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Correlation bounds, mixing and m-dependence under random time-varying network distances with an application to Cox-processes
... For the identifiability of model (1), in what follows we set β n (t) = 0 as in Yan et al. (2016a). Now, we discuss the differences between our model and those proposed by Perry and Wolfe (2013), Kreiß et al. (2019) and Kreiß (2021). In model (1), if we multiply by an indicator function of the receiver set of sender i and set β j (t) = 0 (j ∈ [n]) and γ(t) = γ, then it reduces to Perry and Wolfe's model. ...
... In addition, Perry and Wolfe (2013) treated the baseline intensity function as a nuisance parameter and focused on estimating γ. In model (1), if α i (t) = α(t), i ∈ [n], and β j (t) = β(t), j ∈ [n], then after transforming the baseline function to λ 0,ij (t) = exp{α(t) + β(t)}, it becomes the model proposed by Kreiß et al. (2019) and Kreiß (2021). One drawback of the model in Kreiß et al. (2019) and Kreiß (2021) is that they set the same degree parameters for all nodes which neglects the effect of degree heterogeneity in real-world networks. ...
... In model (1), if α i (t) = α(t), i ∈ [n], and β j (t) = β(t), j ∈ [n], then after transforming the baseline function to λ 0,ij (t) = exp{α(t) + β(t)}, it becomes the model proposed by Kreiß et al. (2019) and Kreiß (2021). One drawback of the model in Kreiß et al. (2019) and Kreiß (2021) is that they set the same degree parameters for all nodes which neglects the effect of degree heterogeneity in real-world networks. In model (1), our interest is on estimating not only γ(t), but also the 2n node-specified parameters α i (t) and β j (t). ...
Continuous time network data have been successfully modeled by multivariate counting processes, in which the intensity function is characterized by covariate information. However, degree heterogeneity has not been incorporated into the model which may lead to large biases for the estimation of homophily effects. In this paper, we propose a degree-corrected Cox network model to simultaneously analyze the dynamic degree heterogeneity and homophily effects for continuous time directed network data. Since each node has individual-specific in- and out-degree effects in the model, the dimension of the time-varying parameter vector grows with the number of nodes, which makes the estimation problem non-standard. We develop a local estimating equations approach to estimate unknown time-varying parameters, and establish consistency and asymptotic normality of the proposed estimators by using the powerful martingale process theories. We further propose test statistics to test for trend and degree heterogeneity in dynamic networks. Simulation studies are provided to assess the finite sample performance of the proposed method and a real data analysis is used to illustrate its practical utility.
In statistical network analysis it is common to observe so called interaction data. Such data is characterized by the actors who form the vertices of a network. These are able to interact with each other along the edges of the network. One usually assumes that the edges in the network are randomly formed and dissolved over the observation horizon. In addition covariates are observed and the interest is to model the impact of the covariates on the interactions. In this paper we develop a framework to test if a non-parametric form of the baseline intensity allows for more flexibility than a baseline which is parametrically dependent on system-wide covariates (i.e. covariates which take the same value for all individuals, e.g. time). This allows to test if certain seasonality effects can be explained by simple covariates like the time. The procedure is applied to modeling the baseline intensity in a bike-sharing network by using weather and time information.
We use LASSO methods to shrink, select, and estimate the high-dimensional network linking the publicly traded subset of the world's top 150 banks, 2003–2014. We characterize static network connectedness using full-sample estimation and dynamic network connectedness using rolling-window estimation. Statically, we find that global bank equity connectedness has a strong geographic component, whereas country sovereign bond connectedness does not. Dynamically, we find that equity connectedness increases during crises, with clear peaks during the Great Financial Crisis and each wave of the subsequent European Debt Crisis, and with movements coming mostly from changes in cross-country as opposed to within-country bank linkages.
