We provide a computationally and statistically efficient method for estimating the parameters of a stochastic covariance model observed on a regular spatial grid in any number of dimensions. Our proposed method, which we call the Debiased Spatial Whittle likelihood, makes important corrections to the well-known Whittle likelihood to account for large sources of bias caused by boundary effects and aliasing. We generalise the approach to flexibly allow for significant volumes of missing data including those with lower-dimensional substructure, and for irregular sampling boundaries. We build a theoretical framework under relatively weak assumptions which ensures consistency and asymptotic normality in numerous practical settings including missing data and non-Gaussian processes. We also extend our consistency results to multivariate processes. We provide detailed implementation guidelines which ensure the estimation procedure can be conducted in O(n log n) operations, where n is the number of points of the encapsulating rectangular grid, thus keeping the computational scalability of Fourier and Whittle-based methods for large data sets. We validate our procedure over a range of simulated and real-world settings, and compare with state-of-the-art alternatives, demonstrating the enduring practical appeal of Fourier-based methods, provided they are corrected by the procedures developed in this paper.
The estimation of parameters in the frequency spectrum of a seasonally persistent stationary stochastic process is addressed. For seasonal persistence associated with a pole in the spectrum located away from frequency zero, a new Whittle-type likelihood is developed that explicitly acknowledges the location of the pole. This Whittle likelihood is a large sample approximation to the distribution of the periodogram over a chosen grid of frequencies, and constitutes an approximation to the time-domain likelihood of the data, via the linear transformation of an inverse discrete Fourier transform c…
We study the asymptotic behaviour of needlets-based approximate maximum likelihood estimators for the spectral parameters of Gaussian and isotropic spherical random fields. We prove consistency and asymptotic Gaussianity, in the high-frequency limit, thus generalizing earlier results by Durastanti et al. (2011) based upon standard Fourier analysis on the sphere. The asymptotic results are then illustrated by an extensive Monte Carlo study.
In this paper, using spectral theory of Hilbertian operators, we study ARMA Gaussian processes indexed by graphs. We extend Whittle maximum likelihood estimation of the parameters for the corresponding spectral density and show their asymptotic optimality.
We consider random sample splitting for estimation and inference in high dimensional generalized linear models, where we first apply the lasso to select a submodel using one subsample and then apply the debiased lasso to fit the selected model using the remaining subsample. We show that, no matter including a prespecified subset of regression coefficients or not, the debiased lasso estimation of the selected submodel after a single splitting follows a normal distribution asymptotically. Furthermore, for a set of prespecified regression coefficients, we show that a multiple splitting procedure …
Empirical likelihood approach is one of non-parametric statistical methods, which is applied to the hypothesis testing or construction of confidence regions for pivotal unknown quantities. This method has been applied to the case of independent identically distributed random variables and second order stationary processes. In recent years, we observe heavy-tailed data in many fields. To model such data suitably, we consider symmetric scalar and multivariate $α$-stable linear processes generated by infinite variance innovation sequence. We use a Whittle likelihood type estimating function in th…
The Whittle likelihood is a widely used and computationally efficient pseudo-likelihood. However, it is known to produce biased parameter estimates for large classes of models. We propose a method for de-biasing Whittle estimates for second-order stationary stochastic processes. The de-biased Whittle likelihood can be computed in the same $\mathcal{O}(n\log n)$ operations as the standard approach. We demonstrate the superior performance of the method in simulation studies and in application to a large-scale oceanographic dataset, where in both cases the de-biased approach reduces bias by up to…
Max-stable processes are a common choice for modelling spatial extreme data as they arise naturally as the infinite-dimensional generalisation of multivariate extreme value theory. Statistical inference for such models is complicated by the intractability of the multivariate density function. Nonparametric, composite likelihood-based, and Bayesian approaches have been proposed to address this difficulty. More recently, a simulation-based approach using approximate Bayesian computation (ABC) has been employed for estimating parameters of max-stable models. ABC algorithms rely on the evaluation …
Deep candidate generation (DCG) that narrows down the collection of relevant items from billions to hundreds via representation learning has become prevalent in industrial recommender systems. Standard approaches approximate maximum likelihood estimation (MLE) through sampling for better scalability and address the problem of DCG in a way similar to language modeling. However, live recommender systems face severe exposure bias and have a vocabulary several orders of magnitude larger than that of natural language, implying that MLE will preserve and even exacerbate the exposure bias in the long…
Warrington College of Business, University of Florida（佛罗里达大学沃灵顿商学院）；Leonard N. Stern School of Business, New York University（纽约大学伦纳德·N·斯特恩商学院）；School of Management, Xi’an Jiaotong University（西安交通大学管理学院）；Shanghai University of Finance and Economics（上海财经大学）
Data-driven sequential decision has found a wide range of applications in modern operations management, such as dynamic pricing, inventory control, and assortment optimization. Most existing research on data-driven sequential decision focuses on designing an online policy to maximize the revenue. However, the research on uncertainty quantification on the underlying true model function (e.g., demand function), a critical problem for practitioners, has not been well explored. In this paper, using the problem of demand function prediction in dynamic pricing as the motivating example, we study the…
