Lévy-based growth models

Kristjana Ýr Jónsdóttir, Jürgen Schmiegel, and Eva B. Vedel Jensen

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In the present paper, we give a condensed review, for the nonspecialist reader, of a new modelling framework for spatio-temporal processes, based on Lévy theory. We show the potential of the approach in stochastic geometry and spatial statistics by studying Lévy-based growth modelling of planar objects. The growth models considered are spatio-temporal stochastic processes on the circle. As a by product, flexible new models for space–time covariance functions on the circle are provided. An application of the Lévy-based growth models to tumour growth is discussed.

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Bernoulli, Volume 14, Number 1 (2008), 62-90.

First available in Project Euclid: 8 February 2008

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growth models Lévy basis spatio-temporal modelling tumour growth


Jónsdóttir, Kristjana Ýr; Schmiegel, Jürgen; Jensen, Eva B. Vedel. Lévy-based growth models. Bernoulli 14 (2008), no. 1, 62--90. doi:10.3150/07-BEJ6130.

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  • Alt, W. (1999). Statistics and dynamics of cellular shape changes. In, On Growth and Form: Spatio-temporal Pattern Formation in Biology (M.A.J. Chaplain, G.D. Singh and J.C. McLachlan, eds.) 287–307. Chichester: Wiley.
  • Barndorff-Nielsen, O. and Schmiegel, J. (2004). Lévy based spatial–temporal modelling, with applications to turbulence., Russian Math. Surveys 59 63–90.
  • Barndorff-Nielsen, O. and Thorbjørnsen, S. (2003). A connection between classical and free infinite divisability. Technical Report 2003-7, MaPhySto, Univ. Aarhus, Denmark.
  • Bramson, M. and Griffeath, D. (1981). On the Williams–Bjerknes tumour growth model. I., Ann. Probab. 9 173–185.
  • Brix, A. (1998). Spatial and spatio-temporal models for weed abundance. Ph.D. thesis, Royal Veterinary and Agricultural Univ., Copenhagen.
  • Brix, A. (1999). Generalized Gamma measures and shot-noise Cox processes., Adv. in Appl. Probab. 31 929–953.
  • Brix, A. and Chadoeuf, J. (2002). Spatio-temporal modelling of weeds by shot-noise G Cox processes., Biom. J. 44 83–99.
  • Brix, A. and Diggle, P.J. (2001). Spatiotemporal prediction for log-Gaussian Cox processes., J. Roy. Statist. Soc. Ser. B 63 823–841.
  • Brix, A. and Møller, J. (2001). Space–time multitype log Gaussian Cox processes with a view to modelling weeds., Scand. J. Statist. 28 471–488.
  • Brú, A., Pastor, J.M., Fernaud, I., Brú, I., Melle, S. and Berenguer, C. (1998). Super-rough dynamics on tumour growth., Phys. Rev. Lett. 81 4008–4011.
  • Calder, I. (1986). A stochastic model of rainfall interception., J. Hydrology 89 65–71.
  • Cantalapiedra, I., Lacasta, A., Auguet, C., Peñaranda, A. and Ramirez-Piscina, L. (2001). Pattern formation modelling of bacterial colonies. In, Branching in Nature 359–364. EDP Sciences, Springer.
  • Cressie, N. (1991a). Modelling growth with random sets. In, Spatial Statistics and Imaging (A. Possolo and C.A. Hayward, eds.) 31–45. IMS Lecture Notes Monogr. Ser. 20. IMS, Hayward, CA.
  • Cressie, N. (1991b)., Statistics for Spatial Data. New York: Wiley.
  • Cressie, N. and Hulting, F. (1992). A spatial statistical analysis of tumor growth., J. Amer. Statist. Assoc. 87 272–283.
  • Cressie, N. and Laslett, G.M. (1987). Random set theory and problems of modelling., SIAM Rev. 29 557–574.
  • Deijfen, M. (2003). Asymptotic shape in a continuum growth model., Adv. in Appl. Probab. 35 303–318.
  • Delsanto, P., Romano, A., Scalerandi, M. and Pescarmona, G. (2000). Analysis of a “phase transition” from tumor growth to latency., Phys. Rev. E 62 2547–2554.
  • Durrett, R. and Liggett, T. (1981). The shape of the limit set in Richardson’s growth model., Ann. Probab. 9 186–193.
  • Feideropoulou, G. and Pesquet-Popescu, B. (2004). Stochastic modelling of the spatio-temporal wavelet coefficients and applications to quality enhancement and error concealment., EURASIP JASP 12 1931–1942.
