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A systematic methodology for empirical modeling of non-linear state space systems

机译：非线性状态空间系统经验建模的系统方法

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In this paper the authors formulate a theoretical framework for the empirical modelling of non-linear state space systems. The classification of non-linear system data, selection of model structure and order, system parameterisation, stationarity of the data, handling of outliers and noise in the data, parameter estimation and model validation can all be addressed with established, though loosely associated numerical techniques, often referred to as nonlinear process modelling. Relatively few researchers in system identification are comfortable with the application of these numerical techniques, such as time series embedding, surrogate data methods, non-linear stationarity, Lyapunov exponents for chaotic processes and nonlinear predictability. The authors reinterpret some of the above non-linear empirical concepts against the established background for linear state space system identification. Hereby we lay a basis for a systematic methodology to address empirical modelling of non-linear process dynamics, which can be implemented in a non-linear system identification toolbox. In particular, we apply surrogate data methods for the classification of data as stochastic or deterministic. For deterministic data, we embed the individual observations of the process and separate the embedding variables by non-linear factor analysis to arrive at a state space parameterisation of the system. The separation function makes no prior assumptions about the probability distributions of the observations and is robust against dynamic and measurement noise. An ensemble learning technique is used to estimate the parameters of the separation function. After parameterisation of the system a multiple-layer perceptron neural network maps the time evolution of the state vector onto the observations, one sample step ahead. In this manner, the dynamics of the process are captured. Model order is established against the Schwarz information criterion, formulated for multidimensional observations as a function of the model order and modelling error. Model validation is performed against the R~2 statistic, as well as in terms of free-run prediction performance.
机译：在本文中，作者制定了非线性状态空间系统的实证建模的理论框架。非线性系统数据的分类，选择模型结构和顺序，系统参数化，数据的平静性，数据的处理和数据中的异常值和噪声，都可以通过建立来解决，尽管松散相关的数值技术，通常被称为非线性过程建模。系统识别中的研究人员对这些数值技术的应用感到舒适，例如时间序列嵌入，代理数据方法，非线性保同性，Lyapunov指数用于混沌过程和非线性可预测性。作者将一些上述非线性经验概念重新诠释到用于线性状态空间系统识别的已建立的背景。因此，我们为系统方法进行了基础，以解决非线性过程动态的经验建模，可以在非线性系统识别工具箱中实现。特别是，我们将替代数据方法应用于数据作为随机或确定性的数据分类。对于确定性数据，我们嵌入了该过程的各个观察，并通过非线性因子分析将嵌入变量分开以到达系统的状态空间参数。分离功能没有关于观察的概率分布的现有假设，并且对动态和测量噪声具有稳健。合奏学习技术用于估计分离功能的参数。在系统的参数化之后，将状态向量的多层的Perceptron神经网络映射到观察到观察到，一个样本步骤。以这种方式，捕获该过程的动态。模型顺序是针对Schwarz信息标准建立的，用于作为模型顺序和建模误差的函数的多维观察。模型验证是针对R〜2统计而执行的，以及自由运行预测性能。

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