Max-linear Models Research Articles

Extreme Value Statistics for Evolving Random Networks

Our objective is to survey recent results concerning the evolution of random networks and related extreme value statistics, which are a subject of interest due to numerous applications. Our survey concerns the statistical methodology but not the structure of random networks. We focus on the problems arising in evolving networks mainly due to the heavy-tailed nature of node indices. Tail and extremal indices of the node influence characteristics like in-degrees, out-degrees, PageRanks, and Max-linear models arising in the evolving random networks are discussed. Related topics like preferential and clustering attachments, community detection, stationarity and dependence of graphs, information spreading, finding the most influential leading nodes and communities, and related methods are surveyed. This survey tries to propose possible solutions to unsolved problems, like testing the stationarity and dependence of random graphs using known results obtained for random sequences. We provide a discussion of unsolved or insufficiently developed problems like the distribution of triangle and circle counts in evolving networks, or the clustering attachment and the local dependence of the modularity, the impact of node or edge deletion at each step of evolution on extreme value statistics, among many others. Considering existing techniques of community detection, we pay attention to such related topics as coloring graphs and anomaly detection by machine learning algorithms based on extreme value theory. In order to understand how one can compute tail and extremal indices on random graphs, we provide a structured and comprehensive review of their estimators obtained for random sequences. Methods to calculate the PageRank and PageRank vector are shortly presented. This survey aims to provide a better understanding of the directions in which the study of random networks has been done and how extreme value analysis developed for random sequences can be applied to random networks.

Tail Index Estimation of PageRanks in Evolving Random Graphs

Random graphs are subject to the heterogeneities of the distributions of node indices and their dependence structures. Superstar nodes to which a large proportion of nodes attach in the evolving graphs are considered. In the present paper, a statistical analysis of the extremal part of random graphs is considered. We used the extreme value theory regarding sums and maxima of non-stationary random length sequences to evaluate the tail index of the PageRanks and max-linear models of superstar nodes in the evolving graphs where existing nodes or edges can be deleted or not. The evolution is provided by a linear preferential attachment. Our approach is based on the analysis of maxima and sums of the node PageRanks over communities (block maxima and block sums), which can be independent or weakly dependent random variables. By an empirical study, it was found that tail indices of the block maxima and block sums are close to the minimum tail index of representative series extracted from the communities. The tail indices are estimated by data of simulated graphs.

Max-linear models in random environment

We extend previous work of max-linear models on finite directed acyclic graphs to infinite graphs as well as random graphs, and investigate their relations to classical percolation theory, more particularly the impact of Bernoulli bond percolation on such models. We show that the critical probability of percolation on the oriented square lattice graph Z2 describes a phase transition in the obtained model. Focus is on the dependence introduced by this graph into the max-linear model. We discuss natural applications in communication networks, in particular, concerning the propagation of influences.

Discrete max-linear Bayesian networks

Discrete max-linear Bayesian networks are directed graphical models specified by the same recursive structural equations as max-linear models but with discrete innovations. When all of the random variables in the model are binary, these models are isomorphic to the conjunctive Bayesian network (CBN) models of Beerenwinkel, Eriksson, and Sturmfels. Many of the techniques used to study CBN models can be extended to discrete max-linear models and similar results can be obtained. In particular, we extend the fact that CBN models are toric varieties after linear change of coordinates to all discrete max-linear models.

Recursive max-linear models with propagating noise

Recursive max-linear vectors model causal dependence between node variables by a structural equation model, expressing each node variable as a max-linear function of its parental nodes in a directed acyclic graph (DAG) and some exogenous innovation. For such a model, there exists a unique minimum DAG, represented by the Kleene star matrix of its edge weight matrix, which identifies the model and can be estimated. For a more realistic statistical modeling we introduce some random observational noise. A probabilistic analysis of this new noisy model reveals that the unique minimum DAG representing the distribution of the non-noisy model remains unchanged and identifiable. Moreover, the distribution of the minimum ratio estimators of the model parameters at their left limits are completely determined by the distribution of the noise variables up to a positive constant. Under a regular variation condition on the noise variables we prove that the estimated Kleene star matrix converges to a matrix of independent Weibull entries after proper centering and scaling.

Max-linear regression models with regularization

Motivated by the newly developed max-linear competing copula factor models and max-stable nonlinear time series models, we propose a new class of max-linear regression models to take advantages of easy interpretable features embedded in linear regression models. It can be seen that linear relation is a special case of max-linear relation. We develop an EM algorithm based maximum likelihood estimation procedure. The consistency and asymptotics of the estimators for parameters are proved. To advance max-linear models to deal with high dimensional predictors, we adopt the common strategy of regularization in the high dimensional regression literature. We demonstrate the broad applicability of max-linear models using simulation examples and real applications in econometric and business modeling. The results, in terms of predictability, show a significant improvement compared with solely using regular regression models and other existing machine learning methods. The results enhance our understanding of the relationship between the response variable and the predictors, and among the predictors as well.

$k$-means clustering of extremes

The $k$-means clustering algorithm and its variant, the spherical $k$-means clustering, are among the most important and popular methods in unsupervised learning and pattern detection. In this paper, we explore how the spherical $k$-means algorithm can be applied in the analysis of only the extremal observations from a data set. By making use of multivariate extreme value analysis we show how it can be adopted to find "prototypes" of extremal dependence and we derive a consistency result for our suggested estimator. In the special case of max-linear models we show furthermore that our procedure provides an alternative way of statistical inference for this class of models. Finally, we provide data examples which show that our method is able to find relevant patterns in extremal observations and allows us to classify extremal events.

