Directed graph representation of a Markov chain model to study global earthquake sequencing leads to a time series of state-to-state transition probabilities that includes the spatio-temporally linked recurrent events in the record-breaking sense. A state refers to a configuration comprised of zones with either the occurrence or non-occurrence of an earthquake in each zone in a pre-determined time interval. Since the time series is derived from non-linear and non-stationary earthquake sequencing, we use known analysis methods to glean new information. We apply decomposition procedures such as ensemble empirical mode decomposition (EEMD) to study the state-to-state fluctuations in each of the intrinsic mode functions. We subject the intrinsic mode functions, derived from the time series using the EEMD, to a detailed analysis to draw information content of the time series. Also, we investigate the influence of random noise on the data-driven state-to-state transition probabilities. We consider a second aspect of earthquake sequencing that is closely tied to its time-correlative behaviour. Here, we extend the Fano factor and Allan factor analysis to the time series of state-to-state transition frequencies of a Markov chain. Our results support not only the usefulness of the intrinsic mode functions in understanding the time series but also the presence of power-law behaviour exemplified by the Fano factor and the Allan factor.

Earthquake sequencing has been the subject of detailed research
(Nava et al., 2005; Ünal and Çelebioğlu, 2011; Ünal et
al., 2014; Telesca et al., 2001, 2009, 2011; Telesca and Lovallo, 2008;
Cavers and Vasudevan, 2013, 2015; Vasudevan and Cavers, 2012, 2013) both in
the regional and global sense in recent years. Nava et al. (2005) have
introduced the Markov chain model to study the earthquake sequencing in a
seismogenically active region where the region is partitioned into zones. The
functionality of the method is determined by the characteristics of the
state-to-state transitions where each state is described by the earthquake
occupancy of the zones. In particular, for a given number of zones,

One consequence of the approach taken by Cavers and Vasudevan (2015) and
Vasudevan and Cavers (2013) is that it results in a time series of
state-to-state transition frequencies of the modified Markov chain model,

Earthquake sequencing may be considered a non-linear and non-stationary process (Kanamori, 2003; Telesca et al., 2001, 2009, 2011; Telesca and Lovallo, 2008; Flores-Marquez and Valverde-Esparza, 2012). In earthquake sequencing, earthquakes are viewed as part of a point process, with earthquake events occurring at some random locations in time. This means that the earthquake sequencing is dictated by the set of event times, and can also be expressed by the set of time intervals between events. The time series of earthquakes for any time interval can be analysed in many ways (Bohnenstiehl et al., 2001; Telesca et al., 2001, 2009, 2011; Telesca and Lovallo, 2008).

We postulate here that the non-linear and non-stationary behaviour in the time series should also be present in the time series of the state-to-state transition frequencies derived from earthquake sequencing. Hence, we consider the approaches of Telesca et al. (2001, 2009, 2011) and Telesca and Lovallo (2008) to be appropriate for a study here.

Non-linear and non-stationary time series have been examined in recent years with a method known as empirical mode decomposition (EMD) and the intrinsic mode functions derived from this are useful in this regard (Huang et al., 1998). The present time series of state-to-state transition frequencies is suited for such a study.

In general, the time series has non-zero amplitudes for the state-to-state transition frequencies (Cavers and Vasudevan, 2015). In this particular case, there are instances where there are no earthquakes exceeding the magnitude of 5.6 in all zones for one or more time steps. This introduces “intermittency” in the time series.

However, because of the presence of intermittency in it, an ensemble approach to empirical mode decomposition, EEMD (Wu and Huang, 2004, 2009; Flandrin et al., 2004, 2005) is applied here. The intermittency problem is handled with the addition of random noise to the time series before carrying out the EEMD (Wu and Huang, 2009). We examine the criteria used for the selection of the added noise and the ensemble number for the EEMD.

