y, Grorud A (1980) Analyse factorielle des matrices carrees non symetriques. In: Diday E et al (eds) Data analysis and informatics. North Holland, Amsterdam, pp 263–276

    Google Scholar 

  • France SL, Carroll JD (2011) Two-way multidimensional scaling: a review. IEEE Trans Syst Man Cybern C Appl Rev 41(5):644–661

    Article  Google Scholar 

  • Freeman LC (1997) Uncovering organizational hierarchies. Comput Math Org Theory 3:5–18

    Article  MATH  Google Scholar 

  • Fujiwara H (1980) Hitaisho sokudo to toshitsusei keisuu o mochiita kurasuta bunsekiho [Methods for cluster analysis using asymmwric measures and homogeneity coefficient]. Kodo Keiryogaku [Jpn J Behav] 7(2):12–21 (in Japanese)

    Article  Google Scholar 

  • Gabriel KR (1971) The biplot graphic display of matrices with application to principal component analysis. Biometrika 58(3):453–467

    Article  MathSciNet  MATH  Google Scholar 

  • Goodman LA (1979) Simple models for the analysis of association in cross classifications having ordered categories. J Am Stat Assoc 74:537–552

    Article  MathSciNet  Google Scholar 

  • Gower JC (1977) The analysis of asymmetry and orthogonality. In: Barra JR et al (eds) Recent developments in statistics. North Holland, Amsterdam, pp 109–123

    Google Scholar 

  • Gower JC (2008) Asymmetry analysis: the place of models. In: Shigemasu K, Okada A et al (eds) New trends in psychometrics. Universal Academy, Tokyo, pp 79–86

    Google Scholar 

  • Gower JC (2014) Skew symmetry in retrospect. Adv Data Anal Classif. https://doi.org/10.1007/s11634-014-0181-7

    Google Scholar 

  • Gower JC, Hand DJ (1996) Biplots. Chapman and Hall, London

    MATH  Google Scholar 

  • Gower JC, Groenen PJF, van de Velden M (2010) Area biplots. J Comput Graph Stat 19(1):46–61. https://doi.org/10.1198/jcgs.2010.07134

    Article  MathSciNet  Google Scholar 

  • Gower JC, Lubbe S, Le Roux N (2011) Understanding biplots. Wiley, Chichester

    Book  Google Scholar 

  • Greenacre M (2000) Correspondence analysis of square asymmetric matrices. Appl Stat 49:297–310

    MathSciNet  MATH  Google Scholar 

  • Greenacre MJ (2010) Biplots in practice. BBVA Foundation, Madrid

    Google Scholar 

  • Greenacre MJ, Groenen PJF (2016) Weighted euclidean biplots. J Classif 33:442–459

    Article  MathSciNet  MATH  Google Scholar 

  • Groenen PJF, Borg I (2014) Past, present and future of multidimensional scaling. In: Blasius J, Greenacre M (eds) Visualization and verbalization of data. Chapman and Hall/CRC, London, pp 95–116

    Google Scholar 

  • Harshmann RA (1978) Models for analysis of asymmetrical relationships among N objects or stimuli. Paper presented at the first joint meeting of the psychometric society and the society for mathematical psychology, McMaster University, Hamilton, Ontario

  • Harshmann RA (1981) DEDICOM analysis of skew-symmetric data. Part I: theory. Unpublished technical memorandum, Bell Laboraries, Murray Hill

  • Harshmann RA, Green PE, Wind Y, Lundy ME (1982) A model for the analysis of asymmetric data in marketing research. Mark Sci 1:204–242

    Article  Google Scholar 

  • Heiser WJ (1989) Order invariant unfolding analysis under smoothness restrictions. In: de Soete G, Feger H, Klauer KC (eds) New developments in psychological choice modeling. North-Holland, Amsterdam, pp 3–31

    Chapter  Google Scholar 

  • Heiser WJ, Busing FMTA (2004) Multidimensional scaling and unfolding of symmetric and asymmetric proximity relations. In: Kaplan D (ed) The sage handbook of quantitative methodology for the social sciences. Sage Publications, London, pp 25–48

    Google Scholar 

  • Heiser WJ, de Leeuw J (1981) Multidimensional mapping of preference data. Math Sci Hum 19(73):39–96

    MathSciNet  Google Scholar 

  • Holman EW (1979) Monotonic models for asymmetric proximities. J Math Psychol 20:1–15

    Article  MATH  Google Scholar 

  • Horan CB (1969) Multidimensional scaling: combining observations when individuals have different perceptual structures. Psychometrika 34:319–165

    Article  Google Scholar 

  • Hubert L (1973) Min and max hierarchical clustering using asymmetric similarity measures. Psychometrika 38:63–72

