Lajos Horváth, Gregory Rice
The cumulative sum (CUSUM) process is often used in change point analysis to detect changes in the mean of sequentially observed data. We provide a full description of the asymptotic distribution of $L^p, 1\leq p <\infty$, functionals of the weighted CUSUM process for time series under general conditions.
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Functional Sieve Bootstrap for the Partial Sum Process with Application to Change-Point Detection
本文提出函数筛子自助法（FSB）估计部分和过程分布，并用于基于CUSUM统计量的函数时间序列均值变点检验，证明其渐近有效性和有限样本性能。
Multi-Scale CUSUM Tests for Time Dependent Spherical Random Fields
本文研究了时间相关球面随机场调和域结构性突变检验的渐近行为，证明了样本球面调和系数波动的泛函中心极限定理及 CUSUM 检验的一致性，并应用于 NCEP 全球温度数据，发现气候变化不仅影响全球平均温度，还影响不同尺度的空间波动性质。
We propose new tests to detect a change in the mean of a time series. Like many existing tests, the new ones are based on the CUSUM process. Existing CUSUM tests require an estimator of a scale parameter to make them asymptotically distribution free under the no change null hypothesis. Even if the observations are independent, the estimation of the scale parameter is not simple since the estimator for the scale parameter should be at least consistent under the null as well as under the alternative. The situation is much more complicated in case of dependent data, where the empirical spectral d…
In this paper, we present an efficient statistical method (denoted as "Adaptive Resources Allocation CUSUM") to robustly and efficiently detect the hotspot with limited sampling resources. Our main idea is to combine the multi-arm bandit (MAB) and change-point detection methods to balance the exploration and exploitation of resource allocation for hotspot detection. Further, a Bayesian weighted update is used to update the posterior distribution of the infection rate. Then, the upper confidence bound (UCB) is used for resource allocation and planning. Finally, CUSUM monitoring statistics to de…
The q-weighted CUSUM and their corresponding estimator are well known statistics for change-point detection and estimation. They have the difficulty that the performance is highly dependent on the location of the change. An adaptive estimator with data-driven weights is presented to overcome this problem, and it is shown that the corresponding adaptive change-point tests are valid.
Quickest Change Detection Using Mismatched CUSUM
本文针对最快变化检测问题，提出失配CUSUM形式的停止规则，基于隐马尔可夫模型推导线性函数类下的最优解，建立条件独立模型的渐近理论，数值验证了近似的准确性。
Change-Point Detection for Heterogeneous High-Dimensional Functional Time Series
针对高维函数时间序列变点检测中传统方法的信号衰减问题，提出Energy-PE统计量并结合聚类算法，经数值和股票应用验证其功效更优且可解释。
Robust Change-Point Detection for Functional Time Series Based on $U$-Statistics and Dependent Wild Bootstrap
本文将 Wilcoxon 两样本检验推广至函数时间序列，提出利用完整函数信息且抗异常值的稳健变点检验，并通过 Hilbert 空间 $U$-统计量极限定理与相依野自助法确定渐近分布和临界值。
The analysis of record-breaking events is of interest in fields such as climatology, hydrology or anthropology. In connection with the record occurrence, we propose three distribution-free statistics for the changepoint detection problem. They are CUSUM-type statistics based on the upper and/or lower record indicators observed in a series. Using a version of the functional central limit theorem, we show that the CUSUM-type statistics are asymptotically Kolmogorov distributed. The main results under the null hypothesis are based on series of independent and identically distributed random variab…
As an example for the fast calculation of distributional parameters of Gaussian processes, we propose a new Monte Carlo algorithm for the computation of quantiles of the supremum norm of weighted Brownian bridges. As it is known, the corresponding distributions arise asymptotically for weighted CUSUM statistics for change-point detection. The new algorithm employs an adaptive (sequential) time discretization for the trajectories of the Brownian bridge. A simulation study shows that the new algorithm by far outperforms the standard approach, which employs a uniform time discretization.
We study the detection of change-points in time series. The classical CUSUM statistic for detection of jumps in the mean is known to be sensitive to outliers. We thus propose a robust test based on the Wilcoxon two-sample test statistic. The asymptotic distribution of this test can be derived from a functional central limit theorem for two-sample U-statistics. We extend a theorem of Csorgo and Horvath to the case of dependent data.
