Skip to main content
SpringerLink
Log in

On piecewise polynomial regression under general dependence conditions, with an application to calcium-imaging data

  • Published:
Sankhya B Aims and scope Submit manuscript

Abstract

Motivated by the analysis of glomerular time series extracted from calcium-imaging data, asymptotic theory for piecewise polynomial and spline regression with partially free knots and residuals exhibiting three types of dependence structures (long memory, short memory and anti-persistence) is considered. Unified formulas based on fractional calculus are derived for subordinated residual processes in the domain of attraction of a Hermite process. The results are applied to testing for the effect of a neurotransmitter on the response of olfactory neurons in honeybees to odorant stimuli.

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

Similar content being viewed by others

References

  • Beran, J. (1991). M-estimators of location for data with slowly decaying serial correlations. J. Amer. Statist. Assoc., 86, 704–708.

    MATH  MathSciNet  Google Scholar 

  • Beran, J. (1994). Statistics for long-memory processes. Chapman and Hall, London.

    MATH  Google Scholar 

  • Beran, J. and Feng, Y. (2001). Local polynomial estimation with a FARIMA–GARCH error process. Bernoulli, 7, 733–750.

    Article  MATH  MathSciNet  Google Scholar 

  • Beran, J. and Feng, Y. (2002a). SEMIFAR models - a semiparametric framework for modelling trends, long-range dependence and nonstationarity. Comput. Stat. Data Anal., 40, 393–419.

    Article  MATH  MathSciNet  Google Scholar 

  • Beran, J. and Feng, Y. (2002b). Data driven bandwidth choice for SEMIFAR models. J. Comput. Graph. Statist., 11, 690–713.

    Article  MathSciNet  Google Scholar 

  • Beran, J. and Feng, Y. (2002c). Local polynomial fitting with long memory, short memory andantipersistent errors. Ann. Inst. Statist. Math., 54, 291–311.

    Article  MATH  MathSciNet  Google Scholar 

  • Beran, J. and Feng, Y. (2007). Weighted averages and local polynomial estimation for fractional linear ARCH processes. J. Stat. Theory Pract., 1, 149–166.

    Article  MATH  MathSciNet  Google Scholar 

  • Beran J. and Ghosh S. (1998). Root-n-consistent estimation in partial linear models with long-memory errors. Scand. J. Stat., 25, 345–357.

    Article  MATH  MathSciNet  Google Scholar 

  • Beran, J. and Weiershäuser, A. (2011). On spline regression under Gaussian subordination with long memory. J. Multivariate Anal., 102, 315–335.

    Article  MATH  MathSciNet  Google Scholar 

  • Bingham, N.H., Goldie, C.M. and Teugels, J.L. (1987). Regular variation. Cambridge University Press.

  • De Boor, C. (2001). A practical guide to splines. Springer, New York.

    MATH  Google Scholar 

  • Brodsky, B.E. and Darkhosky, B.S. (2000). Non-parametric statistical diagnosis: problems and methods. Springer, New York.

    Book  Google Scholar 

  • Chen, J. and Gupta, A.K. (2000). Parametric statistical change point analysis (Oberwolfach seminars). Birkhäuser, Basel.

    Book  Google Scholar 

  • Csörgö, S. and Horvath, L. (1998). Limit theorems in change-point analysis. Wiley, New York.

    Google Scholar 

  • Csörgö, S. and Mielniczuk, J. (1995). Nonparametric regression under long-range dependent normal errors. Ann. Statist., 23, 1000–1014.

    Article  MATH  MathSciNet  Google Scholar 

  • Csörgö, S. and Mielniczuk, J. (1999). Random-design regression under long-range dependent errors. Bernoulli, 5, 209–224.

    Article  MATH  MathSciNet  Google Scholar 

  • Dahlhaus, R. (1995). Efficient location and regression estimation for long range dependent regression models. Ann. Statist., 23, 1029–1047.

    Article  MATH  MathSciNet  Google Scholar 

  • Davydov, J.A. (1970). The invariance principle for stationary processes. Theory Probab. Appl., 15, 487–498.

    Article  Google Scholar 

  • Deo, R.S. (1997). Asymptotic theory for certain regression models with long memory errors. J. Time Series Anal., 18, 385–393.

    Article  MATH  MathSciNet  Google Scholar 

  • Diggle, P.J. and Hutchinson, M.F. (1989). On spline smoothing with autocorrelated errors. Aust. J. Stat., 31, 166–182.

