Estimation Of Covariance Function Research Articles

On the frequency domain composite likelihood methods for estimating space-time covariance functions for large datasets

Covariance function plays very important roles in modeling and in prediction of spatial data. For instance, many statistical inferences such as maximum likelihood estimation and best linear unbiased prediction involve covariance matrices which must be inverted and this evaluation is slow when the number of observations is large, and the complexity increases even further in a space-time context. In this paper, we propose approximation methods based on three types of frequency domain composite likelihoods for the estimation of spatial-temporal covariance functions: paired, differenced, and conditional composite likelihoods to overcome this problem. The proposed likelihoods involve computationally efficient and matrix-free Gaussian likelihood approximations that have been successful in fitting parametric space-time covariance functions for massive datasets. We conduct a simulation study to evaluate the performance of the proposed estimation methods from statistical and computational viewpoints. The frequency domain spectral composite likelihood estimates demonstrate better performance compared to the observation domain composite likelihoods. Furthermore, our numerical examples show that paired and conditional frequency domain composite likelihoods work better than differenced frequency domain composite likelihood from statistical point of view.

On estimation of covariance function for functional data with detection limits

In many studies on disease progression, biomarkers are restricted by detection limits, hence informatively missing. Current approaches ignore the problem by just filling in the value of the detection limit for the missing observations for the estimation of the mean and covariance function, which yield inaccurate estimation. Inspired by our recent work [Liu and Houwing-Duistermaat (2022), ‘Fast Estimators for the Mean Function for Functional Data with Detection Limits’, Stat, e467.] in which novel estimators for mean function for data subject to detection limit are proposed, in this paper, we will propose a novel estimator for the covariance function for sparse and dense data subject to a detection limit. We will derive the asymptotic properties of the estimator. We will compare our method to the standard method, which ignores the detection limit, via simulations. We will illustrate the new approach by analysing biomarker data subject to a detection limit. In contrast to the standard method, our method appeared to provide more accurate estimates of the covariance. Moreover its computation time is small.

Dynamical BLUP modeling of reaction norm evolution, accommodating changing environments, overlapping generations, and multivariate data

AbstractFor theoretical studies, reaction norm evolution in a changing environment can be modeled by means of the multivariate breeder's equation, with the reaction norm parameters treated as traits in their own right. This is, however, not a feasible approach for the use of field data, where the intercept and slope values are not available. An alternative approach is to use infinite‐dimensional characters and smooth covariance function estimates found by, e.g., random regression. This is difficult because of the need to find, for example, polynomial basis functions that fit the data reasonably well over time, and because reaction norms in multivariate cases are correlated, such that they cannot be modeled independently. Here, I present an alternative approach based on a multivariate linear mixed model of any order, with dynamical incidence and residual covariance matrices that reflect the changing environment. From such a mixed model follows a dynamical BLUP model for the estimation of the individual reaction norm parameter values at any given parent generation, and for updating of the mean reaction norm parameter values from generation to generation by means of Robertson's secondary theorem of natural selection. This will, for example, make it possible to disentangle the microevolutionary and plasticity components in climate change responses. The BLUP model incorporates the additive genetic relationship matrix in the usual way, and overlapping generations can easily be accommodated. Additive genetic and environmental model parameters are assumed to be known and constant, but it is discussed how they can be estimated by means of a prediction error method. The identifiability by the use of field or laboratory data containing environmental, phenotypic, fitness, and additive genetic relationship data is an important feature of the proposed model.

Adaptive Functional Thresholding for Sparse Covariance Function Estimation in High Dimensions

Covariance function estimation is a fundamental task in multivariate functional data analysis and arises in many applications. In this article, we consider estimating sparse covariance functions for high-dimensional functional data, where the number of random functions p is comparable to, or even larger than the sample size n. Aided by the Hilbert–Schmidt norm of functions, we introduce a new class of functional thresholding operators that combine functional versions of thresholding and shrinkage, and propose the adaptive functional thresholding estimator by incorporating the variance effects of individual entries of the sample covariance function into functional thresholding. To handle the practical scenario where curves are partially observed with errors, we also develop a nonparametric smoothing approach to obtain the smoothed adaptive functional thresholding estimator and its binned implementation to accelerate the computation. We investigate the theoretical properties of our proposals when p grows exponentially with n under both fully and partially observed functional scenarios. Finally, we demonstrate that the proposed adaptive functional thresholding estimators significantly outperform the competitors through extensive simulations and the functional connectivity analysis of two neuroimaging datasets. Supplementary materials for this article are available online.

