Increasingly, statisticians are faced with the task of analyzing complex data that are non-Euclidean and specifically do not lie in a vector space. To address the need for statistical methods for such data, we introduce the concept of Fréchet regression. This is a general approach to regression when responses are complex random objects in a metric space and predictors are in R p , achieved by extending the classical concept of a Fréchet mean to the notion of a conditional Fréchet mean. We develop generalized versions of both global least squares regression and local weighted least squares smoothing. The target quantities are appropriately defined population versions of global and local regression for response objects in a metric space. We derive asymptotic rates of convergence for the corresponding fitted regressions using observed data to the population targets under suitable regularity conditions by applying empirical process methods. For the special case of random objects that reside in a Hilbert space, such as regression models with vector predictors and functional data as responses, we obtain a limit distribution. The proposed methods have broad applicability. Illustrative examples include responses that consist of probability distributions and correlation matrices, and we demonstrate both global and local Fréchet regression for demographic and brain imaging data. Local Fréchet regression is also illustrated via a simulation with response data which lie on the sphere.
Functional data that are nonnegative and have a constrained integral can be considered as samples of one-dimensional density functions. Such data are ubiquitous. Due to the inherent constraints, densities do not live in a vector space and, therefore, commonly used Hilbert space based methods of functional data analysis are not applicable. To address this problem, we introduce a transformation approach, mapping probability densities to a Hilbert space of functions through a continuous and invertible map. Basic methods of functional data analysis, such as the construction of functional modes of variation, functional regression or classification, are then implemented by using representations of the densities in this linear space. Representations of the densities themselves are obtained by applying the inverse map from the linear functional space to the density space. Transformations of interest include log quantile density and log hazard transformations, among others. Rates of convergence are derived for the representations that are obtained for a general class of transformations under certain structural properties. If the subject-specific densities need to be estimated from data, these rates correspond to the optimal rates of convergence for density estimation. The proposed methods are illustrated through simulations and applications in brain imaging.
We asked how team dynamics can be captured in relation to function by considering games in the first round of the NBA 2010 play-offs as networks. Defining players as nodes and ball movements as links, we analyzed the network properties of degree centrality, clustering, entropy and flow centrality across teams and positions, to characterize the game from a network perspective and to determine whether we can assess differences in team offensive strategy by their network properties. The compiled network structure across teams reflected a fundamental attribute of basketball strategy. They primarily showed a centralized ball distribution pattern with the point guard in a leadership role. However, individual play-off teams showed variation in their relative involvement of other players/positions in ball distribution, reflected quantitatively by differences in clustering and degree centrality. We also characterized two potential alternate offensive strategies by associated variation in network structure: (1) whether teams consistently moved the ball towards their shooting specialists, measured as “uphill/downhill” flux, and (2) whether they distributed the ball in a way that reduced predictability, measured as team entropy. These network metrics quantified different aspects of team strategy, with no single metric wholly predictive of success. However, in the context of the 2010 play-offs, the values of clustering (connectedness across players) and network entropy (unpredictability of ball movement) had the most consistent association with team advancement. Our analyses demonstrate the utility of network approaches in quantifying team strategy and show that testable hypotheses can be evaluated using this approach. These analyses also highlight the richness of basketball networks as a dataset for exploring the relationships between network structure and dynamics with team organization and effectiveness.
Background and Purpose— There is increasing evidence that higher systolic blood pressure variability (SBPV) may be associated with poor outcome in patients with intracerebral hemorrhage (ICH). We explored the association between SBPV and in-hospital ICH outcome. Methods— We collected 10-years of consecutive data of spontaneous ICH patients at 2 healthcare systems. Demographics, medical history, laboratory tests, computed tomography scan data, in-hospital treatments, and neurological and functional assessments were recorded. Blood pressure recordings were extracted up to 24 hours postadmission. SBPV was measured using SD, coefficient of variation, successive variation (SV), range and 1 novel index termed functional SV. The effects of SBPV on the functional outcome at discharge were evaluated by multivariate logistic and ordinal regression analyses for dichotomous and trichotomous modified Rankin Scale categorizations, respectively. In secondary analyses, associations between SBPV, history of hypertension, and hematoma expansion were explored. Results— The analysis included 762 subjects. All 5 SBPV indices were significantly associated with the probability of unfavorable outcome (modified Rankin Scale score, 4–6) in logistic models. In ordinal models, SD, coefficient of variation, range, and functional SV were found to have a significant effect on the probabilities of poor (modified Rankin Scale score, 3–4) and severe/death (modified Rankin Scale score, 5–6) outcomes. Normotensive patients had significantly lower mean SBPV compared with the untreated-hypertension cohort for all SBPV indices and compared with treated-hypertension patients for 3 out of 5 SBPV indices. Lower mean SBPV of treated-hypertension subjects compared with untreated-hypertension subjects was only detected in the SV and functional SV indices ( P =0.045). None of the SBPV indices were significantly associated with the probability of hematoma expansion. Conclusions— Higher SBPV in the first 24 hours of admission was associated with unfavorable in-hospital outcome among ICH patients. Further prospective studies are warranted to understand any cause-effect relationship and whether controlling for SBPV may improve the ICH outcome.
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