Another aspect of the study here is to ask a question if the time series resulting from a directed graph representation of the Markov chain model of earthquake sequences exhibits power-law statistics similar to a description of fractal stochastic point processes (Telesca et al., 2001, 2009, 2011) to model the time-occurrence sequence of seismic events. Quantifying the earthquake sequencing in terms of its fractal properties was done by means of the Fano factor and the Allan factor (Allan, 1966; Barnes and Allan, 1966; Lowen and Teich, 1993, 1995; Thurner et al., 1997; Telesca et al., 2001, 2009, 2011; Flores-Marquez and Valverde-Esparza, 2012; Serinaldi and Kilsby, 2013). Since the fractal properties of the time series studied here has never been investigated, we calculate the Fano factor and the Allan factor for the purpose of quantitative analysis.

The remainder of the paper is divided into three sections. In the next section, we show how the time series of the state-to-state transition frequencies for a modified Markov chain model as described in Cavers and Vasudevan (2015) is generated. In the following section, we describe the EEMD procedure used and the analysis of the results that accrue from this procedure. We extend the approaches of Telesca et al. (2001, 2009, 2011) and Telesca and Lovallo (2008) to calculate the Fano factor and the Allan factor with a view to study the fractal properties of the time series. In the last section, we discuss the results of the analysis methods and draw certain inferences about the state-to-state transition frequencies.

A

A graph representation of earthquake sequencing with arcs (with
weights

Kagan et al. (2010) partitioned the shallow (

Tectonic zone identifier, tectonic zone and the number of
earthquakes considered for

Zone and state definition used in the construction of a directed graph of a Markov chain. “0” and “1” refer to the no occurrence or occurrence of an earthquake for a given zone. For five zones, there are 32 states.

For a Markov chain structure given earlier for the five zones, the
computation of transition frequencies and hence, transition probabilities,
depend on the chosen time interval,

Each sample in the time series shown in Fig. 2a represents a “zone-configuration” state (Table 2). By definition, a zone-configuration has no zone or some zones or all zones highlighted by an earthquake or more in the optimally chosen time interval. Going from one sample to the next does not only represent going from one state to the next but also shows the amplitude fluctuation between them. The adjacent states could represent the same zone-configuration or different zone-configurations. The time series deduced from using the present approach with the five-zones marks the state-to-state fluctuations arising out of the fluctuations of oscillations or earthquake occurrences in the five-zones. We present in the following two analysis methods to glean an insight into the characteristics of the time series.

For non-linear and non-stationary time series, the method of empirical mode decomposition (EMD) has been recently proposed as an adaptive time-frequency analysis method (Huang et al., 1998, 1999) to decompose the original data into a basis set of intrinsic mode functions. Since the process that leads to the state-to-state transition frequency sequence or time series is inherently non-linear and non-stationary, it is appropriate to apply the EMD to this data to understand the behaviour of the intrinsic mode functions. The time series (Fig. 2a) reveals the fluctuations in the state-to-state transition frequencies arising out of varying occupancy of the zones from one time interval to the next. A situation would easily arise when two or three successive state-to-state transitions do not have earthquake occurrences in any of the zones studied. This would translate into intermittency in the time series. Recent studies (Flandrin et al., 2004, 2005; Gledhill, 2003; Wu and Huang, 2004, 2009) support the idea of carrying out noise-added analyses with the EMD. The noise-added analyses involves multiple realizations of added noises to the time series in question, leading to the ensemble EMD (EEMD), as proposed by Wu and Huang (2004, 2009).

In the EEMD, the signal or the time series in question with the added Gaussian white noise, denoted as one trial, would populate the whole time-frequency space uniformly with the constituting component of different scales. Since the noise added in each trial is different, the ensemble mean of the noise cancels out and, hence, the signal resides in the intrinsic mode functions generated from the EEMD (Wu and Huang, 2009).

The time series of state-to-state transition frequencies of the modified
Markov chain model,

Ensemble empirical mode decomposition of the time series.

Hilbert–Huang amplitude spectrum of the intrinsic functions.