Risk-Adjusted learning curve assessment using comparative probability metrics
本文提出一种基于比较概率指标的风险调整外科学习曲线评估方法，通过估计而非假设检验评估受训者表现，具有更高可解释性，并在结直肠手术数据中验证了有效性。
The paper studies the asymptotic behaviour of weighted functionals of long-range dependent data over increasing observation windows. Various important statistics, including sample means, high order moments, occupation measures can be given by these functionals. It is shown that in the discrete sampling case additive functionals have the same asymptotic distribution as the corresponding integral functionals for the continuous functional data case. These results are applied to obtain non-central limit theorems for weighted additive functionals of random fields. As the majority of known results c…
We prove moment inequalities for a class of functionals of i.i.d. random fields. We then derive rates in the central limit theorem for weighted sums of such randoms fields via an approximation by $m$-dependent random fields.
Power of Weighted Test Statistics for Structural Change in Time Series
研究常见变点检验的功效与变点位置的关系，基于加权U统计量最大值，发现在时间区间两端变化时加权检验功效更强，中心处减弱。
Many experiments record sequential trajectories where each trajectory consists of oscillations and fluctuations around zero. Such trajectories can be viewed as zero-mean functional data. When there are structural breaks (on the sequence of trajectories) in higher order moments, it is not always easy to spot these by mere visual inspection. Motivated by this challenging problem in brain signal analysis, we propose a detection and testing procedure to find the change point in functional covariance. The detection procedure is based on the cumulative sum statistics (CUSUM). The classical testing p…
Laplacian U-Processes for Multiple Change-Point Detection in Dependent Text Networks: An Application to Historical Chinese Articles
提出WCO框架检测弱相依文本网络变点，在《新青年》语料中定位1919年11月变点，与五四运动语言转型吻合。
Many popular robust estimators are $U$-quantiles, most notably the Hodges-Lehmann location estimator and the $Q_n$ scale estimator. We prove a functional central limit theorem for the sequential $U$-quantile process without any moment assumptions and under weak short-range dependence conditions. We further devise an estimator for the long-run variance and show its consistency, from which the convergence of the studentized version of the sequential $U$-quantile process to a standard Brownian motion follows. This result can be used to construct CUSUM-type change-point tests based on $U$-quantile…
This paper studies methods for testing and estimating change-points in the covariance structure of a high-dimensional linear time series. The assumed framework allows for a large class of multivariate linear processes (including vector autoregressive moving average (VARMA) models) of growing dimension and spiked covariance models. The approach uses bilinear forms of the centered or non-centered sample variance-covariance matrix. Change-point testing and estimation are based on maximally selected weighted cumulated sum (CUSUM) statistics. Large sample approximations under a change-point regime …
A Functional Central Limit Theorem for Locally Stationary Time Series in Banach Spaces
本文为Banach空间中的局部平稳时间序列建立了函数中心极限定理，不依赖type-2或cotype假设，并应用于自归一化CUSUM变点检测。
A large class of goodness-of-fit test statistics based on sup-functionals of weighted empirical processes is proposed and studied. The weight functions employed are Erdős-Feller-Kolmogorov-Petrovski upper-class functions of a Brownian bridge. Based on the result of M. Csörgő, S. Csörgő, Horváth, and Mason obtained for this type of test statistics, we provide the asymptotic null distribution theory for the class of tests in hand, and present an algorithm for tabulating the limit distribution functions under the null hypothesis. A new family of nonparametric confidence bands is constructed for t…
Network Weighted Functional Regression: a method for modeling dependencies between functional data in a network
本文提出网络加权函数回归（NWFR）模型，通过显式建模网络结构来处理网络设定下的函数数据，并开发了函数保形预测程序生成免分布预测区间，显著提升了点预测准确度及预测区间的有效性与精确度。
We introduce a rank-based bent linear regression with an unknown change point. Using a linear reparameterization technique, we propose a rank-based estimate that can make simultaneous inference on all model parameters, including the location of the change point, in a computationally efficient manner. We also develop a score-like test for the existence of a change point, based on a weighted CUSUM process. This test only requires fitting the model under the null hypothesis in absence of a change point, thus it is computationally more efficient than likelihood-ratio type tests. The asymptotic pro…
The problem of estimating a linear functional based on observational data is canonical in both the causal inference and bandit literatures. We analyze a broad class of two-stage procedures that first estimate the treatment effect function, and then use this quantity to estimate the linear functional. We prove non-asymptotic upper bounds on the mean-squared error of such procedures: these bounds reveal that in order to obtain non-asymptotically optimal procedures, the error in estimating the treatment effect should be minimized in a certain weighted $L^2$-norm. We analyze a two-stage procedure …
Score-Based Change-Point Detection and Region Localization for Spatio-Temporal Point Processes
本文提出一种基于评分的时空点过程变化点检测方法，能够实时定位变化发生的空间区域，提供理论保证和实证验证。
We provide asymptotic results for the distribution of weighted nonlinear functionals of Gaussian field with long-range dependence. We also show that integral functionals and the corresponding additive functionals have same distributions under certain assumptions. The result is applied to integrals over a multidimensional rectangle with a constant weight function.
