the Creative Commons Attribution 4.0 License.

the Creative Commons Attribution 4.0 License.

# Hidden-state modeling of a cross-section of geoelectric time series data can provide reliable intermediate-term probabilistic earthquake forecasting in Taiwan

### Haoyu Wen

### Hong-Jia Chen

### Chien-Chih Chen

### Massimo Pica Ciamarra

### Siew Ann Cheong

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 *discrimination power* measure that we defined. Next, we
compare the discrimination power of empirical HS TSs against those of 400 simulated HS TSs and then
organized the statistical significance values from this cellular-level
hypothesis testing of the forecasting skill obtained into grid maps of
*discrimination reliability*. Having found such significance values to be high for many grid cells for
all stations, we proceeded with a statistical hypothesis test of the
forecasting skill at the global level to find high statistical significance
across large parts of the hyperparameter spaces of most stations. We
therefore concluded that geoelectric TSs indeed contain earthquake-related
information and the HMM approach is capable of extracting this
information for earthquake forecasting.

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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 *alarm* with high accuracy and
reliability indicating the time, location, and magnitude of the next large
EQ (Geller et al., 1997), whereas the term forecast is a statement
about the probability of EQs within the specified spatial–temporal window
(Ismail-Zadeh, 2013). Till this day, it is extremely difficult to
make accurate and specific EQ predictions (Geller et al., 1997).
However, the forecasting of EQs is a far more tractable task: a method that
performs better than random guesses (the null hypothesis) is recognized as
having *predictive power *or *predictive skill* (prediction and forecast used as synonyms here) (Kagan,
1997). In this paper, we will also use the two terms interchangeably.

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 *M*≥5 EQs in Japan between 2000 and 2009. Intermediate-term EQ
forecasting can, for example, help local authorities prioritize inspections
and reinforcements of old buildings over the construction of new ones.

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 *M*_{L}≥ 5 EQs, where *M*_{L} is the Richter
magnitude scale. Through statistical analyses, they found significant
correlations between geoelectric anomalies and these large EQs. They then
developed an EQ forecasting algorithm named GEMSTIP to extract TIPs for
future EQs. TIPs were identified through differences in the distributions of
skewness and kurtosis from those found during normal periods. Moreover,
Jiang et al. (2020) investigated the geoelectric signals before, during,
and after EQs by the shifting correlation method and found that the lateral
and vertical electrical resistivity variation and subsurface conductors
might amplify SESs, which agreed with the findings by Sarlis et
al. (1999) and Huang and Lin (2010).

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 (*C*), variance (*V*), skewness (*S*), and kurtosis
(*K*). During initial explorations, we computed the TSs of these indexes
(see Sect. 2.2 for computation details) on geoelectric TSs given by the 20
stations over the 7-year period of January 2012–December 2018 (see Sect. 2.1 for
data details). We then aggregated the distribution of the indexes' values
within different time-to-failure (TTF, i.e., time remaining to the next
EQ) intervals. In Fig. 2, we show the normalized frequency distributions of
*C*, *V*, *S*, and *K* computed from the KAOH station at different TTFs (using
0.9 d intervals) for *M*_{L}≥4 EQs within 2^{∘} longitude and latitude
of the KAOH station. In this figure, we see bands of darker-colored pixels
across the TTFs. Specifically, for *C*, *V*, and *S*, there are sudden
shifts in the average position of the bands, suggesting that there are two
regimes (short TTFs and long TTFs) where the geoelectric fields show
qualitatively different behaviors. For all statistical indexes, we find the
darkest pixels concentrated in the long-TTF regime, whereas in the short-TTF
regime, the pixels show a lower variability in their intensities. We suspect
that this second phenomenon is the result of fewer samples at longer TTFs.

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 *C*, *V*, *S*, and *K*. Here we chose to use
only two, instead of more, HSs because
two-state HMMs have already been
successfully applied to model regimes with different EQ frequencies using EQ
catalogs as the only inputs (Yip et al., 2018; Chambers et al., 2012).
Thereafter, for each monitoring station, we obtained the TS of posterior HS
probability, or HS TS, using the TSs of *C*, *V*, *S*, and *K* and the
Baum–Welch algorithm (BWA). We then partitioned the time periods under study
according to the HS TSs and investigated whether these HS TSs that are
obtained purely from geoelectric data can separate time periods of high
versus low EQ (*M*_{L}≥3) probabilities, with high statistical
confidence.

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 *C*,
*V*, *S*, and *K* from them. We then explain how an HMM and the Baum–Welch algorithm works, before applying them to our problem. We also explain why we
did not estimate individual HMMs from the index TSs of *C*, *V*, *S*, and *K*
but one HMM for each station from an observation TS aggregating *C*, *V*,
*S*, and *K* through *k*-means clustering. At the end of this section, we present
our procedures for quantifying how informative the HSs are against EQ
activities, by defining and analyzing EQ grid maps, EQ frequencies, and EQ
frequency ratios (*R*_{F}). In Sect. 3, we first used the *R*_{F} grid map
of 1 of the 20 stations to illustrate how we can compare a discrimination
power (*D*) grid map against 400 simulated grid maps of *D* to obtain the
discrimination reliability (*R*_{D}) grid map, which comprises cellular-level
statistical significances that the HSs are useful for EQ forecasting. We
then performed significance tests to verify that the HSs' forecasting power
is also significant at the global level, using a metric of the global
confidence level (GCL) that we defined. To end Sect. 3, we explored how
robust the GCL values are across the hyperparameter space and clarified
how we chose the optimal hyperparameters for each station. Finally, we
conclude in Sect. 4.

## 2.1 Data description

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^{∘}. For the
purpose of this study, we used the geoelectric TSs provided by GEMS with
the same time span as the EQ catalog data, which is from January 2012 to
December 2018. We downsampled the data to 0.5 Hz and used these in
subsequent analyses.

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 (*M*_{c}), defined as
the lowest magnitude above which all EQs are reliably detected, is
approximately between 2 and 3 (Chen et al., 2012; Mignan et al., 2011).
Chen et al. (2012) showed the temporal variation in *M*_{c}, while
Mignan et al. (2011) provided the spatial information of that. In this
study, for a conservative estimate, we took the completeness magnitude of
3 and analyzed EQs with *M*_{L}≥3, during the period from January 2012 to
December 2018 in the area of 119.5–122.5^{∘} E and
21.5–25.5^{∘} N, as shown in Fig. 1, in which the locations of
strong events with *M*_{L}≥6 are marked. Some of these events were
destructive. For instance, at 03:57 on 6 February 2016 (UTC+8), an
*M*_{L} 6.6 EQ occurred in the southern part of Taiwan (22.92^{∘} N, 120.54^{∘} E). This event struck at a depth of around 14.6 km
(Chen
et al., 2017; Lee et al., 2016; Pan et al., 2019). Such a comparatively
shallow depth caused more intensities on the surface and resulted in
widespread damage which included 117 deaths and over 500 wounded.

