Statistical estimation of a growth-fragmentation model observed on a genealogical tree

Bernoulli - Tập 21 Số 3 - 2015
Marie Doumic1,2, Marc Hoffmann3, Nathalie Krell4, Lydia Robert5
1LJLL - Laboratoire Jacques-Louis Lions (Université Pierre et Marie Curie, Boîte courrier 187 - 75252 Paris Cedex 05 - France)
2MAMBA - Modelling and Analysis for Medical and Biological Applications (France)
3CEREMADE - CEntre de REcherches en MAthématiques de la DEcision (Place du Maréchal de Lattre de Tassigny 75775 - Paris Cedex 16 - France)
4IRMAR - Institut de Recherche Mathématique de Rennes (Campus de Beaulieu, bâtiments 22 et 23, 263 avenue du Général Leclerc, CS 74205 35042 RENNES Cédex - France)
5MICALIS - MICrobiologie de l'ALImentation au Service de la Santé (F-78350 JOUY-EN-JOSAS - France)

Tóm tắt

Từ khóa


Tài liệu tham khảo

[9] Chauvin, B., Rouault, A. and Wakolbinger, A. (1991). Growing conditioned trees. <i>Stochastic Process. Appl.</i> <b>39</b> 117–130.

[11] Douc, R., Moulines, E. and Rosenthal, J.S. (2004). Quantitative bounds on convergence of time-inhomogeneous Markov chains. <i>Ann. Appl. Probab.</i> <b>14</b> 1643–1665.

[18] Fort, G., Moulines, E. and Priouret, P. (2011). Convergence of adaptive and interacting Markov chain Monte Carlo algorithms. <i>Ann. Statist.</i> <b>39</b> 3262–3289.

[1] Baccelli, F., McDonald, D.R. and Reynier, J. (2002). A mean-field model for multiple TCP connections through a buffer implementing RED. <i>Performance Evaluation</i> <b>49</b> 77–97.

[2] Balagué, D., Canizo, J. and Gabriel, P. (2013). Fine asymptotics of profiles and relaxation to equilibrium for growth-fragmentation equations with variable drift rates. <i>Kinetic and Related Models</i> <b>6</b> 219–243.

[3] Banks, H.T., Sutton, K.L., Thompson, W.C., Bocharov, G., Roosec, D., Schenkeld, T. and Meyerhanse, A. (2011). Estimation of cell proliferation dynamics using CFSE data. <i>Bull. Math. Biol.</i> <b>73</b> 116–150.

[4] Bansaye, V. (2008). Proliferating parasites in dividing cells: Kimmel’s branching model revisited. <i>Ann. Appl. Probab.</i> <b>18</b> 967–996.

[5] Bansaye, V., Delmas, J.-F., Marsalle, L. and Tran, V.C. (2011). Limit theorems for Markov processes indexed by continuous time Galton–Watson trees. <i>Ann. Appl. Probab.</i> <b>21</b> 2263–2314.

[6] Baxendale, P.H. (2005). Renewal theory and computable convergence rates for geometrically ergodic Markov chains. <i>Ann. Appl. Probab.</i> <b>15</b> 700–738.

[7] Bertoin, J. (2006). <i>Random Fragmentation and Coagulation Processes. Cambridge Studies in Advanced Mathematics</i> <b>102</b>. Cambridge: Cambridge Univ. Press.

[10] Cloez, B. (2011). Limit theorems for some branching measure-valued processes. Available at <a href="arXiv:1106.0660v2">arXiv:1106.0660v2</a>.

[12] Doumic, M., Hoffmann, M., Reynaud-Bouret, P. and Rivoirard, V. (2012). Nonparametric estimation of the division rate of a size-structured population. <i>SIAM J. Numer. Anal.</i> <b>50</b> 925–950.

[13] Doumic, M., Maia, P. and Zubelli, J.P. (2010). On the calibration of a size-structured population model from experimental data. <i>Acta Biotheor.</i> <b>58</b> 405–413.

[14] Doumic, M., Perthame, B. and Zubelli, J.P. (2009). Numerical solution of an inverse problem in size-structured population dynamics. <i>Inverse Problems</i> <b>25</b> 1–22.

[15] Doumic, M. and Tine, L.M. (2012). Estimating the division rate for the growth-fragmentation equation. <i>J. Math. Biol.</i> <b>67</b> 69–103.

[16] Doumic, M. and Gabriel, P. (2010). Eigenelements of a general aggregation–fragmentation model. <i>Math. Models Methods Appl. Sci.</i> <b>20</b> 757–783.

