Covariation of a time series and a point process

1982 ◽  
Vol 19 (03) ◽  
pp. 609-618 ◽  
Author(s):  
J. S. Willie
Keyword(s):  
Stochastic Process ◽  
Time Series ◽  
Point Process ◽  
Statistical Properties ◽  
Ordinary Time

Let {X(t), N(t)}, – ∞< t <∞, be a stationary bivariate stochastic process where X(t) is an ordinary time series and N(t) is an orderly point process counting the number of points in (0, t]. Suppose values of {X(t), N(t)} are available for 0< t ≦T and let σ1, σ2, ···, σ N (T) denote the jump points of N(t) in (0, T]. For |v| < T, define m T 12(υ)=∑ X (σj+υ)/T and μ T 12(υ)=∑ X (σ j +υ)/∑1 where all summations are over indices j such that 0<σ j , σ j +υ≦T for some σj. The functions M T 12(υ) and μ T 12(υ) are often useful in analyzing the covariation of the time series and point process. In this paper, we shall develop some statistical properties of the functions M T 12(υ) and μ T 12(υ) and discuss some specific situations where it is useful to consider these functions.

Covariation of a time series and a point process

1982 ◽  
Vol 19 (3) ◽  
pp. 609-618 ◽  
Author(s):  
J. S. Willie
Keyword(s):  
Stochastic Process ◽  
Time Series ◽  
Point Process ◽  
Statistical Properties ◽  
Ordinary Time

Let {X(t), N(t)}, – ∞< t <∞, be a stationary bivariate stochastic process where X(t) is an ordinary time series and N(t) is an orderly point process counting the number of points in (0, t]. Suppose values of {X(t), N(t)} are available for 0< t ≦T and let σ1, σ2, ···, σ N(T) denote the jump points of N(t) in (0, T]. For |v| < T, define mT12(υ)=∑ X (σj+υ)/T and μT12(υ)=∑ X (σj+υ)/∑1 where all summations are over indices j such that 0<σj, σj+υ≦T for some σj. The functions MT12(υ) and μT12(υ) are often useful in analyzing the covariation of the time series and point process. In this paper, we shall develop some statistical properties of the functions MT12(υ) and μT12(υ) and discuss some specific situations where it is useful to consider these functions.

Measuring the association of a time series and a point process

1982 ◽  
Vol 19 (3) ◽  
pp. 597-608 ◽  
Author(s):  
J. S. Willie
Keyword(s):  
Time Series ◽  
Point Process ◽  
Intensity Function ◽  
Asymptotic Distributions ◽  
The Other ◽  
Stationary Case ◽  
Conditional Intensity ◽  
A Value ◽  
Conditional Intensity Function ◽  
Ordinary Time

We consider a bivariate stochastic process where one component is an ordinary time series and the other is a point process. In the stationary case, a useful measure of the association of the time series and the point process is provided by a conditional intensity function, ṙ11(x;u), which gives the intensity with which events occur near time t given that the time series takes on a value x at time t + u. In this paper we consider the estimation of the function ṙ11(x;u) and certain related functions that are also useful in partially characterizing the degree of interdependence of the time series and the point process. Histogram and smoothed histogram-type estimates are proposed and asymptotic distributions of these estimates are derived. We also discuss an application of the estimation theory to the analysis of some data from a neurophysiological study.

Measuring the association of a time series and a point process

1982 ◽  
Vol 19 (03) ◽  
pp. 597-608
Author(s):  
J. S. Willie
Keyword(s):  
Time Series ◽  
Point Process ◽  
Intensity Function ◽  
Asymptotic Distributions ◽  
The Other ◽  
Stationary Case ◽  
Conditional Intensity ◽  
A Value ◽  
Conditional Intensity Function ◽  
Ordinary Time

We consider a bivariate stochastic process where one component is an ordinary time series and the other is a point process. In the stationary case, a useful measure of the association of the time series and the point process is provided by a conditional intensity function, ṙ 11(x;u), which gives the intensity with which events occur near time t given that the time series takes on a value x at time t + u. In this paper we consider the estimation of the function ṙ 11(x;u) and certain related functions that are also useful in partially characterizing the degree of interdependence of the time series and the point process. Histogram and smoothed histogram-type estimates are proposed and asymptotic distributions of these estimates are derived. We also discuss an application of the estimation theory to the analysis of some data from a neurophysiological study.

