Browse
Submit Data
Databases
API
Help

Dataset Information

11 Views

0 Connections

0 Citations

0 Reanalyses

0 Downloads

Omics score: 0

Visualizing rank deficient models: a row equation geometry of rank deficient matrices and constrained-regression.

ABSTRACT: Situations often arise in which the matrix of independent variables is not of full column rank. That is, there are one or more linear dependencies among the independent variables. This paper covers in detail the situation in which the rank is one less than full column rank and extends this coverage to include cases of even greater rank deficiency. The emphasis is on the row geometry of the solutions based on the normal equations. The author shows geometrically how constrained-regression/generalized-inverses work in this situation to provide a solution in the face of rank deficiency.

SUBMITTER: O'Brien RM

PROVIDER: S-EPMC3378569 | biostudies-literature | 2012

REPOSITORIES: biostudies-literature

ACCESS DATA

Json Xml

Publications

Visualizing rank deficient models: a row equation geometry of rank deficient matrices and constrained-regression.

O'Brien Robert M RM

PloS one 20120619 6

Situations often arise in which the matrix of independent variables is not of full column rank. That is, there are one or more linear dependencies among the independent variables. This paper covers in detail the situation in which the rank is one less than full column rank and extends this coverage to include cases of even greater rank deficiency. The emphasis is on the row geometry of the solutions based on the normal equations. The author shows geometrically how constrained-regression/generali ...[more]

PMID: 22723906

Similar Datasets

Estimation of Ordinary Differential Equation Parameters Using Constrained Local Polynomial Regression.

Project description:We propose a new method to use a constrained local polynomial regression to estimate the unknown parameters in ordinary differential equation models with a goal of improving the smoothing-based two-stage pseudo-least squares estimate. The equation constraints are derived from the differential equation model and are incorporated into the local polynomial regression in order to estimate the unknown parameters in the differential equation model. We also derive the asymptotic bias and variance of the proposed estimator. Our simulation studies show that our new estimator is clearly better than the pseudo-least squares estimator in estimation accuracy with a small price of computational cost. An application example on immune cell kinetics and trafficking for influenza infection further illustrates the benefits of the proposed new method.

| S-EPMC4577067 | biostudies-literature

Integrative multi-view regression: Bridging group-sparse and low-rank models.

Project description:Multi-view data have been routinely collected in various fields of science and engineering. A general problem is to study the predictive association between multivariate responses and multi-view predictor sets, all of which can be of high dimensionality. It is likely that only a few views are relevant to prediction, and the predictors within each relevant view contribute to the prediction collectively rather than sparsely. We cast this new problem under the familiar multivariate regression framework and propose an integrative reduced-rank regression (iRRR), where each view has its own low-rank coefficient matrix. As such, latent features are extracted from each view in a supervised fashion. For model estimation, we develop a convex composite nuclear norm penalization approach, which admits an efficient algorithm via alternating direction method of multipliers. Extensions to non-Gaussian and incomplete data are discussed. Theoretically, we derive non-asymptotic oracle bounds of iRRR under a restricted eigenvalue condition. Our results recover oracle bounds of several special cases of iRRR including Lasso, group Lasso, and nuclear norm penalized regression. Therefore, iRRR seamlessly bridges group-sparse and low-rank methods and can achieve substantially faster convergence rate under realistic settings of multi-view learning. Simulation studies and an application in the Longitudinal Studies of Aging further showcase the efficacy of the proposed methods.

| S-EPMC6849205 | biostudies-literature

L2RM: Low-rank Linear Regression Models for High-dimensional Matrix Responses.

Project description:The aim of this paper is to develop a low-rank linear regression model (L2RM) to correlate a high-dimensional response matrix with a high dimensional vector of covariates when coefficient matrices have low-rank structures. We propose a fast and efficient screening procedure based on the spectral norm of each coefficient matrix in order to deal with the case when the number of covariates is extremely large. We develop an efficient estimation procedure based on the trace norm regularization, which explicitly imposes the low rank structure of coefficient matrices. When both the dimension of response matrix and that of covariate vector diverge at the exponential order of the sample size, we investigate the sure independence screening property under some mild conditions. We also systematically investigate some theoretical properties of our estimation procedure including estimation consistency, rank consistency and non-asymptotic error bound under some mild conditions. We further establish a theoretical guarantee for the overall solution of our two-step screening and estimation procedure. We examine the finite-sample performance of our screening and estimation methods using simulations and a large-scale imaging genetic dataset collected by the Philadelphia Neurodevelopmental Cohort (PNC) study.

| S-EPMC7781207 | biostudies-literature

Bayesian Generalized Low Rank Regression Models for Neuroimaging Phenotypes and Genetic Markers.

