Identity Covariance Research Articles

On Fundamental Limits of Joint Sparse Support Recovery Using Certain Correlation Priors

This paper provides new probabilistic guarantees for recovering the common support of jointly sparse vectors in multiple measurement vector (MMV) models. In recent times, Bayesian approaches for sparse signal recovery (such as sparse Bayesian learning and correlation-aware LASSO) have shown preliminary evidence that under appropriate conditions (such as access to ideal covariance matrix of the measurements or certain restrictive orthogonality condition on the signals), it is possible to recover supports of size ( $K$ ) larger than the dimension ( $M$ ) of each measurement vector. However, no results exist that characterize the probability with which this can be achieved for a finite number of measurement vectors ( $L$ ). This paper bridges this gap by formulating the support recovery problem in terms of a multiple hypothesis testing framework. Chernoff-type upper bounds on the probability of error are established, and new sufficient conditions are derived that guarantee its exponential decay with respect to $L$ even when $K=O(M^2)$ . Our sufficient conditions are based on the properties of the so-called Khatri–Rao product of the measurement matrix and reveal the importance of a sampler design. Negative results are also established indicating that when $K$ exceeds a certain threshold (in terms of $M$ ), there will exist a class of measurement matrices for which any support recovery algorithm will fail. Using results from the geometric probability, we characterize the probability with which a randomly generated measurement matrix will belong to this class and show that this probability tends to 1 asymptotically in the size ( $N$ ) of the sparse vectors.

Projection pursuit in high dimensions

Projection pursuit is a classical exploratory data analysis method to detect interesting low-dimensional structures in multivariate data. Originally, projection pursuit was applied mostly to data of moderately low dimension. Motivated by contemporary applications, we here study its properties in high-dimensional settings. Specifically, we analyze the asymptotic properties of projection pursuit on structureless multivariate Gaussian data with an identity covariance, as both dimension p and sample size n tend to infinity, with [Formula: see text] Our main results are that (i) if [Formula: see text] then there exist projections whose corresponding empirical cumulative distribution function can approximate any arbitrary distribution; and (ii) if [Formula: see text], not all limiting distributions are possible. However, depending on the value of γ, various non-Gaussian distributions may still be approximated. In contrast, if we restrict to sparse projections, involving only a few of the p variables, then asymptotically all empirical cumulative distribution functions are Gaussian. And (iii) if [Formula: see text], then asymptotically all projections are Gaussian. Some of these results extend to mean-centered sub-Gaussian data and to projections into k dimensions. Hence, in the "small n, large p" setting, unless sparsity is enforced, and regardless of the chosen projection index, projection pursuit may detect an apparent structure that has no statistical significance. Furthermore, our work reveals fundamental limitations on the ability to detect non-Gaussian signals in high-dimensional data, in particular through independent component analysis and related non-Gaussian component analysis.

Increased Left Ventricular Mass Index Is Associated With Compromised White Matter Microstructure Among Older Adults.

BackgroundLeft ventricular (LV) hypertrophy is associated with cerebrovascular disease and cognitive decline. Increased LV mass index is a subclinical imaging marker that precedes overt LV hypertrophy. This study relates LV mass index to white matter microstructure and cognition among older adults with normal cognition and mild cognitive impairment.Methods and ResultsVanderbilt Memory & Aging Project participants free of clinical stroke, dementia, and heart failure (n=318, 73±7 years, 58% male, 39% mild cognitive impairment) underwent brain magnetic resonance imaging, cardiac magnetic resonance, and neuropsychological assessment. Voxelwise analyses related LV mass index (g/m2) to diffusion tensor imaging metrics. Models adjusted for age, sex, education, race/ethnicity, Framingham Stroke Risk Profile, cognitive diagnosis, and apolipoprotein E–ε4 status. Secondary analyses included a LV mass index×diagnosis interaction term with follow‐up models stratified by diagnosis. With identical covariates, linear regression models related LV mass index to neuropsychological performances. Increased LV mass index related to altered white matter microstructure (P<0.05). In models stratified by diagnosis, associations between LV mass index and diffusion tensor imaging were present among mild cognitive impairment participants only (P<0.05). LV mass index was related only to worse visuospatial memory performance (β=−0.003, P=0.036), an observation that would not withstand correction for multiple testing.ConclusionsIn the absence of prevalent heart failure and clinical stroke, increased LV mass index corresponds to altered white matter microstructure, particularly among older adults with clinical symptoms of prodromal dementia. Findings highlight the potential link between subclinical LV remodeling and cerebral white matter microstructure vulnerability.

