Geoelectric time series (TS) have long been studied for their
potential for probabilistic earthquake forecasting, and a recent model
(GEMSTIP) directly used the skewness and kurtosis of geoelectric TS to
provide times of increased probability (TIPs) for earthquakes for several
months in the future. We followed up on this work by applying the hidden Markov
model (HMM) to the correlation, variance, skewness, and kurtosis TSs to
identify two hidden states (HSs) with different distributions of these
statistical indexes. More importantly, we tested whether these HSs could
separate time periods into times of higher/lower earthquake probabilities.
Using 0.5 Hz geoelectric TS data from 20 stations across Taiwan over 7 years, we first computed the statistical index TSs and then applied the
Baum–Welch algorithm with multiple random initializations to obtain a
well-converged HMM and its HS TS for each station. We then divided the map
of Taiwan into a 16-by-16 grid map and quantified the forecasting skill,
i.e., how well the HS TS could separate times of higher/lower earthquake
probabilities in each cell in terms of a

Earthquakes (EQs) are one of the most destructive natural hazards that can befall us, with the potential to take many human lives and cause serious damage to economies and environments. It is imperative for us to work towards better forecasting/prediction capabilities against EQs, to inform pre-EQ evacuation and post-EQ relief, as well as expediting critical reinforcement works for selected buildings and infrastructures. To achieve this goal, the scientific community has done much work discovering precursors and models that are useful for the forecasting/prediction of EQs.

First, let us clarify that in the seismological community, the terms
“prediction” and “forecast” are often used interchangeably
(Kagan, 1997; Ismail-Zadeh, 2013). When they are distinguished, the
term prediction emphasizes the issuing of an

If we categorize EQ forecasting methods according to their timescales, we can organize them into three categories: long-term (decades ahead), intermediate-term (a few years ahead), and short-term (days or a few months ahead) (Peresan et al., 2005; Kanamori, 2003). EQ forecasting at different timescales serves different purposes. For a region of interest, long-term EQ forecasting aims to estimate the probabilities of large EQs in the next decades or more. In most past studies, the primary input data were from the historical EQ catalog, which allowed statistical modeling of the occurrence times of large- and medium-sized EQs (Kagan and Jackson, 1994; Sykes, 1996; Papazachos et al., 1987; Papadimitriou, 1993; Papazachos et al., 1997), assuming that EQs' occurrences in the same spatial area follow a Poisson process of a relatively constant rate. One such example is the probabilistic seismic hazard assessment (PSHA) first established by Cornell in 1968 (Cornell, 1968). This became a popular method for long-term seismic hazard assessment implemented in many countries (Tavakoli and Ghafory-Ashtiany, 1999; Petersen, 1996; Meletti et al., 2008; Vilanova and Fonseca, 2007; Nath and Thingbaijam, 2012; Wang et al., 2016). In this method, we take into account both historical EQ catalog information and ground motion characteristics for the modeling of energy attenuation over spatial distances, thus providing a map of seismic hazard rates that varies across location for the next 50 years. Long-term EQ forecasting such as PSHA can be valuable for location-specific seismic risk evaluation, thereby providing guidelines or criteria for local construction projects. For example, a building that is expected to last 100 years must be able to withstand 10 large EQs of the magnitude that occurs once every 10 years locally. What long-term EQ forecasting cannot do is tell people how to do things differently at any time.

For intermediate-term EQ forecasting, the aim is to detect deviations of EQ
rates from their long-term values to assess increased probabilities of EQs
within the next 1 to 10 years. For example, if a region usually has a
magnitude 6 EQ every 10 years and 15 years have passed without one, the
region would be in a state of increased probability. A famous example of
intermediate-term EQ forecasting is the M8 algorithm (Kossobokov et
al., 2002; Peresan et al., 2005; Keilis-Borok, 1996), developed by Healy
et al. (1992). The M8 algorithm used the EQ catalog as input and returned
as output the time of increased probability (TIP) for EQs of magnitude 7.5
and above for the next 1 year. Another example is the CN algorithm
(Peresan et al., 2005; Keilis-Borok, 1996) developed by Keilis-Borok
and Rotwain (1990), which also took the EQ catalog as input to produce as
output the TIP for strong EQs (defined specifically for different regions)
within the next half year to a few years. In the literature, we also found the
self-organizing spinodal (SOS) model (Chen, 2003; Rundle et al., 2000),
which used the increased activity of medium-sized EQs as precursors to large
EQs that could occur within the next several years or decades. Finally, one
of the more successful methods at this timescale is pattern informatics
(Nanjo et al., 2006), which was demonstrated to be effective at
predicting

