Guaranteed non-asymptotic confidence regions in system identification

In this paper we consider the problem of constructing confidence regions for the parameters of identified models of dynamical systems. Taking a major departure from the previous literature on the subject, we introduce a new approach called ‘Leave-out Sign-dominant Correlation Regions’ (LSCR) which delivers confidence regions with guaranteed probability. All results hold rigorously true for any finite number of data points and no asymptotic theory is involved. Moreover, prior knowledge on the noise affecting the data is reduced to a minimum. The approach is illustrated on several simulation examples, showing that it delivers practically useful confidence sets with guaranteed probabilities.     

Non-asymptotic confidence regions for model parameters in the presence of unmodelled dynamics

This paper deals with the problem of constructing confidence regions for the parameters of truncated series expansion models. The models are represented using orthonormal basis functions, and we extend the ‘Leave-out Sign-dominant Correlation Regions’ (LSCR) algorithm such that non-asymptotic confidence regions for the parameters can be constructed in the presence of unmodelled dynamics. The constructed regions have guaranteed probability of containing the true parameters for any finite number of data points. The algorithm is first developed for FIR models and then extended to models with generalized orthonormal basis functions. The usefulness of the developed approach is demonstrated for FIR and Laguerre models in simulation examples.     

Generalization of the incenter subdivision scheme

We introduce a new interpolatory subdivision scheme generalizing the incenter subdivision [8]. The proposed scheme is equipped with a shape controlling tension parameter, is Hermitian, and reproduces circles from non-uniform samples. We prove that for any value of the free parameter the limit curve is G¹ continuous. The scheme is shape preserving and avoids undesirable oscillations by producing curves with a finite number of inflection points at the regions where the control polygon suggests a change of convexity. Several examples are presented demonstrating the properties of the scheme.     

Fuzzy confidence intervals for mean of Gaussian fuzzy random variables

A new approach to construct the two-sided and one-sided fuzzy confidence intervals for the fuzzy parameter is introduced, based on normal fuzzy random variables. Fuzzy data, that are observations of normal fuzzy random variables, are used in constructing such fuzzy confidence intervals. We invoke usual methods of finding confidence intervals for parameters obtained form h-level sets of fuzzy parameter to construct fuzzy confidence intervals. The crisp data that are used in constructing these confidence intervals come form h-level sets of fuzzy observations. Combining such confidence intervals yields a fuzzy set of the class of all fuzzy parameters, which is called the fuzzy confidence interval.Then, a criterion is proposed to determine the degree of membership of every fuzzy parameter in the introduced fuzzy confidence interval. A numerical example is provided to clarify the proposed method. Finally, the advantages of the proposed method with respect to some common methods are discussed.     

Model selection for support vector machines via uniform design

The problem of choosing a good parameter setting for a better generalization performance in a learning task is the so-called model selection. A nested uniform design (UD) methodology is proposed for efficient, robust and automatic model selection for support vector machines (SVMs). The proposed method is applied to select the candidate set of parameter combinations and carry out a k-fold cross-validation to evaluate the generalization performance of each parameter combination. In contrast to conventional exhaustive grid search, this method can be treated as a deterministic analog of random search. It can dramatically cut down the number of parameter trials and also provide the flexibility to adjust the candidate set size under computational time constraint. The key theoretic advantage of the UD model selection over the grid search is that the UD points are “far more uniform”and “far more space filling” than lattice grid points. The better uniformity and space-filling phenomena make the UD selection scheme more efficient by avoiding wasteful function evaluations of close-by patterns. The proposed method is evaluated on different learning tasks, different data sets as well as different SVM algorithms.     

Spectral Methods for Substantial Fractional Differential Equations

In this paper, a non-polynomial spectral Petrov–Galerkin method and its associated collocation method for substantial fractional differential equations are proposed, analyzed, and tested. We modify a class of generalized Laguerre polynomials to form our trial basis and test basis. After a proper scaling of these bases, our Petrov–Galerkin method results in diagonal and well-conditioned linear systems for certain types of fractional differential equations. In the meantime, we provide superconvergence points of the Petrov–Galerkin approximation for associated fractional derivative and function value of true solution. Additionally, we present explicit fractional differential collocation matrices based upon Laguerre–Gauss–Radau points. It is noteworthy that the proposed methods allow us to adjust a parameter in the basis according to different given data to maximize the convergence rate. All these findings have been proved rigorously in our convergence analysis and confirmed in our numerical experiments.     

