Dimension Reduction Methodology Research Articles

Reliability estimation using univariate dimension reduction and extended generalised lambda distribution

This paper presents an analytical approach for structural reliability analysis without requiring the calculation of most probable point of failure. Initially, the primary statistical moments of a multi-dimensional performance function are estimated using the Univariate Dimension-Reduction (UDR) methodology based on additive decomposition of the limit state function. Through moment matching, the UDR-based estimated moments are then used to fit the parameters of Extended Generalised Lambda Distribution (EGLD), and finally the probability of failure is calculated. To evaluate the accuracy and efficiency of the UDR + EGLD approach in comparison to the traditional First-Order Reliability Method (FORM) and direct Monte Carlo Simulation (MCS), five example problems involving nonlinear limit state functions are examined. The results show that UDR + EGLD offers nearly the same level of accuracy as MCS with superior efficiency to FORM. However, UDR + EGLD appears to have tail sensitivity, which limits its application to problems with moderate levels of reliability.

Functional data analytic approach of modeling ECG T-wave shape to measure cardiovascular behavior

The T-wave of an electrocardiogram (ECG) represents the ventricular repolarization that is critical in restoration of the heart muscle to a pre-contractile state prior to the next beat. Alterations in the T-wave reflect various cardiac conditions; and links between abnormal (prolonged) ventricular repolarization and malignant arrhythmias have been documented. Cardiac safety testing prior to approval of any new drug currently relies on two points of the ECG waveform: onset of the Q-wave and termination of the T-wave; and only a few beats are measured. Using functional data analysis, a statistical approach extracts a common shape for each subject (reference curve) from a sequence of beats, and then models the deviation of each curve in the sequence from that reference curve as a four-dimensional vector. The representation can be used to distinguish differences between beats or to model shape changes in a subject's T-wave over time. This model provides physically interpretable parameters characterizing T-wave shape, and is robust to the determination of the endpoint of the T-wave. Thus, this dimension reduction methodology offers the strong potential for definition of more robust and more informative biomarkers of cardiac abnormalities than the QT (or QT corrected) interval in current use.

Kernel dimension reduction in regression

We present a new methodology for sufficient dimension reduction (SDR). Our methodology derives directly from the formulation of SDR in terms of the conditional independence of the covariate $X$ from the response $Y$, given the projection of $X$ on the central subspace [cf. J. Amer. Statist. Assoc. 86 (1991) 316--342 and Regression Graphics (1998) Wiley]. We show that this conditional independence assertion can be characterized in terms of conditional covariance operators on reproducing kernel Hilbert spaces and we show how this characterization leads to an $M$-estimator for the central subspace. The resulting estimator is shown to be consistent under weak conditions; in particular, we do not have to impose linearity or ellipticity conditions of the kinds that are generally invoked for SDR methods. We also present empirical results showing that the new methodology is competitive in practice.

Dimension Reduction in Regressions With Exponential Family Predictors

We present first methodology for dimension reduction in regressions with predictors that, given the response, follow one-parameter exponential families. Our approach is based on modeling the conditional distribution of the predictors given the response, which allows us to derive and estimate a sufficient reduction of the predictors. We also propose a method of estimating the forward regression mean function without requiring an explicit forward regression model. Whereas nearly all existing estimators of the central subspace are limited to regressions with continuous predictors only, our proposed methodology extends estimation to regressions with all categorical or a mixture of categorical and continuous predictors. Supplementary materials including the proofs and the computer code are available from the JCGS website.

Partial moment-based sufficient dimension reduction

In this paper, a partial moment-based sufficient dimension reduction methodology is developed. This extends and generalizes some existing methods and enables us to test predictor contributions. All tests in the proposed method are done with chi-squared distributions.

Kinetic Modeling of Large‐Scale Reaction Systems

This review gives a brief account of basic ideas underlying approaches to analysis and modeling of large‐scale reaction systems. The emphasis is on model simplification and mechanism/dimension reduction via heuristic concepts and formal mathematical techniques. Among the key topics discussed are: top‐down and bottom‐up modeling approaches, graph/matrix representation of chemical reactions, mechanistic vs. pathways models, quantitative structure‐reactivity relationships, mathematical reduction of dimensionality, high‐fidelity surrogate models, continuum approximation, lumping of nonlinear kinetics, overall behavior/kinetics of many reactions, effect of pore diffusion, steady state multiplicity and stability. Some common features of dimension reduction methodologies are noted. Areas where further work will be valuable are identified.

Response dimension reduction for the conditional mean in multivariate regression

Sufficient dimension reduction methodologies in regression have been developed in the past decade, focusing mostly on predictors. Here, we propose a methodology to reduce the dimension of the response vector in multivariate regression, without loss of information about the conditional mean. The asymptotic distributions of dimension test statistics are chi-squared distributions, and an estimate of the dimension reduction subspace is asymptotically efficient. Moreover, the proposed methodology enables us to test response effects for the conditional mean. Properties of the proposed method are studied via simulation.