Project description:We propose a Bayesian generalized low rank regression model (GLRR) for the analysis of both high-dimensional responses and covariates. This development is motivated by performing searches for associations between genetic variants and brain imaging phenotypes. GLRR integrates a low rank matrix to approximate the high-dimensional regression coefficient matrix of GLRR and a dynamic factor model to model the high-dimensional covariance matrix of brain imaging phenotypes. Local hypothesis testing is developed to identify significant covariates on high-dimensional responses. Posterior computation proceeds via an efficient Markov chain Monte Carlo algorithm. A simulation study is performed to evaluate the finite sample performance of GLRR and its comparison with several competing approaches. We apply GLRR to investigate the impact of 1,071 SNPs on top 40 genes reported by AlzGene database on the volumes of 93 regions of interest (ROI) obtained from Alzheimer's Disease Neuroimaging Initiative (ADNI).

| S-EPMC4208701 | biostudies-literature

Robust reduced-rank regression.

Project description:In high-dimensional multivariate regression problems, enforcing low rank in the coefficient matrix offers effective dimension reduction, which greatly facilitates parameter estimation and model interpretation. However, commonly used reduced-rank methods are sensitive to data corruption, as the low-rank dependence structure between response variables and predictors is easily distorted by outliers. We propose a robust reduced-rank regression approach for joint modelling and outlier detection. The problem is formulated as a regularized multivariate regression with a sparse mean-shift parameterization, which generalizes and unifies some popular robust multivariate methods. An efficient thresholding-based iterative procedure is developed for optimization. We show that the algorithm is guaranteed to converge and that the coordinatewise minimum point produced is statistically accurate under regularity conditions. Our theoretical investigations focus on non-asymptotic robust analysis, demonstrating that joint rank reduction and outlier detection leads to improved prediction accuracy. In particular, we show that redescending [Formula: see text]-functions can essentially attain the minimax optimal error rate, and in some less challenging problems convex regularization guarantees the same low error rate. The performance of the proposed method is examined through simulation studies and real-data examples.

| S-EPMC5793675 | biostudies-literature

Constrained Fourth Order Latent Differential Equation Reduces Parameter Estimation Bias for Damped Linear Oscillator Models.

Project description:Second order linear differential equations can be used as models for regulation since under a range of parameter values they can account for return to equilibrium as well as potential oscillations in regulated variables. One method that can estimate parameters of these equations from intensive time series data is the method of Latent Differential Equations (LDE). However, the LDE method can exhibit bias in its parameters if the dimension of the time delay embedding and thus the width of the convolution kernel is not chosen wisely. This article presents a simulation study showing that a constrained fourth order Latent Differential Equation (FOLDE) model for the second order system almost completely eliminates bias as long as the width of the convolution kernel is less than two thirds the period of oscillations in the data. The FOLDE model adds two degrees of freedom over the standard LDE model but significantly improves model fit.

| S-EPMC7518521 | biostudies-literature

Bayesian longitudinal low-rank regression models for imaging genetic data from longitudinal studies.

Project description:To perform a joint analysis of multivariate neuroimaging phenotypes and candidate genetic markers obtained from longitudinal studies, we develop a Bayesian longitudinal low-rank regression (L2R2) model. The L2R2 model integrates three key methodologies: a low-rank matrix for approximating the high-dimensional regression coefficient matrices corresponding to the genetic main effects and their interactions with time, penalized splines for characterizing the overall time effect, and a sparse factor analysis model coupled with random effects for capturing within-subject spatio-temporal correlations of longitudinal phenotypes. Posterior computation proceeds via an efficient Markov chain Monte Carlo algorithm. Simulations show that the L2R2 model outperforms several other competing methods. We apply the L2R2 model to investigate the effect of single nucleotide polymorphisms (SNPs) on the top 10 and top 40 previously reported Alzheimer disease-associated genes. We also identify associations between the interactions of these SNPs with patient age and the tissue volumes of 93 regions of interest from patients' brain images obtained from the Alzheimer's Disease Neuroimaging Initiative.

| S-EPMC5368019 | biostudies-literature

Intrinsic Regression Models for Positive-Definite Matrices With Applications to Diffusion Tensor Imaging.