Non-asymptotic confidence ellipsoids for the least-squares estimate

In this paper, we consider the finite sample properties of least-squares system identification, and derive non-asymptotic confidence ellipsoids for the estimate. The shape of the confidence ellipsoids is similar to the shape of the ellipsoids derived using asymptotic theory, but unlike asymptotic theory, they are valid for a finite number of data points. The probability that the estimate belongs to a certain ellipsoid has a natural dependence on the volume of the ellipsoid, the data generating mechanism, the model order and the number of data points available.     

PEAS: A toolbox to assess the accuracy of estimated parameters in environmental models

This paper presents a Matlab™ toolbox to assess the accuracy of the estimated parameters of environmental models, based on their approximate confidence regions. Before describing the application, the underlying theory is briefly recalled to familiarize the reader with the numerical methods involved. The software, named PEAS as an acronym for Parameter Estimation Accuracy Software, performs both the estimation and the accuracy analysis, using a user-friendly graphical interface to minimize the required programming. The user is required to specify the model structure according to the Matlab/Simulink™ syntax, supply the experimental data, provide an initial parameter guess and select an estimation method. PEAS provides several model assessment tools, in addition to parameter estimation, such as error function plotting, trajectory sensitivity, Monte Carlo analysis, all useful to assess the adequacy of the experimental data to the estimation problem. After the parameters have been estimated, the reliability assessment is performed: approximate and exact confidence regions are computed and a confidence test is produced. The Monte Carlo analysis is available for approximate accuracy assessment whenever the model structure prevents the application of the confidence regions method. The software, which is freely available for research purposes, is demonstrated here with two examples: a dynamical and an algebraic model. In both cases, software usage and outputs are presented and commented. The examples show how the user is guided through the application of the methods and how warning messages are returned if the estimation does not satisfy the accuracy criteria.     

Simplified parameter quantization procedure for adaptive estimation

Optimum Kalman filter design often requires estimation of the true value of an unknown parameter vector. In Magill's adaptive procedure, the parameter space must be quantized. An accurate estimate of the true value requires fine quantization, but this results in an unreasonable number of elemental filters. Iterative techniques that require only binary quantization of each unknown parameter are proposed. This reduces the number of elemental filters without sacrificing accuracy of the parameter estimate.     

3D registration of laser range scenes by coincidence of coarse binary cubes

Scene registration of a pair of three-dimensional (3D) range images is a 6D optimization problem usually required in mobile robotics. This problem is different from object registration, since all scan directions and depths may contain relevant data, and because farther regions are sampled with lower densities. The paper proposes an efficient scene matching method based on the concept of coarse binary cubes. An integer objective function is defined as the number of coincident cubes between both scans. This is a metric of the degree of overlap that does not employ point distances. Its value is obtained without actually using any 3D grid data structure, with a computational complexity of order O(n), where n represents the number of laser points. This objective function is optimized with a globalized version of the downhill Simplex algorithm to avoid local minima. Experimental results are presented from indoor and outdoor environments with different degrees of structuring. The effect of cube size and the number of vertices on registration performance has been analyzed. Besides, experiments show that the proposed method achieves similar accuracy as iterative closest points (ICP) and normal distribution transform (NDT), while it improves both computation time and robustness against initial misalignments.     

范数有界型参数不确定性系统的概率鲁棒H∞控制方法

利用确定性鲁棒控制方法对参数摄动的最坏情况进行研究,设计出的控制器具有较大的保守性和较高的控制成本.针对这一问题,建立范数有界型参数不确定性系统模型,分析系统性能的置信概率与参数不确定性随机向量的范数边界之间的关系,提出一种基于概率估计的鲁棒H∞控制方法,该方法能以有限的迭代步数给出一个与系统性能置信概率相关的鲁棒H∞控制器的可行解.最后通过仿真实例验证了所提出方法的有效性.     

Guaranteed characterization of exact non-asymptotic confidence regions as defined by LSCR and SPS

In parameter estimation, it is often desirable to supplement the estimates with an assessment of their quality. A new family of methods proposed by Campi et al. for this purpose is particularly attractive, as it makes it possible to obtain exact, non-asymptotic confidence regions under mild assumptions on the noise distribution. A bottleneck of this approach, however, is the numerical characterization of these confidence regions. So far, it has been carried out by gridding, which provides no guarantee as to its results and is only applicable to low dimensional spaces. This paper shows how interval analysis can contribute to removing this bottleneck.     

