函数型数据回归分析综述

随着计算机储存能力和在线观测技术的提高,当今数据越来越多的以曲线和图像的形式存在.曲线和图像数据两个最显著的特征是高维和相邻数据间高度相关.这些特征使得传统的多元统计分析方法不再适合,而函数型数据在处理曲线和图像数据中具有无可比拟的优势.近年来各种各样的函数型数据分析方法得以发展,其中包括数据的对齐、主成分分析、回归、分类、聚类等.本文主要介绍函数型数据回归分析研究的起源、发展及最新进展.具体地,本文首先介绍函数型数据的概念;其次介绍函数型主成分分析方法;再次着重介绍函数型回归模型的估计、变量选择和检验方法;最后将简要探讨函数型数据未来的可能发展方向.     

Statistical Inference with Fractional Brownian Motion

We give a test between two complex hypothesis; namely we test whether a fractional Brownian motion (fBm) has a linear trend against a certain non-linear trend. We study some related questions, like goodness-of-fit test and volatility estimation in these models.     

Asymptotic expansions of the distributions of some test statistics for Gaussian ARMA processes

Let {X_t} be a Gaussian ARMA process with spectral density f_θ(λ), where θ is an unknown parameter. The problem considered is that of testing a simple hypothesis H:θ = θ₀ against the alternative A:θ ≠ θ₀. For this problem we propose a class of tests , which contains the likelihood ratio (LR), Wald (W), modified Wald (MW) and Rao (R) tests as special cases. Then we derive the χ² type asymptotic expansion of the distribution of T up to order n⁻¹, where n is the sample size. Also we derive the χ² type asymptotic expansion of the distribution of T under the sequence of alternatives A_n: θ = θ₀ + /√n, ε > 0. Then we compare the local powers of the LR, W, MW, and R tests on the basis of their asymptotic expansions.     

A brief remark on van der Waerden spaces

We demonstrate that Martin's axiom for -centered notions of forcing implies the existence of a van der Waerden space that is not a Hindman space. Our proof is an adaptation of the one given by M. Kojman and S. Shelah that such a space exists if one assumes the continuum hypothesis to be true.

     

A spectral interpretation for the explicit formula in the theory of prime numbers

A spectral interpretation for the poles and zeros of the L-function of algebraic number fields is given by Meyer. As Meyer works with Schwartz spaces which are not Hilbert spaces, the information on the location of zeros of the L-function is lost. In 1999, A. Connes gave a spectral interpretation for the critical zeros the Riemann zeta function. He works with Hilbert spaces. In this paper, we show that a variant of Connes’ trace formula is essentially equal to the explicit formula of A. Weil.     

A unifying local-semilocal convergence analysis and applications for two-point Newton-like methods in Banach space

We provide a local as well as a semilocal convergence analysis for two-point Newton-like methods in a Banach space setting under very general Lipschitz type conditions. Our equation contains a Fréchet differentiable operator F and another operator G whose differentiability is not assumed. Using more precise majorizing sequences than before we provide sufficient convergence conditions for Newton-like methods to a locally unique solution of equation F(x)+G(x)=0. In the semilocal case we show under weaker conditions that our error estimates on the distances involved are finer and the information on the location of the solution at least as precise as in earlier results. In the local case a larger radius of convergence is obtained. Several numerical examples are provided to show that our results compare favorably with earlier ones. As a special case we show that the famous Newton-Kantorovich hypothesis is weakened under the same hypotheses as the ones contained in the Newton-Kantorovich theorem.     

On a Riemann hypothesis analogue for selfdual weight enumerators of genus less than 3

In this contribution the Riemann hypothesis analogue presented by Iwan Duursma is considered. Whether the analogue for a given selfdual code holds depends on its Hamming weight enumerator. A necessary and sufficient condition is provided in the case of genus less than 3 for the homogeneous polynomials in two variables invariant under the MacWilliams transform. Also the case of half integral genus is studied and similar results are obtained.     

On Newton’s method for solving equations containing Fréchet-differentiable operators of order at least two

We provide sufficient conditions for the semilocal convergence of Newton’s method to a locally unique solution of a nonlinear operator equation containing operators that are Fréchet-differentiable of order at least two, in a Banach space setting. Numerical examples are also provided to show that our results apply to solve nonlinear equations in cases earlier ones cannot [J.M. Gutiérrez, A new semilocal convergence theorem for Newton’s method, J. Comput. Appl. Math. 79(1997) 131-145; Z. Huang, A note of Kantorovich theorem for Newton iteration, J. Comput. Appl. Math. 47 (1993) 211-217; F.A. Potra, Sharp error bounds for a class of Newton-like methods, Libertas Mathematica 5 (1985) 71-84].     

On the asymptotic existence of Hadamard matrices

It is conjectured that Hadamard matrices exist for all orders 4t (t>0). However, despite a sustained effort over more than five decades, the strongest overall existence results are asymptotic results of the form: for all odd natural numbers k, there is a Hadamard matrix of order k²[a+blog₂k], where a and b are fixed non-negative constants. To prove the Hadamard Conjecture, it is sufficient to show that we may take a=2 and b=0. Since Seberry's ground-breaking result, which showed that we may take a=0 and b=2, there have been several improvements where b has been by stages reduced to 3/8. In this paper, we show that for all ?>0, the set of odd numbers k for which there is a Hadamard matrix of order k²2+[?log₂k] has positive density in the set of natural numbers. The proof adapts a number-theoretic argument of Erdos and Odlyzko to show that there are enough Paley Hadamard matrices to give the result.     

Decision-making in testing process performance with fuzzy data

Over the years, numerous process capability indices (PCIs) have been proposed to the manufacturing industry to provide numerical measures of process performance. Most research efforts have focused on developing and investigating PCIs that assess process capability by precise measurements of output quality. However, real observations of continuous quantities are not precise numbers; in practice, they are more or less imprecise. Since observations of continuous random variables are imprecise the values of related test statistics become imprecise. Therefore, decision rules for statistical tests have to be adapted to this situation. This article presents a set of confidence intervals that produces triangular fuzzy numbers for the estimation of C_pk index using Buckley’s approach with some modification. Additionally, a three-decision testing rule and step-by-step procedure are developed to assess process performance based on fuzzy critical values and fuzzy p-values. This concept is also illustrated with an example for testing process performance.