The main objective of this paper is to introduce and illustrate relational event models, a new class of statistical models for the analysis of time-stamped data with complex temporal and relational dependencies. We outline the main differences between recently proposed relational event models and more conventional network models based on the graph-theoretic formalism typically adopted in empirical studies of social networks. Our main contribution involves the definition and implementation of a marked point process extension of currently available models. According to this approach, the sequence of events of interest is decomposed into two components: (a) event time and (b) event destination. This decomposition transforms the problem of selection of event destination in relational event models into a conditional multinomial logistic regression problem. The main advantages of this formulation are the possibility of controlling for the effect of event-specific data and a significant reduction in the estimation time of currently available relational event models. We demonstrate the empirical value of the model in an analysis of interhospital patient transfers within a regional community of health care organizations. We conclude with a discussion of how the models we presented help to overcome some the limitations of statistical models for networks that are currently available.
Dynamic probabilistic networks are a compact representation of complex
stochastic processes. In this paper we examine how to learn the structure of a
DPN from data. We extend structure scoring rules for standard probabilistic
networks to the dynamic case, and show how to search for structure when some of
the variables are hidden. Finally, we examine two applications where such a
technology might be useful: predicting and classifying dynamic behaviors, and
learning causal orderings in biological processes. We provide empirical results
that demonstrate the applicability of our methods in both domains.
We present a general principle for estimating a regression function
nonparametrically, allowing for a wide variety of data filtering, for example,
repeated left truncation and right censoring. Both the mean and the median
regression cases are considered. The method works by first estimating the
conditional hazard function or conditional survivor function and then
integrating. We also investigate improved methods that take account of model
structure such as independent errors and show that such methods can improve
performance when the model structure is true. We establish the pointwise
asymptotic normality of our estimators.
This article incorporates a political decision process into an urban land use model to predict the likely location of a public good. It fills an important gap in the literature by modeling the endogenous location of open space. The article compares open space decisions made under a majority-rules voting scheme with welfare-improving criterion and finds households tied to a location in space compete against each other for public goods located nearer them. Significant differences emerge between the two decision criteria, indicating that requiring referenda for open space decisions is likely to lead to inefficient outcomes. Specifically, many open space votes are likely to fail that would lead to welfare improvements, and any open space decisions that do pass will require amenities larger than needed to achieve the social optimum. The more dispersed and large the population, the larger is the gap between the socially efficient level and the level needed for a public referendum to pass.
The Cox regression model for censored survival data specifies that covariates have a proportional effect on the hazard function of the life-time distribution of an individual. In this paper we discuss how this model can be extended to a model where covariate processes have a proportional effect on the intensity process of a multivariate counting process. This permits a statistical regression analysis of the intensity of a recurrent event allowing for complicated censoring patterns and time dependent covariates. Furthermore, this formulation gives rise to proofs with very simple structure using martingale techniques for the asymptotic properties of the estimators from such a model. Finally an example of a statistical analysis is included.
We introduce a new kernel hazard estimator in a nonparametric model where the stochastic hazard depends on the current value of time and on the current value of a time dependent covariate or marker. We establish the pointwise and global convergence of our estimator.
A semiparametric hazard model with parametrized time but general covariate dependency is formulated and analyzed inside the framework of counting process theory. A profile likelihood principle is introduced for estimation of the parameters: the resulting estimator is -consistent, asymptotically normal and achieves the semiparametric efficiency bound. An estimation procedure for the nonparametric part is also given and its asymptotic properties are derived. We provide an application to mortality data.
In this paper we discuss weak dependence and mixing properties of some popular models. We also develop some of their econometric applications. Autoregressive models, autoregressive conditional heteroskedasticity (ARCH) models, and bilinear models are widely used in econometrics. More generally, stationary Markov modeling is often used. Bernoulli shifts also generate many useful stationary sequences, such as autoregressive moving average (ARMA) or ARCH( ) processes. For Volterra processes, mixing properties obtain given additional regularity assumptions on the distribution of the innovations.We recall associated probability limit theorems and investigate the nonparametric estimation of those sequences.We first thank the editor for the huge amount of additional editorial work provided for this review paper. The efficiency of the numerous referees was especially useful. The error pointed out in Hall and Horowitz (1996) was the origin of the present paper, and we thank the referees for asking for a more detailed treatment of a correct proof for this paper in Section 2.3. Also we thank Marc Henry and Rafal Wojakowski for a very careful rereading of the paper. An anonymous referee has been particularly helpful in the process of revision of the paper. The authors thank him for his numerous suggestions of improvement, including important results on negatively associated sequences and a thorough update in standard English.