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 ${f}_{\mathrm{1}}={\mathrm{10}}^{-\mathrm{4.0}}$ and ${f}_{\mathrm{2}}={\mathrm{10}}^{-\mathrm{1.75}}$ Hz respectively. These lower and upper cutoff frequencies were determined by Chen et al. (2021) to give the optimal signal-to-noise ratio.

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.

## 2.2 Computation of index TSs of *C*, *V*, *S*, and *K*

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 (*C*, *V*,
*S*, *K*). For each station, we therefore obtained up to eight index TSs, four for
each direction (NS and EW). Starting from the 0.5 Hz geoelectric TS, we
computed one index point for every non-overlapping time window of length
*L*_{w} geoelectric TS data points. Later in Sect. 3.5, we will discuss in
detail how we chose the optimal *L*_{w} individually for each station in the
parameter space that we tested: {0.02, 0.03, 0.04, 0.05, 0.1,
0.2, 0.25} (d). As can be noticed from Fig. 12, 11 out of 20 stations'
optimal choice was *L*_{w}=0.02 or *L*_{w}=0.03 d, which we suppose
can be a good compromise between timely monitoring of state shifts and
updating at a comfortable frequency for the human decision makers. Potential
decisions that such an update frequency may enable include the forward
deployment of relief materials such as backup generators, portable water
treatment units, tents, medical supplies, and refresher training of emergency
response teams, as well as administrative prioritizing of re-certification
works for buildings and structures in regions where more EQs are expected
soon.

Next, we present the definitions for each index. Within each time window,
let us write the geoelectric field as ${\left\{{X}_{n}\right\}}_{n=\mathrm{1},\mathrm{\dots},{L}_{\mathrm{w}}}$. The correlation *C* that we used in
this paper is the lag-1 Pearson autocorrelation of ${\left\{{D}_{n}={X}_{n+\mathrm{1}}-{X}_{n}\right\}}_{n=\mathrm{1},\mathrm{\dots},{L}_{\mathrm{w}}-\mathrm{1}}$, which is the
*difference sequence* of ${\left\{{X}_{n}\right\}}_{n=\mathrm{1},\mathrm{\dots},{L}_{\mathrm{w}}}$. Mathematically,

where *E* is the expectation, *μ*_{D} is the mean of ${\left\{{D}_{n}\right\}}_{n=\mathrm{1},\mathrm{\dots},{L}_{\mathrm{w}}-\mathrm{1}}$, and *σ*_{D} is the standard
deviation of ${\left\{{D}_{n}\right\}}_{n=\mathrm{1},\mathrm{\dots},{L}_{\mathrm{w}}-\mathrm{1}}$. The
range of *C* is [$-\mathrm{1},\mathrm{1}$], and *C* measures how fast the TS relaxes back
to the equilibrium. If *C* is close to 1, *X* tends to increase or
decrease persistently; if *C* is around 0, *X* is equivalent to
random walks; and if *C* is close to −1, every increase in *X* would tend
to be followed by a similar decrease.

The variance *V* of ${\left\{{X}_{n}\right\}}_{n=\mathrm{1},\mathrm{\dots},{L}_{\mathrm{w}}}$ is
the sequence's second standard central moment. It is a positive number that
measures how drastically the values in the sequence are different from each
other, with higher values indicating higher difference. It is defined as

where *μ*_{X} is the mean of ${\left\{{X}_{n}\right\}}_{n=\mathrm{1},\mathrm{\dots},{L}_{\mathrm{w}}}$. Additionally, we observed astronomically extreme values in the *V*
TSs for most stations, which were caused by unknown technical errors, and we
therefore considered them outliers that have to be removed for consistent
data quality. We discuss how we removed them in detail in Supplement Sect. S1. From here onwards, the *V* TSs will always refer to
those after the outlier-removal process.

The skewness *S* of ${\left\{{X}_{n}\right\}}_{n=\mathrm{1},\mathrm{\dots},{L}_{\mathrm{w}}}$, or
the sequence's third standard central moment, is defined as

where *σ*_{X} is the standard deviation of ${\left\{{X}_{n}\right\}}_{n=\mathrm{1},\mathrm{\dots},{L}_{\mathrm{w}}}$. It is a real number measuring how
asymmetric the distribution of ${\left\{{X}_{n}\right\}}_{n=\mathrm{1},\mathrm{\dots},{L}_{\mathrm{w}}}$ is about the mean. For a perfectly symmetric distribution such as
the normal distribution, the skewness is 0. A positive skewness means the
distribution has a longer tail to the right, and a negative skewness means
the distribution has a longer tail to the left.

The kurtosis *K* of ${\left\{{X}_{n}\right\}}_{n=\mathrm{1},\mathrm{\dots},{L}_{\mathrm{w}}}$, or
the sequence's fourth standard central moment, is defined as

It is a real number measuring how frequently extreme values (values very far
from the mean) appear in the distribution. The higher the number, the more
frequently extreme values can be found. As a reference, the kurtosis of the
normal distribution is *K*=3. If *K*>3, we say that the distribution is
*leptokurtic*, meaning the distribution has fatter tails and more frequent extreme values
compared to the normal distribution. If *K*<3, the distribution is said to
be *platykurtic*, meaning the distribution has thinner tails and extreme values appear
less frequently compared to the normal distribution.

## 2.3 Estimation of the HMM using the Baum–Welch algorithm

A Markov model is a stochastic model that can be used to describe a system
whose future state *s*_{t+1} is drawn from a set of *L* states ${\left\{{S}_{l}\right\}}_{l=\mathrm{1},\mathrm{\dots},L}$ with probabilities ${p}_{j\leftarrow i}=P\left({s}_{t+\mathrm{1}}={S}_{j}\mathrm{|}{s}_{t}={S}_{i}\right)$
conditioned by its current state *s*_{t}. The probabilities *p*_{j←i} can be organized into a transition matrix **A**, where
$\mathbf{A}\left(i,j\right)={p}_{j\leftarrow i}$. The HMM is an extension of
the Markov model, with the additional property that the system state *s*_{t}
is not explicitly known, hence the word “hidden” in the name. Instead,
what can be observed from an HMM at any time *t* is an observable *o*_{t}
drawn from a size-*Q* observable set ${\left\{{O}_{q}\right\}}_{q=\mathrm{1},\mathrm{\dots},Q}$. Just as in a Markov model, the future
state *s*_{t+1} of an HMM is drawn from the set ${\left\{{S}_{l}\right\}}_{l=\mathrm{1},\mathrm{\dots},L}$ with probabilities *p*_{j←i}
(similarly conditioned by the current state *s*_{t}) taken from the
transition matrix **A**. At time *t*, the observable *o*_{t} is
emitted with a probability $P({o}_{t}={O}_{q}\mathrm{|}{s}_{t}={S}_{l})$ that depends on
which HS *s*_{t}=*S*_{l} the system is in. These probabilities can be
organized into an *L*×*Q* emission matrix **B**, where
$\mathbf{B}\left(l,k\right)=P({o}_{t}={O}_{q}\mathrm{|}{s}_{t}={S}_{l})$.
Additionally, we call the HS probability distributions at the initial time ${\mathit{\pi}}_{\mathrm{0}}=\left[P\left({S}_{\mathrm{1}}\right),P\left({S}_{\mathrm{2}}\right),\mathrm{\dots},P\left({S}_{L}\right)\phantom{\rule{0.125em}{0ex}}\right]$. With this, we have
fully specified the HMM: the sets of HSs ${\left\{{S}_{l}\right\}}_{l=\mathrm{1},\mathrm{\dots},L}$ and observations ${\left\{{O}_{q}\right\}}_{q=\mathrm{1},\mathrm{\dots},Q}$ as well as the model parameters that are
collectively called $\mathit{\lambda}=\left(\mathbf{A},\mathbf{B},{\mathit{\pi}}_{\mathrm{0}}\right)$.