    Article  MATH  MathSciNet  Google Scholar 

  • Dobrushin, R.L. and Major, P. (1979). Non-central limit theorems for nonlinear functionals of Gaussian fields. Z. Wahrsch. Verw. Gebiete, 50, 27–52.

    Article  MATH  MathSciNet  Google Scholar 

  • Doukhan, P., Oppenheim, G. and Taqqu, M.S. (2002). Theory and application of long-range dependence. Birkhäuser, Boston.

    Google Scholar 

  • Eubank, R.L. (1999). Nonparametric regression and spline smoothing, 2nd edition. Marcel Dekker, New York.

    MATH  Google Scholar 

  • Feder, P.I. (1975). On asymptotic distribution theory in segmented regression problems - identified case. Ann. Statist., 3, 49–83.

    Article  MATH  MathSciNet  Google Scholar 

  • Fox, R. And Taqqu, M.S. (1986). Large-sample properties of parameter estimates for strongly dependent stationary Gaussian time series. Ann. Statist., 14, 517–532.

    Article  MATH  MathSciNet  Google Scholar 

  • Galizia, C.G. and Menzel, R. (2001). The role of glomeruli in the neural representation of odors: results from optical recording studies. J. Insect Physiol., 47, 115–129.

    Article  Google Scholar 

  • Galizia, C.G. and Kimmerle, B. (2004). Physiological and morphological characterization of honeybee olfactory neurons combining electrophysiology, calcium imaging and confocal microscopy. J. Comput. Physiol. A Neuroethol. Sens. Neural Behav. Physiol., 190, 21–38.

    Article  Google Scholar 

  • Gallant, A.R. (1974). The theory of nonlinear regression as it relates to segmented polynomial regression with estimated join points. Mimeograph Series No. 925, Institute of Statistics, North Carolina State University, Raleigh.

  • Gallant, A.R. and Goebel, J.J. (1975). Nonlinear regression with autoregrressive errors. Insitute of Statistics Mimeograph Series No. 986. Institute of Statistics, North Carolina State University, Raleigh.

  • Gao, J.T. and Anh V.V. (1999). Semiparametric regression under long-range dependent errors. J. Statist. Plann. Inference., 80, 37–57.

    Article  MATH  MathSciNet  Google Scholar 

  • Giraitis, L. and Surgailis, D. (1985). CLT and other limit theorems for functionals of Gaussian processes. Z. Wahrsch. Verw. Gebiete, 70, 191–212.

    Article  MATH  MathSciNet  Google Scholar 

  • Giraitis, L. and Surgailis, D. (1990). A central limit theorem for quadratic forms in strongly dependent linear variables and application to asymptotical normality of Whittle’s estimate. Probab. Theory Related Fields, 86, 87–104.

    Article  MATH  MathSciNet  Google Scholar 

  • Gradshteyn, I.S. and Rhyzhik, I.M. (1965). Tables of integrals, series and products. Academic Press.

  • Granger, C.W.J. and Joyeux, R. (1980). An introduction to long-memory time series. J. Time Ser. Anal., 1, 15–30.

    Article  MATH  MathSciNet  Google Scholar 

  • Green, P.J. and Silverman, B.W. (1994). Nonparametric regression and generalized linear models: a roughness penalty approach. Chapman and Hall, London.

    Book  MATH  Google Scholar 

  • Grohmann, L., Wolfgang Blenau, Erber, J., Ebert, P.R., Strünker, T. and Baumann, A. (2003). Molecular and functional characterization of an octopamine receptor from honeybee (Apis mellifera) brain. J. Neurochemistry, 86, 725–735.

    Article  Google Scholar 

  • Hall, P. and Hart, J.D. (1990). Nonparametric regression with long-range dependence. Stoch. Proc. Appl., 36, 339–351.

    Article  MATH  MathSciNet  Google Scholar 

  • Hall, P., Jing, B.-Y. and Lahiri, S. N. (1998). On the sampling window method for long-range dependent data. Statist. Sinica, 8, 1189–1204.

    MATH  MathSciNet  Google Scholar 

  • Hammer, M. (1993). An identified neuron mediates the unconditioned stimulus in associative olfactory learning in honeybees. Nature, 366, 59–63.

