Canonical correlation for principal components of time series

With contemporary data collection capacity, data sets containing large numbers of different multivariate time series relating to a common entity (e.g., fMRI, financial stocks) are becoming more prevalent. One pervasive question is whether or not there are patterns or groups of series within the larger data set (e.g., disease patterns in brain scans, mining stocks may be internally similar but themselves may be distinct from banking stocks). There is a relatively large body of literature centered on clustering methods for univariate and multivariate time series, though most do not utilize the time dependencies inherent to time series. This paper develops an exploratory data methodology which in addition to the time dependencies, utilizes the dependency information between S series themselves as well as the dependency information between p variables within the series simultaneously while still retaining the distinctiveness of the two types of variables. This is achieved by combining the principles of both canonical correlation analysis and principal component analysis for time series to obtain a new type of covariance/correlation matrix for a principal component analysis to produce a so-called “principal component time series”. The results are illustrated on two data sets.     

基于医学影像数据的大脑功能连接方法研究

本文采用不同的方法对比研究了精神病患者与 正常人大脑功能连接网络中的显著差异. 首先选用皮尔逊相关系数, 偏相关系数, 小波相关系数构造三种不同的大脑功能连接网络, 其次运用风险差检验出具有显著差异的功能连接, 最后进行比较和分析. 结果表明: 用不同方法得到的大脑功能连接网络中存在公共的异常连接. 在三种相关网络中本文均检验出病人的楔前叶与后扣带回这条连接较正常人而言功能减弱. 更值得注意的是, 一些公共异常连接的风险差值符号相反, 即病人大脑功能连接网络中该功能连接较正常人而言既增强又减弱, 表明在构造统计量寻找大脑功能异常连接时需作统计检验才能使结果更具说服力.     

On first-order definitions of subgraph isomorphism properties

Let φ(F) be the property of containing (as a subgraph) an isomorphic copy of a graph F. It is easy to show that this property cannot be defined in a first-order language by a sentence with a quantifier depth (or variable width) strictly less than the number of vertices in F. Nevertheless, such a definition exists in some classes of graphs. Three classes of graphs are considered: connected graphs with a large number of vertices, graphs with large treewidth, and graphs with high connectivity.     

On Eccentric Connectivity Index of Eccentric Graph of Regular Dendrimer

The eccentric connectivity index \(\xi ^c(G)\) of a connected graph G is defined as \(\xi ^c(G) =\sum _{v \in V(G)}{deg(v) e(v)},\) where deg(v) is the degree of vertex v and e(v) is the eccentricity of v. The eccentric graph, \(G_e\), of a graph G has the same set of vertices as G,  with two vertices u, v adjacent in \(G_e\) if and only if either u is an eccentric vertex of v or v is an eccentric vertex of u. In this paper, we obtain a formula for the eccentric connectivity index of the eccentric graph of a regular dendrimer. We also derive a formula for the eccentric connectivity index for the second iteration of eccentric graph of regular dendrimer.     

Tikhonov–Phillips regularizations in linear models with blurred design

The paper deals with recovering an unknown vector β ∈ ? ^p based on the observations Y = Xβ + ? ξ and Z = X + σζ, where X is an unknown n × p matrix with n ≥ p, ξ ∈ ? ^p is a standard white Gaussian noise, ζ is an n × p matrix with i.i.d. standard Gaussian entries, and ?, σ ∈ ?⁺ are known noise levels. It is assumed that X has a large condition number and p is large. Therefore, in order to estimate β, the simple Tikhonov–Phillips regularization (ridge regression) with a data-driven regularization parameter is used. For this estimation method, we study the effect of noise σζ on the quality of recovering Xβ using concentration inequalities for the prediction error.     

Resolution of the Wavefront Set Using General Continuous Wavelet Transforms

We consider the problem of characterizing the wavefront set of a tempered distribution \(u\in \mathcal {S}'(\mathbb {R}^{d})\) in terms of its continuous wavelet transform, where the latter is defined with respect to a suitably chosen dilation group \(H\subset \mathrm{GL}(\mathbb {R}^{d})\). In this paper we develop a comprehensive and unified approach that allows to establish characterizations of the wavefront set in terms of rapid coefficient decay, for a large variety of dilation groups. For this purpose, we introduce two technical conditions on the dual action of the group H, called microlocal admissibility and (weak) cone approximation property. Essentially, microlocal admissibility sets up a systematic relationship between the scales in a wavelet dilated by \(h\in H\) on one side, and the matrix norm of h on the other side. The (weak) cone approximation property describes the ability of the wavelet system to adapt its frequency-side localization to arbitrary frequency cones. Together, microlocal admissibility and the weak cone approximation property allow the characterization of points in the wavefront set using multiple wavelets. Replacing the weak cone approximation by its stronger counterpart gives rise to single wavelet characterizations. We illustrate the scope of our results by discussing—in any dimension \(d\ge 2\)—the similitude, diagonal and shearlet dilation groups, for which we verify the pertinent conditions. As a result, similitude and diagonal groups can be employed for multiple wavelet characterizations, whereas for the shearlet groups a single wavelet suffices. In particular, the shearlet characterization (previously only established for \(d=2\)) holds in arbitrary dimensions.     

