Finite dimensional filters with non-linear drift IX Construction of finite dimensional estimation algebras of non-maximal rank

The idea of using estimation algebra to construct finite-dimensional nonlinear filters was first proposed by Brockett and Mitter independently. It has proven to be an invaluable tool in the study of nonlinear filtering problem. In 1983, Brockett proposed to classify all finite-dimensional estimation algebras. In this paper, we give the construction of finite-dimensional estimation algebras of non-maximal rank. These non-maximal rank finite-dimensional estimation algebras play an important role in Brockett's classification problem.     

Structure theorem for five-dimensional estimation algebras

The problem of classification of finite-dimensional estimation algebras was formally proposed by Brockett in his lecture at International Congress of Mathematicians in 1983. Due to the difficulty of the problem, in the early 1990s Brockett suggested that one should understand the low-dimensional estimation algebras first. In this paper we give classification of estimation algebras of dimension at most five. Although the classification of finite-dimensional estimation algebra of maximal rank was completed by Yau and his coworkers Chen, Chiou, Hu, Wong and Wu; the problem of classification of non-maximal rank finite-dimensional estimation algebra is still wide open except for the case of state space dimension 2. Hopefully, the result of this paper will shed some light on the non-maximal rank estimation algebras.     

A note on estimation algebras on nonlinear filtering theory

The idea of using estimation algebras to construct finite dimensional nonlinear filters was first proposed by Brockett and Mitter independently. It turns out that the concept of estimation algebra plays a crucial role in the investigation of finite dimensional nonlinear filters. In his talk at the International Congress of Mathematicians in 1983, Brockett proposed to classify all finite dimensional estimation algebras. In this paper we consider some filtering systems. In a special filtering system: (1) We have some structure results. (2) For any arbitrary finite dimensional state space, under the condition that the drift term is a linear vector field plus a gradient vector field, we classify all finite dimensional estimation algebras with maximal rank. (3) We classify all finite dimensional estimation algebras with maximal rank if the dimension of the state space is less than or equal to three. A more general filtering system is considered. The above three results can be ‘used’ locally. Therefore from the algebraic point of view, we have now understood generically some finite dimensional filters.     

Classification of four-dimensional estimation algebras

Of central importance in nonlinear filtering theory is the classification of finite-dimensional estimation algebras, through which we can construct recursive filters. We classify all t-dimensional estimation algebras, t⩽4, for arbitrary state space dimension     

Classification of finite-dimensional estimation algebras of maximal rank with arbitrary state–space dimension and mitter conjecture

In the late seventies, the concept of the estimation algebra of a filtering system was introduced. It was proven to be an invaluable tool in the study of non-linear filtering problems. In the early eighties, Brockett proposed to classify finite dimensional estimation algebras and Mitter conjectured that all functions in finite dimensional estimation algebras are necessarily polynomials of total degree at most one. Despite the massive effort in understanding the finite dimensional estimation algebras, the 20 year old problem of Brockett and Mitter conjecture remains open. In this paper, we give a classification of finite dimensional estimation algebras of maximal rank and solve the Mitter conjecture affirmatively for finite dimensional estimation algebras of maximal rank. In particular, for an estimation algebra E of maximal rank, we give a necessary and sufficient conditions for E to be finite dimensional in terms of the drift f_i (x) and observation h_j (x). As an important corollary, we show that the number of statistics needed to compute the conditional density of the state given the observation {y(s):0?≤?s?≤?t} by the algebraic method is n where n is the dimension of the state.     

Finite-dimensional filters with nonlinear drift,VI: Linear structure of Ω

Ever since the concept of estimation algebra was first introduced by Brockett and Mitter independently, it has been playing a crucial role in the investigation of finite-dimensional nonlinear filters. Researchers have classified all finite-dimensional estimation algebras of maximal rank with state space less than or equal to three. In this paper we study the structure of quadratic forms in a finite-dimensional estimation algebra. In particular, we prove that if the estimation algebra is finite dimensional and of maximal rank, then the Ω=(∂f _j /∂x _i −∂f _i /∂x _j )matrix, wheref denotes the drift term, is a linear matrix in the sense that all the entries in Ω are degree one polynomials. This theorem plays a fundamental role in the classification of finite-dimensional estimation algebra of maximal rank. This research was supported by Army Research Office Grants DAAH 04-93-0006 and DAAH 04-1-0530.     

