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Sparse Estimation of Polynomial and Rational Dynamical Models

机译：多项式和有理动力学模型的稀疏估计

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摘要

In many practical situations, it is highly desirable to estimate an accurate mathematical model of a real system using as few parameters as possible. At the same time, the need for an accurate description of the system behavior without knowing its complete dynamical structure often leads to model parameterizations describing a rich set of possible hypotheses; an unavoidable choice, which suggests sparsity of the desired parameter estimate. An elegant way to impose this expectation of sparsity is to estimate the parameters by penalizing the criterion with the “norm” of the parameters. Due to the non-convex nature of the -norm, this penalization is often implemented as solving an optimization program based on a convex relaxation (e.g., /LASSO, nuclear norm, ). Two difficulties arise when trying to apply these methods: (1) the need to use cross-validation or some related technique for choosing the values of regularization parameters associated with the penalty; and (2) the requirement that the (unpenalized) cost function must be convex. To address the first issue, we propose a new technique for sparse linear regression called SPARSEVA, with close ties with the LASSO (least absolute shrinkage and selection operator), which provides an automatic tuning of the amount of regularization. The second difficulty, which imposes a severe constraint on the types of model structures or estimation methods on which the relaxation can be applied, is addressed by combining SPARSEVA and the Steigl- tz-McBride method. To demonstrate the advantages of the proposed approach, a solid theoretical analysis and an extensive simulation study are provided.
机译：在许多实际情况下，非常需要使用尽可能少的参数来估计实际系统的精确数学模型。同时，对系统行为的准确描述而不需要知道其完整的动力学结构的需求常常导致模型参数化，描述大量可能的假设。这是不可避免的选择，这表明所需参数估计的稀疏性。施加稀疏期望的一种优雅方法是通过用参数的“范数”对准则进行惩罚来估计参数。由于-norm的非凸性质，因此这种惩罚通常是在求解基于凸松弛的优化程序（例如/ LASSO，核规范）时实施的。尝试应用这些方法时会遇到两个困难：（1）需要使用交叉验证或一些相关技术来选择与惩罚相关的正则化参数的值； （2）（无罚）成本函数必须是凸的。为了解决第一个问题，我们提出了一种稀疏线性回归的新技术SPARSEVA，该技术与LASSO（最小绝对收缩和选择算子）密切相关，该技术可自动调整正则化量。第二个难题是对可应用松弛的模型结构或估算方法的类型施加了严格的约束，这是通过结合SPARSEVA和Steigltz-McBride方法来解决的。为了证明所提出方法的优点，提供了扎实的理论分析和广泛的仿真研究。

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