Automatic induction of projection pursuit indices

Skewness-Based Projection Pursuit as an Eigenvector Problem in Scale Mixtures of Skew-Normal Distributions

This paper addresses the projection pursuit problem assuming that the distribution of the input vector belongs to the flexible and wide family of multivariate scale mixtures of skew normal distributions. Under this assumption, skewness-based projection pursuit is set out as an eigenvector problem, described in terms of the third order cumulant matrix, as well as an eigenvector problem that involves the simultaneous diagonalization of the scatter matrices of the model. Both approaches lead to dominant eigenvectors proportional to the shape parametric vector, which accounts for the multivariate asymmetry of the model; they also shed light on the parametric interpretability of the invariant coordinate selection method and point out some alternatives for estimating the projection pursuit direction. The theoretical findings are further investigated through a simulation study whose results provide insights about the usefulness of skewness model-based projection pursuit in the statistical practice.

An Enhanced Visualization Method to Aid Behavioral Trajectory Pattern Recognition Infrastructure for Big Longitudinal Data

Big longitudinal data provide more reliable information for decision making and are common in all kinds of fields. Trajectory pattern recognition is in an urgent need to discover important structures for such data. Developing better and more computationally-efficient visualization tool is crucial to guide this technique. This paper proposes an enhanced projection pursuit (EPP) method to better project and visualize the structures (e.g. clusters) of big high-dimensional (HD) longitudinal data on a lower-dimensional plane. Unlike classic PP methods potentially useful for longitudinal data, EPP is built upon nonlinear mapping algorithms to compute its stress (error) function by balancing the paired weights for between and within structure stress while preserving original structure membership in the high-dimensional space. Specifically, EPP solves an NP hard optimization problem by integrating gradual optimization and non-linear mapping algorithms, and automates the searching of an optimal number of iterations to display a stable structure for varying sample sizes and dimensions. Using publicized UCI and real longitudinal clinical trial datasets as well as simulation, EPP demonstrates its better performance in visualizing big HD longitudinal data.

Hybrid Hierarchical Clustering — Piecewise Aggregate Approximation, with Applications

Piecewise Aggregate Approximation (PAA) provides a powerful yet computationally efficient tool for dimensionality reduction and Feature Extraction (FE) on large datasets compared to previously reported and well-used FE techniques, such as Principal Component Analysis (PCA). Nevertheless, performance can degrade as a result of either regional information insufficiency or over-segmentation, and because of this, additional relatively complex modifications have subsequently been reported, for instance, Adaptive Piecewise Constant Approximation (APCA). To recover some of the simplicity of the original PAA, whilst addressing the known problems, a distance-based Hierarchical Clustering (HC) technique is now proposed to adjust PAA segment frame sizes to focus segment density on information rich data regions. The efficacy of the resulting hybrid HC-PAA methodology is demonstrated using two application case studies viz. fault detection on industrial gas turbines and ultrasonic biometric face identification. Pattern recognition results show that the extracted features from the hybrid HC-PAA provide additional benefits with regard to both cluster separation and classification performance, compared to traditional PAA and APCA alternatives. The method is therefore demonstrated to provide a robust and readily implemented algorithm for rapid FE and identification for datasets.

Sequential projection pursuit with kernel matrix update and symbolic model selection

This paper proposes a novel way for generating reliable low-dimensional features with improved class separability in a kernel-induced feature space. The feature projections rely on a very efficient sequential projection pursuit method, adapted to support nonlinear projections using a new kernel matrix update scheme. This enables the gradual removal of structure from the space of residual dimensions to allow the recovery of multiple projections. An adaptive kernel function is employed to unfold different types of data characteristics. We follow a holistic model selection procedure that, together with the optimal projections, dimensionality, and kernel parameters, additionally optimizes symbolically the projection index that controls the actual measurement of the data interestingness without user interaction. We tackle the underlying complex bi-level optimization model as a mixture of evolutionary and gradient search. The effectiveness of the proposed algorithm over existing approaches is demonstrated with benchmark evaluations and comparisons.

