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点过程的主成分分析AI 解读

We introduce a novel statistical framework for the analysis of replicated point processes that allows for the study of point pattern variability at a population level. By treating point process realizations as random measures, we adopt a functional analysis perspective and propose a form of functional Principal Component Analysis (fPCA) for point processes. The originality of our method is to base our analysis on the cumulative mass functions of the random measures, which…

函数型图形套索AI 解读

We consider the problem of recovering conditional independence relationships between p jointly distributed Hilbertian random elements given n realizations thereof. We operate in the sparse high-dimensional regime, where n ≪ p and no element is related to more than d ≪ p other elements. In this context, we propose an infinite-dimensional generalization of the graphical lasso. We prove model selection consistency under natural assumptions and extend many classical results to…

通过移位部分迹估计随机曲面的协方差AI 解读

The problem of covariance estimation for replicated surface-valued processes is examined from the functional data analysis perspective. Considerations of statistical and computational efficiency often compel the use of separability of the covariance, even though the assumption may fail in practice. We consider a setting where the covariance structure may fail to be separable locally—either due to noise contamination or due to the presence of a nonseparable short-range…

协方差算子秩的检验AI 解读

How can we discern whether the covariance operator of a stochastic process is of reduced rank, and if so, what its precise rank is? And how can we do so at a given level of confidence? This question is central to a great deal of methods for functional data, which require low-dimensional representations whether by functional PCA or other methods. The difficulty is that the determination is to be made on the basis of i.i.d. replications of the process observed discretely and…

协方差核的补全AI 解读

We consider the problem of positive-semidefinite continuation: extending a partially specified covariance kernel from a subdomain Ω of a rectangular domain I×I to a covariance kernel on the entire domain I×I. For a broad class of domains Ω called serrated domains, we are able to present a complete theory. Namely, we demonstrate that a canonical completion always exists and can be explicitly constructed. We characterise all possible completions as suitable perturbations of…

CovNet：多维定义域函数型数据的协方差网络AI 解读

Abstract Covariance estimation is ubiquitous in functional data analysis. Yet, the case of functional observations over multidimensional domains introduces computational and statistical challenges, rendering the standard methods effectively inapplicable. To address this problem, we introduce Covariance Networks (CovNet) as a modelling and estimation tool. The CovNet model is universal—it can be used to approximate any covariance up to desired precision. Moreover, the model…

通过矩阵补全进行函数型数据分析AI 解读

Functional data analyses typically proceed by smoothing, followed by functional PCA. This paradigm implicitly assumes that rough variation is due to nuisance noise. Nevertheless, relevant functional features such as time-localised or short scale fluctuations may indeed be rough relative to the global scale, but still smooth at shorter scales. These may be confounded with the global smooth components of variation by the smoothing and PCA, potentially distorting the parsimony…

函数型时间序列动态差异的检测与定位：分子生物物理案例AI 解读

Motivated by the problem of inferring the molecular dynamics of DNA in solution, and linking them with its base-pair composition, we consider the problem of comparing the dynamics of functional time series (FTS), and of localizing any inferred differences in frequency and along curvelength. The approach we take is one of Fourier analysis, where the complete second-order structure of the FTS is encoded by its spectral density operator, indexed by frequency and curvelength.…

点过程的幅度变异与相位变异AI 解读

We develop a canonical framework for the study of the problem of registration of multiple point processes subjected to warping, known as the problem of separation of amplitude and phase variation. The amplitude variation of a real random function $\{Y(x):x\in[0,1]\}$ corresponds to its random oscillations in the $y$-axis, typically encapsulated by its (co)variation around a mean level. In contrast, its phase variation refers to fluctuations in the $x$-axis, often caused by…

函数空间中平稳时间序列的傅里叶分析AI 解读

We develop the basic building blocks of a frequency domain framework for drawing statistical inferences on the second-order structure of a stationary sequence of functional data. The key element in such a context is the spectral density operator, which generalises the notion of a spectral density matrix to the functional setting, and characterises the second-order dynamics of the process. Our main tool is the functional Discrete Fourier Transform (fDFT). We derive an…

多元核函数的非参数构造AI 解读

We propose a nonparametric method for constructing multivariate kernels tuned to the configuration of the sample, for density estimation in , d moderate. The motivation behind the approach is to break down the construction of the kernel into two parts: determining its overall shape and then its global concentration. We consider a framework that is essentially nonparametric, as opposed to the usual bandwidth matrix parameterization. The shape of the kernel to be employed is…

高斯随机函数的二阶比较与 DNA 微环几何AI 解读

Given two samples of continuous zero-mean iid Gaussian processes on [0,1], we consider the problem of testing whether they share the same covariance structure. Our study is motivated by the problem of determining whether the mechanical properties of short strands of DNA are significantly affected by their base-pair sequence; though expected to be true, had so far not been observed in three-dimensional electron microscopy data. The testing problem is seen to involve aspects…