Diwei Zhou

NON-EUCLIDEAN STATISTICS FOR COVARIANCE MATRICES, WITH APPLICATIONS TO DIFFUSION TENSOR IMAGING

The statistical analysis of covariance matrices occurs in m any important applications, e.g. in diffusion tensor imaging or longitudinal data analysis. We consider the situation where it is of interest to estimate an average covariance matrix, describe its anisotropy and to carry out principal geodesic analysis of covariance matrices. In medical image analysis a particular type of covariance matrix arises in diffusion weighted imaging called a diffusion tensor. The diffusion tensor is a 3 × 3 covariance matrix which is estimated at each voxel in the brain, and is obtained by fittin g a physically-motivated model on measurements from the Fourier transform of the molecule displacement density (Basser et al., 1994). A strongly anisotropic diffusion tensor indicates a strong direction of white matter fibre tracts, and plots of measures of anisotropy are very useful t o neurologists. A measure that is very commonly used in diffusion tensor imaging is Fractional Anisotropy

Regularisation, interpolation and visualisation of diffusion tensor images using non-Euclidean statistics.

Practical statistical analysis of diffusion tensor images is considered, and we focus primarily on methods that use metrics based on Euclidean distances between powers of diffusion tensors. First, we describe a family of anisotropy measures based on a scale invariant power-Euclidean metric, which are useful for visualisation. Some properties of the measures are derived and practical considerations are discussed, with some examples. Second, we discuss weighted Procrustes methods for diffusion tensor imaging interpolation and smoothing, and we compare methods based on different metrics on a set of examples as well as analytically. We establish a key relationship between the principal-square-root-Euclidean metric and the size-and-shape Procrustes metric on the space of symmetric positive semi-definite tensors. We explain, both analytically and by experiments, why the size-and-shape Procrustes metric may be preferred in practical tasks of interpolation, extrapolation and smoothing, especially when observed tensors are degenerate or when a moderate degree of tensor swelling is desirable. Third, we introduce regularisation methodology, which is demonstrated to be useful for highlighting features of prior interest and potentially for segmentation. Finally, we compare several metrics in a data set of human brain diffusion-weighted magnetic resonance imaging, and point out similarities between several of the non-Euclidean metrics but important differences with the commonly used Euclidean metric.

Competitive algorithms for unbounded one-way trading

In the one-way trading problem, a seller has L units of product to be sold to a sequence ? of buyers u 1 , u 2 , ? , u ? arriving online and he needs to decide, for each u i , the amount of product to be sold to u i at the then-prevailing market price p i . The objective is to maximize the sellers revenue. We note that all previous algorithms for the problem need to impose some artificial upper bound M and lower bound m on the market prices, and the seller needs to know either the values of M and m, or their ratio M / m , at the outset.This paper gives a one-way trading algorithm that does not impose any bounds on market prices and whose performance guarantee depends directly on the input. In particular, we give a class of one-way trading algorithms such that for any positive integer h and any positive number ?, we have an algorithm A h , ? that has competitive ratio O ( log ? r * ( log ( 2 ) ? r * ) ? ( log ( h - 1 ) ? r * ) ( log ( h ) ? r * ) 1 + ? ) if the value of r * = p * / p 1 , the ratio of the highest market price p * = max i ? p i and the first price p 1 , is large and satisfies log ( h ) ? r * 1 , where log ( i ) ? x denotes the application of the logarithm function i times to x; otherwise, A h , ? has a constant competitive ratio ? h . We also show that our algorithms are near optimal by showing that given any positive integer h and any one-way trading algorithm A, we can construct a sequence of buyers ? with log ( h ) ? r * 1 such that the ratio between the optimal revenue and the revenue obtained by A is ? ( log ? r * ( log ( 2 ) ? r * ) ? ( log ( h - 1 ) ? r * ) ( log ( h ) ? r * ) ) . A special case of the one-way trading is also studied, in which the L units of product are comprised of L items, each of which must be sold atomically (or equivalently, the amount of product sold to each buyer must be an integer).Furthermore, a complementary problem to the one-way trading problem, say, the one-way buying problem, is studied in this paper. In the one-way buying problem, a buyer wants to purchase one unit of product through a sequence of n sellers v 1 , v 2 , ? , v n arriving online, and she needs to decide the fraction to purchase from each v i at the then-prevailing market price p i . Her objective is to minimize the cost. The optimal competitive algorithms whose performance guarantees depend only on the lowest market price p * = min i ? p i , and one of M and ?, the price fluctuation ratio, are presented.

The project insurance option in infrastructure procurement

Purpose – Project insurance is designed to get over the perceived deficiencies of the conventional insurance practice. It involves the entire project supply chain being insured under a single policy taken out by the project owner. The purpose of this article is to report the outcomes of a study aimed at developing understanding of project insurance practice and how it compares with the conventional system.Design/methodology/approach – The study consisted of a questionnaire survey across four sectors of the construction industry.Findings – Direct experience of project insurance is still very patchy. They also raise doubt whether project insurance offers significant benefits to the supply chain members with direct responsibility for designing or executing projects. The main advantages of project insurance over the traditional fragmented insurance products reported by respondents were: avoidance of litigation to determine which member of the project supply chain should ultimately be liable when loss or damag...

Cluster Analysis of Diffusion Tensor Fields with Application to the Segmentation of the Corpus Callosum

Accurate segmentation of the Corpus Callosum (CC) is an important aspect of clinical medicine and is used in the diagnosis of various brain disorders. In this paper, we propose an automated method for two and three dimensional segmentation of the CC using diffusion tensor imaging. It has been demonstrated that Hartigans K-means is more efficient than the traditional Lloyd algorithm for clustering. We adapt Hartigans K-means to be applicable for use with the metrics that have a f -mean (e.g. Cholesky, root Euclidean and log Euclidean). Then we applied the adapted Hartigans K-means, using Euclidean, Cholesky, root Euclidean and log Euclidean metrics along with Procrustes and Riemannian metrics (which need numerical solutions for mean computation), to diffusion tensor images of the brain to provide a segmentation of the CC. The log Euclidean and Riemannian metrics provide more accurate segmentations of the CC than the other metrics as they present the least variation of the shape and size of the tensors in the CC for 2D segmentation. They also yield a full shape of the splenium for the 3D segmentation.

Competitive analysis of interrelated price online inventory problems with demands

This paper investigates interrelated price online inventory problems, in which decisions as to when and how much of a product to replenish must be made in an online fashion to meet some demand even without a concrete knowledge of future prices. The objective of the decision maker is to minimize the total cost while meeting the demands. Two different types of demand are considered carefully, that is, demands which are linearly and exponentially related to price. In this paper, the prices are online, with only the price range variation known in advance, and are interrelated with the preceding price. Two models of price correlation are investigated, namely, an exponential model and a logarithmic model. The corresponding algorithms of the problems are developed, and the competitive ratios of the algorithms are derived as the solutions by use of linear programming. doi:10.1017/S144618111700013X

Modeling Diffusion Directions of Corpus Callosum

Diffusion Tensor Imaging (DTI) has been used to study the characteristics of Multiple Sclerosis (MS) in the brain. The von Mises-Fisher distribution (vmf) is a probability distribution for modeling directional data on the unit hypersphere. In this paper we modeled the diffusion directions of the Corpus Callosum (CC) as a mixture of vmf distributions for both MS subjects and healthy controls. Higher diffusion concentration around the mean directions and smaller sum of angles between the mean directions are observed on the normal-appearing CC of the MS subjects as compared to the healthy controls.