Hawkes process model with a time-dependent background rate and its application to high-frequency financial data.

A Hawkes process model with a time-varying background rate is developed for analyzing the high-frequency financial data. In our model, the logarithm of the background rate is modeled by a linear model with a relatively large number of variable-width basis functions, and the parameters are estimated by a Bayesian method. Our model can capture not only the slow time variation, such as in the intraday seasonality, but also the rapid one, which follows a macroeconomic news announcement. By analyzing the tick data of the Nikkei 225 mini, we find that (i) our model is better fitted to the data than the Hawkes models with a constant background rate or a slowly varying background rate, which have been commonly used in the field of quantitative finance; (ii) the improvement in the goodness-of-fit to the data by our model is significant especially for sessions where considerable fluctuation of the background rate is present; and (iii) our model is statistically consistent with the data. The branching ratio, which quantifies the level of the endogeneity of markets, estimated by our model is 0.41, suggesting the relative importance of exogenous factors in the market dynamics. We also demonstrate that it is critically important to appropriately model the time-dependent background rate for the branching ratio estimation.

Forecasting large aftershocks within one day after the main shock.

Forecasting the aftershock probability has been performed by the authorities to mitigate hazards in the disaster area after a main shock. However, despite the fact that most of large aftershocks occur within a day from the main shock, the operational forecasting has been very difficult during this time-period due to incomplete recording of early aftershocks. Here we propose a real-time method for efficiently forecasting the occurrence rates of potential aftershocks using systematically incomplete observations that are available in a few hours after the main shocks. We demonstrate the method's utility by retrospective early forecasting of the aftershock activity of the 2011 Tohoku-Oki Earthquake of M9.0 in Japan. Furthermore, we compare the results by the real-time data with the compiled preliminary data to examine robustness of the present method for the aftershocks of a recent inland earthquake in Japan.

Information transmission using non-poisson regular firing.

In many cortical areas, neural spike trains do not follow a Poisson process. In this study, we investigate a possible benefit of non-Poisson spiking for information transmission by studying the minimal rate fluctuation that can be detected by a Bayesian estimator. The idea is that an inhomogeneous Poisson process may make it difficult for downstream decoders to resolve subtle changes in rate fluctuation, but by using a more regular non-Poisson process, the nervous system can make rate fluctuations easier to detect. We evaluate the degree to which regular firing reduces the rate fluctuation detection threshold. We find that the threshold for detection is reduced in proportion to the coefficient of variation of interspike intervals.

Optimizing time histograms for non-Poissonian spike trains.

The time histogram is a fundamental tool for representing the inhomogeneous density of event occurrences such as neuronal firings. The shape of a histogram critically depends on the size of the bins that partition the time axis. In most neurophysiological studies, however, researchers have arbitrarily selected the bin size when analyzing fluctuations in neuronal activity. A rigorous method for selecting the appropriate bin size was recently derived so that the mean integrated squared error between the time histogram and the unknown underlying rate is minimized (Shimazaki & Shinomoto, 2007 ). This derivation assumes that spikes are independently drawn from a given rate. However, in practice, biological neurons express non-Poissonian features in their firing patterns, such that the spike occurrence depends on the preceding spikes, which inevitably deteriorate the optimization. In this letter, we revise the method for selecting the bin size by considering the possible non-Poissonian features. Improvement in the goodness of fit of the time histogram is assessed and confirmed by numerically simulated non-Poissonian spike trains derived from the given fluctuating rate. For some experimental data, the revised algorithm transforms the shape of the time histogram from the Poissonian optimization method.

Deciphering elapsed time and predicting action timing from neuronal population signals.

The proper timing of actions is necessary for the survival of animals, whether in hunting prey or escaping predators. Researchers in the field of neuroscience have begun to explore neuronal signals correlated to behavioral interval timing. Here, we attempt to decode the lapse of time from neuronal population signals recorded from the frontal cortex of monkeys performing a multiple-interval timing task. We designed a Bayesian algorithm that deciphers temporal information hidden in noisy signals dispersed within the activity of individual neurons recorded from monkeys trained to determine the passage of time before initiating an action. With this decoder, we succeeded in estimating the elapsed time with a precision of approximately 1 s throughout the relevant behavioral period from firing rates of 25 neurons in the pre-supplementary motor area. Further, an extended algorithm makes it possible to determine the total length of the time-interval required to wait in each trial. This enables observers to predict the moment at which the subject will take action from the neuronal activity in the brain. A separate population analysis reveals that the neuronal ensemble represents the lapse of time in a manner scaled relative to the scheduled interval, rather than representing it as the real physical time.

Optimal observation time window for forecasting the next earthquake.

We report that the accuracy of predicting the occurrence time of the next earthquake is significantly enhanced by observing the latest rate of earthquake occurrences. The observation period that minimizes the temporal uncertainty of the next occurrence is on the order of 10 hours. This result is independent of the threshold magnitude and is consistent across different geographic areas. This time scale is much shorter than the months or years that have previously been considered characteristic of seismic activities.

Can distributed delays perfectly stabilize dynamical networks?

Signal transmission delays tend to destabilize dynamical networks leading to oscillation, but their dispersion contributes oppositely toward stabilization. We analyze an integrodifferential equation that describes the collective dynamics of a neural network with distributed signal delays. With the Gamma distributed delays less dispersed than exponential distribution, the system exhibits reentrant phenomena, in which the stability is once lost but then recovered as the mean delay is increased. With delays dispersed more highly than exponential, the system never destabilizes.

Reverberating activity in a neural network with distributed signal transmission delays.

It is known that an identical delay in all transmission lines can destabilize the macroscopic stationarity of a neural network, causing oscillation. We analyze the collective dynamics of a network whose transmission delays are distributed in time. Here, a neuron is modeled as a discrete-time threshold element that responds in an all-or-nothing manner to a linear sum of signals that arrive after delays assigned to individual transmission lines. Even though transmission delays are distributed in time, a whole network exhibits a single collective oscillation with a period close to the average transmission delay. The collective oscillation cannot only be a simple alternation of the consecutive firing and resting, but also arbitrarily sequenced series of firing and resting, reverberating in a certain period of time. Moreover, the system dynamics can be made quasiperiodic or chaotic by changing the distribution of delays.