Incorporating covariate into mean and covariance function estimation of functional data under a general weighing scheme

This paper develops the estimation method of mean and covariance functions of functional data with additional covariate information. With the strength of both local linear smoothing modeling and general weighing scheme, we are able to explicitly characterize the mean and covariance functions with incorporating covariate for irregularly spaced and sparsely observed longitudinal data, as typically encountered in engineering technology or biomedical studies, as well as for functional data which are densely measured. Theoretically, we establish the uniform convergence rates of the estimators in the general weighing scheme. Monte Carlo simulation is conducted to investigate the finite-sample performance of the proposed approach. Two applications including the children growth data and white matter tract dataset obtained from Alzheimer's Disease Neuroimaging Initiative study are also provided.

Optimal Estimation of Large Functional and Longitudinal Data by Using Functional Linear Mixed Model

The estimation of large functional and longitudinal data, which refers to the estimation of mean function, estimation of covariance function, and prediction of individual trajectory, is one of the most challenging problems in the field of high-dimensional statistics. Functional Principal Components Analysis (FPCA) and Functional Linear Mixed Model (FLMM) are two major statistical tools used to address the estimation of large functional and longitudinal data; however, the former suffers from a dramatically increasing computational burden while the latter does not have clear asymptotic properties. In this paper, we propose a computationally effective estimator of large functional and longitudinal data within the framework of FLMM, in which all the parameters can be automatically estimated. Under certain regularity assumptions, we prove that the mean function estimation and individual trajectory prediction reach the minimax lower bounds of all nonparametric estimations. Through numerous simulations and real data analysis, we show that our new estimator outperforms the traditional FPCA in terms of mean function estimation, individual trajectory prediction, variance estimation, covariance function estimation, and computational effectiveness.

Asymptotic normality for kernel weighted averages estimation

This paper presents a set of general normality results for kernel weighted averages. We extend existing results in literature for independent data a s in Yao (2017) to stationary dependent longitudinal data. The asymptotic properties of the proposed weighted averages are investigated under \(\alpha\)-mixing conditions. These results are useful for covariance function estimation based on nonparametric kernel method.

Online Estimation for Functional Data

Functional data analysis has attracted considerable interest and is facing new challenges, one of which is the increasingly available data in a streaming manner. In this article we develop an online nonparametric method to dynamically update the estimates of mean and covariance functions for functional data. The kernel-type estimates can be decomposed into two sufficient statistics depending on the data-driven bandwidths. We propose to approximate the future optimal bandwidths by a sequence of dynamically changing candidates and combine the corresponding statistics across blocks to form the updated estimation. The proposed online method is easy to compute based on the stored sufficient statistics and the current data block. We derive the asymptotic normality and, more importantly, the relative efficiency lower bounds of the online estimates of mean and covariance functions. This provides insight into the relationship between estimation accuracy and computational cost driven by the length of candidate bandwidth sequence. Simulations and real data examples are provided to support such findings. Supplementary materials for this article are available online.

Framelet block thresholding estimator for sparse functional data

Nonparametric estimation of mean and covariance functions based on discretely observed data is important in functional data analysis. In this paper, we propose a framelet block thresholding method for the case of sparsely observed functional data. The procedure is easily implemented and the resultant estimators are represented as explicit B-spline expressions. For sparsely observed functional data, we establish, under some mild conditions but without knowing the smoothness parameter, convergence rates of mean integrated squared errors for mean and covariance estimators respectively. In particular, the mean estimator attains minimax optimal rate. The simulated and real data examples are provided to offer empirical support of the theoretical properties. Compared to the existing methods, the proposed method outperforms in adapting automatically to local variations.

Evaluation and Numerical Simulation of Music Education Informationization Based on the Local Linear Regression Method

The study of this theoretical problem enables sparse or dense functional data, including educational information evaluation data. The choice of different weights is subjected to principal component analysis. The evaluation of music education informatization level mainly evaluates the status quo of music education informatization development, provides a basis for formulating and adjusting music education informatization development policies, and provides support for educational decision-making, to promote the sustainable and balanced development of music education informatization. The evaluation of music education informatization has become the key promotion work of music education informatization at this stage. This paper studies the convergence rate of functional principal components based on the local linear method under general weighting conditions. First, we introduce the related research on the estimation of mean and covariance function under general weighting. Secondly, for principal functional components under general weighting, namely, eigenvalues and eigen functions, the text gives the corresponding estimated values and derives its strong uniform convergence rate. Finally, the convergence rate was verified by simulation research. The estimation methods and conclusions of this article enrich the research of functional linear regression models and will help analyze the complex and changeable problems encountered in the application of music education information.