Short-term EQ forecasting uses a variety of methods to forecast the time, place, and magnitude of a specific large EQ. Here we commonly find methods using the EQ catalog as input data and apply machine learning approaches (Asim et al., 2017; Reyes et al., 2013), as well as hidden Markov model (HMM) approaches (Yip et al., 2018; Chambers et al., 2012). For example, in Chambers et al. (2012) an HMM was trained to track the waiting time between EQs with magnitudes above 4 in southern California and western Nevada (Yip et al., 2018), giving EQ forecasts for up to 10 d in the future. Apart from using EQ catalog data, there are an increasing variety of methods using other data inputs, such as the widely used seismic electric signals (SESs) (Uyeda et al., 2000; Varotsos et al., 2002, 2013, 2017; Varotsos and Lazaridou, 1991; Varotsos et al., 1993), to look for EQ precursors in the form of abnormal changes to the geoelectric potential. In addition to looking for specific SES-type precursors, we also found papers using methods such as artificial neural networks (ANNs) (Moustra et al., 2011), Fisher information (Telesca et al., 2005a, 2009;), and multi-fractal analysis (Telesca et al., 2005b) directly on geoelectric time series (TS) data to make short-term EQ forecasting. Other data that can be used include the combination of geoelectric and magnetic data (Kamiyama et al., 2016; Sarlis, 2018), GPS crustal movements (Kamiyama et al., 2016; Wang and Bebbington, 2013), electromagnetics of the atmosphere (Hayakawa and Hobara, 2010), and lithosphere dynamics (Shebalin et al., 2006). Short-term EQ forecasting can guide emergency responses such as evacuations and preemptive relief efforts, although it is usually not reliable enough based on our current level of understanding.

Among all these precursors, our recent research interest has been in the
potential use of geoelectric TSs for EQ forecasting (Chen and Chen, 2016;
Chen et al., 2020; Jiang et al., 2020; Telesca et al., 2014; Chen et al.,
2017). In 2016 and 2017, Chen and his colleagues (Chen and Chen, 2016;
Chen et al., 2017) analyzed the data of 20 geoelectric stations in Taiwan
(Fig. 1) and studied the association between skewness and kurtosis of the
geoelectric data and

Map of the spatial distributions of seismicity and geoelectric
stations (green triangles) in Taiwan. In this figure, past EQs with

Inspired by these findings, in this paper we wanted to take a closer look at
the relationship between the EQ times and statistical indexes of geoelectric
TSs, namely correlation (

Heatmaps of normalized probability density functions of

To overcome this problem, which is created by superimposing the index TSs of
different lengths between EQs, we decided to discover such regimes directly
from the geoelectric TSs by using HMMs. The HMM is well known for being
data-driven, enabling us to search for and use more general statistical features
beyond limited templates that we currently know (Beyreuther and
Wassermann, 2008). Additionally, its explicit incorporation of the time
dimension into the model is a distinct advantage for providing holistic and
time-sensitive representations, especially in the application of EQ
forecasting (Beyreuther and Wassermann, 2008). In our HMM, we defined two
hidden states (HSs) as the high-level representations of geoelectricity,
featuring unique distributions of

The goal of this investigation is to decide whether the hidden Markov modeling of geoelectric TSs could provide features (i.e., HS TSs) of true forecasting skill for intermediate-term EQ forecasting. Therefore, we are more concerned with statistical significance than with evaluating the exact forecasting accuracy or with the forecasting of specific EQs. In this regard, we also note that the same HMM approach described in this paper can be applied to many other geophysical high-frequency time series data, such as geomagnetic or GPS ground movement data, even though we only used geoelectric data as the input of the HMM, to show that the underlying seismic dynamics is indeed clearly separable into distinct regimes of higher versus lower seismic activities (as supported by Yip et al., 2018; Chambers et al., 2012).