Max-linear models on directed acyclic graphs

We consider a new structural equation model in which all random variables can be written as a max-linear function of their parents and independent noise variables. For the corresponding graph we assume that it is a directed acyclic graph. We show that the model is max-linear and detail the relation between the weights of the structural equation model and the max-linear coefficients. We characterize all max-linear models which are generated by this structural equation model. This leads to the presentation of a max-linear structural equation model as the solution of a fixed point equation and to a unique minimal DAG describing the relationships between the variables.The model structure introduces an order between the random variables, which yields certain model reductions, represented by subgraphs of the DAG which we call order DAGs. This results also in a reduced form for the regular conditional distributions compared to previous representations.

Tail dependence of recursive max-linear models with regularly varying noise variables

Recursive max-linear structural equation models with regularly varying noise variables are considered. Their causal structure is represented by a directed acyclic graph (DAG). The problem of identifying a recursive max-linear model and its associated DAG from its matrix of pairwise tail dependence coefficients is discussed. For example, it is shown that if a causal ordering of the associated DAG is additionally known, then the minimum DAG representing the recursive structural equations can be recovered from the tail dependence matrix. For a relevant subclass of recursive max-linear models, identifiability of the associated minimum DAG from the tail dependence matrix and the initial nodes is shown. Algorithms find the associated minimum DAG for the different situations. Furthermore, given a tail dependence matrix, an algorithm outputs all compatible recursive max-linear models and their associated minimum DAGs.

On generalized max-linear models in max-stable random fields

Abstract In practice, it is not possible to observe a whole max-stable random field. Therefore, we propose a method to reconstruct a max-stable random field in C([0, 1]k) by interpolating its realizations at finitely many points. The resulting interpolating process is again a max-stable random field. This approach uses a generalized max-linear model. Promising results have been established in the k = 1 case of Falk et al. (2015). However, the extension to higher dimensions is not straightforward since we lose the natural order of the index space.

Properties of extremal dependence models built on bivariate max-linearity

Bivariate max-linear models provide a core building block for characterizing bivariate max-stable distributions. The limiting distribution of marginally normalized component-wise maxima of bivariate max-linear models can be dependent (asymptotically dependent) or independent (asymptotically independent). However, for modeling bivariate extremes they have weaknesses in that they are exactly max-stable with no penultimate form of convergence to asymptotic dependence, and asymptotic independence arises if and only if the bivariate max-linear model is independent. In this work we present more realistic structures for describing bivariate extremes. We show that these models are built on bivariate max-linearity but are much more general. In particular, we present models that are dependent but asymptotically independent and others that are asymptotically dependent but have penultimate forms. We characterize the limiting behavior of these models using two new different angular measures in a radial–angular representation that reveal more structure than existing measures.

On generalized max-linear models and their statistical interpolation

We propose a method to generate a max-stable process in C[0, 1] from a max-stable random vector in R d by generalizing the max-linear model established by Wang and Stoev (2011). For this purpose, an interpolation technique that preserves max-stability is proposed. It turns out that if the random vector follows some finite-dimensional distribution of some initial max-stable process, the approximating processes converge uniformly to the original process and the pointwise mean-squared error can be represented in a closed form. The obtained results carry over to the case of generalized Pareto processes. The introduced method enables the reconstruction of the initial process only from a finite set of observation points and, thus, a reasonable prediction of max-stable processes in space becomes possible. A possible extension to arbitrary dimensions is outlined.

On generalized max-linear models and their statistical interpolation

We propose a method to generate a max-stable process in C[0, 1] from a max-stable random vector in Rd by generalizing the max-linear model established by Wang and Stoev (2011). For this purpose, an interpolation technique that preserves max-stability is proposed. It turns out that if the random vector follows some finite-dimensional distribution of some initial max-stable process, the approximating processes converge uniformly to the original process and the pointwise mean-squared error can be represented in a closed form. The obtained results carry over to the case of generalized Pareto processes. The introduced method enables the reconstruction of the initial process only from a finite set of observation points and, thus, a reasonable prediction of max-stable processes in space becomes possible. A possible extension to arbitrary dimensions is outlined.

Conditional sampling for spectrally discrete max-stable random fields

Max-stable random fields play a central role in modeling extreme value phenomena. We obtain an explicit formula for the conditional probability in general max-linear models, which include a large class of max-stable random fields. As a consequence, we develop an algorithm for efficient and exact sampling from the conditional distributions. Our method provides a computational solution to the prediction problem for spectrally discrete max-stable random fields. This work offers new tools and a new perspective to many statistical inference problems for spatial extremes, arising, for example, in meteorology, geology, and environmental applications.

Conditional sampling for spectrally discrete max-stable random fields

Max-stable random fields play a central role in modeling extreme value phenomena. We obtain an explicit formula for the conditional probability in general max-linear models, which include a large class of max-stable random fields. As a consequence, we develop an algorithm for efficient and exact sampling from the conditional distributions. Our method provides a computational solution to the prediction problem for spectrally discrete max-stable random fields. This work offers new tools and a new perspective to many statistical inference problems for spatial extremes, arising, for example, in meteorology, geology, and environmental applications.