1,720,978 research outputs found
Hawkes and INAR(∞) processes
No full textAbstractIn this paper, we discuss integer-valued autoregressive time series (INAR), Hawkes point processes, and their interrelationship. Besides presenting structural analogies, we derive a convergence theorem. More specifically, we generalize the well-known INAR(p), p∈N, time series model to a corresponding model of infinite order: the INAR(∞) model. We establish existence, uniqueness, finiteness of moments, and give formulas for the autocovariance function as well as for the joint moment-generating function. Furthermore, we derive a branching-process–as well as an AR(∞)–and an MA(∞) representation for the model. We compare Hawkes process properties with their INAR(∞) counterparts. Given a Hawkes process N, in the main theorem of the paper we construct an INAR(∞)-based family of point processes and prove its convergence to N. This connection between INAR and Hawkes models will be relevant in applications
Perspectives on Hawkes Processes
Full text linkThis thesis addresses Hawkes point processes in seven scientific papers. We build theoretical bridges between Hawkes processes and other mathematical concepts—such as time series, branching random walks, or graph theory. In Paper A, we represent monotype Hawkes processes as limits of time-series based point processes. We examine the corresponding time series, the integer-valued autoregressive (INAR) time series of infinite order, in some detail. Furthermore, we point out structural analogies between Hawkes processes and INAR time series. In Paper B, we represent multitype Hawkes processes as type/space projections of certain branching random walks. This representation allows to generalize the convergence result from Paper A to the multitype case. Furthermore, it opens the door to generalizations of Hawkes processes that might be interesting in applications. In Paper C, we introduce a nonparametric estimation procedure for multitype Hawkes processes: we discretize Hawkes-process data. From Paper A and Paper B, we know that the resulting bin-count sequences can be approximated by INAR time series. Thus, we estimate the INAR parameters by standard methods and¡ retranslate the results into the point process world. In Paper D, we represent multitype Hawkes processes as directed weighted graphs. These ‘Hawkes graphs’ summarize the branching structure of a Hawkes process in a compact, yet meaningful way. We point out how the graphical perspective is also fertile mathematically, implementation-wise, and pedagogically. Furthermore, we apply the estimation method from Paper C to infer the Hawkes graph from large datasets. We pay special attention to computational issues. In Paper E, we apply the methods and concepts from Paper C and Paper D to limit-order-book data. In particular, we extend our estimation procedure to the marked case. The various estimation results allow insights into market microstructure. In Paper F, we give the results of a simulation study, where we compare our estimation procedure with maximum-likelihood estimation. Finally, in Paper G, we consider a certain critical case of the monotype Hawkes process. We study the critical Hawkes process by applying results from critical cluster fields, renewal theory, and regular variation. We discuss a possible Poisson embedding and a Palm version of the critical Hawkes process. Our methods give possible directions for the open discussion of multitype critical Hawkes processes as well as of critical INAR times series
Palm versions of Hawkes processes
Full text linkThis brief paper identifies the Palm distribution of a linear Hawkes process. The textbook example for Palm distributions is the Palm version of a stationary Poisson process that corresponds to the original process plus a point in zero. The present result generalizes this example in a more complex but nevertheless tractable way. As a next step, we derive the intensity measure of the Palm version of a Hawkes process and show how it could be used for estimation. Finally, we discuss further possible applications to the theory of Hawkes processes
An estimation procedure for the Hawkes process
No full textIn this paper, we present a nonparametric estimation procedure for the multivariate Hawkes point process. The timeline is cut into bins and—for each component process—the number of points in each bin is counted. As a consequence of earlier results in Kirchner [Stoch. Process. Appl., 2016, 162, 2494–2525], the distribution of the resulting ‘bin-count sequences’ can be approximated by an integer-valued autoregressive model known as the (multivariate) INAR(p) model. We represent the INAR(p) model as a standard vector-valued linear autoregressive time series with white-noise innovations (VAR(p)). We establish consistency and asymptotic normality for conditional least-squares estimation of the VAR(p), respectively, the INAR(p) model. After appropriate scaling, these time-series estimates yield estimates for the underlying multivariate Hawkes process as well as corresponding variance estimates. The estimates depend on a bin-size and a support s. We discuss the impact and the choice of these parameters. All results are presented in such a way that computer implementation, e.g. in R, is straightforward. Simulation studies confirm the effectiveness of our estimation procedure. In the second part of the paper, we present a data example where the method is applied to bivariate event-streams in financial limit-order-book data. We fit a bivariate Hawkes model on the joint process of limit and market order arrivals. The analysis exhibits a remarkably asymmetric relation between the two component processes: incoming market orders excite the limit-order flow heavily whereas the market-order flow is hardly affected by incoming limit orders. For the estimated excitement functions, we observe power-law shapes, inhibitory effects for lags under 0.003 s, second periodicities and local maxima at 0.01, 0.1 and 0.5 s
Hawkes model specification for limit order books
No full textThis paper discusses Hawkes modeling of order arrivals in limit order books. We model the flow of market orders, limit orders, and cancelations by a self- and crossexciting multitype marked Hawkes process with state-dependent baseline intensities. The marks carry the order sizes and the state of the book is summarized by the ‘limit-order-book imbalance’. We specify the model very carefully – with few a priori assumptions: we select the non-zero excitements (the ‘Hawkes skeleton’), the shape of the decay kernels, and the shape of the impact functions in a nonparametric manner. Furthermore, we show that our data exhibit perfect bid–ask symmetry. We observe that the imbalance of the order book explains the probability for a bid (ask) market order – given the occurrence of a market order – in a perfectly linear manner. Thus, we include a term involving the imbalance in the baseline intensity of the process. We calibrate the specified parametric model by maximum likelihood estimation and discuss the results. Finally, we apply the fitted model in order to estimate the conditional distribution of the next order type. This opens the door to order-type prediction
Algorithmische Alchemie – die sozio-technische Reproduktion sozialer Ungleichheit im Bildungssystem
Full text linkAlgorithmische Systeme versprechen, die Bewertung schulischer Leistung zu objektivieren und Bildungsangebote zu individualisieren. Statt subjektiver Urteile von Lehrpersonen sollen Algorithmen entscheiden, wie Freitextaufgaben und Essays zu bewerten sind und wie Aufgaben zugeteilt werden. Doch die Intransparenz der Technik erzeugt unbemerkt neue Formen der Ungleichheit. Die Datensätze und ihre algorithmische Auswertung unterliegen einem algorithmic bias, der bestehende Bildungsungleichheiten nicht bloß reproduziert, sondern sogar verstärkt. Wir diskutieren diesen Umstand anhand zweier Anwendungsfälle: adaptive Lernsysteme und automatisierte Bewertung. Wir betrachten die angewandten algorithmischen Systeme im Kontext ihrer praktischen Wirkzusammenhänge und plädieren abschließend für eine Aufwertung der pädagogischen Profession angesichts der geschilderten Risiken
Going Beyond Counting First Authors in Author Co-citation Analysis
Full text linkThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
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