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 $P({s}_{t}={S}_{l}\mathrm{|}{\left\{{o}_{t}\right\}}_{t=\mathrm{1},\mathrm{\dots},T}\mathit{\lambda})$. More often than not, the model
parameter *λ* is unknown and has to be simultaneously estimated as
well. One of the most common ways to do this is the Baum–Welch algorithm
(BWA) (Zhang et al., 2014; Oudelha and Ainon, 2010; Yang et al., 1995;
Bilmes, 1998), which belongs to the family of expectation maximization
methods (Bilmes, 1998). Starting from randomly initialized model
parameters *λ*, the algorithm runs recursively to maximize the
likelihood of the model given the observation TS. When the algorithm
converges, we will obtain a set of estimated model parameters
$\stackrel{\mathrm{\u0303}}{\mathit{\lambda}}=\left(\stackrel{\mathrm{\u0303}}{\mathbf{A}},\stackrel{\mathrm{\u0303}}{\mathbf{B}},{\stackrel{\mathrm{\u0303}}{\mathit{\pi}}}_{\mathrm{0}}\right)$, as well as a posterior probability $P({s}_{t}={S}_{l}\mathrm{|}{\left\{{o}_{t}\right\}}_{t=\mathrm{1},\mathrm{\dots},T}\stackrel{\mathrm{\u0303}}{\mathit{\lambda}})$ TS. We include more
details on the BWA in Sect. 2.5. Additionally, for readers who want an
intuitive demonstration of how the HMM and BWA work, we have included a simulation
of a simple HMM and its BWA application in Sect. S2.

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.

## 2.4 Hidden Markov modeling and inputs to the BWA

In the context of this study, we assume for simplicity two seismicity states
of the earth crust beneath each station. These are our HSs {*S*_{1}, *S*_{2}} since they cannot be directly observed. What we can observe
directly are the geoelectric TSs for each station. Our goal is to
reconstruct the HS TSs so that the distributions of indexes (*C*, *V*, *S*, *K*)
of the geoelectric TSs in *S*_{1} and *S*_{2} are as different as possible.
To do this, we computed four index TSs each for NS and EW geoelectric fields
using the procedure described in Sect. 2.2 and organized them into a TS of
8-dimensional feature vectors ${\mathit{F}}_{t}=\left({C}_{\mathrm{NS},t},{V}_{\mathrm{NS},t},{S}_{\mathrm{NS},t},\phantom{\rule{0.125em}{0ex}}{K}_{\mathrm{NS},t},{C}_{\mathrm{EW},t},{V}_{\mathrm{EW},t},{S}_{\mathrm{EW},t},{K}_{\mathrm{EW},t}\right)$.
The values of each of the indexes are continuously distributed, but the
standard BWA requires discrete observations ${\left\{{O}_{q}\right\}}_{q=\mathrm{1},\mathrm{\dots},Q}$ as input. In this section, we discuss
possible ways to convert *F*_{t} into discrete observations for the
BWA and why we chose one particular method for implementation.

One way to do so would be to model each component of *F*_{t} as
samples drawn from known distributions, such as a normal distribution or a
gamma distribution. Unfortunately, as we can see from Fig. 3 (introduced in
the next paragraph), none of the known distributions fit the empirical data
well. Alternatively, we can discretize the components of *F*_{t} by
binning them. In other words, we represent the distribution of each
component with a histogram, with a specific choice of the number of bins (50
for example). This will effectively convert the continuous values of each
component of *F*_{t} into discrete values, such as integer labels
from 1 to 50 if we use 50 bins. Let us write the discretized
*F*_{t} as ${\stackrel{\mathrm{\u203e}}{\stackrel{\mathrm{\u203e}}{\mathit{F}}}}_{t}=\left({\stackrel{\mathrm{\u203e}}{\stackrel{\mathrm{\u203e}}{C}}}_{\mathrm{NS},t},{\stackrel{\mathrm{\u203e}}{\stackrel{\mathrm{\u203e}}{V}}}_{\mathrm{NS},t},{\stackrel{\mathrm{\u203e}}{\stackrel{\mathrm{\u203e}}{S}}}_{\mathrm{NS},t},{\stackrel{\mathrm{\u203e}}{\stackrel{\mathrm{\u203e}}{K}}}_{\mathrm{NS},t},{\stackrel{\mathrm{\u203e}}{\stackrel{\mathrm{\u203e}}{C}}}_{\mathrm{EW},t},{\stackrel{\mathrm{\u203e}}{\stackrel{\mathrm{\u203e}}{V}}}_{\mathrm{EW},t},{\stackrel{\mathrm{\u203e}}{\stackrel{\mathrm{\u203e}}{S}}}_{\mathrm{EW},t},\phantom{\rule{0.125em}{0ex}}{\stackrel{\mathrm{\u203e}}{\stackrel{\mathrm{\u203e}}{K}}}_{\mathrm{EW},t}\right)$.

If we do this for the TSs of individual components, such as the TS of ${\stackrel{\mathrm{\u203e}}{\stackrel{\mathrm{\u203e}}{C}}}_{\mathrm{NS},t}$, and use them as inputs for the BWA, we will obtain one HS TS for each of the eight components. In Fig. 3, we show (a) the estimated emission matrix $\stackrel{\mathrm{\u0303}}{B}$ in Fig. 3a, c, e, and g and (b) the posterior probability TSs in Fig. 3b, d, f, and h for four components: ${\stackrel{\mathrm{\u203e}}{\stackrel{\mathrm{\u203e}}{C}}}_{\mathrm{NS},t}$, ${\stackrel{\mathrm{\u203e}}{\stackrel{\mathrm{\u203e}}{V}}}_{\mathrm{NS},t}$, ${\stackrel{\mathrm{\u203e}}{\stackrel{\mathrm{\u203e}}{S}}}_{\mathrm{NS},t}$, and ${\stackrel{\mathrm{\u203e}}{\stackrel{\mathrm{\u203e}}{K}}}_{\mathrm{NS},t}$ of the KAOH station. These posterior probability TSs are different, which is not what we desire. Therefore, instead of this, we would like to use all eight components in ${\stackrel{\mathrm{\u203e}}{\stackrel{\mathrm{\u203e}}{\mathit{F}}}}_{t}$ as a single input to the BWA to obtain a single HS TS for each station.