For the sake of our readers, we organize our “Data and methods” in Sect. 2,
“Results and discussions” in Sect. 3, and Conclusions in Sect. 4. In Sect. 2,
we provide information on the EQ catalog; the geoelectric TSs; and how we
pre-processed the latter and subsequently computed the index TSs of

The 1 Hz geoelectric TSs data used in this paper were provided by the 20
monitoring stations located across Taiwan (see Fig. 1), which are
collectively named the Geoelectric Monitoring System (GEMS). The spacing between
stations is generally 50 km. The geoelectric data here are the
self-potential data, which are the natural electric potential differences in
the earth, measured by dipoles placed 1–4 km apart within each station.
Each station can output two sets of high-frequency geoelectric TSs,
measuring separately the NS direction and the EW direction. Depending on the
spatial constraints of some stations, the azimuths of the dipoles might
deviate from the exact NS or EW directions by 10–40

The HMMs that we will show in Sect. 3 partitioned the 20 geoelectric TSs
into two HSs, distinguished by the local statistics of their geoelectric
fields. We believe these HSs can also exhibit different seismicity within
their time durations. To check this, we used EQ catalog data compiled by the
Central Weather Bureau (CWB), in charge of monitoring EQs in the region of
Taiwan (Shin et al., 2013). The CWB seismic network is highly dense and
provides an abundant set of waveform data. Due to the considerable EQs recorded,
the seismotectonics of Taiwan is well depicted, showing the complicated
subduction between the Philippine Sea and Eurasian plates
(Kuo-Chen et al., 2012; Yi-Ben, 1986). Despite
the dense seismic network, the EQ catalog was shown to be incomplete at
small magnitudes due to the detection threshold of seismic instruments and
the coverage of networks (Fischer and Bachura, 2014; Nanjo et al., 2010;
Rydelek and Sacks, 1989). In Taiwan, the completeness magnitude (

In the latest update of the GEMSTIP model, Chen et al. (2021) found
that by applying a specific bandpass filter to the geoelectric TS, the model
became better at anticipating EQs using the skewness and kurtosis TSs. The
filter they used is the third-order Butterworth bandpass filter with lower and
higher cutoff frequencies of

Similarly to the GEMSTIP model, our hidden Markov modeling also searched for EQ-related information in skewness and kurtosis TSs computed from the geoelectric TS; we conveniently utilized the insight from Chen et al. (2021) and applied the same Butterworth filter to our geoelectric TS data before computing the index TSs. This filter was applied using the scipy.signal (v1.4.1) package in Python (v3.6.5), with instructions from the SciPy Cookbook (2012), which also demonstrated a clear working example of the Butterworth bandpass filter that readers can refer to.

For each station, there are two geoelectric TSs (NS and EW) of frequency
0.5 Hz. Each geoelectric TS will produce four statistical index TSs (

Next, we present the definitions for each index. Within each time window,
let us write the geoelectric field as

The variance

The skewness

The kurtosis

A Markov model is a stochastic model that can be used to describe a system
whose future state

In common real-world applications of the HMM, the question is to estimate the
probability distributions of the HS TS given the observation TS and the
model parameter, namely

HMMs are traditionally applied in fields such as speech recognition (Palaz et al., 2019; Novoa et al., 2018; Chavan and Sable, 2013; Abdel-Hamid and Jiang, 2013), bioinformatics, and anomaly detection (Qiao et al., 2002; Joshi and Phoha, 2005; Cho and Park, 2003). It has also been used for short-term EQ forecasting, using observations from EQ catalogs (Yip et al., 2018; Chambers et al., 2012; Ebel et al., 2007), as well as GPS measurements of ground deformations (Wang and Bebbington, 2013). To the best of our knowledge, there is no past HMM study on geoelectric TSs for EQ forecasting. In this paper, we argue that the HMM is an objective tool because the HSs were estimated only from the geoelectric TSs and thereafter validated against the EQ catalog. We believe this statistical procedure limits the bias that we could introduce into our prediction model when we optimized the model. This will be even clearer by the end of Sect. 2.5 where we summarize the entire procedure.

