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Register a new userInference of dynamic systems from noisy and sparse data via manifold-constrained Gaussian processes
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Added by
Shihao Yang
Publication date
2020
fields
Mathematical Statistics
and research's language is
English
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Parameter estimation for nonlinear dynamic system models, represented by ordinary differential equations (ODEs), using noisy and sparse data is a vital task in many fields. We propose a fast and accurate method, MAGI (MAnifold-constrained Gaussian process Inference), for this task. MAGI uses a Gaussian process model over time-series data, explicitly conditioned on the manifold constraint that derivatives of the Gaussian process must satisfy the ODE system. By doing so, we completely bypass the need for numerical integration and achieve substantial savings in computational time. MAGI is also suitable for inference with unobserved system components, which often occur in real experiments. MAGI is distinct from existing approaches as we provide a principled statistical construction under a Bayesian framework, which incorporates the ODE system through the manifold constraint. We demonstrate the accuracy and speed of MAGI using realistic examples based on physical experiments.
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In this paper, we propose a two-stage method called Spline Assisted Partial Differential Equation involved Model Identification (SAPDEMI) to efficiently identify the underlying partial differential equation (PDE) models from the noisy data. In the first stage -- functional estimation stage -- we employ the cubic spline to estimate the unobservable derivatives, which serve as candidates included the underlying PDE models. The contribution of this stage is that, it is computational efficient because it only requires the computational complexity of the linear polynomial of the sample size, which achieves the lowest possible order of complexity. In the second stage -- model identification stage -- we apply Least Absolute Shrinkage and Selection Operator (Lasso) to identify the underlying PDE models. The contribution of this stage is that, we focus on the model selections, while the existing literature mostly focuses on parameter estimations. Moreover, we develop statistical properties of our method for correct identification, where the main tool we use is the primal-dual witness (PDW) method. Finally, we validate our theory through various numerical examples.
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