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 $D={\mathrm{50}}^{\mathrm{8}}\approx \mathrm{3.91}\times {\mathrm{10}}^{\mathrm{13}}$ possible observations, meaning the emission matrix would be of
dimensions 3.91×10^{13} by 2. Reducing the number of bins to just
10 for each index, we still have *D*=10^{8} possible observations. This
latter space is still too large for the BWA to search through exhaustively
in a reasonable amount of time, even though we feel 10 bins for each index
may already be too coarse and likely to miss subtle details. Furthermore,
with so many possible observations, we expect the emission probabilities to
be significantly different from 0 only for a very small subset of the *D*
possible observations.

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 *k*-means clustering algorithm (Gupta et al.,
2010; Wen et al., 2006; Dash et al., 2011), which is commonly used for
discretizing continuous vectors such as *F*_{t}. We chose to use
the *k*-means clustering for discretizing *F*_{t} because of its low
computational cost as well as its reliability in grouping similar feature
vectors in the feature space. In so doing, we created a discrete feature
space with reasonable size as high-level labels of different geoelectric
dynamics. The mathematical details of *k*-means clustering can be found in
Sect. S3.

The indexes *C*_{NS,t}, *V*_{NS,t}, *S*_{NS,t}, and *K*_{NS,t} have highly
disparate dynamic ranges and should not be directly combined into a feature
vector. Therefore, before the clustering, we first standardized our indexes
by dividing them by their respective standard deviations. The purpose of
this step is to ensure the weights associated with each index during the
*k*-means clustering are equal so as not to bias our search for features with
high dynamic range. Mathematically, the feature vector of standardized
indexes at time *t*, ${\mathit{F}}_{t}^{\prime}$, can be written as

We then implemented *k*-means clustering using the scikit-learn package (v0.23.1) in
Python (v3.6.5), on the sequence of feature vectors ${\mathit{F}}_{t}^{\prime}$ covering
the time period from January 2012 to December 2018. The choice of the
*number of clusters* *Q* was determined as part of the hyperparameter optimization, described in
Sect. 3.5. In this way, we matched each ${\mathit{F}}_{t}^{\prime}$ to a discrete
label *o*_{t}→*O*_{q} (where *q* is an integer from 1 to *Q*) to obtain
the TS of discrete observations $\left\{{o}_{\mathrm{1}},{o}_{\mathrm{2}},\mathrm{\dots},{o}_{t},\mathrm{\dots},{o}_{T}\right\}$ for each station as its input to the BWA.

## 2.5 Implementation of BWA

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 $\left(\mathbf{A},\mathbf{B},\mathit{\pi}\right)$. Since we had no prior
knowledge on the model parameters, we initialized parameters $\left({\mathbf{A}}_{\mathrm{0}},{\mathbf{B}}_{\mathrm{0}},{\mathit{\pi}}_{\mathrm{0}}\right)$ randomly. After this, we iterated BWA's expectation maximization
steps 30 times, starting with iteration index *i*=1. Each iteration
comprises the forward procedure, the backward procedure, and the update.
In Sect. S4, we present the mathematical details of
how the forward procedure, the backward procedure, and the update are performed.

As the iteration goes, the BWA improves the likelihood of observing the
input observation TS ${o}_{\mathrm{1}},{o}_{\mathrm{2}},\mathrm{\dots},{o}_{T}$ given the model
parameters $\left({\mathbf{A}}_{i},{\mathbf{B}}_{i},{\mathit{\pi}}_{i}\right)$, which converges when the improvements on the posterior probability
$P({o}_{\mathrm{1}},{o}_{\mathrm{2}},\mathrm{\dots},{o}_{T}\mathrm{|}\left({\mathbf{A}}_{i},{\mathbf{B}}_{i},{\mathit{\pi}}_{i}\right))$ become minimal. In practice, we found that 30 iterations were long
enough for most models to converge. We therefore obtained the estimated
model parameters $\left(\stackrel{\mathrm{\u0303}}{\mathbf{A}},\stackrel{\mathrm{\u0303}}{\mathbf{B}},\stackrel{\mathrm{\u0303}}{\mathit{\pi}}\right)=\left({\mathbf{A}}_{\mathrm{30}},{\mathbf{B}}_{\mathrm{30}},{\mathit{\pi}}_{\mathrm{30}}\right)$, as well as the posterior probability TS of $\mathit{P}({s}_{t}={S}_{l}\mathrm{|}{o}_{\mathrm{1}},{o}_{\mathrm{2}},\mathrm{\dots},{o}_{T},\stackrel{\mathrm{\u0303}}{\mathbf{A}},\stackrel{\mathrm{\u0303}}{\mathbf{B}},\stackrel{\mathrm{\u0303}}{\mathit{\pi}})$ for both HSs and all *t* values, which we
write in short form as ${\mathit{P}}_{\mathrm{1}}=\left(P\left({s}_{\mathrm{1}}={S}_{\mathrm{1}}\right),P\left({s}_{\mathrm{2}}={S}_{\mathrm{1}}\right),\mathrm{\dots},P\left({s}_{T}={S}_{\mathrm{1}}\right)\right)$ and ${\mathit{P}}_{\mathrm{2}}=\left(P\left({s}_{\mathrm{1}}={S}_{\mathrm{2}}\right),P\left({s}_{\mathrm{2}}={S}_{\mathrm{2}}\right),\mathrm{\dots},P\left({s}_{T}={S}_{\mathrm{2}}\right)\right)$. Here, we noted that BWA assigns the indexing of HSs
randomly; therefore, the *S*_{1} of one station is not guaranteed to be
equivalent to the *S*_{1} of another station.

We cannot simply do the above BWA estimation once to obtain $\left(\stackrel{\mathrm{\u0303}}{\mathbf{A}},\stackrel{\mathrm{\u0303}}{\mathbf{B}},\stackrel{\mathrm{\u0303}}{\mathit{\pi}}\right)$ because the BWA converges to local optima instead of the global optimum in the model parameter space (Bilmes, 1998; Yang et al., 2017; Larue et al., 2011). Also, the initial parameters have a significant influence on the local optimum where the BWA converges. In order to obtain a global optimum result within a reasonable computation time, we ran 15 BWA estimations in parallel for each station, with different random initial parameters. For each station, we then chose the model with the highest model score given by $P\left({o}_{\mathrm{1}},{o}_{\mathrm{2}},\mathrm{\dots},{o}_{T}\mathrm{|}\left(\stackrel{\mathrm{\u0303}}{\mathbf{A}},\stackrel{\mathrm{\u0303}}{\mathbf{B}},\stackrel{\mathrm{\u0303}}{\mathit{\pi}}\right)\right)$ for subsequent analysis. Later in Fig. 4a, we also show all 15 HMMs to demonstrate how consistent the converged models are. We can write the posterior probability TS of this model as ${\stackrel{\mathrm{\u0303}}{\mathit{P}}}_{\mathrm{1}}=(P\left({s}_{\mathrm{1}}={S}_{\mathrm{1}}\right)$, $P\left({s}_{\mathrm{2}}={S}_{\mathrm{1}}\right),\mathrm{\dots},P\left({s}_{T}={S}_{\mathrm{1}}\right)\mathrm{|}{o}_{\mathrm{1}},{o}_{\mathrm{2}},\mathrm{\dots},{o}_{T}$, $\left(\stackrel{\mathrm{\u0303}}{\mathbf{A}},\stackrel{\mathrm{\u0303}}{\mathbf{B}},\stackrel{\mathrm{\u0303}}{\mathit{\pi}}\right))$.