Using time-varying models to estimate post-transplant survival in pediatric liver transplant recipients.

PurposeTo distinguish clinical factors that have time-varying (as opposed to constant) impact upon patient and graft survival among pediatric liver transplant recipients.MethodsUsing national data from 2002 through 2013, we examined potential clinical and demographic covariates using Gray’s piecewise constant time-varying coefficients (TVC) models. For both patient and graft survival, we estimated univariable and multivariable Gray’s TVC, retaining significant covariates based on backward selection. We then estimated the same specification using traditional Cox proportional hazards (PH) models and compared our findings.ResultsFor patient survival, covariates included recipient diagnosis, age, race/ethnicity, ventilator support, encephalopathy, creatinine levels, use of living donor, and donor age. Only the effects of recipient diagnosis and donor age were constant; effects of other covariates varied over time. We retained identical covariates in the graft survival model but found several differences in their impact.ConclusionThe flexibility afforded by Gray’s TVC estimation methods identify several covariates that do not satisfy constant proportionality assumptions of the Cox PH model. Incorporating better survival estimates is critical for improving risk prediction tools used by the transplant community to inform organ allocation decisions.

Mainlobe maintenance using shrinkage estimator method

A mainlobe maintenance method based on shrinkage estimator is presented here to promote the adaptive digital beamforming performance when there exists mainlobe jamming (MLJ). First, block matrix preprocessing (BMP) method is applied to suppress the MLJ. Then, the linear combination of estimated covariance matrix and identity matrix is optimised to generate more accurate estimation of the covariance matrix. After that, the improved covariance matrix is utilised to generate the adaptive weights to suppress the sidelobe jamming. Finally, the simulation shows that the proposed method is capable of eliminating peak offset of mainlobe and high sidelobes introduced by BMP and provides robustness against finite data samples effects. Accordingly, it outperforms noise whitening with error compensation, diagonal loading, and robust covariance matrix reconstruction in output SINR.

Haemoglobin glycation index and risk for diabetes-related complications in the Action in Diabetes and Vascular Disease: Preterax and Diamicron Modified Release Controlled Evaluation (ADVANCE) trial

Aims/hypothesisPrevious studies have suggested that the haemoglobin glycation index (HGI) can be used as a predictor of diabetes-related complications in individuals with type 1 and type 2 diabetes. We investigated whether HGI was a predictor of adverse outcomes of intensive glucose lowering and of diabetes-related complications in general, using data from the Action in Diabetes and Vascular Disease: Preterax and Diamicron Modified Release Controlled Evaluation (ADVANCE) trial.MethodsWe studied participants in the ADVANCE trial with data available for baseline HbA1c and fasting plasma glucose (FPG) (n = 11,083). HGI is the difference between observed HbA1c and HbA1c predicted from a simple linear regression of HbA1c on FPG. Using Cox regression, we investigated the association between HGI, both categorised and continuous, and adverse outcomes, considering treatment allocation (intensive or standard glucose control) and compared prediction of HGI and HbA1c.ResultsIntensive glucose control lowered mortality risk in individuals with high HGI only (HR 0.74 [95% CI 0.61, 0.91]; p = 0.003), while there was no difference in the effect of intensive treatment on mortality in those with high HbA1c. Irrespective of treatment allocation, every SD increase in HGI was associated with a significant risk increase of 14–17% for macrovascular and microvascular disease and mortality. However, when adjusted for identical covariates, HbA1c was a stronger predictor of these outcomes than HGI.Conclusions/interpretationHGI predicts risk for complications in ADVANCE participants, irrespective of treatment allocation, but no better than HbA1c. Individuals with high HGI have a lower risk for mortality when on intensive treatment. Given the discordant results and uncertain relevance beyond HbA1c, clinical use of HGI in type 2 diabetes cannot currently be recommended.