Correlation in the heart rate data

1998 ◽  
Vol 2 ◽  
pp. 141-148
Author(s):  
J. Ulbikas ◽  
A. Čenys ◽  
D. Žemaitytė ◽  
G. Varoneckas
Keyword(s):  
Nonlinear Dynamics ◽  
Experimental Data ◽  
Heart Rate ◽  
Time Series ◽  
Cardiovascular Function ◽  
Statistical Properties ◽  
Rate Data ◽  
Heart Rate Data ◽  
Dynamical Nature ◽  
Analysis Of Time Series

Variety of methods of nonlinear dynamics have been used for possibility of an analysis of time series in experimental physiology. Dynamical nature of experimental data was checked using specific methods. Statistical properties of the heart rate have been investigated. Correlation between of cardiovascular function and statistical properties of both, heart rate and stroke volume, have been analyzed. Possibility to use a data from correlations in heart rate for monitoring of cardiovascular function was discussed.

Earthquake clusters expected from bare statistics: How bursts and swarms emerge from exogenous and epidemic aftershock processes.

2021 ◽  
Author(s):  
Jordi Baro
Keyword(s):  
Heat Flow ◽  
Solid Earth ◽  
Point Process ◽  
Branching Process ◽  
Aftershock Sequence ◽  
Statistical Properties ◽  
Etas Model ◽  
Stress Changes ◽  
Earthquake Clusters ◽  
Statistical Laws

&lt;p&gt;Earthquake catalogs exhibit strong spatio-temporal correlations. As such, earthquakes are often classified into clusters of correlated activity. Clusters themselves are traditionally classified in two different kinds: (i) bursts, with a clear hierarchical structure between a single strong mainshock, preceded by a few foreshocks and followed by a power-law decaying aftershock sequence, and (ii) swarms, exhibiting a non-trivial activity rate that cannot be reduced to such a simple hierarchy between events.&amp;#160;&lt;/p&gt;&lt;p&gt;The Epidemic Aftershock Sequence (ETAS) model is a linear Hawkes point process able to reproduce earthquake clusters from empirical statistical laws [Ogata, 1998]. Although not always explicit, the ETAS model is often interpreted as the outcome of a background activity driven by external forces and a Galton-Watson branching process with one-to-one causal links between events [Saichev et al., 2005]. Declustering techniques based on field observations [Baiesi &amp; Paczuski, 2004] can be used to infer the most likely causal links between events in a cluster. Following this method, Zaliapin and Ben&amp;#8208;Zion (2013) determined the statistical properties of earthquake clusters characterizing bursts and swarms, finding a relationship between the predominant cluster-class and the heat flow in seismic regions.&lt;/p&gt;&lt;p&gt;Here, I show how the statistical properties of clusters are related to the fundamental statistics of the underlying seismogenic process, modeled in two point-process paradigms [Bar&amp;#243;, 2020].&lt;/p&gt;&lt;p&gt;The classification of clusters into bursts and swarms appears naturally in the standard ETAS model with homogeneous rates and are determined by the average branching ratio (nb) and the ratio between exponents &amp;#945; and b characterizing the production of aftershocks and the distribution of magnitudes, respectively. The scale-free ETAS model, equivalent to the BASS model [Turcotte, et al., 2007], and usual in cold active tectonic regions, is imposed by &amp;#945;=b and reproduces bursts. In contrast, by imposing &amp;#945;&lt;0.5b, we recover the properties of swarms, characteristic of regions with high heat flow.&amp;#160;&lt;/p&gt;&lt;p&gt;Alternatively, the same declustering methodology applied to a non-homogeneous Poisson process with a non-factorizable intensity, i.e. in absence of causal links, recovers swarms with &amp;#945;=0, i.e. a Poisson Galton-Watson process, with similar statistical properties to the ETAS model in the regime &amp;#945;&lt;0.5b.&lt;/p&gt;&lt;p&gt;Therefore, while bursts are likely to represent actual causal links between events, swarms can either denote causal links with low &amp;#945;/b ratio or variations of the background rate caused by exogenous processes introducing local and transient stress changes. Furthermore, the redundancy in the statistical laws can be used to test the hypotheses posed by the ETAS model as a memory&amp;#8208;less branching process.&amp;#160;&lt;/p&gt;&lt;p&gt;References:&lt;/p&gt;&lt;ul&gt;&lt;li&gt; &lt;p&gt;Baiesi, M., &amp; Paczuski, M. (2004). &lt;em&gt;Physical Review E&lt;/em&gt;, 69, 66,106. doi:10.1103/PhysRevE.69.066106.&lt;/p&gt; &lt;/li&gt; &lt;li&gt; &lt;p&gt;Bar&amp;#243;, J. (2020).&amp;#160; &lt;em&gt;Journal of Geophysical Research: Solid Earth,&lt;/em&gt; 125, e2019JB018530. doi:10.1029/2019JB018530.&lt;/p&gt; &lt;/li&gt; &lt;li&gt; &lt;p&gt;Ogata, Y. (1998) &lt;em&gt;Annals of the Institute of Statistical Mathematics,&lt;/em&gt; 50(2), 379&amp;#8211;402. doi:10.1023/A:1003403601725.&lt;/p&gt; &lt;/li&gt; &lt;li&gt; &lt;p&gt;Saichev, A., Helmstetter, A. &amp; Sornette, D. (2005) &lt;em&gt;Pure appl. geophys.&lt;/em&gt; 162, 1113&amp;#8211;1134. doi:10.1007/s00024-004-2663-6.&lt;/p&gt; &lt;/li&gt; &lt;li&gt; &lt;p&gt;Turcotte, D. L., Holliday, J. R., and Rundle, J. B. (2007), &lt;em&gt;Geophys. Res. Lett.&lt;/em&gt;, 34, L12303, doi:10.1029/2007GL029696.&lt;/p&gt; &lt;/li&gt; &lt;li&gt; &lt;p&gt;Zaliapin, I., and Ben&amp;#8208;Zion, Y. (2013), &lt;em&gt;J. Geophys. Res. Solid Earth&lt;/em&gt;, 118, 2865&amp;#8211; 2877, doi:10.1002/jgrb.50178.&lt;/p&gt; &lt;/li&gt; &lt;/ul&gt;