  • Fewster, R. (2003). A spatiotemporal stochastic process model for species spread., Biometrics 59 640–649.
  • Gratzer, G., Canham, C., Dieckmann, U., Fischer, A., Iwasa, Y., Law, R., Lexer, M., Sandman, H., Spies, T., Splechtna, B. and Szwagrzyk, L. (2004). Spatio-temporal development of forests – current trends in field methods and models., Oikos 107 3–15.
  • Hellmund, G. (2005). Lévy driven Cox processes with a view to modelling tropical rain forests. Master thesis, Dept. Mathematical Sciences, Univ., Aarhus.
  • Hellmund, G., Prokešová, M. and Jensen, E.B.V. (2007). Spatial and spatio-temporal Lévy based Cox point processes., Submitted.
  • Hobolth, A. and Jensen, E. (2000). Modelling stochastic changes in curve shape, with an application to cancer diagnostics., Adv. in Appl. Probab. 32 344–362.
  • Hobolth, A., Pedersen, J. and Jensen, E.B.V. (2003). A continuous parametric shape model., Ann. Inst. Statist. Math. 55 227–242.
  • Jensen, E.B.V., Jónsdóttir, K.Y., Schmiegel, J. and Barndorff-Nielsen, O.E. (2006). Spatio-temporal modelling – with a view to biological growth. In, Statistical Methods of Spatio-Temporal Systems (B. Finkenstadt and V. Isham, eds.) 45–73. Boca Raton: Chapman & Hall/CRC.
  • Jónsdóttir, K.Ý. and Jensen, E.B.V. (2005). Gaussian radial growth., Image Analysis Stereology 24 117–126.
  • Kallenberg, O. (1989)., Random Measures, 4th ed. Berlin: Akademie Verlag.
  • Kansal, A.R., Torquato, S., Harsh, G.R., Chiocca, E.A. and Deisboeck, T.S. (2000). Simulated brain tumor growth dynamics using a three-dimensional cellular automaton., J. Theor. Biol. 203 367–382.
  • Kwapien, S. and Woyczynski, W. (1992)., Random Series and Stochastic Integrals: Single and Multiple. Boston: Birkhäuser.
  • Lee, T. and Cowan, R. (1994). A stochastic tessellation of digital space. In, Mathematical Morphology and Its Applications to Image Processing (J. Serra, ed.) 217–224. Dordrecht: Kluwer.
  • Lovejoy, S., Schertzer, D. and Watson, B. (1992). Radiative transfer and multifractal clouds: Theory and applications., International Radiation Symposium 92 108–111.
  • Møller, J. (2003). Shot noise Cox processes., Adv. in Appl. Probab. 35 614–640.
  • Pang, N. and Tzeng, W. (2004). Anomalous scaling of superrough growing surfaces: From correlation functions to residual local interfacial widths and scaling exponents., Phys. Rev. E 70 (036115).
  • Peirolo, R. and Scalerandi, M. (2004). Markovian model of growth and histologic progression in prostate cancer., Phys. Rev. E 70 (011902).
  • Prokešová, M., Hellmund, G. and Jensen, E.B.V. (2006). On spatio-temporal Lévy based Cox processes. In, Proceedings of S4G, International Conference on Stereology, Spatial Statistics and Stochastic Geometry 111–116. Union of Czech Mathematics and Physicists.
  • Qi, A.S., Zheng, X., Du, C.Y. and An, B.S. (1993). A cellular automaton model of cancerous growth., J. Theor. Biol. 161 1–12.
  • Rajput, B. and Rosinski, J. (1989). Spectral representations of divisible processes., Probab. Theory Related Fields 82 451–487.
  • Richardson, D. (1973). Random growth in a tessellation., Proc. Cambridge Philos. Soc. 74 515–528.
  • Schmiegel, J. (2006). Self-scaling tumor growth., Phys. A 367 509–524.
  • Schmiegel, J., Barndorff-Nielsen, O. and Eggers, H. (2005). A class of spatio-temporal and causal stochastic processes, with application to multiscaling and multifractality., South African J. Science 101 513–519.
  • Schmiegel, J., Cleve, J., Eggers, H., Pearson, B. and Greiner, M. (2004). Stochastic energy-cascade model for 1 + 1 dimensional fully developed turbulence., Phys. Lett. A 320 247–253.
  • Sornette, D. and Ouillon, G. (2005). Multifractal scaling of thermally activated rupture processes., Phys. Rev. Lett. 94 (038501).
  • Steel, G.G. (1977)., Growth Kinetics of Tumours. Oxford: Clarendon Press.
  • Stein, M. (2005). Space–time covariance functions., J. Amer. Statist. Assoc. 100 310–321.
  • Wolpert, R.L. and Ickstadt, K. (1998). Poisson/Gamma random field models for spatial statistics., Biometrika 85 251–267.