Clark-Ocone Formula for Generalized Functionals of Discrete-Time Normal Noises

The Clark-Ocone formula in the theory of discrete-time chaotic calculus holds only for square integrable functionals of discrete-time normal noises. In this paper, we aim at extending this formula to generalized functionals of discrete-time normal noises. Let Z be a discrete-time normal noise that has the chaotic representation property. We first prove a result concerning the regularity of generalized functionals of Z. Then, we use the Fock transform to define some fundamental operators on generalized functionals of Z and apply the abovementioned regularity result to prove the continuity of these operators. Finally, we establish the Clark-Ocone formula for generalized functionals of Z and show its application results, which include the covariant identity result and the variant upper bound result for generalized functionals of Z.

Estimation-Equivalent and Dispersion-Equivalent Error Covariance Matrices for the General Linear Model

SYNOPTIC ABSTRACTWe give a new, very concise derivation of an explicit characterization representation of the general nonnegative-definite error covariance matrix for a Gauss-Markov model, such that the best linear unbiased estimator is identical to the least-squares estimator. Our characterization derivation is very concise, and we use only elementary matrix properties in the proof. We also characterize the general symmetric nonnegative-definite error covariance matrix of a Gauss-Markov model, such that the covariance matrices of the best linear unbiased estimator, the least squares estimator, and the independently and identically-distributed least-squares estimator have identical covariance structures.

An Improved DOA Estimation Approach Using Coarray Interpolation and Matrix Denoising.

Co-prime arrays can estimate the directions of arrival (DOAs) of sources with sensors, and are convenient to analyze due to their closed-form expression for the locations of virtual lags. However, the number of degrees of freedom is limited due to the existence of holes in difference coarrays if subspace-based algorithms such as the spatial smoothing multiple signal classification (MUSIC) algorithm are utilized. To address this issue, techniques such as positive definite Toeplitz completion and array interpolation have been proposed in the literature. Another factor that compromises the accuracy of DOA estimation is the limitation of the number of snapshots. Coarray-based processing is particularly sensitive to the discrepancy between the sample covariance matrix and the ideal covariance matrix due to the finite number of snapshots. In this paper, coarray interpolation based on matrix completion (MC) followed by a denoising operation is proposed to detect more sources with a higher accuracy. The effectiveness of the proposed method is based on the capability of MC to fill in holes in the virtual sensors and that of MC denoising operation to reduce the perturbation in the sample covariance matrix. The results of numerical simulations verify the superiority of the proposed approach.

A note on the approximate admissibility of regularized estimators in the Gaussian sequence model

We study the problem of estimating an unknown vector $\theta $ from an observation $X$ drawn according to the normal distribution with mean $\theta $ and identity covariance matrix under the knowledge that $\theta $ belongs to a known closed convex set $\Theta $. In this general setting, Chatterjee (2014) proved that the natural constrained least squares estimator is “approximately admissible” for every $\Theta $. We extend this result by proving that the same property holds for all convex penalized estimators as well. Moreover, we simplify and shorten the original proof considerably. We also provide explicit upper and lower bounds for the universal constant underlying the notion of approximate admissibility.

