Spectra of some self-exciting and mutually exciting point processes

Abstract

SUMMAKYIn recent years methods of data analysis for point processes have received some attention, for example, by Cox & Lewis (1966) and Lewis (1964). In particular Bartlett (1963a, b) has introduced methods of analysis based on the point spectrum. Theoretical models are relatively sparse. In this paper the theoretical properties of a class of processes with particular reference to the point spectrum or corresponding covariance density functions are discussed. A particular result is a self-exciting process with the same second-order properties as a certain doubly stochastic process. These are not distinguishable by methods of data analysis based on these properties.at University College London on July 1, 2014 http://biomet.oxfordjournals.org/ Downloaded from J -oo Applying (2), we find, for r > 0, li(j) = A<7(T)+ g{T-v)(i{v)dv.

Uncovering and Predicting the Dynamic Process of Collective Attention with Survival Theory

“…Without loss of generality, we have 0 =  t 0  ≤  t 1  ≤  t 2  ≤ ... ≤  t i  ≤ ... ≤  t N  ≤  T . In this report, we model its dynamic process of attentions using a self-excited Hawkes process 37 , incorporating three key ingredients simultaneously: (1) attractiveness of an item, characterizing its inherent competitiveness against other items; (2) a reinforcement mechanism based on sum of previous attention triggers, capturing the well-known “richer-get-richer” phenomenon; (3) a general temporal relaxation function corresponding to the aging effect, characterizing time-dependent attractiveness of individual items. Taken these three factors together, for an individual item d , we model its dynamics of attentions characterized by the rate function λ ( t ) aswhere μ is the intrinsic attractiveness of the item, φ ( τ ) is the relaxation function that characterizes the temporal inhomogeneity due to the aging effect.…”

Detecting correlations among functional-sequence motifs

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“…In Ref. [9] the seismic activity (i.e., the probability of a seismic event) is described by a point process with stochastic intensity that includes self-excitation as well as * d.pirino@sssup.it 1 For example, if the events under study are earthquake shocks, S t can be a measure of their magnitude.…”

STDP in oscillatory recurrent networks: theoretical conditions for desynchronization and applications to deep brain stimulation

“…This technical step has been achieved by Hawkes (1971) and used by Gilson et al (2009a,b) in the context of recurrent network with STDP. The present study extends the work of Gilson et al (2009b) to the case of oscillatory inputs and highlights the conditions for which the network elicits bistability.…”