The Whittle likelihood is widely used for Bayesian nonparametric estimation of the spectral density of stationary time series. However, the loss of efficiency for non-Gaussian time series can be substantial. On the other hand, parametric methods are more powerful if the model is well-specified, but may fail entirely otherwise. Therefore, we suggest a nonparametric correction of a parametric likelihood taking advantage of the efficiency of parametric models while mitigating sensitivities through a nonparametric amendment. Using a Bernstein-Dirichlet prior for the nonparametric spectral correcti…
Spectral subsampling MCMC was recently proposed to speed up Markov chain Monte Carlo (MCMC) for long stationary univariate time series by subsampling periodogram observations in the frequency domain. This article extends the approach to multivariate time series using a multivariate generalisation of the Whittle likelihood. To assess the computational gains from spectral subsampling in challenging problems, a multivariate generalisation of the autoregressive tempered fractionally integrated moving average model (ARTFIMA) is introduced and some of its properties derived. Bayesian inference based…
A class of Fourier based statistics for irregular spaced spatial data is introduced, examples include, the Whittle likelihood, a parametric estimator of the covariance function based on the $L_{2}$-contrast function and a simple nonparametric estimator of the spatial autocovariance which is a non-negative function. The Fourier based statistic is a quadratic form of a discrete Fourier-type transform of the spatial data. Evaluation of the statistic is computationally tractable, requiring $O(nb^{})$ operations, where $b$ are the number Fourier frequencies used in the definition of the statistic a…
In this paper, we develop a method for estimating and clustering two-dimensional spectral density functions (2D-SDFs) for spatial data from multiple subregions. We use a common set of adaptive basis functions to explain the similarities among the 2D-SDFs in a low-dimensional space and estimate the basis coefficients by maximizing the Whittle likelihood with two penalties. We apply these penalties to impose the smoothness of the estimated 2D-SDFs and the spatial dependence of the spatially-correlated subregions. The proposed technique provides a score matrix, that is comprised of the estimated …
Traditional statistical methods are faced with new challenges due to streaming data. The major challenge is the rapidly growing volume and velocity of data, which makes storing such huge datasets in memory impossible. The paper presents an online inference framework for regression parameters in high-dimensional semiparametric single-index models with unknown link functions. The proposed online procedure updates only the current data batch and summary statistics of historical data instead of re-accessing the entire raw data set. At the same time, we do not need to estimate the unknown link func…
Based on a novel dynamic Whittle likelihood approximation for locally stationary processes, a Bayesian nonparametric approach to estimating the time-varying spectral density is proposed. This dynamic frequency-domain based likelihood approximation is able to depict the time-frequency evolution of the process by utilizing the moving periodogram previously introduced in the bootstrap literature. The posterior distribution is obtained by updating a bivariate extension of the Bernstein-Dirichlet process prior with the dynamic Whittle likelihood. Asymptotic properties such as sup-norm posterior con…
The generalized linear models (GLM) have been widely used in practice to model non-Gaussian response variables. When the number of explanatory features is relatively large, scientific researchers are of interest to perform controlled feature selection in order to simplify the downstream analysis. This paper introduces a new framework for feature selection in GLMs that can achieve false discovery rate (FDR) control in two asymptotic regimes. The key step is to construct a mirror statistic to measure the importance of each feature, which is based upon two (asymptotically) independent estimates o…
Empirical likelihood method has been applied to dependent observations by Monti (1997) through the Whittle's estimation method. Similar asymptotic distribution of the empirical likelihood ratio statistic for stationary time series has been derived to construct the confidence regions for the parameters. However, required numerical problem of computing profile empirical likelihood function which involves constrained maximization has no solution sometimes, which leads to the drawbacks of using the original version of the empirical likelihood ratio. In this paper, we propose an adjusted empirical …
Many physical datasets are generated by collections of instruments that make measurements at regular time intervals. For such regular monitoring data, we extend the framework of half-spectral covariance functions to the case of nonstationarity in space and time and demonstrate that this method provides a natural and tractable way to incorporate complex behaviors into a covariance model. Further, we use this method with fully time-domain computations to obtain bona fide maximum likelihood estimators---as opposed to using Whittle-type likelihood approximations, for example---that can still be co…
Spectral subsampling MCMC for Lévy-driven continuous-time ARMA models with expensive likelihood contributions
Human Technology Institute, University of Technology Sydney（悉尼科技大学人类技术研究所）；School of Mathematical and Physical Sciences, University of Technology Sydney（悉尼科技大学数理科学学院）；School of Economics, University of New South Wales（新南威尔士大学经济管理学院）；Guanghua School of Management, Peking University（北京大学光华管理学院）
While much work has explored probabilistic graphical models for independent data, less attention has been paid to time series. The goal in this setting is to determine conditional independence relations between entire time series, which for stationary series, are encoded by zeros in the inverse spectral density matrix. We take a Bayesian approach to structure learning, placing priors on (i) the graph structure and (ii) spectral matrices given the graph. We leverage a Whittle likelihood approximation and define a conjugate prior---the hyper complex inverse Wishart---on the complex-valued and gr…
Debiased machine learning for combining probability and non-probability survey data
MRC Biostatistics Unit, University of Cambridge（剑桥大学医学研究委员会生物统计学单元）；Department of Genetics, University of Cambridge（剑桥大学生物学系）；Institute for Global Health, University College London（伦敦大学学院全球健康研究所）