New classes of finite-dimensional nonlinear filters

We derive sufficient conditions for the estimation algebras of two special classes of nonlinear estimation systems to be finite dimensional. By carrying out the Brockett—Wei—Norman program new classes of finite-dimensional filters are obtained. These systems can be regarded respectively as extensions of the Kalman—Bucy filters and the Benes filters.     

Characterization of a subclass of finite-dimensional estimation algebras with maximal rank application to filtering

Finite-dimensional estimation Lie algebras play a crucial role in the study of finite-dimensional filters for partially observed stochastic process. When the dynamics noise is Gaussian we can characterize the so-called estimation Lie algebras with maximal rank in terms of the observation functions (necessarily affine) and the drift (necessarily a sum of a skew-symmetric linear term and a gradient vector field, with a functional relationship), under the assumption that the estimation algebra has one and only one operator of order greater or equal to two in any of its basis.     

Poincaré–Birkhoff–Witt theorem for Leibniz -algebras

We construct the universal enveloping algebra of a Leibniz n-algebra and we prove that the category of modules over this algebra is equivalent to the category of representations.We also give a proof of the Poincaré–Birkhoff–Witt theorem for universal enveloping algebras of finite-dimensional Leibniz n-algebras using Gröbner bases in a free associative algebra.     

Performance assessment,diagnosis, and optimal selection of non-linear state filters

Non-linear state filters of different approximations and capabilities allow for real-time estimation of unmeasured states in non-linear stochastic processes. It is well known that the performance of non-linear filters depends on the underlying numerical and statistical approximations used in their design. Despite the theoretical and practical interest in evaluating the performance of non-linear filtering methods, it remains one of the most complex problems in the area of state estimation. We propose the use of posterior Cramér–Rao lower bound (PCRLB) or mean square error (MSE) inequality as a filtering performance benchmark. Using the PCRLB inequality, we develop assessment and diagnosis tools for monitoring and evaluating the performance of non-linear filters. Using the PCRLB inequality-based performance assessment tool, an optimal non-linear filter switching strategy is proposed for state estimation in general non-linear systems. The non-linear filter switching strategy is an optimal performance strategy, which maintains high filtering performance under all operating conditions. The complex, high dimensional integrals involved in the computation of the PCRLB inequality-based non-linear filter assessment and diagnosis tools are approximated using sequential Monte-Carlo (SMC) methods. The utility and efficacy of the developed tools are illustrated through a numerical example.     

Exact filters for Newton–Raphson parameter estimation algorithms for continuous-time partially observed stochastic systems

This paper presents explicit finite-dimensional filters for implementing Newton–Raphson (NR) parameter estimation algorithms. The models which exhibit nonlinear parameter dependence are stochastic, continuous-time and partially observed. The implementation of the NR algorithm requires evaluation of the log-likelihood gradient and the Fisher information matrix. Fisher information matrices are important in bounding the estimation error from below, via the Cramer–Rao bound. The derivations are based on relations between incomplete and complete data, likelihood, gradient and Hessian likelihood functions, which are derived using Girsanov's measure transformations.     

Quadratically constrained information theoretic analysis

Recently Zhang and Brockett extended the framework of‘minimum discrimination information’ (MDI) estimation techniques to include quadratic constraints. They claimed their approach was quite different from the usual Lagrange duality theory. We show that the dual problem obtained by Zhang and Brockett is actually a geometric dual. Hence the quadratically constrained MDI estimation can be enriched by the theory of generalized geometric programming.     