scholarly journals Time series analysis of trial-to-trial variability of MEG power spectrum during rest state, unattented listening and frequency-modulated tones classification

2021 ◽  
Author(s):  
Lech Kipiński ◽  
Wojciech Kordecki
Keyword(s):  
Stochastic Process ◽  
Time Series ◽  
Power Spectrum ◽  
Spectral Density ◽  
Cognitive Task ◽  
Clinical Neuroscience ◽  
Rest State ◽  
Short Time ◽  
Low Complex ◽  
Dominant Frequencies

AbstractThe nonstationarity of EEG/MEG signals is important for understanding the functioning of human brain. From the previous research we know that even very short, i.e. 250—500ms MEG signals are variance-nonstationary. The covariance of stochastic process is mathematically associated with its spectral density, therefore we investigate how the spectrum of such nonstationary signals varies in time.We analyze the data from 148-channel MEG, that represent rest state, unattented listening and frequency-modulated tones classification. We transform short-time MEG signals to the frequency domain using the FFT algorithm and for the dominant frequencies 8—12 Hz we prepare the time series representing their trial-to-trial variability. Then, we test them for level- and trend-stationarity, unit root, heteroscedasticity and gaussianity and based on their properties we propose the ARMA-modelling for their description.The analyzed time series have the weakly stationary properties independently of the functional state of brain and localization. Only their small percentage, mostly related to the cognitive task, still presents nonstationarity. The obtained mathematical models show that the spectral density of analyzed signals depends on only 2—3 previous trials.The presented method has limitations related to FFT resolution and univariate models, but it is not computationally complicated and allows to obtain a low-complex stochastic models of the EEG/MEG spectrum variability.Although the physiological short-time MEG signals are in principle nonstationary in time domain, its power spectrum at the dominant frequencies varies as weakly stationary stochastic process. Described technique has the possible applications in prediction of the EEG/MEG spectral properties in theoretical and clinical neuroscience.

Sign in / Sign up

Export Citation Format

Share Document