    Article  MATH  Google Scholar 

  • Johannesson M (1997) Modelling asymmetric similarity with prominence. Lund University Cognitive Studies—LUCS 55 (ISSN: 1101–8453 D)

  • Johannesson M (2000) Modelling asymmetric similarity with prominence. Br J Math Stat Psychol 53:121–139

    Article  Google Scholar 

  • Johnson SC (1967) Hierarchical clustering schemes. Psychometrika 32:241–254

    Article  MATH  Google Scholar 

  • Kiers HAL (1993) An alternating least squares algorithm for PARAFAC2 and DEDICOM3. Comput Stat Data Anal 16:103–118

    Article  MATH  Google Scholar 

  • Kiers HAL, Takane Y (1993) Constrained DEDICOM. Psychometrika 58:339–355

    Article  MathSciNet  MATH  Google Scholar 

  • Kiers HAL, Takane Y (1994) A generalization of GIPSCAL for the analysis of asymmetric data. J Classif 11:79–99

    Article  MATH  Google Scholar 

  • Krumhansl CL (1978) Concerning the applicability of geometric models to similarity data: the interrelationship between similarity and spatial density. Psychol Rev 85:445–463

    Article  Google Scholar 

  • Kruskal JB (1964) Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis. Psychometrika 29:1–29

    Article  MathSciNet  MATH  Google Scholar 

  • Kruskal JB, Carmone FJ Jr (1971) How to use M-D-SCAL (Version 5M) and other useful information. Multidimensional Scaling Program Package of Bell Laboratories, Bell Laboratories, Murray Hill

    Google Scholar 

  • Kruskal JB, Carroll JD (1969) Geometrical models and badness-of-fit functions. In: Krishnaiah PR (ed) Multivariate analysis, vol 2. Academic, New York, pp 639–671

    Google Scholar 

  • Lance GN, Williams WT (1966) A generalized sorting strategy for computer classifications. Nature 212:218

    Article  Google Scholar 

  • Lance GN, Williams WT (1967) A general theory of classificatory sorting strategies: 1 hierarchical systems. Comput J 9:373–380

    Article  Google Scholar 

  • Mair P, De Leeuw J, Borg I, Groenen PJF (2017) Package SMACOF, cran.r-project.org

  • Muñoz A, Gonzalez J (2012) Hierarchical latent semantic class extraction using asymmetric term similarities. Behaviormetrika 39(1):91–109

    Article  Google Scholar 

  • Nosofsky RM (1991) Stimulus bias, asymmetric similarity, and classification. Cogn Psychol 23:94–140

    Article  Google Scholar 

  • Okada A (1988) An analysis of intergenerational occupational mobility by asymmetric multidimensional scaling. In: Jansen MGH, Schuur WH (eds) The many faces of multivariate analysis: proceedings of the SMABS-88 conference, vol 1. RION, Institute for Educational Research, University of Groningen, Groningen, pp 1–15

    Google Scholar 

  • Okada A, Imaizumi T (1987) Non-metric multidimensional scaling of asymmetric similarities. Behaviormetrika 21:81–96

    Article  Google Scholar 

  • Okada A, Imaizumi T (1997) Asymmetric multidimensional scaling of two-mode three-way proximities. J Classif 14:195–224

    Article  MATH  Google Scholar 

  • Okada A, Imaizumi T (2000) Two-mode three-way asymmetric multidimensional scaling with constraints on asymmetry. In: Decker R, Gaul W (eds) Classification and information processing at the turn of the millennium. Springer, Berlin, pp 52–59

    Chapter  Google Scholar 

  • Okada A, Imaizumi T (2002a) Multidimensional scaling with different orientations of dimensions for symmetric and asymmetric relationships. In: Nishisato S, Baba Y, Bozdogan H, Kanefuji K (eds) Measurement and multivariate analysis. Springer, Tokyo, pp 97–106

    Chapter  Google Scholar 

  • Okada A, Imaizumi T (2002b) A generalization of two-mode three-way asymmetric multidimensional scaling. In: Gaul W, Ritter G (eds) Classification, automation, and new media. Springer, Berlin, pp 113–122

    Google Scholar 

  • Okada A, Imaizumi T (2003) Two-mode three-way nonmetric multidimensional scaling with different directions of asymmetry for different sources. In: Yanai H, Okada A, Shigemasu K, Kano Y, Meulman JJ (eds) New developments in psychometrics. Springer, Tokyo, pp 495–502