We consider functionals which are weighted averages of the avoidance function of a Poisson process. Using the approach to Stein's method based on Malliavin calculus for Poisson functionals we provide explicit bounds for the Wasserstein distance between these standardized functionals and the standard normal distribution. Our approach relies on closed-form expressions for the action of some Malliavin type operators on avoidance functionals of Poisson processes. As a result we provide Berry-Esseen bounds in the CLT for the volume of the union of balls of a fixed radius around random Poisson cente…
The paper investigates isotropic random fields for which the spectral density is unbounded at some frequencies. Limit theorems for weighted functionals of these random fields are established. It is shown that for a wide class of functionals, which includes the Donsker scheme, the limit is not affected by singularities at non-zero frequencies. For general schemes, in contrast to the Donsker line, we demonstrate that the singularities at non-zero frequencies play a role even for linear functionals.
We consider a generalization of the weighted random ball model. The model is driven by a random Poisson measure with a product heavy tailed intensity measure. Such a model typically represents the transmission of a network of stations with a fading effect. In a previous article, the authors proved the convergence of the finite-dimensional distributions of related generalized random fields under various scalings and in the particular case when the fading function is the indicator function of the unit ball. In this paper, tightness and functional convergence are investigated. Using suitable mome…
Stability of Hölder regularity and weighted functional inequalities
本文研究度量测度空间上的对称Dirichlet形式，引入跳跃测度尾部条件和加权泛函不等式的新表述，建立了caloric和harmonic函数Hölder正则性的稳定等价刻画，并应用于证明具有边界爆炸行为的对称Markov过程的caloric函数的Hölder连续性。
A functional central limit theorem for weighted occupancy processes of the Karlin model
该论文为 Karlin 模型的加权占用过程建立了泛函中心极限定理，并应用于 $(\alpha,\theta)$-座位中餐馆过程诱导的随机排列极限定理。
Let $M$ be a discrete-time normal martingale that has the chaotic representation property. Then, from the space of square integrable functionals of $M$, one can construct generalized functionals of $M$. In this paper, by using a type of weights, we introduce a class of continuous linear operators acting on generalized functionals of $M$, which we call generalized weighted number (GWN) operators. We prove that GWN operators can be represented in terms of generalized annihilation and creation operators (acting on generalized functionals of $M$). We also examine commutation relations between a GW…
Functional Inequalities for doubly weighted Brownian Motion with Sticky-Reflecting Boundary Diffusion
本文研究了具有粘性反射边界的流形上双重加权布朗运动的Poincaré和Logarithmic Sobolev常数上界，采用能量交互和加权Reilly公式进行插值分析，同时得到Steklov特征值和边界迹算子的上界。
We introduce the notion of a weighted lift zonoid and show that, for properly chosen weights v, the ordering condition on a measure μ, formulated in terms of the weighted lift zonoids of this measure, leads to certain functional inequalities for this measure, such as non-linear extensions of Bobkov's shift inequality and weighted inverse log-Sobolev inequality. The choice of the weight K, involved in our version of the inverse log-Sobolev inequality, differs substantially from those available in the literature, and requires the weight v, involved into the definition of the weighted lift zonoid…
Global universal approximation of functional input maps on weighted spaces
本文提出了一种在加权空间上实现函数输入映射的神经网络结构，并利用加权Stone-Weierstrass定理证明了其在连续函数和路径空间函数逼近中的全局通用逼近能力，同时探讨了其在不确定性量化中的应用。
Quantum Bernoulli noises (QBN, for short) are the family of annihilation and creation operators acting on Bernoulli functionals, which satisfy a canonical anti-commutation relation (CAR) in equal-time. In this paper, by using QBN, we first introduce a class of self-adjoint operators acting on Bernoulli functionals, which we call the weighted number operators. We then make clear spectral decompositions of these operators, and establish their commutation relations with the annihilation as well as the creation operators. We also obtain a necessary and sufficient condition for a weighted number op…
Law of the Iterated Logarithm for Weighted Sums of Functionals of Long-Memory Gaussian Sequences
研究长记忆平稳高斯序列的加权非线性泛函，利用Hermite混沌确定几乎必然尺度，得到线性情形的方差归一化LIL常数和非线性情形的加权泛函聚类集与变分公式。
We derive normal approximation bounds in the Wasserstein distance for sums of weighted U-statistics, based on a general distance bound for functionals of independent random variables of arbitrary distributions. Those bounds are applied to normal approximation for the combined weights of subgraphs in the Erdős-Rényi random graph, extending the graph counting results of [1] to the setting of graph weighting. Our approach relies on a general stochastic analytic framework for functionals of independent random sequences.
