Modelling the extremes of seasonal viruses and hospital congestion: The example of flu in a Swiss hospital

Viruses causing flu or milder coronavirus colds are often referred to as ‘seasonal viruses’ as they tend to subside in warmer months. In other words, meteorological conditions tend to impact the activity of viruses, and this information can be exploited for the operational management of hospitals. In this study, we use 3 years of daily data from one of the biggest hospitals in Switzerland and focus on modelling the extremes of hospital visits from patients showing flu-like symptoms and the number of positive flu cases. We propose employing a discrete generalized Pareto distribution for the number of positive and negative cases. Our modelling framework allows for the parameters of these distributions to be linked to covariate effects, and for outlying observations to be dealt with via a robust estimation approach. Because meteorological conditions may vary over time, we use meteorological and not calendar variations to explain hospital charge extremes, and our empirical findings highlight their significance. We propose a measure of hospital congestion and a related tool to estimate the resulting CaRe (Charge-at-Risk-estimation) under different meteorological conditions. The relevant numerical computations can be easily carried out using the freely available GJRM R package. The empirical effectiveness of the proposed method is assessed through a simulation study

An efficient semiparametric maxima estimator of the extremal index

The extremal index $\theta$, a measure of the degree of local dependence in the extremes of a stationary process, plays an important role in extreme value analyses. We estimate $\theta$ semiparametrically, using the relationship between the distribution of block maxima and the marginal distribution of a process to define a semiparametric model. We show that these semiparametric estimators are simpler and substantially more efficient than their parametric counterparts. We seek to improve efficiency further using maxima over sliding blocks. A simulation study shows that the semiparametric estimators are competitive with the leading estimators. An application to sea-surge heights combines inferences about $\theta$ with a standard extreme value analysis of block maxima to estimate marginal quantiles.Comment: 17 pages, 7 figures. Minor edits made to version 1 prior to journal publication. The final publication is available at Springer via http://dx.doi.org/10.1007/s10687-015-0221-

Extreme Value Theory versus traditional GARCH approaches applied to financial data: a comparative evaluation

Although stock prices fluctuate, the variations are relatively small and are frequently assumed to be normally distributed on a large time scale. But sometimes these fluctuations can become determinant, especially when unforeseen large drops in asset prices are observed that could result in huge losses or even in market crashes. The evidence shows that these events happen far more often than would be expected under the generalised assumption of normally distributed financial returns. Thus it is crucial to model distribution tails properly so as to be able to predict the frequency and magnitude of extreme stock price returns. In this paper we follow the approach suggested by McNeil and Frey in 2000 and combine GARCH-type models with the extreme value theory to estimate the tails of three financial index returns ¿ S&P 500, FTSE 100 and NIKKEI 225 ¿ representing three important financial areas in the world. Our results indicate that EVT-based conditional quantile estimates are more accurate than those from conventional GARCH models assuming normal or Student¿s t distribution innovations when doing not only in-sample but also out-of-sample estimation. Moreover, these results are robust to alternative GARCH model specifications. The findings of this paper should be useful to investors in general, since their goal is to be able to forecast unforeseen price movements and take advantage of them by positioning themselves in the market according to these predictions

Hierarchy of Temporal Responses of Multivariate Self-Excited Epidemic Processes

We present the first exact analysis of some of the temporal properties of multivariate self-excited Hawkes conditional Poisson processes, which constitute powerful representations of a large variety of systems with bursty events, for which past activity triggers future activity. The term "multivariate" refers to the property that events come in different types, with possibly different intra- and inter-triggering abilities. We develop the general formalism of the multivariate generating moment function for the cumulative number of first-generation and of all generation events triggered by a given mother event (the "shock") as a function of the current time $t$. This corresponds to studying the response function of the process. A variety of different systems have been analyzed. In particular, for systems in which triggering between events of different types proceeds through a one-dimension directed or symmetric chain of influence in type space, we report a novel hierarchy of intermediate asymptotic power law decays $\sim 1/t^{1-(m+1)\theta}$ of the rate of triggered events as a function of the distance $m$ of the events to the initial shock in the type space, where $0 < \theta <1$ for the relevant long-memory processes characterizing many natural and social systems. The richness of the generated time dynamics comes from the cascades of intermediate events of possibly different kinds, unfolding via a kind of inter-breeding genealogy.Comment: 40 pages, 8 figure

Non-parametric kernel estimation for symmetric Hawkes processes. Application to high frequency financial data

We define a numerical method that provides a non-parametric estimation of the kernel shape in symmetric multivariate Hawkes processes. This method relies on second order statistical properties of Hawkes processes that relate the covariance matrix of the process to the kernel matrix. The square root of the correlation function is computed using a minimal phase recovering method. We illustrate our method on some examples and provide an empirical study of the estimation errors. Within this framework, we analyze high frequency financial price data modeled as 1D or 2D Hawkes processes. We find slowly decaying (power-law) kernel shapes suggesting a long memory nature of self-excitation phenomena at the microstructure level of price dynamics.Comment: 6 figure

Generating Functions and Stability Study of Multivariate Self-Excited Epidemic Processes

We present a stability study of the class of multivariate self-excited Hawkes point processes, that can model natural and social systems, including earthquakes, epileptic seizures and the dynamics of neuron assemblies, bursts of exchanges in social communities, interactions between Internet bloggers, bank network fragility and cascading of failures, national sovereign default contagion, and so on. We present the general theory of multivariate generating functions to derive the number of events over all generations of various types that are triggered by a mother event of a given type. We obtain the stability domains of various systems, as a function of the topological structure of the mutual excitations across different event types. We find that mutual triggering tends to provide a significant extension of the stability (or subcritical) domain compared with the case where event types are decoupled, that is, when an event of a given type can only trigger events of the same type.Comment: 27 pages, 8 figure

Spectral Density Regression for Bivariate Extremes

We introduce a density regression model for the spectral density of a bivariate extreme value distribution, that allows us to assess how extremal dependence can change over a covariate. Inference is performed through a double kernel estimator, which can be seen as an extension of the Nadaraya–Watson estimator where the usual scalar responses are replaced by mean constrained densities on the unit interval. Numerical experiments with the methods illustrate their resilience in a variety of contexts of practical interest. An extreme temperature dataset is used to illustrate our methods