[17] Engler, H., Prüss, J. and Webb, G.F. (2006). Analysis of a model for the dynamics of prions. II. <i>J. Math. Anal. Appl.</i> <b>324</b> 98–117.

[19] Gobet, E., Hoffmann, M. and Reiß, M. (2004). Nonparametric estimation of scalar diffusions based on low frequency data. <i>Ann. Statist.</i> <b>32</b> 2223–2253.

[20] Goldenshluger, A. and Lepski, O. (2011). Bandwidth selection in kernel density estimation: Oracle inequalities and adaptive minimax optimality. <i>Ann. Statist.</i> <b>39</b> 1608–1632.

[21] Haas, B. (2003). Loss of mass in deterministic and random fragmentations. <i>Stochastic Process. Appl.</i> <b>106</b> 245–277.

[22] Harris, S.C. and Roberts, M.I. (2012). The many-to-few lemma and multiple spines. Available at <a href="arXiv:1106.4761v3">arXiv:1106.4761v3</a>.

[23] Kaern, M., Elston, T.C., Blake, W.J. and Collins, J.J. (2005). Stochasticity in gene expression: From theories to phenotypes. <i>Nat. Rev. Genet.</i> <b>6</b> 451–464.

[24] Kubitschek, H.E. (1969). Growth during the bacterial cell cycle: Analysis of cell size distribution. <i>Biophys. J.</i> <b>9</b> 792–809.

[25] Laurençot, P. and Perthame, B. (2009). Exponential decay for the growth-fragmentation/cell-division equation. <i>Commun. Math. Sci.</i> <b>7</b> 503–510.

[26] Metz, J.A.J. and Diekmann, O., eds. (1986). <i>The Dynamics of Physiologically Structured Populations. Lecture Notes in Biomathematics</i> <b>68</b>. Berlin: Springer. Papers from the colloquium held in Amsterdam, 1983.

[28] Michel, P. (2006). Existence of a solution to the cell division eigenproblem. <i>Math. Models Methods Appl. Sci.</i> <b>16</b> 1125–1153.

[30] Niethammer, B. and Pego, R.L. (1999). Non-self-similar behavior in the LSW theory of Ostwald ripening. <i>J. Stat. Phys.</i> <b>95</b> 867–902.

[31] Pakdaman, K., Perthame, B. and Salort, D. (2012). Adaptation and fatigue model for neuron networks and large time asymptotics in a nonlinear fragmentation equation. Available at <a href="http://hal.upmc.fr/hal-00794841">http://hal.upmc.fr/hal-00794841</a>.

[33] Perthame, B. and Ryzhik, L. (2005). Exponential decay for the fragmentation or cell-division equation. <i>J. Differential Equations</i> <b>210</b> 155–177.

[34] Perthame, B. and Zubelli, J.P. (2007). On the inverse problem for a size-structured population model. <i>Inverse Problems</i> <b>23</b> 1037–1052.

[35] Stewart, E.J., Madden, R., Paul, G. and Taddei, F. (2005). Aging and death in an organism that reproduces by morphologically symmetric division. <i>PLoS Comput. Biol.</i> <b>3</b> e45.

[36] Sturm, A., Heinemann, M., Arnoldini, M., Benecke, A., Ackermann, M., Benz, M., Dormann, J. and Hardt, W.-D. (2011). The cost of virulence: Retarded growth of <i>Salmonella typhimurium</i> cells expressing type III secretion system 1. <i>PLoS Pathog.</i> <b>7</b> e1002143.

[37] Tan, C., Marguet, P. and You, L. (2009). Emergent bistability by a growth-modulating positive feedback circuit. <i>Nat. Chem. Biol.</i> <b>5</b> 842–848.

[38] Wang, P., Robert, L., Pelletier, J., Dang, W.L., Taddei, F., Wright, A. and Jun, S. (2010). Robust growth of <i>Escherichia coli. Curr. Biol.</i> <b>20</b> 1099–1103.

[8] Cáceres, M.J., Cañizo, J.A. and Mischler, S. (2011). Rate of convergence to an asymptotic profile for the self-similar fragmentation and growth-fragmentation equations. <i>J. Math. Pures Appl.</i> (9) <b>96</b> 334–362.

[27] Meyn, S. and Tweedie, R. (1993). <i>Markov Chains and Stochastic Stability</i>. Berlin: Springer.

[29] Michel, P., Mischler, S. and Perthame, B. (2005). General relative entropy inequality: An illustration on growth models. <i>J. Math. Pures Appl.</i> (9) <b>84</b> 1235–1260.

[32] Perthame, B. (2007). <i>Transport Equations in Biology. Frontiers in Mathematics</i>. Basel: Birkhäuser.