Network Functional Varying Coefficient Model

We consider functional responses with network dependence observed for each individual at irregular time points. To model both the interindividual dependence and within-individual dynamic correlation, we propose a network functional varying coefficient (NFVC) model. The response of each individual is characterized by a linear combination of responses from its connected nodes and its exogenous covariates. All the model coefficients are allowed to be time dependent. The NFVC model adds to the richness of both the classical network autoregression model and the functional regression models. To overcome the complexity caused by the network interdependence, we devise a special nonparametric least-squares-type estimator, which is feasible when the responses are observed at irregular time points for different individuals. The estimator takes advantage of the sparsity of the network structure to reduce the computational burden. To further conduct the functional principal component analysis, a novel within-individual covariance function estimation method is proposed and studied. Theoretical properties of our estimators, which involve techniques related to empirical processes, nonparametrics, functional data analysis and various concentration inequalities, are analyzed. We analyze a social network dataset to illustrate the powerfulness of the proposed procedure.

The covariance function estimation based on the constrained B-spline smoothing method

Covariance functions play a key role in describing signal time-domain characteristics and spatial correlation structures. In recent years, the nonparametric estimation of covariance functions has been studied extensively.However, the positive definiteness cannot be guaranteed for most nonparametric covariance function estimations. In order to overcome the difficulties, in the context of stationary random processes, this paper first studies the asymptotic properties of the unconstrained B-spline covariance function estimator, based on which, the asymptotic 性质 of the constrained B-spline covariance estimator is established. Numerical results of the simulation experiments indicate that the proposed constrained B-spline estimator has an excellent performance of the positive definiteness and AMSE (average mean squared error).Finally, the proposed positive definite covariance estimator is computed for the Canadian temperature data and the Australian electricity demands data.

Analysis of Air Pollution Parameters Using Covariance Function Theory

Abstract The paper analyses the intensity changes of three pollution parameter vectors in space and time. The RGB raster pollution data of the Lithuanian territory used for the research were prepared according to the digital images of the Sentinel-2 Earth satellites. The numerical vectors of environmental pollution parameters CH4 (methane), NO2 (nitrogen dioxide) and for direct comparison O2 (oxygen gas) were used for the calculations. The covariance function theory was used to perform the analysis of intensity changes in digital vectors. Estimates of the covariance functions of the numerical vectors of pollution parameters and O2 or the auto-covariance functions of single vectors are calculated from random functions consisting of arrays of measurement parameters of all parameters vectors. Correlation between parameters vectors depends on the density of parameters and their structure. Estimates of covariance functions were calculated by changing the quantization interval on a time scale and using a compiled computer program using the Matlab procedure package. The probability dependence between the environmental pollution parameter vectors and trace gas of the territory in Lithuania and their change in time scale was determined.

Modeling sparse longitudinal data on Riemannian manifolds.

Modern data collection often entails longitudinal repeated measurements that assume values on a Riemannian manifold. Analyzing such longitudinal Riemannian data is challenging, because of both the sparsity of the observations and the nonlinear manifold constraint. Addressing this challenge, we propose an intrinsic functional principal component analysis for longitudinal Riemannian data. Information is pooled across subjects by estimating the mean curve with local Fréchet regression and smoothing the covariance structure of the linearized data on tangent spaces around the mean. Dimension reduction and imputation of the manifold-valued trajectories are achieved by utilizing the leading principal components and applying best linear unbiased prediction. We show that the proposed mean and covariance function estimates achieve state-of-the-art convergence rates. For illustration, we study the development of brain connectivity in a longitudinal cohort of Alzheimer's disease and normal participants by modeling the connectivity on the manifold of symmetric positive definite matrices with the affine-invariant metric. In a second illustration for irregularly recorded longitudinal emotion compositional data for unemployed workers, we show that the proposed method leads to nicely interpretable eigenfunctions and principal component scores. Data used in preparation of this article were obtained from the Alzheimer's Disease Neuroimaging Initiativedatabase.

Low-Rank Covariance Function Estimation for Multidimensional Functional Data

Multidimensional function data arise from many fields nowadays. The covariance function plays an important role in the analysis of such increasingly common data. In this article, we propose a novel nonparametric covariance function estimation approach under the framework of reproducing kernel Hilbert spaces (RKHS) that can handle both sparse and dense functional data. We extend multilinear rank structures for (finite-dimensional) tensors to functions, which allow for flexible modeling of both covariance operators and marginal structures. The proposed framework can guarantee that the resulting estimator is automatically semipositive definite, and can incorporate various spectral regularizations. The trace-norm regularization in particular can promote low ranks for both covariance operator and marginal structures. Despite the lack of a closed form, under mild assumptions, the proposed estimator can achieve unified theoretical results that hold for any relative magnitudes between the sample size and the number of observations per sample field, and the rate of convergence reveals the phase-transition phenomenon from sparse to dense functional data. Based on a new representer theorem, an ADMM algorithm is developed for the trace-norm regularization. The appealing numerical performance of the proposed estimator is demonstrated by a simulation study and the analysis of a dataset from the Argo project. Supplementary materials for this article are available online.