For each initial condition, the BWA randomly assigns one HS to be *S*_{1}
and the other to be *S*_{2}. To show all 15 HMMs simultaneously in Fig. 4a, we need to standardize *S*_{1} and *S*_{2} across all HMMs. For this
purpose, we set ${\stackrel{\mathrm{\u0303}}{\mathit{P}}}_{\mathrm{1}}$ as the “standard”. For the
remaining 14 posterior probabilities ${\left\{{\mathit{P}}_{\mathrm{1}}^{i}\right\}}_{i=\mathrm{2},\mathrm{\dots},\mathrm{15}}$, we
checked their expected absolute difference, EAD = mean$\left(\mathrm{|}{\stackrel{\mathrm{\u0303}}{\mathit{P}}}_{\mathrm{1}}-{\mathit{P}}_{\mathrm{1}}^{i}\mathit{\}}\right)$, from
${\stackrel{\mathrm{\u0303}}{\mathit{P}}}_{\mathrm{1}}$, whose value ranges from 0 to 1. If EAD > 0.5,
${\mathit{P}}_{\mathrm{1}}^{i}$ is more similar to
${\stackrel{\mathrm{\u0303}}{\mathit{P}}}_{\mathrm{2}}$ than to ${\stackrel{\mathrm{\u0303}}{\mathit{P}}}_{\mathrm{1}}$, and we proceed to
swap the HS indexing for the *i*th HMM by assigning
${\mathit{P}}_{\mathrm{1}}^{i}\left(\mathrm{new}\right)\equiv {\mathit{P}}_{\mathrm{2}}^{i}$ and
${\mathit{P}}_{\mathrm{2}}^{i}\left(\mathrm{new}\right)\equiv {\mathit{P}}_{\mathrm{1}}^{i}$. Otherwise,
${\mathit{P}}_{\mathrm{1}}^{i}$ corresponds to the same HS as
${\stackrel{\mathrm{\u0303}}{\mathit{P}}}_{\mathrm{1}}$, and we leave its HS indexing unchanged. In this
way, we standardized all 15 models so that their *P*_{1}
can be visualized together in Fig. 4a, with the
${\stackrel{\mathrm{\u0303}}{\mathit{P}}}_{\mathrm{1}}$ TSs sorted by their model scores $P\left({o}_{\mathrm{1}},{o}_{\mathrm{2}},\mathrm{\dots},{o}_{T}\mathrm{|}\left(\stackrel{\mathrm{\u0303}}{\mathbf{A}},\stackrel{\mathrm{\u0303}}{\mathbf{B}},\stackrel{\mathrm{\u0303}}{\mathit{\pi}}\right)\right)$ and the optimal model in the first row. In Fig. 4b, we
show the actual posterior probability TS of this optimal model. The figures
of 15 HMMs for all 20 stations are included in Sect. S5.

We summarize the procedures used to obtain ${\stackrel{\mathrm{\u0303}}{\mathit{P}}}_{\mathrm{1}}$,
starting from a pair of geoelectric TSs for each GEMS station in the form of
a flowchart in Fig. 5. It is noteworthy that the full procedure contains
essentially only two hyperparameters: *Q* and *L*_{w}. The figures shown in
the “Results and discussions” section use the optimal hyperparameters, whose
identification procedure will be discussed in detail later in Sect. 3.5.
Additionally, for each station's optimal HMM, we plotted the distribution of
indexes (*C*, *V*, *S*, *K*) at both HSs in Sect. S6.

## 2.6 EQ grid map, EQ frequency, and EQ frequency ratio

Up to this point, we did not incorporate any EQ catalog information into ${\stackrel{\mathrm{\u0303}}{\mathit{P}}}_{\mathrm{1}}$ for each station. Unlike many past EQ studies looking for specific precursory features within the geoelectric data, we made no specific assumptions regarding what these EQ precursors might look like. Instead, we let the BWA search for specific precursory features within the 8-dimensional feature space.

After the hidden Markov modeling, we then checked locally whether *S*_{1} and *S*_{2}
would effectively partition time periods with significantly lower EQ
probabilities from those with significantly higher EQ probabilities. We
think of one HS as a *passive* state (with significantly lower EQ probabilities) and
the other HS as an *active* state (with significantly higher EQ probabilities), but
we cannot call the former *S*_{1} and the latter *S*_{2} because we have not
yet standardized these HS labels across the 20 stations. To do so, we need
to match the HS TS of each station to the EQ catalog to determine the EQ
frequencies of *S*_{1} and *S*_{2} for this station and use *S*_{1} and
*S*_{2} as the HS labels of the active and passive states respectively
(relabeling when necessary). In the remainder of this section, we describe
in detail how this is done.

For each GEMS station we started from ${\stackrel{\mathrm{\u0303}}{\mathit{P}}}_{\mathrm{1}}$ and
classified time periods across the 7 years as belonging to two sets *T*_{1}
and *T*_{2}. The time point *t*_{i} was assigned to *T*_{1} if
${\stackrel{\mathrm{\u0303}}{P}}_{{t}_{i}}\left({S}_{\mathrm{1}}\right)>\mathrm{0.5}$ and to *T*_{2} if
${\stackrel{\mathrm{\u0303}}{P}}_{{t}_{i}}\left({S}_{\mathrm{2}}\right)>\mathrm{0.5}$. After this is done, we checked
how EQs are distributed between *T*_{1} and *T*_{2} for different regions
across Taiwan. For this task, we first made a 16-by-16 grid map of Taiwan
so that EQs within the same grid cell (*i**x*, *i**y*), for *i**x*
and *i**y* in $\left\{\mathrm{0},\mathrm{1},\mathrm{\dots},\mathrm{15}\right\}$, are grouped together
(see Fig. 6).

For each grid cell (*i**x*, *i**y*), we defined the *EQ frequencies* for HSs
*S*_{1} and *S*_{2} as

where *N*_{1} is the number of EQs occurring within *T*_{1}, *N*_{2} is the
number of EQs occurring within *T*_{2}, |*T*_{1}| is the total
duration of *T*_{1} time periods, and |*T*_{2}| is the total
duration of *T*_{2} time periods. From Fig. 6, we see that the spatial
distribution of EQs is highly heterogeneous, so we may find a grid cell with
about 10 EQs but also another grid cell with about 1000 EQs. This tells us
that we should not directly compare the EQ frequencies but should instead
compare the *EQ frequency ratio*, defined as

For any cell containing at least one EQ, the range of its *R*_{F} is
[0,1]. Intuitively, any cell with *R*_{F}<0.5 is a region having lower EQ
frequency in *S*_{1} compared to *S*_{2}; and any cell with *R*_{F}>0.5 is a
region having a higher EQ frequency in *S*_{1} compared to *S*_{2}. For
example, for a cell with *R*_{F}=0.2, *F*_{EQ,1} is only one-quarter of
*F*_{EQ,2}. The *R*_{F} value quantifies how one HS has a higher or lower EQ
frequency than the other. In Sect. 3, we will present how we deep dived into
the spatial–temporal correlations between HS TSs (${\stackrel{\mathrm{\u0303}}{\mathit{P}}}_{\mathrm{1}}$)
and EQ activities for all 20 stations, starting from 20 grid maps of *R*_{F}
values.

