Aihua Xia

Discrete Neuroanatomical Networks Are Associated with Specific Cognitive Abilities in Old Age

There have been many attempts at explaining age-related cognitive decline on the basis of regional brain changes, with the usual but inconsistent findings being that smaller gray matter volumes in certain brain regions predict worse cognitive performance in specific domains. Additionally, compromised white matter integrity, as suggested by white matter hyperintensities or decreased regional white matter fractional anisotropy, has an adverse impact on cognitive functions. The human brain is, however, a network and it may be more appropriate to relate cognitive functions to properties of the network rather than specific brain regions. We report on graph theory-based analyses of diffusion tensor imaging tract-derived connectivity in a sample of 342 healthy individuals aged 72–92 years. The cognitive domains included processing speed, memory, language, visuospatial, and executive functions. We examined the association of these cognitive assessments with both the connectivity of the whole brain network and individual cortical regions. We found that the efficiency of the whole brain network of cortical fiber connections had an influence on processing speed and visuospatial and executive functions. Correlations between connectivity of specific regions and cognitive assessments were also observed, e.g., stronger connectivity in regions such as superior frontal gyrus and posterior cingulate cortex were associated with better executive function. Similar to the relationship between regional connectivity efficiency and age, greater processing speed was significantly correlated with better connectivity of nearly all the cortical regions. For the first time, regional anatomical connectivity maps related to processing speed and visuospatial and executive functions in the elderly are identified.

Changing topological patterns in normal aging using large-scale structural networks.

We examine normal aging from the perspective of topological patterns of structural brain networks constructed from two healthy age cohorts 20 years apart. Based on graph theory, we constructed structural brain networks using 90 cortical and subcortical regions as a set of nodes and the interregional correlations of grey matter volumes across individual brains as edges between nodes, and further analyzed the topological properties of the age-specific networks. We found that the brain structural networks of both cohorts had small-world architecture, and the older cohort (N = 374; mean age = 66.6 years, range 64-68) had lower global efficiency but higher local clustering in the brain structural networks compared with the younger cohort (N = 428; mean age = 46.7, range 44-48). The older cohort had reduced hemispheric asymmetry and lower centrality of certain brain regions, such as the bilateral hippocampus, bilateral insula, left posterior cingulated, and right Heschl gyrus, but that of the prefrontal cortex (PFC) was not different. These structural network differences may provide the basis for changes in functional connectivity and indeed cognitive function as we grow older.

Stein’s method, Palm theory and Poisson process approximation

The framework of Stein’s method for Poisson process approximation is presented from the point of view of Palm theory, which is used to construct Stein identities and define local dependence. A general result (Theorem 2.3) in Poisson process approximation is proved by taking the local approach. It is obtained without reference to any particular metric, thereby allowing wider applicability. A Wasserstein pseudometric is introduced for measuring the accuracy of point process approximation. The pseudometric provides a generalization of many metrics used so far, including the total variation distance for random variables and the Wasserstein metric for processes as in Barbour and Brown [Stochastic Process. Appl. 43 (1992) 9–31]. Also, through the pseudometric, approximation for certain point processes on a given carrier space is carried out by lifting it to one on a larger space, extending an idea of Arratia, Goldstein and Gordon [Statist. Sci. 5 (1990) 403–434]. The error bound in the general result is similar in form to that for Poisson approximation. As it yields the Stein factor 1/λ as in Poisson approximation, it provides good approximation, particularly in cases where λ is large. The general result is applied to a number of problems including Poisson process modeling of rare words in a DNA sequence.

Nonrandom Clusters of Palindromes in Herpesvirus Genomes

Palindromes are symmetrical words of DNA in the sense that they read exactly the same as their reverse complementary sequences. Representing the occurrences of palindromes in a DNA molecule as points on the unit interval, the scan statistics can be used to identify regions of unusually high concentration of palindromes. These regions have been associated with the replication origins on a few herpesviruses in previous studies. However, the use of scan statistics requires the assumption that the points representing the palindromes are independently and uniformly distributed on the unit interval. In this paper, we provide a mathematical basis for this assumption by showing that in randomly generated DNA sequences, the occurrences of palindromes can be approximated by a Poisson process. An easily computable upper bound on the Wasserstein distance between the palindrome process and the Poisson process is obtained. This bound is then used as a guide to choose an optimal palindrome length in the analysis of a collection of 16 herpesvirus genomes. Regions harboring significant palindrome clusters are identified and compared to known locations of replication origins. This analysis brings out a few interesting extensions of the scan statistics that can help formulate an algorithm for more accurate prediction of replication origins.

On metrics in point process approximation

This paper improves some results of Barbour and Brown (1992) on approximation of a point process by a Poisson process. The approximations bound a Wasserstein distance, and here this distance is altered so as to permit the point process and the Poisson process to have different total mass. One approximation in Barbour and Brown (1992) hounds the Wasserstein distance by an average distance between the process and its reduced Palm process. Here the distance used in the bound is reduced so as to permit different total numbers of points and different locations for the points of the process and the reduced Palm process. Explicit relations between the Wasserstein and the Prohorov metrics are also considered

Normal approximation for random sums

In this paper, we adapt the very effective Berry-Esseen theorems of Chen and Shao (2004), which apply to sums of locally dependent random variables, for use with randomly indexed sums. Our particular interest is in random variables resulting from integrating a random field with respect to a point process. We illustrate the use of our theorems in three examples: in a rather general model of the insurance collective; in problems in geometrical probability involving stabilizing functionals; and in counting the maximal points in a two-dimensional region.

Removing logarithms from Poisson process error bounds

We present a new approximation theorem for estimating the error in approximating the whole distribution of a finite-point process by a suitable Poisson process. The metric used for this purpose regards the distributions as close if there are couplings of the processes with the expected average distance between points small in the best-possible matching. In many cases, the new bounds remain constant as the mean of the process increases, in contrast to previous results which, at best, increase logarithmically with the mean. Applications are given to Bernoulli-type point processes and to networks of queues. In these applications the bounds are independent of time and space, only depending on parameters of the system under consideration. Such bounds may be important in analysing properties, such as queueing parameters which depend on the whole distribution and not just the distribution of the number of points in a particular set.

Verifying edges for visual inspection purposes

Abstract In this paper we develop a method for verifying edges for visual inspection purposes. This method characterizes and examines a candidate edge against a given condition which can be designed and modified according to the required specifications of an application. A criterion is established using the edge strength, edge length and the distributions of grey level values of the pixels located in an edge neighbourhood. In consideration of computational speed and the accuracy of the locations of the edges, we first convolve the image with a Gaussian impulse response with a single small smoothing scale factor σ to generate candidate edges. The candidate edges are verified against the criterion and an edge is then retained or discarded on the basis of its local characteristics.

The number of two-dimensional maxima

Let n points be placed uniformly at random in a subset A of the plane. A point is said to be maximal in the configuration if no other point is larger in both coordinates. We show that, for large n and for many sets A, the number of maximal points is approximately normally distributed. The argument uses Steins method, and is also applicable in higher dimensions.

Estimating Stein's constants for compound Poisson approximation

Steins method for compound Poisson approximation was introduced by Barbour, Chen and Loh. One difficulty in applying the method is that the bounds on the solutions of the Stein equation are by no means as good as for Poisson approximation. We show that, for the Kolmogorov metric and under a condition on the parameters of the approximating compound Poisson distribution, bounds comparable with those obtained for the Poisson distribution can be recovered.