Controllability of non-linear impulsive stochastic systems

In this article, we consider the finite-dimensional dynamical control systems described by non-linear impulsive stochastic differential equations. Sufficient conditions for the complete and approximate controllability of non-linear impulsive stochastic systems are formulated and proved under the natural assumption that the corresponding linear system is appropriately controllable.     

Filters of residuated lattices and triangle algebras

An important concept in the theory of residuated lattices and other algebraic structures used for formal fuzzy logic, is that of a filter. Filters can be used, amongst others, to define congruence relations. Specific kinds of filters include Boolean filters and prime filters.In this paper, we define several different filters of residuated lattices and triangle algebras and examine their mutual dependencies and connections. Triangle algebras characterize interval-valued residuated lattices.     

Exact finite-dimensional filters for doubly stochasticauto-regressive processes

In this paper, we derive exact finite-dimensional recursive filters for a class of doubly stochastic auto-regressive (AR) models. We assume that the parameters of the doubly stochastic AR process vary according to a nonlinear polynomial function of a Gaussian state-space process. Apart from being of mathematical interest, these finite-dimensional filters have potential applications in time-series analysis and image-enhanced tracking of maneuvering targets     

Simplex-directed partitioned adaptive filters

A new combination of mathematical programming methods and partitioned adaptive filters is proposed for the simultaneous real-time estimation of the parameters and state of an unknown linear system. The algorithm uses a version of the simplex method of non-linear programming to direct parameter selection for a bank of Kidman filters, thus combining the known convergence characteristics of partitioned adaptive filters with the robustness of this search technique. Motivation for the approach is discussed in light of recently published results concerning convergence of the decision function of the partitioned filters, and a simple example is given.     

Categories of quantum and classical channels

We introduce a construction that turns a category of pure state spaces and operators into a category of observable algebras and superoperators. For example, it turns the category of finite-dimensional Hilbert spaces into the category of finite-dimensional C*-algebras and completely positive maps. In particular, the new category contains both quantum and classical channels, providing elegant abstract notions of preparation and measurement. We also consider nonstandard models that can be used to investigate which notions from algebraic quantum information theory are operationally justifiable.     

Finite-dimensional risk-sensitive filters and smoothers fordiscrete-time nonlinear systems

Finite-dimensional optimal risk-sensitive filters and smoothers are obtained for discrete-time nonlinear systems by adjusting the standard exponential of a quadratic risk-sensitive cost index to one involving the plant nonlinearity. It is seen that these filters and smoothers are the same as those for a fictitious linear plant with the exponential of squared estimation error as the corresponding risk-sensitive cost index. Such finite-dimensional filters do not exist for nonlinear systems in the case of minimum variance filtering and control     

基于改进CNN的年龄和性别识别

人脸图像的年龄和性别识别是人脸分析的重要任务,在真实多变场景下完成识别依然面临挑战。改进深度卷积神经网络（Convolutional Neural Network,CNN）,将首层大尺寸卷积核替换为级联3[×]3卷积核;采用跨连卷积层融合中层和高层抽象特征;加入Batch Normalization（BN）层,设置较高的学习率和较小的Dropout比率;采用1[×]1卷积核与全局平均池化（Global Average Pooling）取代全连接层。实验表明,所提方法与主流的年龄性别识别方法比较具有较好的识别率,在Adience数据集上,年龄识别精度达到89.8%,性别识别精度达到93.3%。     

Detection and classification of non-linearities based on Volterra kernels processing

Modelling and identification of the real systems, which are more or less non-linear, are important activities not only for scientific research but also for real-time applications like fault detection and compensation. In this paper the non-linear system is defined by a serial combination of a linear part followed by a variable static parameterised non-linearity. A structure to detect and classify some non-linearities based on Volterra kernels is proposed. Such kernels can be estimated from the I–O measurements of the system under study. Processing Volterra kernels means estimation followed by classification. Mainly artificial neural networks to obtain a fast answer process the Volterra kernels. The neural methods and the detection structure can be developed to study more complicated non-linear models for fault detection purposes and structural identification of complex non-linear systems. The feasibility of the method is demonstrated using a simulated example.