Fast orthogonal least squares algorithm for efficient subset model selection

Abstract-An efficient implementation of the orthogonal least squares algorithm for subset model selection is derived in this correspondence. Computational complexity of the algorithm is examined and the result shows that this new fast orthogonal least squares algorithm significantly reduces computational requirements. This error reduction ratio provides a criterion for forward subset selection. At the beginning of the 11th stage of the selection procedure, X has been transformed into X”’- ” = [WI... wI,- I xj,’-’)... x::;-’)] and y into y(/’-’), The 11th stage consists of i) For p 5 j 5.If, compute ii) 1

An adaptive orthogonal search algorithm for model subset selection and non-linear system identification

A new adaptive orthogonal search (AOS) algorithm is proposed for model subset selection and non-linear system identification. Model structure detection is a key step in any system identification problem. This consists of selecting significant model terms from a redundant dictionary of candidate model terms, and determining the model complexity (model length or model size). The final objective is to produce a parsimonious model that can well capture the inherent dynamics of the underlying system. In the new AOS algorithm, a modified generalized cross-validation criterion, called the adjustable prediction error sum of squares (APRESS), is introduced and incorporated into a forward orthogonal search procedure. The main advantage of the new AOS algorithm is that the mechanism is simple and the implementation is direct and easy, and more importantly it can produce efficient model subsets for most non-linear identification problems

Improved model identification for non-linear systems using a random subsampling and multifold modelling (RSMM) approach

In non-linear system identification, the available observed data are conventionally partitioned into two parts: the training data that are used for model identification and the test data that are used for model performance testing. This sort of 'hold-out' or 'split-sample' data partitioning method is convenient and the associated model identification procedure is in general easy to implement. The resultant model obtained from such a once-partitioned single training dataset, however, may occasionally lack robustness and generalisation to represent future unseen data, because the performance of the identified model may be highly dependent on how the data partition is made. To overcome the drawback of the hold-out data partitioning method, this study presents a new random subsampling and multifold modelling (RSMM) approach to produce less biased or preferably unbiased models. The basic idea and the associated procedure are as follows. First, generate K training datasets (and also K validation datasets), using a K-fold random subsampling method. Secondly, detect significant model terms and identify a common model structure that fits all the K datasets using a new proposed common model selection approach, called the multiple orthogonal search algorithm. Finally, estimate and refine the model parameters for the identified common-structured model using a multifold parameter estimation method. The proposed method can produce robust models with better generalisation performance

Improved model identification for nonlinear systems using a random subsampling and multifold modelling (RSMM) approach

In nonlinear system identification, the available observed data are conventionally partitioned into two parts: the training data that are used for model identification and the test data that are used for model performance testing. This sort of ‘hold-out’ or ‘split-sample’ data partitioning method is convenient and the associated model identification procedure is in general easy to implement. The resultant model obtained from such a once-partitioned single training dataset, however, may occasionally lack robustness and generalisation to represent future unseen data, because the performance of the identified model may be highly dependent on how the data partition is made. To overcome the drawback of the hold-out data partitioning method, this study presents a new random subsampling and multifold modelling (RSMM) approach to produce less biased or preferably unbiased models. The basic idea and the associated procedure are as follows. Firstly, generate K training datasets (and also K validation datasets), using a K-fold random subsampling method. Secondly, detect significant model terms and identify a common model structure that fits all the K datasets using a new proposed common model selection approach, called the multiple orthogonal search algorithm. Finally, estimate and refine the model parameters for the identified common-structured model using a multifold parameter estimation method. The proposed method can produce robust models with better generalisation performance

Better subset regression

To find efficient screening methods for high dimensional linear regression models, this paper studies the relationship between model fitting and screening performance. Under a sparsity assumption, we show that a subset that includes the true submodel always yields smaller residual sum of squares (i.e., has better model fitting) than all that do not in a general asymptotic setting. This indicates that, for screening important variables, we could follow a "better fitting, better screening" rule, i.e., pick a "better" subset that has better model fitting. To seek such a better subset, we consider the optimization problem associated with best subset regression. An EM algorithm, called orthogonalizing subset screening, and its accelerating version are proposed for searching for the best subset. Although the two algorithms cannot guarantee that a subset they yield is the best, their monotonicity property makes the subset have better model fitting than initial subsets generated by popular screening methods, and thus the subset can have better screening performance asymptotically. Simulation results show that our methods are very competitive in high dimensional variable screening even for finite sample sizes.Comment: 24 pages, 1 figur

Model structure selection using an integrated forward orthogonal search algorithm assisted by squared correlation and mutual information

Model structure selection plays a key role in non-linear system identification. The first step in non-linear system identification is to determine which model terms should be included in the model. Once significant model terms have been determined, a model selection criterion can then be applied to select a suitable model subset. The well known Orthogonal Least Squares (OLS) type algorithms are one of the most efficient and commonly used techniques for model structure selection. However, it has been observed that the OLS type algorithms may occasionally select incorrect model terms or yield a redundant model subset in the presence of particular noise structures or input signals. A very efficient Integrated Forward Orthogonal Search (IFOS) algorithm, which is assisted by the squared correlation and mutual information, and which incorporates a Generalised Cross-Validation (GCV) criterion and hypothesis tests, is introduced to overcome these limitations in model structure selection

Linear Time Feature Selection for Regularized Least-Squares

We propose a novel algorithm for greedy forward feature selection for regularized least-squares (RLS) regression and classification, also known as the least-squares support vector machine or ridge regression. The algorithm, which we call greedy RLS, starts from the empty feature set, and on each iteration adds the feature whose addition provides the best leave-one-out cross-validation performance. Our method is considerably faster than the previously proposed ones, since its time complexity is linear in the number of training examples, the number of features in the original data set, and the desired size of the set of selected features. Therefore, as a side effect we obtain a new training algorithm for learning sparse linear RLS predictors which can be used for large scale learning. This speed is possible due to matrix calculus based short-cuts for leave-one-out and feature addition. We experimentally demonstrate the scalability of our algorithm and its ability to find good quality feature sets.Comment: 17 pages, 15 figure

Sparse kernel density estimation technique based on zero-norm constraint

A sparse kernel density estimator is derived based on the zero-norm constraint, in which the zero-norm of the kernel weights is incorporated to enhance model sparsity. The classical Parzen window estimate is adopted as the desired response for density estimation, and an approximate function of the zero-norm is used for achieving mathemtical tractability and algorithmic efficiency. Under the mild condition of the positive definite design matrix, the kernel weights of the proposed density estimator based on the zero-norm approximation can be obtained using the multiplicative nonnegative quadratic programming algorithm. Using the -optimality based selection algorithm as the preprocessing to select a small significant subset design matrix, the proposed zero-norm based approach offers an effective means for constructing very sparse kernel density estimates with excellent generalisation performance

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

The 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 favourably with the SVM method, in terms of both test accuracy and sparsity, for constructing kernel density estimates