On linkedness in the Cartesian product of graphs

We study linkedness of the Cartesian product of graphs and prove that the product of an a-linked and a b-linked graph is \((a+b-1)\)-linked if the graphs are sufficiently large. Further bounds in terms of connectivity are shown. We determine linkedness of products of paths and products of cycles.     

The general facility location problem with connectivity on trees

In this note we study the general facility location problem with connectivity. We present an O(np ²)-time algorithm for the general facility location problem with connectivity on trees. Furthermore, we present an O(np)-time algorithm for the general facility location problem with connectivity on equivalent binary trees.     

An Algorithm for Approximating the Second Moment of the Normalizing Constant Estimate from a Particle Filter

We propose a new algorithm for approximating the non-asymptotic second moment of the marginal likelihood estimate, or normalizing constant, provided by a particle filter. The computational cost of the new method is O(M) per time step, independently of the number of particles N in the particle filter, where M is a parameter controlling the quality of the approximation. This is in contrast to O(M N) for a simple averaging technique using M i.i.d. replicates of a particle filter with N particles. We establish that the approximation delivered by the new algorithm is unbiased, strongly consistent and, under standard regularity conditions, increasing M linearly with time is sufficient to prevent growth of the relative variance of the approximation, whereas for the simple averaging technique it can be necessary to increase M exponentially with time in order to achieve the same effect. This makes the new algorithm useful as part of strategies for estimating Monte Carlo variance. Numerical examples illustrate performance in the context of a stochastic Lotka–Volterra system and a simple AR(1) model.     

Degree sequences and edge connectivity

For each positive integer k, we give a finite list C(k) of Bondy–Chvátal type conditions on a nondecreasing sequence \(d=(d_1,\dots ,d_n)\) of nonnegative integers such that every graph on n vertices with degree sequence at least d is k-edge-connected. These conditions are best possible in the sense that whenever one of them fails for d then there is a graph on n vertices with degree sequence at least d which is not k-edge-connected. We prove that C(k) is and must be large by showing that it contains p(k) many logically irredundant conditions, where p(k) is the number of partitions of k. Since, in the corresponding classic result on vertex connectivity, one needs just one such condition, this is one of the rare statements where the edge connectivity version is much more difficult than the vertex connectivity version. Furthermore, we demonstrate how to handle other types of edge-connectivity, such as, for example, essential k-edge-connectivity. We prove that any sublist equivalent to C(k) has length at least p(k), where p(k) is the number of partitions of k, which is in contrast to the corresponding classic result on vertex connectivity where one needs just one such condition. Furthermore, we demonstrate how to handle other types of edge-connectivity, such as, for example, essential k-edge-connectivity. Finally, we informally describe a simple and fast procedure which generates the list C(k). Specialized to \(k=3\), this verifies a conjecture of Bauer, Hakimi, Kahl, and Schmeichel, and for \(k=2\) we obtain an alternative proof for their result on bridgeless connected graphs. The explicit list for \(k=4\) is given, too.     

On the <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-consistency of wavelet estimators

This paper deals with the L ^p -consistency of wavelet estimators for a density function based on size-biased random samples. More precisely, we firstly show the L ^p -consistency of wavelet estimators for independent and identically distributed random vectors in R ^d . Then a similar result is obtained for negatively associated samples under the additional assumptions d = 1 and the monotonicity of the weight function.     

Optimal Rates for Regularization of Statistical Inverse Learning Problems

We consider a statistical inverse learning (also called inverse regression) problem, where we observe the image of a function f through a linear operator A at i.i.d. random design points \(X_i\), superposed with an additive noise. The distribution of the design points is unknown and can be very general. We analyze simultaneously the direct (estimation of Af) and the inverse (estimation of f) learning problems. In this general framework, we obtain strong and weak minimax optimal rates of convergence (as the number of observations n grows large) for a large class of spectral regularization methods over regularity classes defined through appropriate source conditions. This improves on or completes previous results obtained in related settings. The optimality of the obtained rates is shown not only in the exponent in n but also in the explicit dependency of the constant factor in the variance of the noise and the radius of the source condition set.     

Assessing the risk of vCJD transmission via surgery: models for uncertainty and complexity

Variant Creutzfeldt–Jakob disease (vCJD) is a fatal degenerative brain disease, which probably passed into humans from cattle infected with bovine spongiform encephalopathy (BSE) or ‘mad cow disease’. Whatever its origins, the question arises of whether vCJD could be spread from person to person. Any such risk is very difficult to quantify, not least because the number of people already infected is unknown. This paper reflects on a study assessing the risk of transmission via instruments used in hospital surgery. The study underpinned a number of policy initiatives, including a £200?m programme to improve instrument decontamination. The methodology had to meet two major challenges. That of dealing with very large and multiple uncertainties was initially addressed using simple spreadsheet-based models to generate alternative scenarios. The complexity in the possible dynamics of vCJD within the population was then tackled. This paper outlines the models built and their use in eliciting expert judgement and informing risk management policy.     