In the context of this study, we assume for simplicity two seismicity states
of the earth crust beneath each station. These are our HSs

One way to do so would be to model each component of

If we do this for the TSs of individual components, such as the TS of

The output of BWA: the emission probability or the probability
mass functions, as well as their posterior HS probability TSs, for

The BWA has no problem dealing with high-dimensional problems, provided the
inputs are discrete. However, this method would work well only if the
overall number of possible observations is small. If we use 50 bins for each
of the eight indexes, there would be

We do not know a priori what the elements of this very small subset are. They may
occur as isolated points in the search space, or they may occur in groups of
closely spaced points. In the continuous feature space, each of these groups
of observations represents a cluster of similar feature vectors. To
determine the number of such clusters and where they occur in the
8-dimensional continuous feature space, we mapped similar feature vectors to
the same label using the

The indexes

In this section, we describe how we implemented the BWA to obtain one HS TS for each station. We start by describing how we initialized and iterated the BWA, as well as how we dealt with local optima in the BWA results by using multiple initializations.

The first step of the BWA is to initialize the HMM parameters

As the iteration goes, the BWA improves the likelihood of observing the
input observation TS

We cannot simply do the above BWA estimation once to obtain

For each initial condition, the BWA randomly assigns one HS to be

The step-by-step data visualization for CHCH.

We summarize the procedures used to obtain

Flowchart summarizing the procedures of obtaining the optimal
posterior probability TS

Up to this point, we did not incorporate any EQ catalog information into

After the hidden Markov modeling, we then checked locally whether

For each GEMS station we started from

A sample EQ grid map with 16 by 16 divisions, in which each cell
measures

For each grid cell

In this section, we present the results obtained for all 20 stations, as well as additional treatments that we felt were necessary to investigate whether the HS TSs have significant forecasting power for EQs.

Once we obtained the

The step-by-step data visualization for CHCH, showing

As can be seen in Fig. 7c, for different regions the HS with higher EQ
activities can be either

The grid maps of EQ frequency ratio

All in all, the findings in this section are important, but we cannot
directly decide whether

We defined the discrimination power

The grid map of discrimination power

In some cells, we find

Since we had the optimal HMMs for the 20 stations, we can test cellular
statistical significance levels indicating that their HSs can indeed separate time
periods of higher/lower EQ probabilities, using

The empirical HS TS and 10 simulated HS TSs, for the stations

In Fig. 11, we show the grid maps of

In the proximity of the LIOQ station located within

The grid map of discrimination reliability

Based on our discoveries regarding the HS–EQ correlations so far, we claim that the
HS TSs can provide usable EQ forecasts for real-world applications. We
understand that for all EQ forecasting, whether short-, medium-, or
long-term, we must specify (a) a time window, (b) a space window, and (c) the magnitudes of EQs expected. We shall next explain how the HS TSs can be
useful for EQ forecasting from these three aspects. (a) Let us consider an
HMM that started out in the passive state, where EQs of all magnitudes are
less frequent compared with the active state. In most stations that we
tested, we noticed that once an active state has persisted for a few weeks,
it is unlikely to switch back to the passive state until a few months have
elapsed. This minimum lifetime found in historical data can be used as a
prediction time window. Based on this timescale, we can say that our HMM
can be useful for short- to medium-term EQ forecasting, depending on
the station of interest. (b) Next, let us consider the grid cells covering
Taiwan. For a given grid cell, it may be satisfactory (

For grid cells with high

Due to the nature of our HSs, we cannot use them to forecast specific EQs or issue evacuation alarms. What the HSs can do, however, is to provide information with forecasting skill to decision makers, in regions where the HS switched from the passive state to the active state convincingly (i.e., the observed active state is persistent and not a temporary fluctuation), to take courses of action that can lower the potential damage with minimal costs. For example, in the passive state, the building inspection authority can prioritize inspection and the issuing of safety permits to new projects over re-inspections of old buildings. With the arrival of an active state that might last a few months to a few years, local authorities would have the incentive to clear up pending re-inspection works so that fewer old buildings are exposed to potential EQ damage. Other than the re-inspection of old buildings, local authorities could also increase the frequency of safety education and drills to vulnerable groups such as students and construction workers to reduce potential injuries or fatalities due to panic or lack of understanding. Additionally, disaster relief services may use the HS's information to re-deploy the stockpile of relief materials, such as food, clothing, tents, and first-aid kits, whenever necessary. In doing so, the stockpile of relief materials can be brought closer to high-risk regions within a convincing active state to be distributed to victims more cost-effectively after a major EQ.

