Earthquake modelling at the country level using aggregated spatio-temporal point processes

The goal of this paper is to derive a hazard map for earthquake occurrences in Pakistan from a catalogue that contains spatial coordinates of shallow earthquakes of magnitude 4.5 or larger aggregated over calendar years. We test relative temporal stationarity by the KPSS statistic and use the inhomogeneous J-function to test for inter-point interactions. We then formulate a cluster model, and de-convolve in order to calculate the hazard map, and verify that no particular year has an undue influence on the map. Within the borders of the single country, the KPSS test did not show any deviation from homogeneity in the spatial intensities. The inhomogeneous J-function indicated clustering that could not be attributed to inhomogeneity, and the analysis of aftershocks showed some evidence of two major shocks instead of one during the 2005 Kashmir earthquake disaster. Thus, the spatial point pattern analysis carried out for these data was insightful in various aspects and the hazard map that was obtained may lead to improved measures to protect the population against the disastrous effects of earthquakes

Parameter estimation of binned Hawkes processes

A key difficulty that arises from real event data is imprecision in the recording of event time-stamps. In many cases, retaining event times with a high precision is expensive due to the sheer volume of activity. Combined with practical limits on the accuracy of measurements, binned data is common. In order to use point processes to model such event data, tools for handling parameter estimation are essential. Here we consider parameter estimation of the Hawkes process, a type of self-exciting point process that has found application in the modeling of financial stock markets, earthquakes and social media cascades. We develop a novel optimization approach to parameter estimation of binned Hawkes processes using a modified Expectation-Maximization algorithm, referred to as Binned Hawkes Expectation Maximization (BH-EM). Through a detailed simulation study, we demonstrate that existing methods are capable of producing severely biased and highly variable parameter estimates and that our novel BH-EM method significantly outperforms them in all studied circumstances. We further illustrate the performance on network flow (NetFlow) data between devices in a real large-scale computer network, to characterize triggering behavior. These results highlight the importance of correct handling of binned data

Distributional Drift Adaptation with Temporal Conditional Variational Autoencoder for Multivariate Time Series Forecasting

Due to the nonstationary nature, the distribution of real-world multivariate time series (MTS) changes over time, which is known as distribution drift. Most existing MTS forecasting models greatly suffer from distribution drift and degrade the forecasting performance over time. Existing methods address distribution drift via adapting to the latest arrived data or self-correcting per the meta knowledge derived from future data. Despite their great success in MTS forecasting, these methods hardly capture the intrinsic distribution changes, especially from a distributional perspective. Accordingly, we propose a novel framework temporal conditional variational autoencoder (TCVAE) to model the dynamic distributional dependencies over time between historical observations and future data in MTSs and infer the dependencies as a temporal conditional distribution to leverage latent variables. Specifically, a novel temporal Hawkes attention mechanism represents temporal factors subsequently fed into feed-forward networks to estimate the prior Gaussian distribution of latent variables. The representation of temporal factors further dynamically adjusts the structures of Transformer-based encoder and decoder to distribution changes by leveraging a gated attention mechanism. Moreover, we introduce conditional continuous normalization flow to transform the prior Gaussian to a complex and form-free distribution to facilitate flexible inference of the temporal conditional distribution. Extensive experiments conducted on six real-world MTS datasets demonstrate the TCVAE's superior robustness and effectiveness over the state-of-the-art MTS forecasting baselines. We further illustrate the TCVAE applicability through multifaceted case studies and visualization in real-world scenarios.Comment: 13 pages, 6 figures, submitted to IEEE Transactions on Neural Networks and Learning Systems (TNNLS

Modelling systemic price cojumps with Hawkes factor models

Instabilities in the price dynamics of a large number of financial assets are a clear sign of systemic events. By investigating portfolios of highly liquid stocks, we find that there are a large number of high-frequency cojumps. We show that the dynamics of these jumps is described neither by a multivariate Poisson nor by a multivariate Hawkes model. We introduce a Hawkes one-factor model which is able to capture simultaneously the time clustering of jumps and the high synchronization of jumps across assets

Modelling Short- and Long-Term Dependencies of Clustered High-Threshold Exceedances in Significant Wave Heights

The peaks-over-threshold (POT) method has a long tradition in modelling extremes in environmental variables. However, it has originally been introduced under the assumption of independently and identically distributed (iid) data. Since environmental data often exhibits a time series structure, this assumption is likely to be violated due to short- and long-term dependencies in practical settings, leading to clustering of high-threshold exceedances. In this paper, we first review popular approaches that either focus on modelling short- or long-range dynamics explicitly. In particular, we consider conditional POT variants and the Mittag–Leffler distribution modelling waiting times between exceedances. Further, we propose a new two-step approach capturing both short- and long-range correlations simultaneously. We suggest the autoregressive fractionally integrated moving average peaks-over-threshold (ARFIMA-POT) approach, which in a first step fits an ARFIMA model to the original series and then in a second step utilises a classical POT model for the residuals. Applying these models to an oceanographic time series of significant wave heights measured on the Sefton coast (UK), we find that neither solely modelling short- nor long-range dependencies satisfactorily explains the clustering of extremes. The ARFIMA-POT approach, however, provides a significant improvement in terms of model fit, underlining the need for models that jointly incorporate short- and long-range dependence to address extremal clustering, and their theoretical justification

Compound Multivariate Hawkes Processes: Large Deviations and Rare Event Simulation

In this paper, we establish a large deviations principle for a multivariate compound process induced by a multivariate Hawkes process with random marks. Our proof hinges on showing essential smoothness of the limiting cumulant of the multivariate compound process, resolving the inherent complication that this cumulant is implicitly characterized through a fixed-point representation. We employ the large deviations principle to derive logarithmic asymptotic results on the marginal ruin probabilities of the associated multivariate risk process. We also show how to conduct rare event simulation in this multivariate setting using importance sampling and prove the asymptotic efficiency of our importance sampling based estimator. The paper is concluded with a systematic assessment of the performance of our rare event simulation procedure