[go: up one dir, main page]
More Web Proxy on the site http://driver.im/ Skip to main content
Springer Nature Link
Log in

An acceleration method for Ten Berge et al.’s algorithm for orthogonal INDSCAL

  • Original Paper
  • Published:
Computational Statistics Aims and scope Submit manuscript

Abstract

INDSCAL (INdividual Differences SCALing) is a useful technique for investigating both common and unique aspects of K similarity data matrices. The model postulates a common stimulus configuration in a low-dimensional Euclidean space, while representing differences among the K data matrices by differential weighting of dimensions by different data sources. Since Carroll and Chang proposed their algorithm for INDSCAL, several issues have been raised: non-symmetric solutions, negative saliency weights, and the degeneracy problem. Orthogonal INDSCAL (O-INDSCAL) which imposes orthogonality constraints on the matrix of stimulus configuration has been proposed to overcome some of these difficulties. Two algorithms have been proposed for O-INDSCAL, one by Ten Berge, Knol, and Kiers, and the other by Trendafilov. In this paper, an acceleration technique called minimal polynomial extrapolation is incorporated in Ten Berge et al.’s algorithm. Simulation studies are conducted to compare the performance of the three algorithms (Ten Berge et al.’s original algorithm, the accelerated algorithm, and Trendafilov’s). Possible extensions of the accelerated algorithm to similar situations are also suggested.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
£29.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (United Kingdom)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

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

    Google Scholar 

  • Bennani Dosse M, Ten Berge JMF (2008) The assumption of proportional components when Candecomp is applied to symmetric matrices in the context of Indscal. Psychometrika 73: 303–307

    Article  MATH  MathSciNet  Google Scholar 

  • Carroll JD, Chang JJ (1970) Individual differences and multidimensional scaling via an N-way generalization of Eckart-Young decomposition. Psychometrika 35: 282–319

    Article  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 the Psychometric Society, Princeton

  • Carroll JD, De Soete G, Pruzansky S (1989) An evaluation of five algorithms for generating an initial configuration for SINDSCAL. J Classif 6: 105–119

    Article  Google Scholar 

  • Clarkson DB (1988) A least squares version of algorithm AS 211: the F-G diagonalization algorithm. Appl Stat-J Roy St C 37: 317–321

    Google Scholar 

  • De Leeuw J, Pruzansky S. (1978) A new computational method to fit the weighted euclidean distance model. Psychometrika 43: 479–490

    Article  MATH  MathSciNet  Google Scholar 

  • Fletcher R (1981) Practical methods of optimization, constrained optimization, vol. 2. Wiley, Chichester

    Google Scholar 

  • Flury B (1988) Common principal components and related multivariate models. Wiley, New York

    MATH  Google Scholar 

  • Flury B, Gautschi W (1986) An algorithm for simultaneous orthogonal transformation of several positive definite symmetric matrices to nearly diagonal form. SIAM J Sci Stat Comp 7: 169–184

    Article  MATH  MathSciNet  Google Scholar 

  • Golub G, van Loan C (1996) Matrix computations, 3rd edn. The Johns Hopkins University Press, London

    MATH  Google Scholar 

  • Harshman 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 of mathematical psychology, Hamilton, ON

  • Helm CE (1964) Multidimensional ratio scaling analysis of perceived color relations. J Opt Soc Am 54: 256–262

    Article  Google Scholar 

  • Jacobowitz D (1975) The acquisition of semantic structures. Unpublished doctoral dissertation, University of North Carolina.

  • Jennrich RI (2001) A simple general procedure for orthogonal rotation. Psychometrika 66: 289–306

    Article  MathSciNet  Google Scholar 

  • Kaiser HF (1958) The varimax criterion for analytic rotation in factor analysis. Psychometrika 23: 187–200

    Article  MATH  Google Scholar 

  • Kiers HAL (1989a) A computational short-cut for INDSCAL with orthonormality constraints on positive semi-definite matrices of low rank. Comput Stat Quart 2: 119–135

    Google Scholar 

  • Kiers HAL (1989b) An alternating least squares algorithm for fitting the two- and three-way DEDICOM model and the IDIOSCAL model. Psychometrika 54: 515–521

    Article  Google Scholar 

  • Kiers HAL (1991) Hierarchical relations among three-way methods. Psychometrika 56: 449–470

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

  • Krijnen WP, Dijkstra TK, Stegeman A (2008) On the non-existence of optimal solutions and the occurrence of “degeneracy” in the Candecomp/Parafac model. Psychometrika 73: 431–439