Reproduction of secondary data in projection pursuit transformation

A challenge when working with multivariate data in a geostatistical context is that the data are rarely Gaussian. Multivariate distributions may include nonlinear features, clustering, long tails, functional boundaries, spikes, and heteroskedasticity. Multivariate transformations account for such features so that they are reproduced in geostatistical models. Projection pursuit as developed for high dimensional data exploration can also be used to transform a multivariate distribution into a multivariate Gaussian distribution with an identity covariance matrix. Its application within a geostatistical modeling context is called the projection pursuit multivariate transform (PPMT). An approach to incorporate exhaustive secondary variables in the PPMT is introduced. With this approach the PPMT can incorporate any number of secondary variables with any number of primary variables. A necessary alteration to the approach to make this numerically practical was the implementation of a continuous probability estimator that relies on Bernstein polynomials for the transformation that takes place in the projections. Stopping criteria were updated to incorporate a bootstrap t test that compares data sampled from a multivariate Gaussian distribution with the data undergoing transformation.

Power analysis to detect treatment effect in longitudinal studies with heterogeneous errors and incomplete data.

&nbsp;S. Usami (2014) describes a method to realistically determine sample size in longitudinal research using a multilevel model. The present research extends the aforementioned work to situations where it is likely that the assumption of homogeneity of the errors across groups is not met and the error term does not follow a scaled identity covariance structure.&nbsp; &nbsp;For this purpose, we followed a procedure based on transforming the variance components of the linear growth model and the parameter related to the treatment effect into specific and easily understandable indices. At the same time, we provide the appropriate statistical machinery&nbsp;for researchers to use when data loss is unavoidable, and changes in the expected value of the observed responses are not linear.&nbsp; &nbsp;The empirical powers based on unknown variance components were virtually the same as the theoretical powers derived from the use of statistically processed indexes.&nbsp; &nbsp;The main conclusion of the study is the accuracy of the proposed method to calculate sample size in the described situations with the stipulated power criteria.

Hospital Variation in Premature Clopidogrel Discontinuation After Drug-Eluting Stent Placement in the Veterans Affairs (VA) Healthcare System.

BackgroundPremature clopidogrel discontinuation after drug‐eluting stent placement is associated with adverse outcomes. Little is known about patient and hospital factors associated with premature discontinuation or whether less variation in premature discontinuation exists in integrated health care systems such as the Veterans Affairs (VA).Methods and ResultsWe evaluated the frequency of premature clopidogrel discontinuation, defined as a gap between clopidogrel refills of ≥90 days during the first 6 months of treatment, among 12 707 patients who received drug‐eluting stents in VA hospitals between 2008 and 2010. We evaluated the association between premature discontinuation and all‐cause mortality and/or acute myocardial infarction, variation in the proportion of premature discontinuation among hospitals, the patient and hospital characteristics associated with premature discontinuation, and the extent to which unexplained hospital characteristics contribute to premature discontinuation. Of the patients, 963 (7.6%) discontinued clopidogrel prematurely. Premature discontinuation was associated with acute myocardial infarction and all‐cause mortality (hazard ratio 1.65, 95% CI 1.37–1.99, P<0.001). The proportion of patients with premature discontinuation varied across hospitals from 0% to 16.5% (P<0.001). We found a median of 24% greater odds of patients with identical covariates with premature discontinuation at one randomly selected hospital compared with another (median odds ratio 1.24, 95% CI 1.17–1.44). Patient factors associated with premature discontinuation included lack of cardiology follow‐up within 30 days of discharge and smaller initial clopidogrel fill.ConclusionOne in 13 patients prematurely discontinued clopidogrel, and variation in discontinuation across hospitals was observed. Patient factors were associated with premature discontinuation that may represent targets for quality improvement.

Mixing of Poisson random measures under interacting transformations

We derive sufficient conditions for the mixing of all orders of interacting transformations of a spatial Poisson point process, under a zero-type condition in probability and a generalized adaptedness condition. This extends a classical result in the case of deterministic transformations of Poisson measures. The approach relies on moment and covariance identities for Poisson stochastic integrals with random integrands.