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.

## 3.1 EQ frequency ratio (*R*_{F}) grid maps

Once we obtained the ${\stackrel{\mathrm{\u0303}}{\mathit{P}}}_{\mathrm{1}}$ TS for each station, the
natural first step of our analysis was to examine the *R*_{F} values for all
cells in the 16-by-16 grid map. We show this procedure for the CHCH station in
Fig. 7, where we visualize the grid maps for *N*_{1} and *N*_{2} in Fig. 7a and b respectively to clearly show how many EQs occurred during
*T*_{1} and *T*_{2}. The resulting *R*_{F} grid map is shown in Fig. 7c,
where there are cells with values close to 0.5 (white-color cells) and
cells with values far from 0.5 (red for close to 0, green for close to 1).
White-color cells are regions whose EQ activities are weakly correlated with
the HSs since the time periods of *S*_{1} and *S*_{2} are not very
different in terms of EQ frequency; whereas red/green cells are regions with
significantly lower/higher EQ frequencies in *S*_{1}.

As can be seen in Fig. 7c, for different regions the HS with higher EQ
activities can be either *S*_{1} or *S*_{2}. This is true not only for the CHCH
station but also for all 20 stations, whose *R*_{F} grid maps are shown in
Fig. 8. Although there is no consistent pattern of any state corresponding
to higher EQ activities globally, we see in Fig. 8 that there are regions
whose *R*_{F} values are far from 0.5 across many stations. This means
that statistically speaking, one of the HSs has higher EQ activities than
the other. In fact, if the active HS has a lot more EQs than the passive HS,
it is also likely that the active HS covers most of the larger EQs (e.g.,
*M*>5), which is a good attribute for potential EQ forecasting applications.
This phenomenon is shown in Sect. S7, where we
visualized the EQ frequency distributions across different magnitudes for
both HSs for three selected cells with the most EQ events.

All in all, the findings in this section are important, but we cannot
directly decide whether *S*_{1} or *S*_{2} is the proxy for increased EQ
probabilities because neither can be associated consistently with the
active or the passive state. Instead, we should understand *S*_{1} and
*S*_{2} as two high-level, fuzzy labels for tectonic dynamics related to EQ
activities in different regions. There can be elements such as rock and soil
formations, the underground water system, and fault lines, forming a complex
dynamical system that influences where and when EQs become active. Concrete mapping between EQ activities and specific elements of the complex
dynamical system would be very difficult, as this would involve
high-resolution subterranean surveys. Nevertheless, we can still measure how
well *S*_{1} and *S*_{2} can partition the time periods so that one HS can
have significantly more EQs than the other. To show this more clearly, we
created grid maps of *discrimination power* *D* and present them in the next section.

## 3.2 Discrimination power (*D*) grid maps

We defined the discrimination power *D* for each cell as

The value of *D* ranges from 0 to 0.5, with 0.5 being the most
discriminative since all EQs are found in one HS and 0 being the least
discriminative since EQ frequencies are identical between the two HSs. We
show the grid maps of *D* for 20 stations in Fig. 9, which are easier to
interpret compared to the grid maps in Fig. 8 where we had to use two
different colors. Intuitively, for a region with *D*=0.25 (not uncommon),
one of its HSs would have an EQ frequency 3 times that of the other HS.
It can be noted that cells around the edge of the map tend to have very high
*D* values because there are very few EQ events in these cells. This is not
a problem as we will take the number of EQs into account later in Sect. 3.3.

In some cells, we find *D* values close to 0.5, which seems to suggest that
the seismicity associated with *S*_{1} is very different from that
associated with *S*_{2}. However, looking at Fig. 9, we see large variations
in *D* values across the cells and more importantly among some neighboring
cells. We therefore wonder whether regions with high *D* values are
statistically significant or the products of random temporal clustering of
EQs (Dieterich, 1994; Frohlich, 1987; Holbrook et al., 2006; Batac and
Kantz, 2014). For example, if all EQs in a cell occurred within a single day
in the 7-year period, any random assignment of HSs would produce the highest
*D* value of 0.5. To address this concern, we investigated the
significance of the grid maps of *D* through statistical tests in the next
section.

## 3.3 Cellular-level significance tests of the forecasting power

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 *D* grid maps shown in Fig. 9. Specifically, for each grid cell and an empirical HS TS we carried out
statistical hypothesis testing using the following null hypothesis: any
random HS TS would achieve the same or higher performance (in terms of the *D*
value). To create random HS TSs for the hypothesis testing, we chose to
directly simulate the HMM using the same model parameters $\left(\stackrel{\mathrm{\u0303}}{\mathbf{A}},\stackrel{\mathrm{\u0303}}{\mathbf{B}},\stackrel{\mathrm{\u0303}}{\mathit{\pi}}\right)$ as the empirical HMM of the
corresponding station. For each hypothesis test of an empirical HS TS
(actual HS TS obtained for each station), we created 400 simulated HS TSs,
which were then used to create 400 grid maps of the discrimination power
*D*. In Fig. 10, we show the empirical HS TSs alongside a random sample of
10 simulated HS TSs for YULI, SHRL, CHCH, and SIHU to illustrate the
simulated counterparts. After this, in each cell, we had one empirical value
of *D* that we can compare against a distribution of 400 simulated values of
*D*. This allows us to compute for each cell the probability that its
empirical *D* value is higher than its simulated counterparts. We named this
quantity the *discrimination reliability* *R*_{D}, defined for each cell in the grid map as

In the language of statistical hypothesis testing, the *p* value for the test
is given by $p=\mathrm{1}-{R}_{\mathrm{D}}$. The value of *R*_{D} ranges from 0 to 1. If
*R*_{D} is close to 1, we are confident that the discrimination power of
the empirical HS TS is statistically significantly high; otherwise, we have
no such confidence.

In Fig. 11, we show the grid maps of *R*_{D} values (as percentages) for all
20 stations. Dark-red cells are regions with *R*_{D} close to 1, and
white and pink cells are regions with *R*_{D} close to 0. From these grid maps,
we can better appreciate the utility of HS TSs across the grid map since
the *R*_{D} value is a statistical significance measure of the HS–EQ
correlation, unlike the discrimination power *D*. To explain this, let us
take the example of LIOQ (upper left of Fig. 11), whose physical
location is marked by the blue star within a dark-red grid cell of
*R*_{D}=0.992. This means that the empirical HS TS performs better than
random guesses (i.e., simulated HS TSs) at separating time periods of
low/high EQ frequencies, with a statistical significance of *p*=0.008. This
means that it is improbable for a simulated HS TS to have such a high *D*,
and therefore the empirical HS TS is unlikely to be a product of random
chance. This is a very strong demonstration of the mutual information
between the HS TS obtained from geoelectric TS and the EQ catalog that was
not used to train the HMM.