From Fig. 11 alone, we have demonstrated the HS TSs are able to separate
time periods of low/high EQ probabilities for regions (cells in the grid
map) with high

In order to answer this question, we need to define a performance metric
that can quantify the performance of each station with a single value,
instead of a grid map of

We carried out this hypothesis test station by station by first computing
the

In Fig. 12, we show the results of our global-level significance tests, for
a choice of

Last but not least, the histograms for each station in Fig. 12 are created
with individually optimized hyperparameters, namely

Histograms (blue) of 400 simulated

Typically, a forecasting model's performance may be sensitive to our choice
of hyperparameters. If possible, we would like to choose hyperparameters
that make the model the most predictive. If there were too many
hyperparameters, this optimization would be challenging in the
high-dimensional search space. Fortunately, there are only two
hyperparameters needed to obtain the HS TS:

For each choice of station and hyperparameter, we followed the same
procedure of computing 1

Heatmaps of

Heatmaps of

To wrap up this section, let us describe how to select the optimal
hyperparameter for each station. We did this in two steps: first, we
selected the hyperparameters with the highest GCL values (1 for many
stations); next, in case of ties, we chose the hyperparameter with the
highest

EQ forecasting is an important research topic because of the potential devastation EQs can cause. As has been pointed out by many past studies, there is a correlation between features within geoelectric TSs and large individual EQs. In those studies, different features of geoelectric TSs were explored for their use of EQ forecasting, among which the GEMSTIP model was the first one to directly use statistic index TSs of geoelectric TSs to produce TIPs for EQ forecasting. Inspired by this, we took a second look at the relationship between these statistic indexes and the timing of EQs and found that there is an abrupt shift in the indexes' distribution along the TTF axis. This suggests that there are at least two distinct geoelectric regimes, which can be modeled and identified using a two-state HMM. This finding is further backed by the knowledge that there can be drastic tectonic configuration changes before and after a large EQ, one important aspect of which being the telluric changes identified in the region around the epicenter of the EQ (Sornette and Sornette, 1990; Tong-En et al., 1999; Orihara et al., 2012; Kinoshita et al., 1989; Nomikos et al., 1997). Therefore, should there be two higher-level tectonic regimes featuring higher/lower EQ frequencies, we would expect to also find two matching geoelectric regimes with contrasting statistical properties, which can be of good utility for EQ forecasting.

Specifically, we modeled the earth crust system as having two HSs
identifiable with distinctive geoelectric features encoded by eight index TSs
from each station. To obtain the HMM for each station, we needed to run the
BWA, which is most convenient to use with a discrete observation TS input.
Therefore, we used

Finally, we showed how we optimized the GCL values through a grid search
in the 2-dimensional hyperparameter space and obtained the optimal
combination of

To the best of our knowledge, while there have been previous applications of
HMMs for earthquake forecasting, this paper is the first to demonstrate the
ability to do so with statistical confidence. As discussed in greater detail
in Sect. 3.3, in real-world scenarios, the HS TSs can be useful for
intermediate-term EQ forecasting either directly (for high-

At this point, we would like to address the issue of out-of-sample testing
(or cross-validation) to support the validity of our model. There are two
ways to do this: (1) split a long time series into a training data set to
calibrate the model and a testing data set to validate the model and (2) use whatever time series data are available to calibrate the model before
collecting more data to validate the model. If the model is statistically
stationary (its parameters do not change with time), both approaches are
acceptable. However, many would agree that an out-of-sample test with
freshly collected data (approach 2) is more impressive, especially if it
is performed in real time. We would certainly like to try this and are writing a
grant application to fund such a validation study. For this paper, however,
we were not even able to use approach 1 because our geoelectric time
series are not long enough. This is especially so if we require that (a) the
validation data are always temporally

The Python codes that we used to produce the results in this paper can be
downloaded at GitHub:

The data set of the index TSs for 20 stations computed using various time
windows

The supplement related to this article is available online at:

SAC and CCC came up with the research motivation; HJC and HW processed the data; SAC and HW analyzed the results; SAC, HW, and HJC drafted the manuscript; all co-authors read the manuscript and suggested revisions.

The contact author has declared that neither they nor their co-authors have any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Chien-Chih Chen has been supported by the Ministry of Science and Technology (Taiwan, grant no. MOST 110-2634-F-008-008) and the Department of Earth Sciences and the Earthquake-Disaster & Risk Evaluation and Management Center (E-DREaM) at the National Central University (Taiwan).

This paper was edited by Filippos Vallianatos and reviewed by two anonymous referees.