Representation of a point process

Wu and Huang (2009) recommended that the ensemble size should be kept large
and the amplitude of the added noise should not be small. We set the ensemble
number for the number of realizations in EEMD large such that the noise
series cancel each other in the final mean of the corresponding IMFs. For the
two parameters, we used an ensemble size of 1000 and added noise with an
amplitude of 0.2 times the standard deviation of the original data. We assume
that the IMFs resulting from the EEMD represent a substantial improvement
over the IMFs of the original EMD in that it utilizes the full advantage of
the statistical characteristics of white noise to perturb the signal in its
true solution neighbourhood, and to cancel itself after serving its purpose
(Wu and Huang, 2009). EEMD results are summarized in Fig. 3a–t with
intrinsic mode function followed by their state-to-state relative weight
matrix derived from the basis set of the intrinsic mode functions of the
time series in a fashion identical to the original time series. By summing
the weights of the recurrence arcs corresponding to occurrences from state

The decomposition of the original time series into intrinsic mode functions
and the trend is dyadic in nature, as shown in Fig. 3. This means that as we
go from the first intrinsic mode function to the second and so on, the
interval increases by a factor of 2 from

In general, the intrinsic mode functions are characterized by (1) a certain number of a pattern of rise and fall of the arc weights and (2) by a systematic decrease in the frequency of the number of such patterns as one goes intrinsic mode function 1 to the intrinsic mode function 9. Since the rise and fall of the arc weights covers the entire catalogue of data, the periodicity that we notice could be intrinsic to earthquake processes.

The Hilbert–Huang amplitude spectrum of the time series, shown in Fig. 4, reveals at least two important features: (1) the temporal fluctuations in amplitudes occur in packets, each packet containing a set of zone to zone interactions. The oscillatory behaviour of packets contains certain periodicity within the earthquake sequence. A periodic trend at low frequencies suggests the role of zone 4 (Trenches) and zone 0 (Intraplate). A higher power at 900 and 950 time interval indicates the importance of zone 4 with earthquakes of larger magnitude prompting a cascade of aftershocks in zone 4 and main shocks in zones that are in close proximity to zone 4. (2) The frequency-dependence of amplitude packets encapsulates the relative importance of the interaction among multiple zones over different time intervals. We interpret them to mean that certain state-to-state transitions involving zone 4 are important over a range of frequencies.

Earthquake occurrences have been modelled to be stochastic point processes
(Thurner et al., 1997; Telesca, 2005; Telesca et al., 2001, 2009 and 2011;
Flores-Marquez and Valverde-Esparza, 2012). One representation of the point
process is to examine the inter-event time intervals. The resulting
inter-event interval probability density function says something about the
behaviour of the times between events. We do not know anything about the
information contained in the relationships among these items. Since
successive events do not occur in constant time intervals, another
representation of a point process is given by dividing the time axis into
equally spaced contiguous counting windows of duration

The Fano factor is a measure of correlation over different timescales
(Thurner et al., 1997). It is defined as the ratio of the variance of the
number of events in a specified counting time

The Allan factor is a relation with the variability of successive counts
(Allan, 1996; Barnes and Allan, 1966). It is the ratio of the variance of
successive counts for a specified counting time

In this paper, we examine both the results of Telesca's approach to the
initial catalogue of the data used and of the new representation of the point
process with a Markov chain model. For the working model, we compute the
state-to-state transition frequencies as described by Nava et al. (2005) and
as applied to global seismicity (Vasudevan and Cavers, 2012; Cavers and
Vasudevan, 2013). Expressions similar to Eqs. (6) and (7) can be derived if
we know the optimal time interval for the Markov chain model. Since we know
the optimal time interval, we introduce a sequence of state-to-state
transition frequencies,

There are a few observations to be made. First,

Here, we seek to understand the correlative behaviour by looking at the two
statistical measures, FF

In our adaptation of the sum of edge weights for the state-to-state
transition frequencies as a new representation of a point-process embedded in
the modified Markov chain here, the arguments of Thurner et al. (1997),
Telesca (2005) and Telesca et al. (2001, 2009, 2011) would apply. This means
that the FF of the modified Markov chain sequence would follow a power law
with the power-law exponent,

