Nonparametric estimation of dynamics of monotone trajectories

We study a class of nonlinear nonparametric inverse problems. Specifically, we propose a nonparametric estimator of the dynamics of a monotonically increasing trajectory defined on a finite time interval. Under suitable regularity conditions, we prove consistency of the proposed estimator and show that in terms of $L^2$-loss, the optimal rate of convergence for the proposed estimator is the same as that for the estimation of the derivative of a trajectory. This is a new contribution to the area of nonlinear nonparametric inverse problems. We conduct a simulation study to examine the finite sample behavior of the proposed estimator and apply it to the Berkeley growth data.

[1]  Hans-Georg Müller,et al.  Functional Data Analysis , 2016 .

[2]  Yanyuan Ma,et al.  Quick and easy one‐step parameter estimation in differential equations , 2014 .

[3]  Hulin Wu,et al.  Sparse Additive Ordinary Differential Equations for Dynamic Gene Regulatory Network Modeling , 2014, Journal of the American Statistical Association.

[4]  Florence Nicol,et al.  Functional principal component analysis of aircraft trajectories , 2013 .

[5]  Jie Peng,et al.  Semiparametric modeling of autonomous nonlinear dynamical systems with application to plant growth , 2011, 1111.7089.

[6]  Roman Vershynin,et al.  Introduction to the non-asymptotic analysis of random matrices , 2010, Compressed Sensing.

[7]  Hulin Wu,et al.  Sieve Estimation of Constant and Time-Varying Coefficients in Nonlinear Ordinary Differential Equation Models by Considering Both Numerical Error and Measurement Error. , 2010, Annals of statistics.

[8]  M. Saunders.  Complex Population Dynamics: A Theoretical/Empirical Synthesis , 2009 .

[9]  Hans-Georg Müller,et al.  Estimating Derivatives for Samples of Sparsely Observed Functions, With Application to Online Auction Dynamics , 2009 .

[10]  Hongyu Zhao,et al.  Asymptotic efficiency and finite-sample properties of the generalized profiling estimation of parameters in ordinary differential equations , 2009, 0903.3400.

[11]  Hulin Wu,et al.  Differential Equation Modeling of HIV Viral Fitness Experiments: Model Identification, Model Selection, and Multimodel Inference , 2009, Biometrics.

[12]  Jiguo Cao,et al.  Estimating a Predator‐Prey Dynamical Model with the Parameter Cascades Method , 2008, Biometrics.

[13]  Wolfgang Jank,et al.  Modeling Price Dynamics in eBay Auctions Using Differential Equations , 2008 .

[14]  Jiguo Cao,et al.  Estimating dynamic models for gene regulation networks , 2008, Bioinform..

[15]  Hulin Wu,et al.  Estimation of time-varying parameters in deterministic dynamic models , 2008 .

[16]  L. Cavalier.  Nonparametric statistical inverse problems , 2008 .

[17]  Hulin Wu,et al.  Efficient Local Estimation for Time-Varying Coefficients in Deterministic Dynamic Models With Applications to HIV-1 Dynamics , 2008 .

[18]  Jiguo Cao,et al.  Parameter estimation for differential equations: a generalized smoothing approach , 2007 .

[19]  N. Brunel.  Parameter estimation of ODE’s via nonparametric estimators , 2007, 0710.4190.

[20]  B. Perthame.  Transport Equations in Biology , 2006 .

[21]  Wolfgang Jank,et al.  Functional Data Analysis in Electronic Commerce Research , 2006, math/0609173.

[22]  P. James McLellan,et al.  Parameter estimation in continuous-time dynamic models using principal differential analysis , 2006, Comput. Chem. Eng..

[23]  Gerard Kerkyacharian,et al.  Wavelet deconvolution in a periodic setting , 2004 .

[24]  Xiaohua Xia,et al.  Estimation of HIV/AIDS parameters , 2003, Autom..

[25]  J. Collins,et al.  Inferring Genetic Networks and Identifying Compound Mode of Action via Expression Profiling , 2003, Science.

[26]  Peter Deuflhard,et al.  Scientific Computing with Ordinary Differential Equations , 2002 .

[27]  S. Milani,et al.  INVITED REVIEW: Kinetic models for normal and impaired growth , 2000, Annals of human biology.

[28]  H Wu,et al.  Population HIV‐1 Dynamics In Vivo: Applicable Models and Inferential Tools for Virological Data from AIDS Clinical Trials , 1999, Biometrics.

[29]  V De Gruttola,et al.  Estimation of HIV dynamic parameters. , 1998, Statistics in medicine.

[30]  H. Müller,et al.  Local Polynomial Modeling and Its Applications , 1998 .

[31]  P. Burman,et al.  Effect of Water Stress on Cortical Cell Division Rates within the Apical Meristem of Primary Roots of Maize , 1997, Plant physiology.

[32]  Jianqing Fan.  Local Polynomial Modelling and Its Applications: Monographs on Statistics and Applied Probability 66 , 1996 .

[33]  D. Donoho.  Nonlinear Solution of Linear Inverse Problems by Wavelet–Vaguelette Decomposition , 1995 .

[34]  Josip Pečarić,et al.  Inequalities Involving Functions and Their Integrals and Derivatives , 1991 .

[35]  Enno Mammen,et al.  Estimating a Smooth Monotone Regression Function , 1991 .

[36]  C. Kelly,et al.  Monotone smoothing with application to dose-response curves and the assessment of synergism. , 1990, Biometrics.

[37]  Ulrich Stadtmüller,et al.  Bandwidth choice and confidence intervals for derivatives of noisy data , 1987 .

[38]  C. J. Stone,et al.  Optimal Global Rates of Convergence for Nonparametric Regression , 1982 .

[39]  J. Varah.  A Spline Least Squares Method for Numerical Parameter Estimation in Differential Equations , 1982 .

[40]  Edward J. Wegman,et al.  Isotonic, Convex and Related Splines , 1980 .

[41]  L. Schumaker,et al.  Local Spline Approximation Methods , 1975 .

[42]  H. MacLean.  Modelling plant growth , 2012 .

[43]  A. Tsybakov,et al.  Block Thresholding and Sharp Adaptive Estimation in Severely Ill-Posed Inverse Problems , 2004 .

[44]  张 景绘.  动力学系统建模 = Modeling of dynamical systems , 2000 .

[45]  J. Ramsay.  Estimating smooth monotone functions , 1998 .

[46]  Jianqing Fan,et al.  Local polynomial modelling and its applications , 1994 .

[47]  H. Müller,et al.  Estimating regression functions and their derivatives by the kernel method , 1984 .

[48]  L. Schumaker.  Spline Functions: Basic Theory , 1981 .

[49]  G. Barón,et al.  A comparative study of the fit of four different functions to longitudinal data of growth in height of Belgian girls. , 1980, Annals of human biology.

[50]  C. R. Deboor,et al.  A practical guide to splines , 1978 .

[51]  R. D. Tuddenham,et al.  Physical growth of California boys and girls from birth to eighteen years. , 1954, Publications in child development. University of California, Berkeley.