Proportional hazard models for survival data, even though popular and numerically handy, suffer from the restrictive assumption that covariate effects are constant over survival time. A number of tests have been proposed to check this assumption. This paper contributes to this area by employing local estimates allowing to fit hazard models in which covariate effects are smoothly varying with time. A formal test is derived to check for proportional hazards against smooth hazards as alternative. The test proves to possess omnibus power in that it is powerful against arbitrary but smooth alternatives. Comparative simulations and two data examples accompany the presentation. Extensions are provided to multiple covariate settings, where the focus of interest is to decide which of the covariate effects vary with time.
We propose new procedures for estimating the component functions in both additive and multiplicative nonparametric marker-dependent hazard models. We work with a full counting process framework that allows for left truncation and right censoring and time-varying covariates. Our procedures are based on kernel hazard estimation as developed by J. P. Nielsen and O. B. Linton [ibid. 23, 1735–1748 (1995; Zbl 0847.62023)] and on the idea of marginal integration. We provide a central limit theorem for the marginal integration estimator. We then define estimators based on finite-step backfitting in both additive and multiplicative cases and prove that these estimators are asymptotically normal and have smaller variance than the marginal integration method.
We establish both uniform and nonuniform error bounds of the Berry-Esseen type in normal approximation under local dependence. These results are of an order close to the best possible if not best possible. They are more general or sharper than many existing ones in the literature. The proofs couple Stein's method with the concentration inequality approach.
We have analyzed the fully-anonymized headers of 362 million messages
exchanged by 4.2 million users of Facebook, an online social network of college
students, during a 26 month interval. The data reveal a number of strong daily
and weekly regularities which provide insights into the time use of college
students and their social lives, including seasonal variations. We also
examined how factors such as school affiliation and informal online friend
lists affect the observed behavior and temporal patterns. Finally, we show that
Facebook users appear to be clustered by school with respect to their temporal
messaging patterns.
This snapshot of the current frontier of statistics and network analysis focuses on the foundational topics of modeling, sampling, and design. Primarily for graduate students and researchers in statistics and closely related fields, emphasis is not only on what has been done, but on what remains to be done.
This paper is concerned with cross-sectional dependence arising because observations are interconnected through an observed network. Following (Doukhan and Louhichi, 1999), we measure the strength of dependence by covariances of nonlinearly transformed variables. We provide a law of large numbers and central limit theorem for network dependent variables. We also provide a method of calculating standard errors robust to general forms of network dependence. For that purpose, we rely on a network heteroskedasticity and autocorrelation consistent (HAC) variance estimator, and show its consistency. The results rely on conditions characterized by tradeoffs between the rate of decay of dependence across a network and network’s denseness. Our approach can accommodate data generated by network formation models, random fields on graphs, conditional dependency graphs, and large functional-causal systems of equations.
We consider models for time-to-event data that allow that an event, e.g., a relapse of a disease, never occurs for a certain percentage p of the population, called the cure rate. We suppose that these data are subject to random right censoring and we model the data using a mixture cure model, in which the survival function of the uncured subjects is left unspecified. The aim is to test whether the cure rate p, as a function of the covariates, satisfies a certain parametric model. To do so, we propose a test statistic that is inspired by a goodness-of-fit test for a regression function due to Härdle & Mammen (1993). We show that the statistic is asymptotically normally distributed under the null hypothesis, that the model is correctly specified, and under local alternatives. A bootstrap procedure is proposed to implement the test. The good performance of the approach is confirmed with simulations. For illustration we apply the test to data on the times between first and second births.
We introduce LASSO‐type regularization for large‐dimensional realized covariance estimators of log‐prices. The procedure consists of shrinking the off‐diagonal entries of the inverse realized covariance matrix towards zero. This technique produces covariance estimators that are positive definite and with a sparse inverse. We name the estimator realized network, since estimating a sparse inverse realized covariance matrix is equivalent to detecting the partial correlation network structure of the daily log‐prices. The large sample consistency and selection properties of the estimator are established. An application to a panel of US blue chip stocks shows the advantages of the estimator for out‐of‐sample GMV asset allocation.
