Orthogonal simple component analysis: A new, exploratory approach

Combining principles with pragmatism, a new approach and accompanying algorithm are presented to a longstanding problem in applied statistics: the interpretation of principal components. Following Rousson and Gasser [53 (2004) 539--555] @p250pt@ the ultimate goal is not to propose a method that leads automatically to a unique solution, but rather to develop tools for assisting the user in his or her choice of an interpretable solution. Accordingly, our approach is essentially exploratory. Calling a vector 'simple' if it has small integer elements, it poses the open question: @p250pt@ What sets of simply interpretable orthogonal axes---if any---are angle-close to the principal components of interest? its answer being presented in summary form as an automated visual display of the solutions found, ordered in terms of overall measures of simplicity, accuracy and star quality, from which the user may choose. Here, 'star quality' refers to striking overall patterns in the sets of axes found, deserving to be especially drawn to the user's attention precisely because they have emerged from the data, rather than being imposed on it by (implicitly) adopting a model. Indeed, other things being equal, explicit models can be checked by seeing if their fits occur in our exploratory analysis, as we illustrate. Requiring orthogonality, attractive visualization and dimension reduction features of principal component analysis are retained.Comment: Published in at http://dx.doi.org/10.1214/10-AOAS374 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

On the Limiting Behaviour of the Fundamental Geodesics of Information Geometry

The Information Geometry of extended exponential families has received much recent attention in a variety of important applications, notably categorical data analysis, graphical modelling and, more specifically, log-linear modelling. The essential geometry here comes from the closure of an exponential family in a high-dimensional simplex. In parallel, there has been a great deal of interest in the purely Fisher Riemannian structure of (extended) exponential families, most especially in the Markov chain Monte Carlo literature. These parallel developments raise challenges, addressed here, at a variety of levels: both theoretical and practical—relatedly, conceptual and methodological. Centrally to this endeavour, this paper makes explicit the underlying geometry of these two areas via an analysis of the limiting behaviour of the fundamental geodesics of Information Geometry, these being Amari’s (+1) and (0)-geodesics, respectively. Overall, a substantially more complete account of the Information Geometry of extended exponential families is provided than has hitherto been the case. We illustrate the importance and benefits of this novel formulation through applications

Computational information geometry in statistics: theory and practice

A broad view of the nature and potential of computational information geometry in statistics is offered. This new area suitably extends the manifold-based approach of classical information geometry to a simplicial setting, in order to obtain an operational universal model space. Additional underlying theory and illustrative real examples are presented. In the infinite-dimensional case, challenges inherent in this ambitious overall agenda are highlighted and promising new methodologies indicated

Neuroanatomical substrates for the volitional regulation of heart rate

The control of physiological arousal can assist in the regulation of emotional state. A subset cortical and subcortical brain regions are implicated in autonomic control of bodily arousal during emotional behaviors. Here, we combined human functional neuroimaging with autonomic monitoring to identify neural mechanisms that support the volitional regulation of heart rate, a process that may be assisted by visual feedback. During functional magnetic resonance imaging (fMRI), 15 healthy adults performed an experimental task in which they were prompted voluntarily to increase or decrease cardiovascular arousal (heart rate) during true, false, or absent visual feedback. Participants achieved appropriate changes in heart rate, without significant modulation of respiratory rate, and were overall not influenced by the presence of visual feedback. Increased activity in right amygdala, striatum and brainstem occurred when participants attempted to increase heart rate. In contrast, activation of ventrolateral prefrontal and parietal cortices occurred when attempting to decrease heart rate. Biofeedback enhanced activity within occipito-temporal cortices, but there was no significant interaction with task conditions. Activity in regions including pregenual anterior cingulate and ventral striatum reflected the magnitude of successful task performance, which was negatively related to subclinical anxiety symptoms. Measured changes in respiration correlated with posterior insula activation and heart rate, at a more lenient threshold, change correlated with insula, caudate, and midbrain activity. Our findings highlight a set of brain regions, notably ventrolateral prefrontal cortex, supporting volitional control of cardiovascular arousal. These data are relevant to understanding neural substrates supporting interaction between intentional and interoceptive states related to anxiety, with implications for biofeedback interventions, e.g., real-time fMRI, that target emotional regulation

An elementary account of Amari's expected geometry

Differential geometry has found fruitful application in statistical inference. In particular, Amari’s (1990) expected geometry is used in higher order asymptotic analysis, and in the study of sufficiency and ancillarity. However, we can see three drawbacks to the use of a differential geometric approach in econometrics and statistics more generally. Firstly, the mathematics is unfamiliar and the terms involved can be difficult for the econometrician to fully appreciate. Secondly, their statistical meaning can be less than completely clear, and finally the fact that, at its core, geometry is a visual subject can be obscured by the mathematical formalism required for a rigorous analysis, thereby hindering intuition. All three drawbacks apply particularly to the differential geometric concept of a non metric affine connection. The primary objective of this paper is to attempt to mitigate these drawbacks in the case of Amari’s expected geometric structure on a full exponential family. We aim to do this by providing an elementary account of this structure which is clearly based statistically, accessible geometrically and visually presented

Analyzing Recent Coronary Heart Disease Mortality Trends in Tunisia between 1997 and 2009.

BACKGROUND: In Tunisia, Cardiovascular Diseases are the leading causes of death (30%), 70% of those are coronary heart disease (CHD) deaths and population studies have demonstrated that major risk factor levels are increasing. OBJECTIVE: To explain recent CHD trends in Tunisia between 1997 and 2009. METHODS: DATA SOURCES: Published and unpublished data were identified by extensive searches, complemented with specifically designed surveys. ANALYSIS: Data were integrated and analyzed using the previously validated IMPACT CHD policy model. Data items included: (i)number of CHD patients in specific groups (including acute coronary syndromes, congestive heart failure and chronic angina)(ii) uptake of specific medical and surgical treatments, and(iii) population trends in major cardiovascular risk factors (smoking, total cholesterol, systolic blood pressure (SBP), body mass index (BMI), diabetes and physical inactivity). RESULTS: CHD mortality rates increased by 11.8% for men and 23.8% for women, resulting in 680 additional CHD deaths in 2009 compared with the 1997 baseline, after adjusting for population change. Almost all (98%) of this rise was explained by risk factor increases, though men and women differed. A large rise in total cholesterol level in men (0.73 mmol/L) generated 440 additional deaths. In women, a fall (-0.43 mmol/L), apparently avoided about 95 deaths. For SBP a rise in men (4 mmHg) generated 270 additional deaths. In women, a 2 mmHg fall avoided 65 deaths. BMI and diabetes increased substantially resulting respectively in 105 and 75 additional deaths. Increased treatment uptake prevented about 450 deaths in 2009. The most important contributions came from secondary prevention following Acute Myocardial Infarction (AMI) (95 fewer deaths), initial AMI treatments (90), antihypertensive medications (80) and unstable angina (75). CONCLUSIONS: Recent trends in CHD mortality mainly reflected increases in major modifiable risk factors, notably SBP and cholesterol, BMI and diabetes. Current prevention strategies are mainly focused on treatments but should become more comprehensive