非线性振动系统参数识别的迭代离散变分方法

针对存在干扰的非线性振动系统,本文提出了一种识别模型参数的时域迭代方法．首先,对于每一组包含随机干扰的测量数据样本,将待辨识参数对时间的导数引入到系统代价函数中,进而利用离散变分原理导出关于待辨识参数的差分方程,并与修改后的系统约束方程一同求解;通过迭代计算使待识别参数从给定的初始值收敛到稳定的真值．然后,对通过n组干扰样本得到的参数识别结果取平均值,并作为最终辨识结果．最后,利用本方法对一个四自由度非线性振动系统的模型参数进行了识别仿真,数值结果证明了该方法的有效性．     

Fast k-nearest neighbors search using modified principal axis search tree

The problem of k-nearest neighbors (kNN) is to find the nearest k neighbors for a query point from a given data set. Among available methods, the principal axis search tree (PAT) algorithm always has good performance on finding nearest k neighbors using the PAT structure and a node elimination criterion. In this paper, a novel kNN search algorithm is proposed. The proposed algorithm stores projection values for all data points in leaf nodes. If a leaf node in the PAT cannot be rejected by the node elimination criterion, data points in the leaf node are further checked using their pre-stored projection values to reject more impossible data points. Experimental results show that the proposed method can effectively reduce the number of distance calculations and computation time for the PAT algorithm, especially for the data set with a large dimension or for a search tree with large number of data points in a leaf node.     

A clustering-based approach for the identification of a class of temporally switched linear systems

The behaviours of hybrid dynamic systems (HDS) are determined by combining continuous variables with discrete switching logic. The identification of a HDS aims to find an accurate model of the system’s dynamics based on its past inputs and outputs. In pattern recognition (PR) methods, each mode is represented by a set of similar patterns that form restricted regions in the feature space. These sets of patterns are called classes. A pattern is a vector built from past inputs and outputs. HDS identification is a challenging problem since it involves the estimation of different sets of parameters without knowing in advance which sections of the measured data correspond to the different modes of the system. Therefore, HDS identification can be achieved by combining two steps: clustering and parameter estimation. In the clustering step, the number of discrete modes (i.e., the classes that input-output data points belong) is estimated. The parameter estimation step finds the parameters of the models that govern the continuous dynamics in each mode. In this paper, an unsupervised PR method is proposed to achieve the clustering step of the identification of temporally switched linear HDS. The determination of the number of modes does not require prior information about the modes or their number.     

Finite-time settling adaptive observer for linear discrete-time systems

This paper deals with a design method for an adaptive scheme which would identify the parameters and observe the state of any unknown single-input single-output linear discrete-time systems using only input-output data. Kreisselmeier's parametrized system [5] is used instead of the original system. Then the parameter identification process and the state observation process are well separated. To accelerate the convergence rate of the estimates, a finite-time settling scheme is proposed. It is shown that the estimates obtained converges to true values at k = 3n ?1, where k is the discrete time and n is the order of system. A numerical example is given to indicate acceptable performance of the proposed scheme.     

An effective density based approach to detect complex data clusters using notion of neighborhood difference

The density based notion for clustering approach is used widely due to its easy implementation and ability to detect arbitrary shaped clusters in the presence of noisy data points without requiring prior knowledge of the number of clusters to be identified. Density-based spatial clustering of applications with noise (DBSCAN) is the first algorithm proposed in the literature that uses density based notion for cluster detection. Since most of the real data set, today contains feature space of adjacent nested clusters, clearly DBSCAN is not suitable to detect variable adjacent density clusters due to the use of global density parameter neighborhood radius N _rad and minimum number of points in neighborhood N _pts . So the efficiency of DBSCAN depends on these initial parameter settings, for DBSCAN to work properly, the neighborhood radius must be less than the distance between two clusters otherwise algorithm merges two clusters and detects them as a single cluster. Through this paper: 1) We have proposed improved version of DBSCAN algorithm to detect clusters of varying density adjacent clusters by using the concept of neighborhood difference and using the notion of density based approach without introducing much additional computational complexity to original DBSCAN algorithm. 2) We validated our experimental results using one of our authors recently proposed space density indexing (SDI) internal cluster measure to demonstrate the quality of proposed clustering method. Also our experimental results suggested that proposed method is effective in detecting variable density adjacent nested clusters.     

Controlling the depth of anesthesia by a novel positive control strategy

In this paper a positive control law is designed for multi-input positive systems that ensures asymptotic tracking of a desired output reference value. This control law can be viewed as a generalization of another one proposed in the literature for the control of the total mass in SISO compartmental systems, but is suitable for a wider class of positive systems. The controller proposed here is applied to the control of the depth of anesthesia (DoA), by means of the administration of propofol and remifentanil, when using a parameter parsimonious Wiener model recently introduced in the literature. Its performance is illustrated by realistic simulations.