Many modern network datasets arise from processes of interactions in a population, such as phone calls, email exchanges, co-authorships, and professional collaborations. In such interaction networks, the edges comprise the fundamental statistical units, making a framework for edge-labeled networks more appropriate for statistical analysis. In this context we initiate the study of edge exchangeable network models and explore its basic statistical properties. Several theoretical and practical features make edge exchangeable models better suited to many applications in network analysis than more common vertex-centric approaches. In particular, edge exchangeable models allow for sparse structure and power law degree distributions, both of which are widely observed empirical properties that cannot be handled naturally by more conventional approaches. Our discussion culminates in the Hollywood model, which we identify here as the canonical family of edge exchangeable distributions. The Hollywood model is computationally tractable, admits a clear interpretation, exhibits good theoretical properties, and performs reasonably well in estimation and prediction as we demonstrate on real network datasets. As a generalization of the Hollywood model, we further identify the vertex components model as a nonparametric subclass of models with a convenient stick breaking construction.
A flexible approach for modeling both dynamic event counting and dynamic link-based networks based on counting processes is proposed, and estimation in these models is studied. We consider nonparametric likelihood based estimation of parameter functions via kernel smoothing. The asymptotic behavior of these estimators is rigorously analyzed by allowing the number of nodes to tend to infinity. The finite sample performance of the estimators is illustrated through an empirical analysis of bike share data.
Presenting tools to aid understanding of asymptotic theory and weakly dependent processes, this book is devoted to inequalities and limit theorems for sequences of random variables that are strongly mixing in the sense of Rosenblatt, or absolutely regular.
The first chapter introduces covariance inequalities under strong mixing or absolute regularity. These covariance inequalities are applied in Chapters 2, 3 and 4 to moment inequalities, rates of convergence in the strong law, and central limit theorems. Chapter 5 concerns coupling. In Chapter 6 new deviation inequalities and new moment inequalities for partial sums via the coupling lemmas of Chapter 5 are derived and applied to the bounded law of the iterated logarithm. Chapters 7 and 8 deal with the theory of empirical processes under weak dependence. Lastly, Chapter 9 describes links between ergodicity, return times and rates of mixing in the case of irreducible Markov chains. Each chapter ends with a set of exercises.
The book is an updated and extended translation of the French edition entitled "Théorie asymptotique des processus aléatoires faiblement dépendants" (Springer, 2000). It will be useful for students and researchers in mathematical statistics, econometrics, probability theory and dynamical systems who are interested in weakly dependent processes.
We propose various self-exciting point process models for the times when e-mails are sent between individuals in a social network. Using an EM-type approach, we fit these models to an e-mail network dataset from West Point Military Academy and the Enron e-mail dataset. We argue that the self-exciting models adequately capture major temporal clustering features in the data and perform better than traditional stationary Poisson models. We also investigate how accounting for diurnal and weekly trends in e-mail activity improves the overall fit to the observed network data. A motivation and application for fitting these self-exciting models is to use parameter estimates to characterize important e-mail communication behaviors such as the baseline sending rates, average reply rates, and average response times. A primary goal is to use these features, estimated from the self-exciting models, to infer the underlying leadership status of users in the West Point and Enron networks.
This monograph is aimed at developing Doukhan/Louhichi's (1999) idea to measure asymptotic independence of a random process. The authors propose various examples of models fitting such conditions such as stable Markov chains, dynamical systems or more complicated models, nonlinear, non-Markovian, and heteroskedastic models with infinite memory. Most of the commonly used stationary models fit their conditions. The simplicity of the conditions is also their strength. The main existing tools for an asymptotic theory are developed under weak dependence. They apply the theory to nonparametric statistics, spectral analysis, econometrics, and resampling. The level of generality makes those techniques quite robust with respect to the model. The limit theorems are sometimes sharp and always simple to apply. The theory (with proofs) is developed and the authors propose to fix the notation for future applications. A large number of research papers deals with the present ideas; the authors as well as numerous other investigators participated actively in the development of this theory. Several applications are still needed to develop a method of analysis for (nonlinear) times series and they provide here a strong basis for such studies.