Automated induction of heterogeneous proximity measures for supervised spectral embedding

Spectral embedding methods have played a very important role in dimensionality reduction and feature generation in machine learning. Supervised spectral embedding methods additionally improve the classification of labeled data, using proximity information that considers both features and class labels. However, these calculate the proximity information by treating all intraclass similarities homogeneously for all classes, and similarly for all interclass samples. In this paper, we propose a very novel and generic method which can treat all the intra- and interclass sample similarities heterogeneously by potentially using a different proximity function for each class and each class pair. To handle the complexity of selecting these functions, we employ evolutionary programming as an automated powerful formula induction engine. In addition, for computational efficiency and expressive power, we use a compact matrix tree representation equipped with a broad set of functions that can build most currently used similarity functions as well as new ones. Model selection is data driven, because the entire model is symbolically instantiated using only problem training data, and no user-selected functions or parameters are required. We perform thorough comparative experimentations with multiple classification datasets and many existing state-of-the-art embedding methods, which show that the proposed algorithm is very competitive in terms of classification accuracy and generalization ability.

Adaptive data embedding framework for multiclass classification

The objective of this paper is the design of an engine for the automatic generation of supervised manifold embedding models. It proposes a modular and adaptive data embedding framework for classification, referred to as DEFC, which realizes in different stages including initial data preprocessing, relation feature generation and embedding computation. For the computation of embeddings, the concepts of friend closeness and enemy dispersion are introduced, to better control at local level the relative positions of the intraclass and interclass data samples. These are shown to be general cases of the global information setup utilized in the Fisher criterion, and are employed for the construction of different optimization templates to drive the DEFC model generation. For model identification, we use a simple but effective bilevel evolutionary optimization, which searches for the optimal model and its best model parameters. The effectiveness of DEFC is demonstrated with experiments using noisy synthetic datasets possessing nonlinear distributions and real-world datasets from different application fields.

Classifiability-based discriminatory projection pursuit

Fisher's linear discriminant (FLD) is one of the most widely used linear feature extraction method, especially in many visual computation tasks. Based on the analysis on several limitations of the traditional FLD, this paper attempts to propose a new computational paradigm for discriminative linear feature extraction, named "classifiability-based discriminatory projection pursuit" (CDPP), which is different from the traditional FLD and its variants. There are two steps in the proposed CDPP: one is the construction of a candidate projection set (CPS), and the other is the pursuit of discriminatory projections. Specifically, in the former step, candidate projections are generated by using the nearest between-class boundary samples, while the latter is efficiently achieved by classifiability-based AdaBoost learning from the CPS. We show that the new "projection pursuit" paradigm not only does not suffer from the limitations of the traditional FLD but also inherits good generalizability from the boundary attribute of candidate projections. Extensive experiments on both synthetic and real datasets validate the effectiveness of CDPP for discriminative linear feature extraction.

Iterative Gaussianization: from ICA to random rotations

Most signal processing problems involve the challenging task of multidimensional probability density function (PDF) estimation. In this paper, we propose a solution to this problem by using a family of rotation-based iterative Gaussianization (RBIG) transforms. The general framework consists of the sequential application of a univariate marginal Gaussianization transform followed by an orthonormal transform. The proposed procedure looks for differentiable transforms to a known PDF so that the unknown PDF can be estimated at any point of the original domain. In particular, we aim at a zero-mean unit-covariance Gaussian for convenience. RBIG is formally similar to classical iterative projection pursuit algorithms. However, we show that, unlike in PP methods, the particular class of rotations used has no special qualitative relevance in this context, since looking for interestingness is not a critical issue for PDF estimation. The key difference is that our approach focuses on the univariate part (marginal Gaussianization) of the problem rather than on the multivariate part (rotation). This difference implies that one may select the most convenient rotation suited to each practical application. The differentiability, invertibility, and convergence of RBIG are theoretically and experimentally analyzed. Relation to other methods, such as radial Gaussianization, one-class support vector domain description, and deep neural networks is also pointed out. The practical performance of RBIG is successfully illustrated in a number of multidimensional problems such as image synthesis, classification, denoising, and multi-information estimation.