In this article, we focus on the estimation of a high-dimensional inverse covariance (i.e., precision) matrix. We propose a simple improvement of the graphical Lasso (glasso) framework that is able to attain better statistical performance without increasing significantly the computational cost. The proposed improvement is based on computing a root of the sample covariance matrix to reduce the spread of the associated eigenvalues. Through extensive numerical results, using both simulated and real datasets, we show that the proposed modification improves the glasso procedure. Our results reveal that the square-root improvement can be a reasonable choice in practice. Supplementary material for this article is available online. ; Andrés M. Alonso gratefully acknowledges financial support from CICYT Grants ECO2012-38442 and CO2015-66593. Francisco J. Nogales and Vahe Avagyan were supported by the Spanish Government through project MTM2013-44902-P.
PURPOSE: The Hortega Study is a prospective study, which investigates novel determinants of selected chronic conditions with an emphasis on cardiovascular health in a representative sample of a general population from Spain. PARTICIPANTS: In 1997, a mailed survey was sent to a random selection of public health system beneficiaries assigned to the University Hospital Rio Hortega's catchment area in Valladolid (Spain) (n=11 423, phase I), followed by a pilot examination in 1999-2000 of 495 phase I participants (phase II). In 2001-2003, the examination of 1502 individuals constituted the Hortega Study baseline examination visit (phase III, mean age 48.7 years, 49% men, 17% with obesity, 27% current smokers). Follow-up of phase III participants (also termed Hortega Follow-up Study) was obtained as of 30 November 2015 through review of health records (9.5% of participants without follow-up information). FINDINGS TO DATE: The Hortega Study integrates baseline information of traditional and non-traditional factors (metabolomic including lipidomic and oxidative stress metabolites, genetic variants and environmental factors, such as metals), with 14 years of follow-up for the assessment of mortality and incidence of chronic diseases. Preliminary analysis of time to event data shows that well-known cardiovascular risk factors are associated with cardiovascular incidence rates, which add robustness to our cohort. FUTURE PLANS: In 2020, we will review updated health and mortality records of this ongoing cohort for a 5-year follow-up extension. We will also re-examine elder survivors to evaluate specific aspects of ageing and conduct geolocation to study additional environmental exposures. Stored biological specimens are available for analysis of new biomarkers. The Hortega Study will, thus, enable the identification of novel factors based on time to event data, potentially contributing to the prevention and control of chronic diseases in ageing populations. ; Public funding agencies (Funds from the Carlos III National Health Institutes from Spain, including National Networks for Research such as CIBEROBN and CIBERDEM, and Regional Funds for Research from the Castilla y Leon county and European Networks such as Ingenious Hypercare) have supported the collection of biological samples and data from study participants (Phases I–III). In Phase IV, the core funding was provided by the FP7 EU project MASCARA, the Carlos III National Health Institutes and INCLIVA Research Institute. In particular, this work was supported by the Strategic Action for Research in Health sciences [CP12/03080, PI15/00071, PI10/0082, PI13/01848, PI14/00874, PI16/01402 and PI11/00726], GRUPOS 03/101; PROMETEO/2009/029 and ACOMP/2013/039 from the Valencia Government, GRS/279/A/08 from Castilla-Leon Government and European Network of Excellence Ingenious Hypercare (EPSS- 037093) from the European Commission; CIBER Fisiopatología Obesidad y Nutrición (CIBEROBN) [CIBER-02-08-2009, CB06/03 and CB12/03/30016] and CIBER de Diabetes y Enfermedades Metabólicas Relacionadas (CIBERDEM). The Strategic Action for Research in Health sciences, CIBEROBN and CIBERDEM are initiatives from Carlos III Health Institute Madrid and the Spanish Ministry of Economy and Competitiveness and co-funded with European Funds for Regional Development (FEDER). The funding bodies had no role in the design of the study and collection, analysis, and interpretation of data and in writing the manuscript ; Sí
