l ₁-l ₁双范数的最优下边界回归模型辨识

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Title:

Optimal lower boundary regression model based on double norms l ₁-l ₁ optimization

作者:

刘小雍, 叶振环

遵义师范学院 工学院，贵州 遵义 563006

Author(s):

LIU Xiaoyong, YE Zhenhuan

College of Engineering, Zunyi Normal University, Zunyi 563006, China

关键词:

${\ell _1}$范数的结构风险最小化; 逼近误差的${\ell _1}$范数; 下边界回归模型; 泛化性能; 建模精度; 最优性; 线性规划

Keywords:

${\ell _1}$-norm-based structural risk minimization; ${\ell _1}$-norm on approximation error; lower boundary regression model; generalization performance; modeling accuracy; optimality; linear programming

分类号:

TP391.1

DOI:

10.11992/tis.201902006

文献标志码:

A

摘要:

考虑到来自传感器测量数据、模型结构以及参数的不确定性等因素，建模由这些因素导致的下边界模型尤为重要。通过将结构风险最小化理论与逼近误差最小化思想相结合，提出了${\ell _1} - {\ell _1}$ 双范数的最优下边界回归模型建模方法。首先，确定满足下边界回归模型的约束条件。其次，将结构风险的${\ell _2}$范数转化为简单的${\ell _1}$范数优化问题，并将回归模型与实际测量数据之间的逼近误差的${\ell _1}$范数融合到结构风险的${\ell _1}$范数优化问题，再应用较简单的线性规划对双范数的优化问题进行求解获取模型参数。最后，通过来自测量数据以及模型参数不确定性的实验分析，论证了提出方法的最优性，体现在：下边界模型的建模精度通过逼近误差的${\ell _1}$范数得到保证；模型结构复杂性在结构风险的${\ell _1}$范数优化条件下得到有效控制，进而提高其泛化性能。

Abstract:

In statistical modeling, regression analysis is a set of statistical processes for estimating the relationships between a dependent variable and one or more independent variables. Considering the uncertainties in the structure and parameters of the model derived from sensor measurement data, a new model called optimal lower boundary model is proposed to remove the uncertainties in parameters and characteristics. The proposed method is a combination of structural risk minimization theory (SRM) and some ideas from approximation error minimization. An optimal lower boundary regression model (LBRM) is presented using ${\ell _1} - {\ell _1}$ double norms optimization. First, constraint conditions subjected to LBRM are defined. Then, ${\ell _2}$-norm optimization based on structural risk is converted into simple ${\ell _1}$-norm optimization so that approximation error between the measurements based on ${\ell _1}$-norm is computed and minimized. Next, LBRM is integrated into ${\ell _1}$-norm optimization (based on structural risk). Thus, simpler linear programming can be applied to the constructed double-norms optimization problem to solve parameters of LBRM. Finally, the proposed method is demonstrated by experiments regarding uncertain measurements and parameters of nonlinear system. It has the following prominent features: modeling accuracy of LBRM can be guaranteed by introducing the ${\ell _1}$-norm minimization on approximation error; model’s structural complexity is under control by ${\ell _1}$-norm optimization based on structural risk, thus the performance of the model can be improved further.