Closed form least-squares solution to 3D symmetric Helmert transformation with rotational invariant covariance structure

Least-squares (LS) solutions to the 3D Helmert and symmetric Helmert transformations with rotational invariant covariance structure are studied in a unified framework. This is an extension of the 3D Helmert transformation with naive identity covariance and the counterpart of the 2D symmetric Helmert transformation with rotational invariant covariance. It is found that the closed form LS solution still exists and that the rotation parameters are still the same between the Helmert and symmetric Helmert transformations.

The behavioral economics of driving after drinking among college drinkers.

Driving after drinking (DAD) among college students is a significant public health concern, yet little is known about specific theoretical risk factors for DAD, beyond drinking level, among college student drinkers. This study had the following aims: (i) to examine the associations between elevated alcohol demand and DAD, (ii) to determine whether demand decreases in response to a hypothetical driving scenario, (iii) to determine whether drivers who report DAD in the past 3months would show less of a reduction in demand in response to the hypothetical driving scenario, and (iv) to determine whether delayed reward discounting (DRD) is associated with DAD. Participants were 419 college students who reported at least 1day of past-month alcohol use. Participants completed 2 alcohol purchase tasks (APTs) that assessed hypothetical alcohol consumption across 17 drink prices with and without a driving scenario, a delay-discounting task, and a series of questions regarding DAD. In logistic regression models that controlled for drinking level, demographics, and sensation seeking, participants reporting higher demand intensity (95% confidence interval [95% CI] [1.04, 2.34]), breakpoint (95% CI [1.23, 2.28]), Omax (95% CI [1.03, 1.53]), and lower elasticity (95% CI [0.15, 1.02]) were more likely to report DAD. Additionally, in analyses of covariance, DAD(+) participants exhibited significantly less of a reduction in demand between the standard and the driving APT (intensity, p<0.01, breakpoint, p=0.05, and Omax , p<0.01). A binary logistic regression model with identical covariates revealed that DRD is not associated with DAD. DAD is associated with elevated/inelastic demand and less sensitivity to a hypothetical driving scenario. Drinkers with elevated demand should be prioritized for DAD intervention efforts.

Across all solid organs, adolescent age recipients have worse transplant organ survival than younger age children: A US national registry analysis.

Univariate analyses suggest that adolescents have worse long-term allograft survival versus younger children across different SOT. This study's objective was to determine whether multivariate analyses of a large national database recording all deceased SOT (KI; LI; HR; LU) also show worse adolescent allograft survival in the different organs. Using data from the national Scientific Registry for Transplant Recipients in the USA for pediatric primary SOT from 1989 to 2010, we calculated median half-lives and constructed K-M graft survival curves. Recipient age at transplant (<12 or adolescent 12-17 yr) was fitted with other identical covariates into multivariate Cox proportional hazards models. In all SOT recipients, unadjusted graft survival curves demonstrated better graft survival for adolescents initially, followed by crossing of the lines, such that adolescent SOT recipients had worse survival after one yr (KI), 4.6 yr (LI), 4.4 yr (HR), and 1.6 yr (LU). Multivariate models of the post-cross period showed a significantly higher AHR for worse graft survival in adolescent age across all four SOTs: AHR 1.400 (KI), 1.958 (LI), 1.414 (HR), and 1.576 (LU). Improving adolescent long-term outcomes across all four organs will be a defining issue in the future.

Gaussian Estimates for the Solutions of Some One-dimensional Stochastic Equations

Using covariance identities based on the Clark-Ocone representation formula we derive Gaussian density bounds and tail estimates for the probability law of the solutions of several types of stochastic differential equations, including Stratonovich equations with boundary condition and irregular drifts, and equations driven by fractional Brownian motion. Our arguments are generally simpler than the existing ones in the literature as our approach avoids the use of the inverse of the Ornstein-Uhlenbeck operator.