    Article  Google Scholar 

  • Hannan, E.J. (1973). Central limit theorems for time series regression. Z. Wahrsch. verw. Geb., 26, 157–170.

    Article  MATH  MathSciNet  Google Scholar 

  • Ho, H.C. and Hsing, T. (1996). On the asymptotic expansion of the empirical process of long-memory moving averages. Ann. Statist., 24, 992–1024.

    Article  MATH  MathSciNet  Google Scholar 

  • Ho, H.C. and Hsing, T. (1997). Limit theorems for functionals of moving averages. Ann. Probab., 25, 1636–1669.

    Article  MATH  MathSciNet  Google Scholar 

  • Hosking, J.R.M. (1981). Fractional differencing. Biometrika, 68, 165–176.

    Article  MATH  MathSciNet  Google Scholar 

  • Hsing, T. (2000). Linear processes, long-range dependence and asymptotic expansions. (English summary) 19th “Rencontres Franco-Belges de Statisticiens” (Marseille, 1998). Stat. Inference Stoch. Process., 3, 19–29.

    Article  MATH  MathSciNet  Google Scholar 

  • Ivanov, A.V. and Leonenko, N.N. (2001). Asymptotic inference for a nonlinear regression with long range dependent errors. Theory Probab. Math. Statist., 63, 65–83.

    MathSciNet  Google Scholar 

  • Ivanov, A.V. and Leonenko, N.N. (2004). Asymptotic theory of non-linear regression with long range dependent errors. Math. Methods Statist., 13, 153–178.

    MATH  MathSciNet  Google Scholar 

  • Kim, J. and Kim, H.J. (2008). Asymptotic results in segmented multiple regression. J. Multivariate Anal., 99, 2016–2038.

    Article  MATH  MathSciNet  Google Scholar 

  • Kohn, R., Ansley, C.F. and Wong, C.M. (1992). Nonparametric spline regression with autoregressive moving average errors. Biometrika, 79, 335–346.

    Article  MATH  MathSciNet  Google Scholar 

  • Koul, H.L. (1996). Asymptotics of M-estimators in non-linear regression with long-range dependent errors. In Athens Conference on Applied Probability and Time Series Volume II: Time Series Analysis in Memory of E.J. Hannan, P.M. Robinson, and M. Rosenblatt (eds.), Lecture Notes in Statistics, Vol 115, pp. 272–290. Springer.

  • Koul, H.L. and Baillie, R.T. (2003). Asymptotics of M-estimators in non-linear regression with long memory designs. Statist. Probab. Lett. 61, 237–252.

    Article  MATH  MathSciNet  Google Scholar 

  • Künsch, H.R., Beran, J. and Hampel, F. (1993). Contrasts under long-range correlations. Ann. Statist., 21, 943–964.

    Article  MATH  MathSciNet  Google Scholar 

  • Lang, G. and Soulier, P. (2000). Convergence de mesures spectrales aléatoires et applications à des principes d’invariance. (French) [Convergence of random spectral measures and applications to invariance principles] 19th “Rencontres Franco-Belges de Statisticiens” (Marseille, 1998). Stat. Inference Stoch. Process., 3, 41–51.

    Article  MATH  MathSciNet  Google Scholar 

  • Liu, J., Wu, S. And Zidek, J.V. (1997). On segmented multivariate regression. Statist. Sinica, 7, 497–525.

    MATH  MathSciNet  Google Scholar 

  • Lowen, S.B. and Teich, M.C. (2005). Fractal based point processes. Wiley, New York.

    Book  MATH  Google Scholar 

  • Maejima, M. And Tudor, C.A. (2007). Wiener integrals with respect to the hermite process and a non-central limit theorem. Stoch. Anal. Appl., 25, 1043–1056.

    Article  MATH  MathSciNet  Google Scholar 

  • Palma, w. (2007). Long-memory time series - theory and methods. Wiley, New York.

    Book  MATH  Google Scholar 

  • Pipiras, V. and Taqqu, M.S. (2000a). Integration questions related to fractional Brownian motion. Probab. Theory Related Fields 118, 251–291.

    Article  MATH  MathSciNet  Google Scholar 

  • Pipiras, V. and Taqqu, M.S. (2000b). Convergence of weighted sums of random variables with long-range dependence. Stoch. Proc. Appl., 90, 157–174.