    Chapter  Google Scholar 

  • Okada A, Imaizumi T (2005a) Joint space model for multidimensional scaling of two-mode three-way asymmetric proximities. In: Baier D, Wernecke K-D (eds) Innovations in classification, data science, and information systems. Springer, Berlin, pp 371–378

    Chapter  Google Scholar 

  • Okada A, Imaizumi T (2005b) External analysis of two-mode three-way asymmetric multidimensional scaling. In: Weihs C, Gaul W (eds) Classification: the ubiquitous challenge. Springer, Berlin, pp 288–295

    Chapter  Google Scholar 

  • Okada A, Imaizumi T (2007) Multidimensional scaling of asymmetric proximities with a dominance point. In: Baier D, Decker R, Lenz H-J (eds) Advances in data analysis. Springer, Berlin, pp 307–318

    Chapter  Google Scholar 

  • Okada A, Iwamoto T (1995) Hitaisho kurasuta bunnsekihou niyoru daigakushinngaku niokeru todoufukennkann no kanren no bunseki [An asymmetric cluster analysis study on university enrolment flow among Japanese prefectures]. Riron to Houhou [Sociol Theory Methods] 10:1–13 (in Japanese)

    Google Scholar 

  • Okada A, Iwamoto T (1996) University enrollment flow among the Japanese prefectures: a comparison before and after the joint first stage achievement test by asymmetric cluster analysis. Behaviormetrika 23:169–185

    Article  Google Scholar 

  • Okada A, Tsurumi H (2012) Asymmetric multidimensional scaling of brand switching among margarine brands. Behaviormetrika 39:111–126

    Article  Google Scholar 

  • Okada A, Tsurumi H (2013) External analysis of asymmetric multidimensional scaling based on singular value decomposition. In: Giudici P, Ingrassia S, Vichi M (eds) Statistical models for data analysis. Springer, Heidelberg, pp 269–278

    Chapter  Google Scholar 

  • Okada A, Tsurumi H (2014) Evaluating the effect of new brand by asymmetric multidimensional scaling. In: Vicari D, Okada A, Ragozini G, Weihs C (eds) Analysis and modeling of complex data in behavioral and social sciences. Springer, Heidelberg, pp 201–209

    Google Scholar 

  • Okada A, Yokoyama S (2015) Asymmetric CLUster analysis based on SKEW-Symmetry: ACLUSKEW. In: Morlini M, Tommaso M, Vichi A (eds) Advance in statistical models for data analysis. Springer, Heidelberg, pp 191–199

    Chapter  Google Scholar 

  • Olszewski D (2011) Asymmetric k-means algorithm. In: Dovnikar A, Lotrič U, Ster B (eds), International conference on adaptive and natural computing algorithms (ICANNGA 2011) part II. Lecture notes in computer science, vol 6594. Springer, Heidelberg, pp 1–10

  • Olszewski D (2012) K-means clustering of asymmetric data. In: Corchado E et al (eds) Hybrid artificial intelligent systems 2012, part I, Lecture notes in computer science, vol. 7208. Springer, Berlin, pp 243–254

  • Olszewski D, Šter B (2014) Asymmetric clustering using the alpha-beta divergence. Pattern Recogn 47:2031–2041

    Article  Google Scholar 

  • Rocci R (2004) A general algorithm to fit constrained DEDICOM models. SMA J Ital Stat Soc 13:139–150

    MathSciNet  MATH  Google Scholar 

  • Rocci R, Bove G (2002) Rotation techniques in asymmetric multidimensional scaling. J Comput Graph Stat 11:405–419

    Article  MathSciNet  Google Scholar 

  • Saburi S, Chino N (2008) A maximum likelihood for asymmetric MDS model. Comput Stat Data Anal 52:4673–4684

    Article  MathSciNet  MATH  Google Scholar 

  • Sagarra M, Busing FMTA, Mar-Molinero C, Rialp J (2014) Assessing the asymmetric effects on branch rivalry of Spanish financial sector restructuring. Adv Data Anal Classif. https://doi.org/10.1007/s11634-014-0186-2

    Google Scholar 

  • Saito T (1986) Multidimensional scaling to explore complex aspects in dissimilarity judgment. Behaviormetrika 20:35–62

    Article  Google Scholar 

  • Saito T (1991) Analysis of asymmetric proximity matrix by a model of distance and additive terms. Behaviormetrika 29:45–60

    Article  Google Scholar 

  • Saito T, Yadohisa H (2005) Data analysis of asymmetric structures: advanced approaches in computational statistics. Marcel Dekker, New York

    MATH  Google Scholar 

  • Shepard RN, Arabie P (1979) Additive clustering: representation of similarities as combination of discrete overlapping properties. Psychol Rev 86:87–123

