Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
Skip to main content

Advertisement

SpringerLink
Log in

Fast computation of spatially adaptive kernel estimates

  • Published:
Statistics and Computing Aims and scope Submit manuscript

Abstract

Kernel smoothing of spatial point data can often be improved using an adaptive, spatially varying bandwidth instead of a fixed bandwidth. However, computation with a varying bandwidth is much more demanding, especially when edge correction and bandwidth selection are involved. This paper proposes several new computational methods for adaptive kernel estimation from spatial point pattern data. A key idea is that a variable-bandwidth kernel estimator for d-dimensional spatial data can be represented as a slice of a fixed-bandwidth kernel estimator in \((d+1)\)-dimensional scale space, enabling fast computation using Fourier transforms. Edge correction factors have a similar representation. Different values of global bandwidth correspond to different slices of the scale space, so that bandwidth selection is greatly accelerated. Potential applications include estimation of multivariate probability density and spatial or spatiotemporal point process intensity, relative risk, and regression functions. The new methods perform well in simulations and in two real applications concerning the spatial epidemiology of primary biliary cirrhosis and the alarm calls of capuchin monkeys.

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
$34.99 /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 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
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13

Similar content being viewed by others

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

References

  • Abramson, I.S.: On bandwidth estimation in kernel estimates—a square root law. Ann. Stat. 10(4), 1217–1223 (1982)

    Article  MATH  Google Scholar 

  • Baddeley, A.J., Turner, R.: spatstat: an R package for analyzing spatial point patterns. J. Stat. Softw. 12(6), 1–42 (2005)

    Article  Google Scholar 

  • Baddeley, A., Rubak, E., Turner, R.: Spatial Point Patterns: Methodology and Applications with R. Chapman and Hall/CRC Press, London (2015)

    MATH  Google Scholar 

  • Bernacchia, A., Pigolotti, S.: Self-consistent method for density estimation. J. R. Stat. Soc. Ser. B 73, 407–422 (2011)

    Article  MathSciNet  Google Scholar 

  • Bithell, J.F.: An application of density estimation to geographical epidemiology. Stat. Med. 9, 691–701 (1990)

    Article  Google Scholar 

  • Bithell, J.F.: Estimation of relative risk functions. Stat. Med. 10, 1745–1751 (1991)

    Article  Google Scholar 

  • Breiman, L., Meisel, W., Purcell, E.: Variable kernel estimates of multivariate densities. Technometrics 19, 135–144 (1977)

    Article  MATH  Google Scholar 

  • Campos, F.A., Fedigan, L.M.: Spatial ecology of perceived predation risk and vigilance behavior in white-faced capuchins. Behav. Ecol. 25, 477–486 (2014)

    Article  Google Scholar 

  • Chaudhuri, P., Marron, J.: Scale space view of curve estimation. Ann. Stat. 28, 408–428 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  • Cooley, J.W., Tukey, J.W.: An algorithm for the machine calculation of complex Fourier series. Math. Comput. 19, 297–301 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  • Davies, T.M., Hazelton, M.L.: Adaptive kernel estimation of spatial relative risk. Stat. Med. 29(23), 2423–2437 (2010)

    MathSciNet  Google Scholar 

  • Davies, T.M., Jones, K., Hazelton, M.L.: Symmetric adaptive smoothing regimens for estimation of the spatial relative risk function. Comput. Stat. Data Anal. 101, 12–28 (2016)

    Article  MathSciNet  Google Scholar 

  • Diggle, P.J.: A kernel method for smoothing point process data. Appl. Stat. 34(2), 138–147 (1985)

    Article  MATH  Google Scholar 

  • Diggle, P.: Nonparametric methods. In: Gelfand, A., Diggle, P., Fuentes, M., Guttorp, P. (eds.) Handbook of Spatial Statistics, Chapter 18, pp. 299–316. CRC Press, Boca Raton (2010)

    Chapter  Google Scholar 

  • Diggle, P.J., Marron, J.S.: Equivalence of smoothing parameter selectors in density and intensity estimation. J. Am. Stat. Assoc. 83, 793–800 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  • Duong, T.: ks: kernel density estimation and kernel discriminant analysis for multivariate data in R. J. Stat. Softw. 21(7), 1–16 (2007)

    Article  Google Scholar 

  • Duong, T., Hazelton, M.L.: Plug-in bandwidth matrices for bivariate kernel density estimation. J. Nonparametr. Stat. 15(1), 17–30 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  • Duong, T., Hazelton, M.L.: Cross-validation bandwidth matrices for multivariate kernel density estimation. Scand. J. Stat. 32, 485–506 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  • Faraway, J.J., Jhun, M.: Bootstrap choice of bandwidth for density estimation. J. Am. Stat. Assoc. 85, 1119–1122 (1990)

    Article  MathSciNet  Google Scholar 

  • Fernando, W.T.P.S., Hazelton, M.L.: Generalizing the spatial relative risk function. Spat. Spatio-Temporal Epidemiol. 8, 1–10 (2014)

    Article  Google Scholar 

  • Gramacki, A., Gramacki, J.: FFT-based fast bandwidth selector for multivariate kernel density estimation. Comput. Stat. Data Anal. 106, 27–45 (2017a)

    Article  MathSciNet  Google Scholar 

  • Gramacki, A., Gramacki, J.: FFT-based fast computation of multivariate kernel density estimators with unconstrained bandwidth matrices. J. Comput. Graph. Stat. 26, 459–462 (2017b)

    Article  MathSciNet  Google Scholar 

  • Hall, P., Marron, J.S.: Variable window width kernel density estimates of probability densities. Probab. Theory Relat. Fields 80, 37–49 (1988)