    Article  MATH  MathSciNet  Google Scholar 

  • Pipiras, V. and Taqqu, M.S. (2003). Fractional calculus and its connect on to fractional Brownian motion. In Long Range Dependence, pp. 166–201. Birkhäuser, Basel.

    Google Scholar 

  • Rein, J., Strauch, M. and Galizia, C.G. (2009). Novel techniques for the exploration of the honeybee antennal lobe (poster abstract). In Proc. of the 8th Meeting of the German Neuroscience Society, Göttingen, Germany, Mar 25–29.

  • Robinson, P.M. (1991). Nonparametric function estimation for long-memory time series. In Nonparametric and Semiparametric Methods in Econometrics and Statistics (W. Barnett, J. Powell and G. Tauchen, eds.), pp. 437–457. Cambridge University Press.

  • Sachse, S. and C. G. Galizia (2002). Role of inhibition for temporal and spatial odor representation in olfactory output neurons: a calcium imaging study. J. Neurophysiol., 87, 1106–1117.

    Google Scholar 

  • Sachse, S., Peele, P., Silbering, A.F., GÜhmann, M. and C. G. Galizia (2006). Role of histamine as a putative inhibitory transmitter in the honeybee antennal lobe. Front. Zool., 3, 22.

    Google Scholar 

  • Samko, S.G., Kilbas, A.A. and Marichev, O.I. (1987). Integrals and derivatives of fractional order and some its applications. In (Nauka i Tehnika, Minsk, 1987) or Fractional Integrals and Derivatives Theory and Applications (Gordon and Breach, New York, 1993).

  • Seber, G.A.F. and Wild, C.J. (2003). Nonlinear regression. Wiley, New York.

    Google Scholar 

  • Strauch, M. and Galizia, C.G. (2008). Registration to a neuroanatomical reference atlas - identifying glomeruli in optical recordings of the honeybee brain. In Proc. of the GCB 2008, September 9–12, 2008, Dresden, Germany, LNI, Vol. 136, pp. 85–95.

  • Surgailis, D. (2003). CLTs for polynomials of linear sequences: Diagram formula with illustrations. In Theory and Applications of Long-range Dependence, (P. Doukhan, G. Oppenheim and M.S. Taqqu eds.), pp. 111–127. Birkhäuser Boston, Boston, MA.

  • Taqqu, M.S. (1975). Weak convergence to fractional Brownian motion and to the Rosenblatt process. Z. Wahrsch. und Verw. Gebiete, 31, 287–302.

    Article  MATH  MathSciNet  Google Scholar 

  • Taqqu, M.S. (1978). A representation for self-similar processes. Stoch. Proc. Appl., 7, 55–64.

    Article  MATH  MathSciNet  Google Scholar 

  • Taqqu, M.S. (2003). Fractional Brownian motion and long range dependence. In Long Range Dependence, pp. 5–38. Birkhäuser, Basel.

  • Wahba, G. (1990). Spline models for observational data. In Regional Conference Series in Applied Mathematics. SIAM.

  • Wang, Y. (1998). Smoothing splines models with correlated random errors. J. Amer. Statist. Assoc., 93, 341–348.

    Article  MATH  Google Scholar 

  • Yajima Y. (1988). On estimation of a regression model with long term errors. Ann. Statist., 16, 791–807.

    Article  MATH  MathSciNet  Google Scholar 

  • Yajima Y. (1991). Asymptotic properties of the LSE in a regression model with long-memory stationary errors. Ann. Statist., 19, 158–177.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Konstanz, 78457, Konstanz, Germany

    Jan Beran & Arno Weiershäuser

  2. Department of Neurobiology, University of Konstanz, 78457, Konstanz, Germany

    C. Giovanni Galizia, Julia Rein & Martin Strauch

  3. School of Life Sciences, Arizona State University Tempe, Arizona, 85287-4501, USA

    Brian H. Smith

Authors
  1. Jan Beran
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Arno Weiershäuser
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. C. Giovanni Galizia
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Julia Rein
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Brian H. Smith
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Martin Strauch
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jan Beran.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Beran, J., Weiershäuser, A., Galizia, C.G. et al. On piecewise polynomial regression under general dependence conditions, with an application to calcium-imaging data. Sankhya B 76, 49–81 (2014). https://doi.org/10.1007/s13571-013-0066-3

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13571-013-0066-3

Keywords

AMS (2000) subject classification.

Search

Navigation