Asymptotic properties of penalized splines for functional data

Penalized spline methods are popular for functional data analysis but their asymptotic properties have not been established. We present a theoretic study of the $L_{2}$ and uniform convergence of penalized splines for estimating the mean and covariance functions of functional data under general settings. The established convergence rates for the mean function estimation are mini-max rate optimal and the rates for the covariance function estimation are comparable to those using other smoothing methods.

Mean and Covariance Estimation for Functional Snippets

We consider estimation of mean and covariance functions of functional snippets, which are short segments of functions possibly observed irregularly on an individual specific subinterval that is much shorter than the entire study interval. Estimation of the covariance function for functional snippets is challenging since information for the far off-diagonal regions of the covariance structure is completely missing. We address this difficulty by decomposing the covariance function into a variance function component and a correlation function component. The variance function can be effectively estimated nonparametrically, while the correlation part is modeled parametrically, possibly with an increasing number of parameters, to handle the missing information in the far off-diagonal regions. Both theoretical analysis and numerical simulations suggest that this hybrid strategy is effective. In addition, we propose a new estimator for the variance of measurement errors and analyze its asymptotic properties. This estimator is required for the estimation of the variance function from noisy measurements. Supplementary materials for this article are available online.

Semiparametric estimation of cross-covariance functions for multivariate random fields.

The prevalence of spatially referenced multivariate data has impelled researchers to develop procedures for joint modeling of multiple spatial processes. This ordinarily involves modeling marginal and cross-process dependence for any arbitrary pair of locations using a multivariate spatial covariance function. However, building a flexible multivariate spatial covariance function that is nonnegative definite is challenging. Here, we propose a semiparametric approach for multivariate spatial covariance function estimation with approximate Matérn marginals and highly flexible cross-covariance functions via their spectral representations. The flexibility in our cross-covariance function arises due to B-spline-based specification of the underlying coherence functions, which in turn allows us to capture nontrivial cross-spectral features. We then develop a likelihood-based estimation procedure and perform multiple simulation studies to demonstrate the performance of our method, especially on the coherence function estimation. Finally, we analyze particulate matter concentrations (PM2.5 ) and wind speed data over the West-North-Central climatic region of the United States, where we illustrate that our proposed method outperforms the commonly used full bivariate Matérn model and the linear model of coregionalization for spatialprediction.

Predictive models for identification of Milky Way Galaxy digital images

In this paper, the theoretical and practical assumptions for identifying digital images in the Milky Way map through the application of image transformation, simulation of the parameter z in the Doppler formula and the theory of covariance functions are discussed. The value of the parameter z in the Doppler formula shows an approximate speed of reciprocal movement of two digital images (objects) in respect of the light speed in vacuum. To calculate the values of the parameter z, vectors transformed by the intensity of pixels of Milky Way digital images and the visible solar spectrum digital images were used. The values of the parameter z made it possible to calculate the speed of reciprocal movement of the Milky Way digital images and the speed of movement of the images in respect of the solar system. To calculate the values of estimates of covariance functions, the algebraic additions of transformed vectors and single vectors were applied. The average value of the parameter z in the Doppler formula is calculated according to the formula developed using the expressions of covariance functions of the algebraic additions of transformed relevant vectors and a single vector. The calculated data were obtained by software developed on the basis of MATLAB programme package operators.

On the estimation of regional covariance functions of TEC variations over Canada

Spatial and temporal variations of Total Electron Content (TEC) can affect GNSS high accuracy positioning. Enhanced estimation of ionospheric variations and their de-correlation can benefit differential and point positioning rapid solutions. Global and regional TEC maps can provide the overall state of ionopsheric variations in space and time domains within their accuracy limits. In this paper, these maps are exploited to retrieve ionospheric variations by means of variograms and their associated covariance functions of TEC residuals over Canadian region during geomagnetically quiet and disturbed conditions. A number of theoretical variogram functions are reviewed for modeling covariance of TEC residuals. The variogram modeling of residuals during a strong geomagnetic storm revealed variances of one order of magnitude larger compared to a rather quiet condition. Variogram models are also used in regional and local kriging interpolation experiments and their performances are evaluated. Global maps of TEC RMS by International GNSS Service and two of its analysis centres are also compared over the Canadian region during a two-year period. Realistic representation of regional variances using estimated variograms when compared to global ionospheric RMS maps are also presented.