    Article  Google Scholar 

  • Takane Y, Jung S, Oshima-Takane Y (2009) Multidimensional scaling. In: Millsap RE, Maydeu-Olivares A (eds) The sage handbook of quantitative methods in psychology. Sage Publications, Los Angeles, pp 219–242

    Chapter  Google Scholar 

  • Takeuchi A, Saito T, Yadohisa H (2007) Asymmetric agglomerative hierarchical clustering algorithms and their evaluations. J Classif 24:123–143

    Article  MathSciNet  MATH  Google Scholar 

  • Tomizawa S, Tahata K (2007) The analysis of symmetry and asymmetry: orthogonality of decomposition of symmetry into quasi-symmetry and marginal symmetry for multi-way tables. J Soc Française de Stat 148:3–36

    MathSciNet  Google Scholar 

  • Tsuchida J, Yadohisa H (2016) Asymmetric multidimensional scaling of \(n\)-mode \(m\)-way categorical data using a log-linear model. Behaviormetrika 44:103–138

    Article  Google Scholar 

  • Tucker LR, Messick WS (1963) An individual differences model for multidimensional scaling. Psychometrika 38:333–368

    Article  Google Scholar 

  • Tversky A (1977) Features of similarity. Psychol Rev 84:327–352

    Article  Google Scholar 

  • Van der Heijden PGM, de Leeuw J (1985) Correspondence analysis used complementary to loglinear analysis. Psychometrika 50:429–447

    Article  MathSciNet  MATH  Google Scholar 

  • Van der Heijden PGM, Mooijart A (1995) Some new log-bilinear models for the analysis of asymmetry in a square contingency table. Sociol Methods Res 24:7–29

    Article  Google Scholar 

  • Van der Heijden PGM, de Falguerolles A, de Leeuw J (1989) A combined approach to contingency table analysis using correspondence analysis and log-linear analysis. Appl Stat 38:249–292

    Article  MathSciNet  MATH  Google Scholar 

  • Vicari D (2014) Classification of asymmetric proximity data. J Classif 31:386–420

    Article  MathSciNet  MATH  Google Scholar 

  • Vicari D (2015) CLUSKEXT: CLUstering model for SKew-symmetric data including EXTernal information. Adv Data Anal Classif. https://doi.org/10.1007/s11634-015-0203-0

    Google Scholar 

  • Weeks DG, Bentler PM (1982) Restricted multidimensional scaling models for asymmetric proximities. Psychometrika 47:201–207

    Article  Google Scholar 

  • Yamaguchi K (1990) Some models for the analysis of asymmetric association in square contingency tables with ordered categories. In: Marsden P (ed) Sociological methodology. Blackwell, Oxford

    Google Scholar 

  • Young FW (1975) An asymmetric Euclidean model for multi-process asymmetric data. Paper presented at U.S.-Japan seminar on MDS, San Diego, pp 79–88

  • Young FW (1987) Weighted distance models. In: Young FW, Hamer RM (eds), Multidimensional scaling: History, theory, and applications. Lawrence Erlbaum, Hillsdale, NJ, pp 117–158

  • Young FW, Hamer RM (1987) Multidimensional scaling: History, theory, and applications. Lawrence Erlbaum, Hillsdale, NJ

    Google Scholar 

  • Young FW (1984a) Scaling. Ann Rev Psychol 35:55–81

    Article  Google Scholar 

  • Young FW (1984b) The general euclidean model. In: Law HG et al (eds) Research methods for multimode data analysis. Praeger, New York, pp 440–465

    Google Scholar 

  • Zielman B (2016) Package asymmetry, cran.r-project.org

  • Zielman B (1991) Three-way scaling of asymmetric proximities. Research report R.R.-91-01, Department of Data Theory, Leiden

  • Zielman B (1993) Two methods for multidimensional analysis of three-way skew-symmetric matrices. Research report R.R.-93-01, Department of Data Theory, Leiden

  • Zielman B, Heiser WJ (1991) Spatial representations of asymmetric proximities. Research report R.R.-91-10, Department of Data Theory, Leiden

  • Zielman B, Heiser WJ (1993) Analysis of asymmetry by a slide-vector. Psychometrika 58:101–114

    Article  MATH  Google Scholar 

  • Zielman B, Heiser WJ (1996) Models for asymmetric proximities. Br J Math Stat Psychol 49:127–146

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Giuseppe Bove.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bove, G., Okada, A. Methods for the analysis of asymmetric pairwise relationships. Adv Data Anal Classif 12, 5–31 (2018). https://doi.org/10.1007/s11634-017-0307-9

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11634-017-0307-9

Keywords

Mathematics Subject Classification

Navigation