Improving the Ensemble-Optimization Method Through Covariance-Matrix Adaptation

Summary Ensemble optimization (referred to throughout the remainder of the paper as EnOpt) is a rapidly emerging method for reservoir-model-based production optimization. EnOpt uses an ensemble of controls to approximate the gradient of the objective function with respect to the controls. Current implementations of EnOpt use a Gaussian ensemble of control perturbations with a constant covariance matrix, and thus a constant perturbation size, during the entire optimization process. The covariance-matrix-adaptation evolutionary strategy is a gradient-free optimization method developed in the “machine learning” community, which also uses an ensemble of controls, but with a covariance matrix that is continually updated during the optimization process. It was shown to be an efficient method for several difficult but small-dimensional optimization problems and was recently applied in the petroleum industry for well location and production optimization. In this study, we investigate the scope to improve the computational efficiency of EnOpt through the use of covariance-matrix adaptation (referred to throughout the remainder of the paper as CMA-EnOpt). The resulting method is applied to the waterflooding optimization of a small multilayer test model and a modified version of the Brugge benchmark model. The controls used are inflow-control-valve settings at predefined time intervals for injectors and producers with undiscounted net present value as the objective function. We compare EnOpt and CMA-EnOpt starting from identical covariance matrices. For the small model, we achieve only slightly higher (0.7 to 1.8%) objective-function values and modest speedups with CMA-EnOpt compared with EnOpt. Significantly higher objective-function values (10%) are obtained for the modified Brugge model. The possibility to adapt the covariance matrix, and thus the perturbation size, during the optimization allows for the use of relatively large perturbations initially, for fast exploration of the control space, and small perturbations later, for more-precise gradients near the optimum. Moreover, the results demonstrate that a major benefit of CMA-EnOpt is its robustness with respect to the initial choice of the covariance matrix. A poor choice of the initial matrix can be detrimental to EnOpt, whereas the CMA-EnOpt performance is near-independent of the initial choice and produces higher objective-function values at no additional computational cost.

Hospital-level variation in angina and mortality at 1 year after myocardial infarction: insights from the Translational Research Investigating Underlying Disparities in Acute Myocardial Infarction Patients' Health Status (TRIUMPH) Registry.

Despite calls to expand measurement of acute myocardial infarction (AMI) outcomes to include symptom burden, little has been done to describe hospital-level variation in this patient-centered outcome, or its association with mortality. Understanding the relationship between symptoms and longer-term mortality could inform the importance of these outcomes for monitoring quality of care. Among 4316 patients with AMI treated at 24 hospitals participating in the Translational Research Investigating Underlying Disparities in Acute Myocardial Infarction Patients' Health Status (TRIUMPH) study, we assessed risk-standardized 1-year symptom burden as measured by the Seattle Angina Questionnaire Angina Frequency Score and mortality attributed to the hospital that provided AMI care. Median odds ratios were used to assess outcome variation and reflect the relative odds of an outcome for 2 patients with identical covariates at different, randomly selected, hospitals. We then evaluated the correlation between hospital-level mortality and angina. Finally, we determined the extent to which variation in mortality and angina was explained by achievement of AMI performance measures. We observed hospital variation in risk-adjusted 1-year mortality (range, 4.9%-8.6%; median odds ratio, 1.30; P=0.01) and angina (range, 17.7%-29.4%; median odds ratio, 1.34; P<0.001). At the hospital level, mortality and angina at 1 year were weakly correlated (r=0.40; 95% confidence interval, 0.00-0.68; P=0.05). Accounting for the quality of AMI care did not attenuate variation in risk-adjusted 1-year mortality or angina. Symptom burden and mortality vary at the hospital level after AMI and are only weakly correlated. These findings suggest that symptom burden should be considered a separate quality domain that is not well captured by current quality metrics.