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Register a new userIdentification of Underlying Dynamic System from Noisy Data with Splines
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Added by
Yujie Zhao
Publication date
2021
fields
Mathematical Statistics
and research's language is
English
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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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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.
In this article, we consider the problem of recovering the underlying trajectory when the longitudinal data are sparsely and irregularly observed and noise-contaminated. Such data are popularly analyzed with functional principal component analysis via the Principal Analysis by Conditional Estimation (PACE) method. The PACE method may sometimes be numerically unstable because it involves the inverse of the covariance matrix. We propose a sparse orthonormal approximation (SOAP) method as an alternative. It estimates the optimal empirical basis functions in the best approximation framework rather than eigen-decomposing the covariance function. The SOAP method avoids estimating the mean and covariance function, which is challenging when the assembled time points with observations for all subjects are not sufficiently dense. The SOAP method avoids the inverse of the covariance matrix, hence the computation is more stable. It does not require the functional principal component scores to follow the Gaussian distribution. We show that the SOAP estimate for the optimal empirical basis function is asymptotically consistent. The finite sample performance of the SOAP method is investigated in simulation studies in comparison with the PACE method. Our method is demonstrated by recovering the CD4 percentage curves from sparse and irregular data in the Multi-center AIDS Cohort Study.
The paper introduces a novel methodology for the identification of coefficients of switched autoregressive linear models. We consider the case when the systems outputs are contaminated by possibly large values of measurement noise. It is assumed that only partial information on the probability distribution of the noise is available. Given input-output data, we aim at identifying switched system coefficients and parameters of the distribution of the noise which are compatible with the collected data. System dynamics are estimated through expected values computation and by exploiting the strong law of large numbers. We demonstrate the efficiency of the proposed approach with several academic examples. The method is shown to be extremely effective in the situations where a large number of measurements is available; cases in which previous approaches based on polynomial or mixed-integer optimization cannot be applied due to very large computational burden.
This paper introduces a novel methodology for the identification of switching dynamics for switched autoregressive linear models. Switching behavior is assumed to follow a Markov model. The systems outputs are contaminated by possibly large values of measurement noise. Although the procedure provided can handle other noise distributions, for simplicity, it is assumed that the distribution is Normal with unknown variance. Given noisy input-output data, we aim at identifying switched system coefficients, parameters of the noise distribution, dynamics of switching and probability transition matrix of Markovian model. System dynamics are estimated using previous results which exploit algebraic constraints that system trajectories have to satisfy. Switching dynamics are computed with solving a maximum likelihood estimation problem. The efficiency of proposed approach is shown with several academic examples. Although the noise to output ratio can be high, the method is shown to be extremely effective in the situations where a large number of measurements is available.
High dimensional B-splines are catching tremendous attentions in fields of Iso-geometry Analysis, dynamic surface reconstruction and so on. However, the actual measured data are usually sparse and nonuniform, which might not meet the requirement of traditional B-spline algorithms. In this paper, we present a novel dynamic surface reconstruction approach, which is a 3-dimensional key points interpolation method (KPI) based on B-spline, aimed at dealing with sparse distributed data. This method includes two stages: a data set generation algorithm based on Kriging and a control point solving method based on key points interpolation. The data set generation method is designed to construct a grided dataset which can meet the requirement of B-spline interpolation, while promisingly catching the trend of sparse data, and it also includes a parameter reduction method which can significantly reduce the number of control points of the result surface. The control points solving method ensures the 3-dimensional B-spline function to interpolate the sparse data points precisely while approximating the data points generated by Kriging. We apply the method into a temperature data interpolation problem. It is shown that the generated dynamic surface accurately interpolates the sparsely distributed temperature data, preserves the dynamic characteristics, with fewer control points than those of traditional B-spline surface interpolation algorithm.
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