Skip to main content
SpringerLink
Log in

Fast algorithms for a space-time concordance measure

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

Abstract

This paper presents a number of algorithms for a recently developed measure of space-time concordance. Based on a spatially explicit version of Kendall’s \(\tau \) the original implementation of the concordance measure relied on a brute force \(O(n^2)\) algorithm which has limited its use to modest sized problems. Several new algorithms have been devised which move this run time to \(O(n log(n) +np)\) where \(p\) is the expected number of spatial neighbors for each unit. Comparative timing of these alternative implementations reveals dramatic efficiency gains in moving away from the brute force algorithms. A tree-based implementation of the spatial concordance is also found to dominate a merge sort implementation.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

Notes

  1. For an overview of this literature see Rey and Le Gallo (2009) and references cited therein.

  2. The decomposition here is with respect to pairs of observations \((i,j)\). Other decompositional approaches in the literature are taken with respect to subsets of the observations \(i=1,2,\ldots ,n\). See for example, D’Agostino and Dardanoni (2009).

  3. GDP per capita is expressed in constant 1995 basis pesos.

  4. Due to their long-run times the values reported in Table 2 for the brute force implementations are based on a single sample for each sample size. Because of this the brute force implementations are omitted from Table 3.

  5. Additional simulations explored the impact of ties on the relative performance of the alternative approaches. Ties do not affect the main quantitative or qualitative findings reported here. Results are available upon request.

References

  • Abdi H (2007) The Kendall rank correlation coefficient. Encyclopedia of Measurement and Statistics Sage, Thousand Oaks, CA, pp 508–510

  • Boyle GE, McCarthy TG (1997) A simple measure of \(\beta \)-convergence. Oxf Bull Econ Stat XXVIII(1):155–211

    Google Scholar 

  • Chapelle O, Sindhwani V, Keerthi S (2008) Optimization techniques for semi-supervised support vector machines. J Mach Learn Res 9:203–233

    MATH  Google Scholar 

  • Cherubini U, Luciano E (2001) Value-at-risk trade-off and capital allocation with copulas. Econ Notes 30(2):235–256

    Article  Google Scholar 

  • Christensen D (2005) Fast algorithms for the calculation of Kendall’s \(\tau \). Comput Stat 20(1):51–62

    Article  MATH  Google Scholar 

  • Cuadrado-Roura J (2009) Regional policy, economic growth and convergence: lessons from the Spanish case. Springer, Berlin

    Google Scholar 

  • D’Agostino M, Dardanoni V (2009) The measurement of rank mobility. J Econ Theory 144(4):1783–1803

    Article  MATH  MathSciNet  Google Scholar 

  • Dardanoni V, Lambert P (2001) Horizontal inequity comparisons. Soc Choice Welf 18(4):799–816

    Article  MATH  MathSciNet  Google Scholar 

  • Dwork C, Kumar R, Naor M, Sivakumar D (2001) Rank aggregation methods for the web. In: Proceedings of the 10th international conference on World Wide Web, ACM, pp 613–622

  • Fagin R, Kumar R, Sivakumar D (2003) Efficient similarity search and classification via rank aggregation. In: Proceedings of the 2003 ACM SIGMOD international conference on Management of data, ACM, pp 301–312

  • Fairbairn D (2006) Measuring map complexity. Cartogr J 43(3):224–238

    Article  Google Scholar 

  • Genest C, Nešlehová J (2011) Estimators based on Kendall’s Tau in multivariate copula models. Aust N Z J Stat 53(2):157–177

    Article  MATH  MathSciNet  Google Scholar 

  • Glick B (1982) A spatial rank-order correlation measure. Geogr Anal 14(2):177–181

    Article  Google Scholar 

  • Jones E, Oliphant T, Peterson P, et al (2001) SciPy: Open source scientific tools for Python. http://www.scipy.org/

  • Kendall MG (1962) Rank correlation methods, 3rd edn. Griffin, London

  • Kepner W, Watts C, Edmonds C, Maingi J, Marsh S, Luna G (2000) A landscape approach for detecting and evaluating change in a semi-arid environment. Environ Monit Assess 64(1):179–195

    Article  Google Scholar 

  • Knight W (1966) A computer method for calculating Kendall’s Tau with ungrouped data. J Am Stat Assoc 61:436–439

    Google Scholar 

  • Kundzewicz Z, Robson A (2004) Change detection in hydrological records: a review of the methodology/revue méthodologique de la détection de changements dans les chroniques hydrologiques. Hydrol Sci J 49(1):1–19

    Google Scholar 

  • MacEachren A (1982) Map complexity: comparison and measurement. Cartogr Geogr Inf Sci 9(1):31–46

    Google Scholar 

  • Nelsen R (1991) Copulas and association. In: DallÁglio G, Kotz S, Aalinetti G (eds) Advances in probability distributions with given marginals. Kluwer, Dordrecth, pp 51–74

    Chapter  Google Scholar 

  • Rey SJ (2004) Spatial dependence in the evolution of regional income distributions. In: Getis A, Múr J, Zoeller H (eds) Spatial econometrics and spatial statistics. Palgrave, Hampshire, pp 193–213

    Google Scholar 

  • Rey SJ, Anselin L (2010) PySAL: a python library of spatial analytical methods. In: Fischer MM, Getis A (eds) Handbook of applied spatial analysis. Springer, Berlin, pp 175–193

    Chapter  Google Scholar 

  • Rey SJ, Le Gallo J (2009) Spatial analysis of economic convergence. In: Mills TC, Patterson K (eds) Handbook of econometrics, vol II. Applied EconometricsPalgrave Macmillan, New York, pp 1251–1290

  • Rey SJ, Sastré-Gutiérrez ML (2010) Interregional inequality dynamics in Mexico. Spatial Econ Anal 5(3):277–298

    Article  Google Scholar 

  • Schweizer B (1991) Thirty years of copulas. In: DallÁglio G, Kotz S, Aalinetti G (eds) Advances in probability distributions with given marginals. Kluwer, Dordrecth, pp 13–50

    Chapter  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Arizona State University, Tempe, AZ, USA

    Sergio J. Rey

Authors
  1. Sergio J. Rey
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sergio J. Rey.

Additional information

This research was funded by NSF Award OCI-1047916, SI2-SSI: CyberGIS Software Integration for Sustained Geospatial Innovation.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Rey, S.J. Fast algorithms for a space-time concordance measure. Comput Stat 29, 799–811 (2014). https://doi.org/10.1007/s00180-013-0461-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00180-013-0461-2

Keywords

Search

Navigation