Dependent phenomena, such as relational, spatial and temporal phenomena, tend to be characterized by local dependence in the sense that units which are close in a well-defined sense are dependent. In contrast with spatial and temporal phenomena, though, relational phenomena tend to lack a natural neighbourhood structure in the sense that it is unknown which units are close and thus dependent. Owing to the challenge of characterizing local dependence and constructing random graph models with local dependence, many conventional exponential family random graph models induce strong dependence and are not amenable to statistical inference. We take first steps to characterize local dependence in random graph models, inspired by the notion of finite neighbourhoods in spatial statistics and M-dependence in time series, and we show that local dependence endows random graph models with desirable properties which make them amenable to statistical inference. We show that random graph models with local dependence satisfy a natural domain consistency condition which every model should satisfy, but conventional exponential family random graph models do not satisfy. In addition, we establish a central limit theorem for random graph models with local dependence, which suggests that random graph models with local dependence are amenable to statistical inference. We discuss how random graph models with local dependence can be constructed by exploiting either observed or unobserved neighbourhood structure. In the absence of observed neighbourhood structure, we take a Bayesian view and express the uncertainty about the neighbourhood structure by specifying a prior on a set of suitable neighbourhood structures. We present simulation results and applications to two real world networks with ‘ground truth’.
This is a concise text developed from lecture notes and ready to be used for a course on the graduate level. The main idea is to introduce the fundamental concepts of the theory while maintaining the exposition suitable for a first approach in the field. Therefore, the results are not always given in the most general form but rather under assumptions that lead to shorter or more elegant proofs. The book has three chapters. Chapter 1 presents basic nonparametric regression and density estimators and analyzes their properties. Chapter 2 is devoted to a detailed treatment of minimax lower bounds. Chapter 3 develops more advanced topics: Pinskers theorem, oracle inequalities, Stein shrinkage, and sharp minimax adaptivity. This book will be useful for researchers and grad students interested in theoretical aspects of smoothing techniques. Many important and useful results on optimal and adaptive estimation are provided. As one of the leading mathematical statisticians working in nonparametrics, the author is an authority on the subject.
The scientific study of networks, including computer networks, social networks, and biological networks, has received an enormous amount of interest in the last few years. The rise of the Internet and the wide availability of inexpensive computers have made it possible to gather and analyze network data on a large scale, and the development of a variety of new theoretical tools has allowed us to extract new knowledge from many different kinds of networks. The study of networks is broadly interdisciplinary and important developments have occurred in many fields, including mathematics, physics, computer and information sciences, biology, and the social sciences. This book brings together the most important breakthroughs in each of these fields and presents them in a coherent fashion, highlighting the strong interconnections between work in different areas. Subjects covered include the measurement and structure of networks in many branches of science, methods for analyzing network data, including methods developed in physics, statistics, and sociology, the fundamentals of graph theory, computer algorithms, and spectral methods, mathematical models of networks, including random graph models and generative models, and theories of dynamical processes taking place on networks.
The natural habitat of most Bayesian methods is data represented by
exchangeable sequences of observations, for which de Finetti's theorem provides
the theoretical foundation. Dirichlet process clustering, Gaussian process
regression, and many other parametric and nonparametric Bayesian models fall
within the remit of this framework; many problems arising in modern data
analysis do not. This expository paper provides an introduction to Bayesian
models of graphs, matrices, and other data that can be modeled by random
structures. We describe results in probability theory that generalize de
Finetti's theorem to such data and discuss the relevance of these results to
nonparametric Bayesian modeling. With the basic ideas in place, we survey
example models available in the literature; applications of such models include
collaborative filtering, link prediction, and graph and network analysis. We
also highlight connections to recent developments in graph theory and
probability, and sketch the more general mathematical foundation of Bayesian
methods for other types of data beyond sequences and arrays.
A stochastic model is proposed for social networks in which the actors in a network are partitioned into subgroups called blocks. The model provides a stochastic generalization of the blockmodel. Estimation techniques are developed for the special case of a single relation social network, with blocks specified a priori. An extension of the model allows for tendencies toward reciprocation of ties beyond those explained by the partition. The extended model provides a one degree-of-freedom test of the model. A numerical example from the social network literature is used to illustrate the methods.