Project description:The aim of this paper is to develop an intrinsic regression model for the analysis of positive-definite matrices as responses in a Riemannian manifold and their association with a set of covariates, such as age and gender, in a Euclidean space. The primary motivation and application of the proposed methodology is in medical imaging. Because the set of positive-definite matrices do not form a vector space, directly applying classical multivariate regression may be inadequate in establishing the relationship between positive-definite matrices and covariates of interest, such as age and gender, in real applications. Our intrinsic regression model, which is a semiparametric model, uses a link function to map from the Euclidean space of covariates to the Riemannian manifold of positive-definite matrices. We develop an estimation procedure to calculate parameter estimates and establish their limiting distributions. We develop score statistics to test linear hypotheses on unknown parameters and develop a test procedure based on a resampling method to simultaneously assess the statistical significance of linear hypotheses across a large region of interest. Simulation studies are used to demonstrate the methodology and examine the finite sample performance of the test procedure for controlling the family-wise error rate. We apply our methods to the detection of statistical significance of diagnostic effects on the integrity of white matter in a diffusion tensor study of human immunodeficiency virus. Supplemental materials for this article are available online.

| S-EPMC2824554 | biostudies-literature

Subspace-constrained approaches to low-rank fMRI acceleration.

Project description:Acceleration methods in fMRI aim to reconstruct high fidelity images from under-sampled k-space, allowing fMRI datasets to achieve higher temporal resolution, reduced physiological noise aliasing, and increased statistical degrees of freedom. While low levels of acceleration are typically part of standard fMRI protocols through parallel imaging, there exists the potential for approaches that allow much greater acceleration. One such existing approach is k-t FASTER, which exploits the inherent low-rank nature of fMRI. In this paper, we present a reformulated version of k-t FASTER which includes additional L2 constraints within a low-rank framework. We evaluated the effect of three different constraints against existing low-rank approaches to fMRI reconstruction: Tikhonov constraints, low-resolution priors, and temporal subspace smoothness. The different approaches are separately tested for robustness to under-sampling and thermal noise levels, in both retrospectively and prospectively-undersampled finger-tapping task fMRI data. Reconstruction quality is evaluated by accurate reconstruction of low-rank subspaces and activation maps. The use of L2 constraints was found to achieve consistently improved results, producing high fidelity reconstructions of statistical parameter maps at higher acceleration factors and lower SNR values than existing methods, but at a cost of longer computation time. In particular, the Tikhonov constraint proved very robust across all tested datasets, and the temporal subspace smoothness constraint provided the best reconstruction scores in the prospectively-undersampled dataset. These results demonstrate that regularized low-rank reconstruction of fMRI data can recover functional information at high acceleration factors without the use of any model-based spatial constraints.

| S-EPMC7611820 | biostudies-literature

Bayes and empirical Bayes methods for reduced rank regression models in matched case-control studies.

Project description:Matched case-control studies are popular designs used in epidemiology for assessing the effects of exposures on binary traits. Modern studies increasingly enjoy the ability to examine a large number of exposures in a comprehensive manner. However, several risk factors often tend to be related in a nontrivial way, undermining efforts to identify the risk factors using standard analytic methods due to inflated type-I errors and possible masking of effects. Epidemiologists often use data reduction techniques by grouping the prognostic factors using a thematic approach, with themes deriving from biological considerations. We propose shrinkage-type estimators based on Bayesian penalization methods to estimate the effects of the risk factors using these themes. The properties of the estimators are examined using extensive simulations. The methodology is illustrated using data from a matched case-control study of polychlorinated biphenyls in relation to the etiology of non-Hodgkin's lymphoma.

| S-EPMC4870158 | biostudies-literature

OmicsDI is part of the ELIXIR infrastructure

OmicsDI is an Elixir interoperability service. Learn more ›

OmicsDI Databases

PRIDE
PeptideAtlas
MassIVE
JPOST Repository
Physiome Model Repository

EGA
EVA
ENA
LINCS
PAXDB
Cell Collective

MetaboLights
Metabolomics Workbench
MetabolomeExpress
GNPS
BioModels
FAIRDOMHub

ArrayExpress
dbGaP
ExpressionAtlas
GEO
NODE

Information

Databases
Help
API
Contact us
Code on GitHub
Terms of use
Submit Data