Predicting the probability of winning sealed bid auctions: a comparison of models

This paper is concerned with predicting the probability of tendering the lowest bid in sealed bid auctions. Four of the leading models from the bidding literature are shown to be subsumed within a generalised model. These models are then tested relative to the equal probability model by an empirical analysis of a large sample of real construction contract bidding data via all-in, one-out and one-on sample frames. A binomial test is used to measure the ability to predict the identity of the lowest bidders and the average logscore is used to measure the ability to predict the probability of each bidder being the lowest. Optimal cut-off criterion values are determined for defining the minimum size of dataset needed for disaggregating bidders. The work also highlights the importance of (1) the treatment of new entrants and general shortage of data on individual bidders, and (2) the treatment of predicted ties.     

Non-homogeneous <Emphasis Type="Italic">Tb</Emphasis> Theorem for Bi-parameter <Emphasis Type="Italic">g</Emphasis>-function

The main result of this paper is a bi-parameter Tb theorem for Littlewood–Paley g-function, where b is a tensor product of two pseudo-accretive function. Instead of the doubling measure, we work with a product measure μ = μ_n × μ_m, where the measures μ_n and μ_m are only assumed to be upper doubling. The main techniques of the proof include a bi-parameter b-adapted Haar function decomposition and an averaging identity over good double Whitney regions. Moreover, the non-homogeneous analysis and probabilistic methods are used again.     

Subspace dual super wavelet and Gabor frames

Due to its potential applications in multiplexing techniques, the study of superframes has interested some researchers. This paper addresses dual super wavelet and Gabor frames in the subspace setting. We obtain a basic-equation characterization for subspace dual super wavelet and Gabor frames. In addition, applying this characterization, we derive a procedure that allows for constructing subspace dual super wavelet frames based on certain subspace dual super Gabor frames, and vice versa. Our results are new even in L²(R;C ^L ) setting.     

The logarithm of the modulus of a holomorphic function as a minorant for a subharmonic function

For an arbitrary subharmonic function not identically equal to ?∞ in a domain D of the complex plane C, we prove the existence of a nonzero holomorphic function in D the logarithm of whose modulus is majorized by locally averaging a subharmonic function with logarithmic additions or even without them in the case D = C.     

Smoothing nonlinear subdivision schemes by averaging

In the theory of linear subdivision algorithms, it is well-known that the regularity of a linear subdivision scheme can be elevated by one order (say, from C ^k to C ^k+1) by composing it with an averaging step (equivalently, by multiplying to the subdivision mask a(z) a (1 + z) factor. In this paper, we show that the same can be done to nonlinear subdivision schemes: by composing with it any nonlinear, smooth, 2-point averaging step, the lifted nonlinear subdivision scheme has an extra order of regularity than the original scheme. A notable application of this result shows that the classical Lane-Riesenfeld algorithm for uniform B-Spline, when extended to Riemannian manifolds based on geodesic midpoint, produces curves with the same regularity as their linear counterparts. (In particular, curvature does not obstruct the nonlinear Lane-Riesenfeld algorithm to inherit regularity from the linear algorithm.) Our main result uses the recently developed technique of differential proximity conditions.     

Detection of Singularities by Discrete Multiscale Directional Representations

One of the most remarkable properties of the continuous curvelet and shearlet transforms is their sensitivity to the directional regularity of functions and distributions. As a consequence of this property, these transforms can be used to characterize the geometry of edge singularities of functions and distributions by their asymptotic decay at fine scales. This ability is a major extension of the conventional continuous wavelet transform which can only describe pointwise regularity properties. However, while in the case of wavelets it is relatively easy to relate the asymptotic properties of the continuous transform to properties of discrete wavelet coefficients, this problem is surprisingly challenging in the case of discrete curvelets and shearlets where one wants to handle also the geometry of the singularity. No result for the discrete case was known so far. In this paper, we derive non-asymptotic estimates showing that discrete shearlet coefficients can detect, in a precise sense, the location and orientation of curvilinear edges. We discuss connections and implications of this result to sparse approximations and other applications.     

Neighbor Connectivity of Two Kinds of Cayley Graphs

In this paper, we determine the neighbor connectivity κNB of two kinds of Cayley graphs: alternating group networks AN _n and star graphs S _n ; and give the exact values of edge neighbor connectivity λ_NB of ANn and Cayley graphs generated by transposition trees Γ _n . Those are κ_NB(AN _n ) = n^?1, λ_NB(AN _n ) = n^?2 and κ_NB(S _n ) = λ_NB(Γ _n ) = n^?1.