In the proximity of the LIOQ station located within 22.55–23.58^{∘} N, we can see a clear pattern of cells with *R*_{D}≥0.9 (dark-red
color), while *R*_{D}≥0.9 occasionally for most cells outside this
general region. This pattern suggests the geoelectric information from LIOQ is approximately local. This is consistent with the logical
requirement for a direct/indirect structural relation between LIOQ and
region X, such as being close to the same subterranean fault line, for the
information at LIOQ to be useful for region X. As a corollary,
information given by LIOQ is less likely to be useful for faraway
regions as they are less likely to have such structural relations with LIOQ. In application scenarios, this means that the state of EQ
probabilities of region X can be estimated using stations closer to the
region. Last but not least, it is also worth mentioning that most cells at
the edge of the map seldom have high *R*_{D} values. This is consistent with
the fact that these cells typically have very few EQ events to provide high
statistical significance.

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 (*R*_{D} being
high enough) for a list of stations. The more stations in this list becoming
persistently active, the more likely large EQs within this grid cell should
occur. This is the spatial window we work with for making predictions. (c) Finally, let us describe how our HMM can help in assessing the magnitudes
of EQs expected. To answer this question, we can examine the distribution of
EQ frequencies across magnitudes 3.0 to 6.0 for both active states and
passive states (in Sect. S7). It turns out that for a
given grid cell with high *R*_{D}, the active state has proportionally more
EQs than the passive state across all magnitudes. Therefore, we expect EQs
of all magnitudes to be more frequent in a positive prediction.

For grid cells with high *R*_{D}, the corresponding HS TS alone is
sufficient to make intermediate-term EQ forecasts. However, we also have
grid cells where none of the 20 stations provide a sufficiently high *R*_{D}
value for intermediate-term EQ forecasting on their own. These HS TSs could
still be useful if we combine all 20 HS TSs as input features for
higher-level forecasting algorithms trained individually for each grid cell.
For example, for any region (grid cell), if we want to decide whether it
currently belongs to the active regime or the passive regime, an algorithm
uses the input from all 20 stations to decide the “local” HS for the given
grid cell. This high-level algorithm can for example comprise weight-based model
averaging (Marzocchi et al., 2012) or decision trees (Asim et al.,
2016). Additionally, the value of *R*_{D} can be helpful for the algorithm
to decide how to weigh the information given by all 20 stations. For
example, we can consider only stations with *R*_{D}≥*R*_{D_min} at the given grid cell. The user-defined threshold
*R*_{D_min} can take on constant values (e.g., 0.9) across
the grid map or be location specific, such as being lower (e.g., 0.8) for
grid cells where few of the 20 stations have *R*_{D}≥0.9. We hope to
explore this in future work.

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.

## 3.4 Global-level significance tests of the forecasting power

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 *R*_{D} values. While the forecasting power of HS TSs in each
of these cells is statistically significant, the more critical among us may
wonder whether some of these cells can be significant purely by chance, even
though there is in reality no persistent correlation between EQs and HSs.
For example, any simulated HS TS in Fig. 10 would have at least a few cells
with high *R*_{D} values. Therefore, in this next section, we will answer
the question of whether these HS TSs indeed contain useful information
about EQs or whether the number of “significant” cells can be explained by a random
null model where the EQs and HSs are mutually uninformative because we test
a large number of cells assuming that they are statistically independent.

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 *R*_{D} values. We start by assuming that all
stations have zero forecasting skill, but as a result of our statistical
test, some cells may still end up with high *R*_{D} by chance. A truly
informative station should have significantly more cells with high *R*_{D}
than random guesses. Taking the number of EQs into consideration, we
further propose that a truly informative station should have significantly
higher EQ counts located in high-performing cells. On the grid map, let us
define cells with *R*_{D}≥*R*_{D_min} as *satisfactory* cells and the
rest as *unsatisfactory* cells, where *R*_{D_min} is the user-defined
threshold that determines how high the *R*_{D} should be in order to be
considered “high-performing”. As mentioned earlier, it is possible to work
out schemes that allow for a regionally acceptable *R*_{D_min}.
Here for simplicity let us consider a scheme with a uniform
*R*_{D_min} across all cells in the grid map. With this
setting we can proceed to define the single-value performance metric for
each station, as the *ratio of EQs in satisfactory cells*, or *R*_{EQS}, as

where *N*_{EQ} is the number of EQs in each cell. This ratio of EQs in satisfactory cells takes on values
$\mathrm{0}\le {R}_{\mathrm{EQS}}\le \mathrm{1}$. Intuitively, if *R*_{EQS}=0.4, it means that given the
*R*_{D_min} value, 40 *%* of all EQs are located within
satisfactory cells and are therefore “forecasted” by the station to the
level required by the user (i.e., *R*_{D_min}). Therefore, to
show that a station has more forecasting power than random guesses, we
proceed to test a given station against the null hypothesis that a random
guess (simulated HS TS) can have the same *R*_{EQS} as the
empirical HS TS or higher.

We carried out this hypothesis test station by station by first computing
the *R*_{EQS} values of a station's empirical HS TS as well as of 400 HS TSs
simulated using the HMM parameters for the given station. We then defined
the *global confidence level* as

Similarly to the *p* value for the cellular-level hypothesis test, the *p* value
for this global-level hypothesis test is given by $p=\mathrm{1}-\mathrm{GCL}$, where the GCL
range is [0,1], and gives the probability that the empirical HS TSs
have higher *R*_{EQS} values than their simulated counterparts. For example, if a
station has GCL =0.99, we can say that given the specified
*R*_{D_min}, we are 99 % confident that the empirical HS TS
yields a higher *R*_{EQS} than its simulated counterparts.

In Fig. 12, we show the results of our global-level significance tests, for
a choice of *R*_{D_min}=0.95, in the form of histograms of
the 400 simulated *R*_{EQS} values compared against the empirical *R*_{EQS}
values. Except for the LIOQ and LISH stations, we can see from Fig. 12 that all
the other stations have GCL values close to 1. This tells us that the
empirical *R*_{EQS} values of the 18 stations are statistically significant.
We also observed that for *R*_{D_min}=0.95, the simulated
*R*_{EQS} values are mostly around (or below) 0.05, meaning that only 5 %
of EQs are located in satisfactory cells by chance. In contrast, the
empirical *R*_{EQS} values are mostly above 0.2, except for TOCH, LIOQ,
PULI, HERM, and LISH. These findings suggest the HS TSs'
EQ forecasting utility to be significant at the global level.

Last but not least, the histograms for each station in Fig. 12 are created
with individually optimized hyperparameters, namely *L*_{w} (length of time
window to compute indexes *C*, *V*, *S*, and *K*, in days) and *Q* (number
of clusters for the *k*-means clustering). The optimal hyperparameter values
for each station are indicated in the titles for each station. Let us
discuss the details of this optimization process in the next section.