A non-linear dimension reduction methodology for generating data-driven stochastic input models

Stochastic analysis of random heterogeneous media (polycrystalline materials, porous media, functionally graded materials) provides information of significance only if realistic input models of the topology and property variations are used. This paper proposes a framework to construct such input stochastic models for the topology and thermal diffusivity variations in heterogeneous media using a data-driven strategy. Given a set of microstructure realizations (input samples) generated from given statistical information about the medium topology, the framework constructs a reduced-order stochastic representation of the thermal diffusivity. This problem of constructing a low-dimensional stochastic representation of property variations is analogous to the problem of manifold learning and parametric fitting of hyper-surfaces encountered in image processing and psychology. Denote by M the set of microstructures that satisfy the given experimental statistics. A non-linear dimension reduction strategy is utilized to map M to a low-dimensional region, A . We first show that M is a compact manifold embedded in a high-dimensional input space R n . An isometric mapping F from M to a low-dimensional, compact, connected set A ⊂ R d ( d ≪ n ) is constructed. Given only a finite set of samples of the data, the methodology uses arguments from graph theory and differential geometry to construct the isometric transformation F : M → A . Asymptotic convergence of the representation of M by A is shown. This mapping F serves as an accurate, low-dimensional, data-driven representation of the property variations. The reduced-order model of the material topology and thermal diffusivity variations is subsequently used as an input in the solution of stochastic partial differential equations that describe the evolution of dependant variables. A sparse grid collocation strategy (Smolyak algorithm) is utilized to solve these stochastic equations efficiently. We showcase the methodology by constructing low-dimensional input stochastic models to represent thermal diffusivity in two-phase microstructures. This model is used in analyzing the effect of topological variations of two-phase microstructures on the evolution of temperature in heat conduction processes.

Sufficient dimension reduction for the conditional mean with a categorical predictor in multivariate regression

Recent sufficient dimension reduction methodologies in multivariate regression do not have direct application to a categorical predictor. For this, we define the multivariate central partial mean subspace and propose two methodologies to estimate it. The first method uses the ordinary least squares. Chi-squared distributed statistics for dimension tests are constructed, and an estimate of the target subspace is consistent and efficient. Moreover, the effects of continuous predictors can be tested without assuming any model. The second method extends Iterative Hessian Transformation to this context. For dimension estimation, permutation tests are used. Simulated and real data examples for illustrating various properties of the proposed methods are presented.

A novel moment-based sufficient dimension reduction approach in multivariate regression

Recently, a moment-based sufficient dimension reduction methodology in multivariate regression, focusing on the first two moments, was introduced. We present in this article a novel approach of the earlier method in roughly the same context. This novel method possesses several desirable properties that the earlier method did not have such as dimension tests with chi-squared distributions, predictor effects test without assuming any model, and so on. Simulated and real data examples are presented for studying various properties of the proposed method and for a numerical comparison to the earlier method.

Optimal sufficient dimension reduction for the conditional mean in multivariate regression

The aim of this article is to develop optimal sufficient dimension reduction methodology for the conditional mean in multivariate regression. The context is roughly the same as that of a related method by Cook & Setodji (2003), but the new method has several advantages. It is asymptotically optimal in the sense described herein and its test statistic for dimension always has a chi-squared distribution asymptotically under the null hypothesis. Additionally, the optimal method allows tests of predictor effects. A comparison of the two methods is provided.

Species reconstruction using pre-image curves

All chemistry dimension-reduction methodologies, such as quasi-steady-state approximation (QSSA), rate-controlled constrained equilibrium (RCCE), and intrinsic low-dimensional manifolds (ILDM), either implicitly or explicitly, provide a means of species reconstruction. Given a reduced description of the chemical composition (e.g., in terms of some “major” species), species reconstruction is a procedure to estimate the full composition. In this paper, we develop and demonstrate a new species reconstruction methodology, which is based on a pre-image curve of the reaction mapping. By definition, in the composition space, if a reaction trajectory is followed from any point on the pre-image curve, then at some time the reduced composition on the trajectory intersects the given reduced composition. The reconstructed composition is taken to be the full composition on the reaction trajectory at this intersection. The method is tested for a case of methane autoignition in which the detailed mechanism has 27 degrees of freedom, and two reduced descriptions are considered with 4 and 6 degrees of freedom. It is shown that the normalized reconstruction errors are very small; they are two orders of magnitude smaller than those in RCCE; and with 6 degrees of freedom they are smaller than those in QSSA with 12 degrees of freedom. The reconstructed compositions (for all reduced compositions) form the reconstructed manifold. It is shown that, to an excellent approximation, this manifold is inertial (i.e., it is composed of reaction trajectories). In comparison to other dimension-reduction methodologies, this is the first approach which determines, locally, points on a low-dimensional inertial manifold. The combination of in situ adaptive tabulation and species reconstruction can be used to reduce substantially the computational cost of applying detailed chemistry to turbulent combustion calculations.