Extending this to FF

Thurner et al. (1997) pointed out
that the sequence of counts, generated by recording the number of events in
successive counting time-windows of certain length, contained information
about the point process depicted by the set of event times. This idea was
further tested in understanding the dynamics of earthquake sequencing
(Telesca et al., 2009, 2011; Flores-Marquez and Valverde-Esparza, 2012), and
in particular, the fractal behaviour of the sequence of counts. We know that
this idea was initially restricted to the sequence of counts for varying
windows of interval times. However, for comparison purposes, we calculated
the Fano factor and the Alan factor for the initial catalogue of data using
Eq. (6) and (7). We include their graphs in Fig. 6a and b. Similar to
observations made by Telesca et al. (2009) with the earthquake data from the
Taiwan region, we find the presence of two distinctly different regions of
scaling behaviour. For small time intervals, we also observe the Poisson
behaviour. Since very poor statistics at time-scales larger than

In our description of the directed graph of the Markov chain model of any earthquake sequencing, regional or global, we stress the significance of the state-to-state transition probabilities for multiple zones that span the sequence of earthquakes over an optimal time window (Cavers and Vasudevan, 2013; Vasudevan and Cavers, 2013). In other words, the edges of the directed graph carry weights. We conjecture that these weights represent a new definition of the point process. Furthermore, a consideration of the earthquake recurrences within each zone and among zones, following the concept of recurrences in the record-breaking sense (Davidsen et al., 2008), leads to an empirically determined distance-dependent weights for the edges. Unlike extending the idea of the sequence of counts where every event occurrence augments the counting value by unity (Thurner et al., 1997; Telesca et al., 2009, 2011; Flores-Marquez and Valverde-Esparza, 2012), we consider the summing of the weights for each edge such that the sum represents a “pulse” for each state-to-state transition. We analyse the resulting time series from the point of view of its Fano factor and Allan factor. There is evidence for fractality of the multi-state modified Markov chain to represent the earthquake sequencing, as is revealed by the power-law scaling behaviour present in the Fano and Allan factors with their respective exponents of 0.27 and 0.30 (Fig. 6c, d). However, it is important to note that the exponents of the power laws in both cases have a smaller value than those observed for the initial catalogue.

Cavers and Vasudevan (2013) interpreted the Markov chain of 32-states for five distinctly different zones to contain the basic combinatoric structure superimposed by the thumb-print of the undulatory structure of the recurrence weights. Since the earthquake sequencing is in general non-linear and non-stationary, we contend that the time series representing the above Markov chain is also non-linear and non-stationary, and is conducive to an ensemble empirical mode decomposition (EEMD) procedure to understand its intrinsic mode functions (IMFs). The ensemble empirical model decomposition of the time series leads to nine intrinsic mode functions and a trend. Each one of the IMFs reveals the amplitude fluctuation of the state-to-state transitions. While there is a commonality in the relative dominance of the subduction-style earthquakes, represented by the top right corner grid of the relative weight matrices (Fig. 3), the presence or absence of certain state-to-state transitions in certain IMFs reveals the importance of integral multiples of the optimal time interval.

A simple observation of the first six or seven IMFs stresses the importance of multiple-zone approach to global seismicity problem in that the earthquake sequencing for the time period we considered has similar oscillatory behaviour of the state-to-state transition probabilities from the point of view of the amplitude scaling and the oscillating period. The growth and decay of oscillations in easily identifiable packets in each IMF following certain periodicity is an intrinsic signature of the role of multiple zones in earthquake sequencing.

The authors would like to express deep gratitude to the department of mathematics and statistics for support and computing time. M. S. Cavers acknowledges the Natural Sciences and Engineering Research Council of Canada for a post-doctoral fellowship during the period of 2010 to 2012 when this research was first initiated. The authors express sincere thanks to Y. Y. Kagan for making the global seismicity data available on the net. They thank Reik Donner of Potsdam Institute for Climate Research, Potsdam and an anonymous referee for constructive criticism of the manuscript and helpful suggestions to improve the original version of the manuscript. Edited by: I. Zaliapin Reviewed by: R. V. Donner and one anonymous referee