    Article  MathSciNet  Google Scholar 

  • Kroonenberg PM (1983) Three-mode principal component analysis. DSWO Press, Leiden

    Google Scholar 

  • Lingoes JC, Roskam EE (1973) A mathematical and empirical analysis of two multidimensional scaling algorithms. Psychometrika Monograph Supplement 19

  • Loisel S, Takane M (2009) Fast indirect robust generalized method of moments. Comput Stat and Data An 53: 3571–3579

    Article  MATH  Google Scholar 

  • Nguyen TK, Oshima-Takane Y (2008) Do 2- to 3-year-old children use functional or shape cues to name objects? Poster presented at the International Society on Infant Studies, Vancouver, BC.

  • Rosenberg S, Kim MP (1975) The method of sorting as a data-gathering procedure in multivariate research. Multivar Behav Res 10: 489–502

    Article  Google Scholar 

  • Schiffman SS, Reynolds ML, Young FW (1981) Introduction to Multidimensional Scaling. Academic Press, New York

    MATH  Google Scholar 

  • Smith DA, Ford WF, Sidi A (1987) Extrapolation methods for vector sequences. SIAM Review 29: 199–233

    Article  MATH  MathSciNet  Google Scholar 

  • Stegeman A (2007) Degeneracy in Candecomp/Parafac explained for p × p × 2 arrays of rank p + 1 or higher. Psychometrika 71: 483–501

    Article  MathSciNet  Google Scholar 

  • Takane Y (2007) Applications of multidimensional scaling in psychometrics. In: Rao CR, Sinharay S (eds) Handbook of statistics: Pyschometrics, vol 26. Elsevier BV, Amsterdam, pp 359–400

    Google Scholar 

  • Takane Y, Zhang Z Algorithms for DEDICOM: acceleration, deceleration, or neither? J Chemometr (in press)

  • Ten Berge JMF (1984) A joint treatment of varimax and the problem of diagonalizing symmetric matrices simultaneously in the least squares sense. Psychometrika 49: 347–358

    Article  MATH  MathSciNet  Google Scholar 

  • Ten Berge JMF, Kiers HAL (1991) Some clarifiation of the CANDECOMP algorithm applied to INDSCAL. Psychometrika 56: 317–326

    Article  MATH  MathSciNet  Google Scholar 

  • Ten Berge JMF, Kiers HAL, De Leeuw J (1988a) Explicit candecomp/parafac solutions for a contrived 2 ×  2 × 2 array of rank three. Psychometrika 53: 579–584

    Article  MATH  MathSciNet  Google Scholar 

  • Ten Berge JMF, Knol DL, Kiers HAL (1988b) A treatment of the orthomax rotation family in terms of diagonalization, and a re-examination of a singular value approach to varimax rotation. Comput Stat Quart 3: 207–217

    Google Scholar 

  • Ten Berge JMF, Kiers HAL, Krijnen WP (1993) Computational solutions for the problem of negative saliences and nonsymmetry in INDSCAL. J Classif 10: 115–124

    Article  MATH  Google Scholar 

  • Trendafilov N (2002) GIPSCAL revisited: a projected gradient approach. Stat Comput 12: 135–145

    Article  MathSciNet  Google Scholar 

  • Trendafilov N (2004) Orthonormality-constrained INDSCAL with nonnegative salience. In: Legan A et al (eds) Computational science and its applications Lecture Notes in Computer Science 3044 Part II. Springer, Berlin, pp 952–960

    Google Scholar 

  • Zhang Z, Takane Y Statistics: multidimensional scaling. In: Baker E, McGaw B, Peterson PP (eds) International encyclopedia of education, 3rd edn. Elsevier, Oxford (in press)

Download references

Author information

Authors and Affiliations

  1. Department of Psychology, McGill University, 1205 Dr. Penfield Ave, Montreal, QC, H3A 1B1, Canada

    Yoshio Takane, Kwanghee Jung & Heungsun Hwang

Authors
  1. Yoshio Takane
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kwanghee Jung
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Heungsun Hwang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yoshio Takane.

Additional information

The work reported in this paper has been supported by SSHRC Research Grant 36952 to the first author, and by NSERC Discovery Grant 290439 to the third author.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Takane, Y., Jung, K. & Hwang, H. An acceleration method for Ten Berge et al.’s algorithm for orthogonal INDSCAL. Comput Stat 25, 409–428 (2010). https://doi.org/10.1007/s00180-010-0184-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00180-010-0184-6

Keywords

Mathematics Subject Classification (2000)

Search

Navigation