The analysis of censored failure times is considered. It is assumed that on each individual are available values of one or more explanatory variables. The hazard function (age‐specific failure rate) is taken to be a function of the explanatory variables and unknown regression coefficients multiplied by an arbitrary and unknown function of time. A conditional likelihood is obtained, leading to inferences about the unknown regression coefficients. Some generalizations are outlined.
Social behavior over short time scales is frequently understood in terms of actions, which can be thought of as discrete events in which one individual emits a behavior directed at one or more other entities in his or her environment (possibly including himself or herself). Here, we introduce a highly flexible framework for modeling actions within social settings, which permits likelihood-based inference for behavioral mechanisms with complex dependence. Examples are given for the parameterization of base activity levels, recency, persistence, preferential attachment, transitive/cyclic interaction, and participation shifts within the relational event framework. Parameter estimation is discussed both for data in which an exact history of events is available, and for data in which only event sequences are known. The utility of the framework is illustrated via an application to dynamic modeling of responder radio communications during the early hours of the World Trade Center disaster.
This paper investigates the problem of density estimation for absolutely regular observations. In a first part, we state two important results: a new variance inequality and a Rosenthal type inequality. This allows us to study the ?
p
-integrated risk, p≧ 2, of a large class of density estimators including kernel or projection estimators. Under the summability condition on the mixing coefficients ∑
k≧ 0
(k+1)
p− 2
β
k
<∞, the rates obtained are those known to be optimal in the independent setting.
Network data often take the form of repeated interactions between senders and
receivers tabulated over time. A primary question to ask of such data is which
traits and behaviors are predictive of interaction. To answer this question, a
model is introduced for treating directed interactions as a multivariate point
process: a Cox multiplicative intensity model using covariates that depend on
the history of the process. Consistency and asymptotic normality are proved for
the resulting partial-likelihood-based estimators under suitable regularity
conditions, and an efficient fitting procedure is described. Multicast
interactions--those involving a single sender but multiple receivers--are
treated explicitly. The resulting inferential framework is then employed to
model message sending behavior in a corporate e-mail network. The analysis
gives a precise quantification of which static shared traits and dynamic
network effects are predictive of message recipient selection.
The objective of the present article is to propose and evaluate a probabilistic approach based on Bayesian networks for modelling non-homogeneous and non-linear gene regulatory processes. The method is based on a mixture model, using latent variables to assign individual measurements to different classes. The practical inference follows the Bayesian paradigm and samples the network structure, the number of classes and the assignment of latent variables from the posterior distribution with Markov Chain Monte Carlo (MCMC), using the recently proposed allocation sampler as an alternative to RJMCMC.
We have evaluated the method using three criteria: network reconstruction, statistical significance and biological plausibility. In terms of network reconstruction, we found improved results both for a synthetic network of known structure and for a small real regulatory network derived from the literature. We have assessed the statistical significance of the improvement on gene expression time series for two different systems (viral challenge of macrophages, and circadian rhythms in plants), where the proposed new scheme tends to outperform the classical BGe score. Regarding biological plausibility, we found that the inference results obtained with the proposed method were in excellent agreement with biological findings, predicting dichotomies that one would expect to find in the studied systems.
Two supplementary papers on theoretical (T) and experi-mental (E) aspects and the datasets used in our study are available from http://www.bioss.ac.uk/associates/marco/supplement/
Very often in survival analysis one has to study martingale integrals where the integrand is not predictable and where the counting process theory of martingales is not directly applicable, as for example in nonparametric and semiparametric applications where the integrand is based on a pilot estimate. We call this the predictability issue in survival analysis. The problem has been resolved by approximations of the integrand by predictable functions which have been justified by ad hoc procedures. We present a general approach to the solution of this problem. The usefulness of the approach is shown in three applications. In particular, we argue that earlier ad hoc procedures do not work in higher-dimensional smoothing problems in survival analysis. Copyright 2007, Oxford University Press.
Statistical Models Based on Counting Processes
- P K Andersen
- Ø Borgan
- R D Gill
- N Keiding
Supplement to “Correlation bounds, mixing and m-dependence under random time-varying network distances with an application to Cox-processes
- A Kreiß
Network dependence and inference
- J Vainora