    Article  MATH  Google Scholar 

  • Hall, P., Marron, J.S., Park, B.U.: Smoothed cross-validation. Probab. Theory Relat. Fields 92, 1–20 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  • Harris, F.: On the use of windows for harmonic analysis with the discrete Fourier transform. Proc. IEEE 66(1), 51–83 (1978)

  • Hu, S., Poskitt, D.S., Zhang, X.: Bayesian adaptive bandwidth kernel density estimation of irregular multivariate distributions. Comput. Stat. Data Anal. 56, 732–740 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  • Jones, M.C.: Simple boundary correction for kernel density estimation. Stat. Comput. 3, 135–146 (1993)

    Article  Google Scholar 

  • Jones, M.C., Marron, J.S., Sheather, S.J.: A brief survey of bandwidth selection for density estimation. J. Am. Stat. Assoc. 91, 401–407 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  • Kelsall, J.E., Diggle, P.J.: Kernel estimation of relative risk. Bernoulli 1, 3–16 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  • Koch, K., Ohser, J., Schladitz, K.: Spectral theory for random closed sets and estimating the covariance via frequency space. Adv. Appl. Probab. 35, 603–613 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  • Loftsgaarden, D.O., Quesenberry, C.P.: A nonparametric estimate of a multivariate density function. Ann. Math. Stat. 36, 1049–1051 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  • Marshall, J.C., Hazelton, M.L.: Boundary kernels for adaptive density estimators on regions with irregular boundaries. J. Multivar. Anal. 101, 949–963 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  • Møller, J., Syversveen, A.R., Waagepetersen, R.P.: Log Gaussian Cox processes. Scand. J. Stat. 25, 451–482 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  • O’Brien, T.A., Kashinath, K., Cavanaugh, N.R., Collins, W.D., O’Brien, J.P.: A fast and objective multidimensional kernel density estimation method: fastKDE. Comput. Stat. Data Anal. 101, 148–160 (2016)

    Article  MathSciNet  Google Scholar 

  • Pinsky, M.: Introduction to Fourier Analysis and Wavelets. No. 102 in Graduate Studies in Mathematics. American Mathematical Society, Providence (2002)

    Google Scholar 

  • Prince, M.I., Chetwynd, A., Diggle, P.J., Jarner, M., Metcalf, J.V., James, O.F.W.: The geographical distribution of primary biliary cirrhosis in a well-defined cohort. Hepatology 34, 1083–1088 (2001)

    Article  Google Scholar 

  • R Core Team: R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna (2017). https://www.R-project.org/

  • Sain, S.R.: Multivariate locally adaptive density estimation. Comput. Stat. Data Anal. 39, 165–186 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  • Sain, S.R., Scott, D.W.: On locally adaptive density estimation. J. Am. Stat. Assoc. 91, 1525–1534 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  • Sain, S.R., Baggerly, K.A., Scott, D.W.: Cross-validation of multivariate densities. J. Am. Stat. Assoc. 89, 807–817 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  • Silverman, B.W.: Kernel density estimation using the fast Fourier transform. Appl. Stat. 31, 93–99 (1982)

    Article  MATH  Google Scholar 

  • Silverman, B.W.: Density Estimation for Statistics and Data Analysis. Chapman and Hall, New York (1986)

    Book  MATH  Google Scholar 

  • Taylor, C.C.: Bootstrap choice of the smoothing parameter in kernel density estimation. Biometrika 76, 705–712 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  • Terrell, G.R.: The maximal smoothing principle in density estimation. J. Am. Stat. Assoc. 85, 470–477 (1990)

    Article  MathSciNet  Google Scholar 

  • Terrell, G.R., Scott, D.W.: Variable kernel density estimation. Ann. Stat. 20, 1236–1265 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  • Wand, M.P.: Fast computation of multivariate kernel estimators. J. Comput. Graph. Stat. 3(4), 433–445 (1994)

    MathSciNet  Google Scholar 

  • Wand, M.P., Jones, M.C.: Comparison of smoothing parameterizations in bivariate kernel density estimation. J. Am. Stat. Assoc. 88, 520–528 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  • Wand, M.P., Jones, M.C.: Kernel Smoothing. Chapman and Hall, Boca Raton (1995)

    Book  MATH  Google Scholar 

  • Zhang, X., King, M.L., Hyndman, R.J.: A Bayesian approach to bandwidth selection for multivariate kernel density estimation. Comput. Stat. Data Anal. 50, 3009–3031 (2006)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors gratefully acknowledge Fernando A. Campos and Linda M. Fedigan for providing the capuchin data and advice related to their original analysis. Two referees and an associate editor are thanked for their constructive feedback.

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Otago, PO Box 56, Dunedin, 9054, New Zealand

    Tilman M. Davies

  2. Department of Mathematics and Statistics, Curtin University, GPO Box U1987, Perth, WA, 6845, Australia

    Adrian Baddeley

  3. Data61, CSIRO, Perth, WA, Australia

    Adrian Baddeley

Authors
  1. Tilman M. Davies
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Adrian Baddeley
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tilman M. Davies.

Additional information

TMD was supported in part by the Royal Society of New Zealand, Marsden Fast-start Grant 15-UOO-092: Smoothing and inference for point process data with applications to epidemiology. AB was supported in part by the Australian Research Council, Discovery Outstanding Researcher Award DP130104470.

Electronic supplementary material

Below is the link to the electronic supplementary material.

Supplementary material 1 (pdf 474 KB)

Supplementary material 2 (gz 18 KB)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Davies, T.M., Baddeley, A. Fast computation of spatially adaptive kernel estimates. Stat Comput 28, 937–956 (2018). https://doi.org/10.1007/s11222-017-9772-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11222-017-9772-4

Keywords

Search

Navigation