A Spline Least Squares Method for Numerical Parameter Estimation in Differential Equations

By James M. Varah. 1982

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Citations in the corpus

Sparse Additive Ordinary Differential Equations for Dynamic Gene Regulatory Network Modeling (2014, methodology): “…(NLS) method (Hemker 1972; Bard 1974; Li, Osborne, and Pravan 2005; Xue, Miao, and Wu 2010), the two-stage smoothing-based estimation method (Varah 1982; Brunel 2008; Chen and Wu 2008a,b; Liang and Wu 2008; Wu, Xue, and Kumar 2012), the principal differential analysis (PDA) and its…”
Network Reconstruction From High-Dimensional Ordinary Differential Equations (2016, methodology): “The two-step collocation procedure ﬁrst proposed by Varah (1982) involves ﬁtting a smoothing estimate ˆ X ( · ; h ) to the observations Y 1 , . . . , Y n in (2) with a smoothing parameter h , and then plugging ˆ X ( · ; h ) and its derivative with respect to t into (1) in order to estimate θ .”
Parameter Estimation in Stochastic Differential Equations (2011, methodology): “The two step method in estimating the structural parameter of the ordinary differential equation (ODE) is initiated in [44].”
√n-consistent parameter estimation for systems of ordinary differential equations : bypassing numerical integration via smoothing (2010, unknown): “If F is linear in θ 1 , . . . , θ p , then as already observed in Varah (1982), see pp. 29 and 31, cf. p. 1262 in Brunel (2008) and p. 1573 in Liang and Wu (2008), this will lead to a weighted linear least squares problem, which can be solved in a routine fashion without using e.g. random search…”
Parameter estimation for differential equations: a generalized smoothing approach (2007, unknown): “Collocation with spline bases was applied to data fltting problems involving an ODE model by Varah (1982) , who suggested a two-stage procedure in which each xi is flrst estimated by data smoothing methods without considering satisfying (1), followed by the minimization of a least squares measure o”
Parameter estimation of ODE’s via nonparametric estimators (2007, unknown): “Nevertheless, Varah (1982) illustrates spline-based method (with natural cubic splines and knots chosen by an expert) on the same model as (24).”
Asymptotic efficiency and finite-sample properties of the generalized profiling estimation of parameters in ordinary differential equations (2009, methodology): “Varah [27] suggested a two-stage procedure where the first step fits the observed data by least squares using cubic spline functions without considering the ODEs, and the second step estimates the parameters by least squares solution of the differential equations sampled at a set of points.”
Optimal rate of direct estimators in systems of ordinary differential equations linear in functions of the parameters (2013, unknown): “, [3, 43]) and generalized profiling ([38]).”
Accelerating Bayesian Inference over Nonlinear Differential Equations with Gaussian Processes (2008, methodology): “A spline-based approach was first suggested in [18] for smoothing experimental data and obtaining derivative estimates, which could then be used to compute a measure of mismatch for derivative values obtained from the system of equations.”
Learning Large Scale Ordinary Differential Equation Systems (2017, methodology): “One example of an inverse collocation method is the gradient matching method (Varah (1982), Brunel (2008)), which minimises the approximate loss function: (7) 1 2 ∑ t∈C ∥∥∥∥dx̂dt (t)− f(x̂(t), θ) ∥∥∥∥2 2 .”