Sparse kernel density construction using orthogonal forward regression with leave-one-out test score and local regularization

This paper presents an efficient construction algorithm for obtaining sparse kernel density estimates based on a regression approach that directly optimizes model generalization capability. Computational efficiency of the density construction is ensured using an orthogonal forward regression, and the algorithm incrementally minimizes the leave-one-out test score. A local regularization method is incorporated naturally into the density construction process to further enforce sparsity. An additional advantage of the proposed algorithm is that it is fully automatic and the user is not required to specify any criterion to terminate the density construction procedure. This is in contrast to an existing state-of-art kernel density estimation method using the support vector machine (SVM), where the user is required to specify some critical algorithm parameter. Several examples are included to demonstrate the ability of the proposed algorithm to effectively construct a very sparse kernel density estimate with comparable accuracy to that of the full sample optimized Parzen window density estimate. Our experimental results also demonstrate that the proposed algorithm compares favorably with the SVM method, in terms of both test accuracy and sparsity, for constructing kernel density estimates.

Construction of tunable radial basis function networks using orthogonal forward selection

An orthogonal forward selection (OFS) algorithm based on leave-one-out (LOO) criteria is proposed for the construction of radial basis function (RBF) networks with tunable nodes. Each stage of the construction process determines an RBF node, namely, its center vector and diagonal covariance matrix, by minimizing the LOO statistics. For regression application, the LOO criterion is chosen to be the LOO mean-square error, while the LOO misclassification rate is adopted in two-class classification application. This OFS-LOO algorithm is computationally efficient, and it is capable of constructing parsimonious RBF networks that generalize well. Moreover, the proposed algorithm is fully automatic, and the user does not need to specify a termination criterion for the construction process. The effectiveness of the proposed RBF network construction procedure is demonstrated using examples taken from both regression and classification applications.

Kernel classifier construction using orthogonal forward selection and boosting with Fisher ratio class separability measure

A greedy technique is proposed to construct parsimonious kernel classifiers using the orthogonal forward selection method and boosting based on Fisher ratio for class separability measure. Unlike most kernel classification methods, which restrict kernel means to the training input data and use a fixed common variance for all the kernel terms, the proposed technique can tune both the mean vector and diagonal covariance matrix of individual kernel by incrementally maximizing Fisher ratio for class separability measure. An efficient weighted optimization method is developed based on boosting to append kernels one by one in an orthogonal forward selection procedure. Experimental results obtained using this construction technique demonstrate that it offers a viable alternative to the existing state-of-the-art kernel modeling methods for constructing sparse Gaussian radial basis function network classifiers. that generalize well.

Radial basis function classifier construction using particle swarm optimisation aided orthogonal forward regression

We develop a particle swarm optimisation (PSO) aided orthogonal forward regression (OFR) approach for constructing radial basis function (RBF) classifiers with tunable nodes. At each stage of the OFR construction process, the centre vector and diagonal covariance matrix of one RBF node is determined efficiently by minimising the leave-one-out (LOO) misclassification rate (MR) using a PSO algorithm. Compared with the state-of-the-art regularisation assisted orthogonal least square algorithm based on the LOO MR for selecting fixednode RBF classifiers, the proposed PSO aided OFR algorithm for constructing tunable-node RBF classifiers offers significant advantages in terms of better generalisation performance and smaller model size as well as imposes lower computational complexity in classifier construction process. Moreover, the proposed algorithm does not have any hyperparameter that requires costly tuning based on cross validation.

Neurofuzzy mixture of experts network parallel learning and model construction algorithms

A connection between a fuzzy neural network model with the mixture of experts network (MEN) modelling approach is established. Based on this linkage, two new neuro-fuzzy MEN construction algorithms are proposed to overcome the curse of dimensionality that is inherent in the majority of associative memory networks and/or other rule based systems. The first construction algorithm employs a function selection manager module in an MEN system. The second construction algorithm is based on a new parallel learning algorithm in which each model rule is trained independently, for which the parameter convergence property of the new learning method is established. As with the first approach, an expert selection criterion is utilised in this algorithm. These two construction methods are equivalent in their effectiveness in overcoming the curse of dimensionality by reducing the dimensionality of the regression vector, but the latter has the additional computational advantage of parallel processing. The proposed algorithms are analysed for effectiveness followed by numerical examples to illustrate their efficacy for some difficult data based modelling problems.

Neurofuzzy network model construction using Bézier-Bernstein polynomial functions

Neurofuzzy modelling systems combine fuzzy logic with quantitative artificial neural networks via a concept of fuzzification by using a fuzzy membership function usually based on B-splines and algebraic operators for inference, etc. The paper introduces a neurofuzzy model construction algorithm using Bezier-Bernstein polynomial functions as basis functions. The new network maintains most of the properties of the B-spline expansion based neurofuzzy system, such as the non-negativity of the basis functions, and unity of support but with the additional advantages of structural parsimony and Delaunay input space partitioning, avoiding the inherent computational problems of lattice networks. This new modelling network is based on the idea that an input vector can be mapped into barycentric co-ordinates with respect to a set of predetermined knots as vertices of a polygon (a set of tiled Delaunay triangles) over the input space. The network is expressed as the Bezier-Bernstein polynomial function of barycentric co-ordinates of the input vector. An inverse de Casteljau procedure using backpropagation is developed to obtain the input vector's barycentric co-ordinates that form the basis functions. Extension of the Bezier-Bernstein neurofuzzy algorithm to n-dimensional inputs is discussed followed by numerical examples to demonstrate the effectiveness of this new data based modelling approach.