## 3.5 Significance levels across the hyperparameter space

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: *L*_{w} and *Q*. In this section, we
show how the model performance (GCL) will vary across the tested
hyperparameter space, as well as how we chose the hyperparameters
[*L*_{w},*Q*], for each station. Due to the high computational cost to test
each combination of *L*_{w} and *Q* (about 40 min per station on a desktop with
4 GHz quad-core i7 processors, 16 GB of RAM, running macOS Mojave 10.14.6),
we performed a coarse grid search over 28 points in the parameter space,
consisting of seven different *L*_{w} values: $\mathit{\{}\mathrm{0.02},\phantom{\rule{0.125em}{0ex}}\mathrm{0.03},\phantom{\rule{0.125em}{0ex}}\mathrm{0.04},\phantom{\rule{0.125em}{0ex}}\mathrm{0.05},\phantom{\rule{0.125em}{0ex}}\mathrm{0.1},\phantom{\rule{0.125em}{0ex}}\mathrm{0.2},\phantom{\rule{0.125em}{0ex}}\mathrm{0.25}\mathit{\}}$ d (or {28.8, 43.2, 57.6, 72, 144, 288, 360} min)
and four different *Q* values $\mathit{\{}\mathrm{30},\phantom{\rule{0.125em}{0ex}}\mathrm{40},\phantom{\rule{0.125em}{0ex}}\mathrm{60},\phantom{\rule{0.125em}{0ex}}\mathrm{80}\mathit{\}}$. We decided on this
search space based on our experience during the model development stage. For
real-world applications, where more computational resources can be invested,
this hyperparameter optimization can be carried out over a larger and finer grid,
in which case better results can be expected.

For each choice of station and hyperparameter, we followed the same
procedure of computing 1 + 400 *R*_{EQS} values, as well as the resulting
GCL value. In Figs. 13 and 14, we show the 20 heatmaps of *R*_{EQS} and
GCL across the hyperparameter space respectively for *R*_{D_min}=0.95. The results shown in Fig. 14 are more intuitive, where we found
that for many stations, the GCL values approach 1 across broad regions
of the hyperparameter space. This can for example be the full hyperparameter
space for the YULI station or a patch within the hyperparameter space for the KUOL
station. There is just one station (LISH) with poor GCL values everywhere
in the hyperparameter space, indicating that there might be exclusive
factors that severely limit LISH's forecasting power. For the other
19 stations, the GCL values are close to 1 across either a large area of
the parameter space or almost the entire parameter space (e.g., YULI, WANL,
ENAN, DABA). This result is compelling and is exactly what we needed for
our goal: to demonstrate the forecasting skill of the HS TS, which does not depend on highly optimized hyperparameters but is valid over a broad range of hyperparameters.

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 *R*_{EQS} as the winner. For example, for the WANL station in Fig. 14,
there are many cells with GCL =1. We therefore proceeded to check the
heatmap for WANL in Fig. 13 and identified the hyperparameter
combination *L*_{w}=0.03 and *Q*=80 as optimal since it has the highest
*R*_{EQS} value. Using this selection procedure, we identified the optimal
hyperparameter for each station and used these individually optimal
hyperparameters to create Figs. 7 to 12. This selection procedure could also
be adapted for real-world applications, when more historical data and
computational power are available, to provide even better model
performances.

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 *k*-means clustering to convert the continuous TS of
8-dimensional index vectors into a discrete observation TS and subsequently
obtained a converged HMM for each station. We then investigated whether
these HS TSs provide informative partitions of EQs, i.e., whether one of the HSs can
be interpreted as a passive state with less frequent EQs and the other
one as an active state with more frequent EQs. For this task, we defined the
EQ frequency ratio (*R*_{F}), which is the frequency of EQs in one of the HSs divided by the
total frequency of the EQs. Using *R*_{F} we further defined the
discrimination power (*D*) to measure how different one HS is from the other HS in terms of
the EQ frequency. We then plotted 16-by-16 grid maps of *R*_{F} and *D* for
all 20 stations and tested the statistical significance of *D* in each
cell by comparing the empirical value against the distribution of *D* from
400 simulated HS TSs to end up with the grid maps of discrimination reliability (*R*_{D}) for all 20
stations. To further investigate the statistical significance level at the
global scale, we defined *R*_{EQS} to measure the percentage of total EQs
located within satisfactory cells, i.e., cells having *R*_{D}≥*R*_{D_min} for a user-specified *R*_{D_min}
value. This *R*_{D_min} value can be easily customized for
different cells, but in this paper, we used a constant *R*_{D_min} value across the grid map for demonstration. By comparing the
*R*_{EQS} value of the empirical model against those of 400 simulated
models, we obtained one global significance value for each station, namely
the global confidence level (GCL). This tells us how confident we can be that information
contained in the empirical HS TSs can be used for EQ forecasting.

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 *L*_{w} and *Q* individually for each station. As a result, among
the 20 stations with optimized hyperparameters, there are 19 stations with
GCL > 0.95, 15 of which have GCL > 0.99. Additionally, the confidence
levels are also robust across the hyperparameter space for most stations.
Based on these positive results, the hidden Markov modeling of the index
TSs computed from geoelectric TSs is indeed a viable way to extract
information that can be useful for EQ forecasting.

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-*R*_{D} cells) or
as input features for higher-level algorithms that take information from all
20 stations (for low-*R*_{D} cells). Beyond our demonstration of extracting
EQ-related information from geoelectric TSs, the HMM approach described in
this paper can also be explored on other high-frequency geophysical data,
such as those from geomagnetic, geochemical, hydrological, and GPS
measurements, for EQ forecasting.

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 *after* the training data and (b) the
validation data are also intermediate term for intermediate-term EQ
forecasting. These two requirements cannot be fulfilled using our limited
7-year data if we want to have a significant number of
validations (e.g., 10 times) to produce confident claims. Therefore, in this paper, we
limited our scope to demonstrating that our model has *forecasting skill*, without quantifying
its *exact forecasting accuracy*. We argue that we have indeed achieved this, without the use of
out-of-sample testing, because in Sect. 3.5, we showed the forecasting skill is statistically significant regardless of the choice of the hyperparameters, for 19 out of the
20 stations that we tested. Furthermore, the statistical hypothesis test has
the advantage of giving rigorous *p* values with moderate computation cost,
through simulating the HMM for multiple null-hypothesis tests.

The Python codes that we used to produce the results in this paper can be downloaded at GitHub: https://github.com/wenhy1111/HMM_Geoelectric_EQ (last access: 3 June 2022; https://doi.org/10.5281/zenodo.6598498, Wen, 2022).

The data set of the index TSs for 20 stations computed using various time
windows (*L*_{w}) is available in a repository and can be accessed
via a DOI link: https://doi.org/10.21979/N9/JSUTCD (Cheong, 2021). For the
0.5 Hz geoelectric TS data for 20 stations, the data are available on request
by contacting Hong-Jia Chen (redhouse6341@gmail.com) or
Chien-Chih Chen (chienchih.chen@g.ncu.edu.tw). The EQ catalogue
data are owned by a third party, the Central Weather Bureau in Taiwan.

The supplement related to this article is available online at: https://doi.org/10.5194/nhess-22-1931-2022-supplement.

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.

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