The objects of our interest are the so-called $A$-permutations, which are permutations whose cycle length lie in a fixed set $A$. They have been extensively studied with respect to the uniform or the Ewens measure. In this paper, we extend some classical results to a more general weighted probability measure which is a natural extension of the Ewens measure and which in particular allows to consider sets $A_n$ depending on the degree $n$ of the permutation. By means of complex analysis arguments and under reasonable conditions on generating functions we study the asymptotic behaviour of classi…
Let $\MM$ be the space of finite measures on a Locally compact Polish space, and let $\BG$ be the Gamma distribution on $\MM$ with intensity measure $ν\in \MM$. Let $\nn^{ext}$ be the extrinsic derivative with tangent bundle $T\MM= \cup_{η\in\MM} L^2(η)$, and let $\GA: T\MM\to T\MM$ be measurable such that $\GA_η$ is a positive definite linear operator on $L^2(η)$ for every $η\in \MM$. Moreover, for a measurable function $V$ on $\MM$, let $\d\BG^V= \e^V\d\BG$. We investigate the Poincaré, weak Poincaré and super Poincaré inequalities for the Dirichlet form $$\EE_{\GA,V}(F,G):= \int_\MM \<\GA_η…
The limit Gaussian distribution of multivariate weighted functionals of nonlinear transformations of Gaussian stationary processes, having multiple singular spectra, is derived, under very general conditions on the weight function. This paper is motivated by its potential applications in nonlinear regression, and asymptotic inference on nonlinear functionals of Gaussian stationary processes with singular spectra.
Weak Functional Inequalities for Perturbed Measures
本文研究了弱泛函不等式在扰动测度中的应用，探讨了弱Poincaré、加权Poincaré、弱对数Sobolev和加权对数Sobolev不等式的问题。
We find exact small deviation asymptotics with respect to weighted Hilbert norm for some well-known Gaussian processes. Our approach does not require the knowledge of eigenfunctions of the covariance operator of a weighted process. Such a peculiarity of the method makes it possible to generalize many previous results in this area. We also obtain new relations connected to exact small deviation asymptotics for a Brownian excursion, a Brownian meander, and Bessel processes and bridges.
Let $L$ be the Euclidean functional with $p$-th power-weighted edges. Examples include the sum of the $p$-th power-weighted lengths of the edges in minimal spanning trees, traveling salesman tours, and minimal matchings. Motivated by the works of Steele, Redmond and Yukich (1994, 1996) have shown that for $n$ i.i.d. sample points $\{X_1,...,X_n\}$ from $[0,1]^d$, $L(\{X_1,...,X_n\})/n^{(d-p)/d}$ converges a.s. to a finite constant. Here we bound the rate of convergence of $EL(\{X_1,...,X_n\})/n^{(d-p)/d}$.
This paper is devoted to the study of probability measures with heavy tails. Using the Lyapunov function approach we prove that such measures satisfy different kind of functional inequalities such as weak Poincaré and weak Cheeger, weighted Poincaré and weighted Cheeger inequalities and their dual forms. Proofs are short and we cover very large situations. For product measures on $\R^n$ we obtain the optimal dimension dependence using the mass transportation method. Then we derive (optimal) isoperimetric inequalities. Finally we deal with spherically symmetric measures. We recover and improve …
Weighted universal approximation of differentiable maps on infinite-dimensional manifolds
通过加权Nachbin定理，将函数输入神经网络的通用逼近定理推广到可微映射，包括导数逼近，并应用于非预期泛函和路径空间泛函的逼近。
Hölder estimates and weak convergences of certain weighted sum processes
本文针对固定阶Wiener混沌中的加权和过程，利用Malliavin分部积分、四阶矩定理和多维Young积分估计，证明了其Hölder连续性和泛函极限定理。
We consider large deviations of empirical measures of diffusion processes. In a first part, we present conditions to obtain a large deviations principle (LDP) for a precise class of unbounded functions. This provides an analogue to the standard Cramér condition in the context of diffusion processes, which turns out to be related to a spectral gap condition for a Witten-Schrödinger operator. Secondly, we study more precisely the properties of the Donsker-Varadhan rate functional associated with the LDP. We revisit and generalize some standard duality results as well as a more original decomposi…
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