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 (Mc), 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 Mc, 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 ML≥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 ML≥6 are marked. Some of these events were destructive. For instance, at 03:57 on 6 February 2016 (UTC+8), an ML 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 f1=10-4.0 and f2=10-1.75 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.

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 Lw geoelectric TS data points. Later in Sect. 3.5, we will discuss in detail how we chose the optimal Lw 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 Lw=0.02 or Lw=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 Xnn=1,…,Lw. The correlation C that we used in this paper is the lag-1 Pearson autocorrelation of Dn=Xn+1-Xnn=1,…,Lw-1, which is the difference sequence of Xnn=1,…,Lw. Mathematically, 1CXn=AC1Dn=EDn-μDDn+1-μDσD2, where E is the expectation, μD is the mean of Dnn=1,…,Lw-1, and σD is the standard deviation of Dnn=1,…,Lw-1. The range of C is [-1,1], 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 Xnn=1,…,Lw 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 2VXn=E(Xn-μX)2, where μX is the mean of Xnn=1,…,Lw. 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 Xnn=1,…,Lw, or the sequence's third standard central moment, is defined as 3SXn=EXn-μXσX3, where σX is the standard deviation of Xnn=1,…,Lw. It is a real number measuring how asymmetric the distribution of Xnn=1,…,Lw 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 Xnn=1,…,Lw, or the sequence's fourth standard central moment, is defined as 4KXn=EXn-μXσX4. 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.

A Markov model is a stochastic model that can be used to describe a system whose future state st+1 is drawn from a set of L states Sll=1,…,L with probabilities pj←i=Pst+1=Sj|st=Si conditioned by its current state st. The probabilities pj←i can be organized into a transition matrix A, where Ai,j=pj←i. The HMM is an extension of the Markov model, with the additional property that the system state st 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 ot drawn from a size-Q observable set Oqq=1,…,Q. Just as in a Markov model, the future state st+1 of an HMM is drawn from the set Sll=1,…,L with probabilities pj←i (similarly conditioned by the current state st) taken from the transition matrix A. At time t, the observable ot is emitted with a probability P(ot=Oq|st=Sl) that depends on which HS st=Sl the system is in. These probabilities can be organized into an L×Q emission matrix B, where Bl,k=P(ot=Oq|st=Sl). Additionally, we call the HS probability distributions at the initial time π0=PS1,PS2,…,PSL. With this, we have fully specified the HMM: the sets of HSs Sll=1,…,L and observations Oqq=1,…,Q as well as the model parameters that are collectively called λ=A,B,π0.

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(st=Sl|ott=1,…,Tλ). 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 λ̃=Ã,B̃,π̃0, as well as a posterior probability P(st=Sl|ott=1,…,Tλ̃) 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.

In the context of this study, we assume for simplicity two seismicity states of the earth crust beneath each station. These are our HSs S1,S2 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 S1 and S2 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 Ft=CNS,t,VNS,t,SNS,t,KNS,t,CEW,t,VEW,t,SEW,t,KEW,t. The values of each of the indexes are continuously distributed, but the standard BWA requires discrete observations Oqq=1,…,Q as input. In this section, we discuss possible ways to convert Ft 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 Ft 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 Ft 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 Ft into discrete values, such as integer labels from 1 to 50 if we use 50 bins. Let us write the discretized Ft as F‾‾t=C‾‾NS,t,V‾‾NS,t,S‾‾NS,t,K‾‾NS,t,C‾‾EW,t,V‾‾EW,t,S‾‾EW,t,K‾‾EW,t.

If we do this for the TSs of individual components, such as the TS of C‾‾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 B̃ in Fig. 3a, c, e, and g and (b) the posterior probability TSs in Fig. 3b, d, f, and h for four components: C‾‾NS,t, V‾‾NS,t, S‾‾NS,t, and K‾‾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 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=508≈3.91×1013 possible observations, meaning the emission matrix would be of dimensions 3.91×1013 by 2. Reducing the number of bins to just 10 for each index, we still have D=108 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 Ft. We chose to use the k-means clustering for discretizing Ft 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 CNS,t, VNS,t, SNS,t, and KNS,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, Ft′, can be written as 5Ft′=CNS,tσCNS,t,VNS,tσVNS,t,SNS,tσSNS,t,KNS,tσKNS,t,CEW,tσCEW,t,VEW,tσVEW,t,SEW,tσSEW,t,KEW,tσKEW,t. 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 Ft′ 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 Ft′ to a discrete label ot→Oq (where q is an integer from 1 to Q) to obtain the TS of discrete observations o1,o2,…,ot,…,oT for each station as its input to the 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 A,B,π. Since we had no prior knowledge on the model parameters, we initialized parameters A0,B0,π0 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 o1,o2,…,oT given the model parameters Ai,Bi,πi, which converges when the improvements on the posterior probability P(o1,o2,…,oT|Ai,Bi,πi) become minimal. In practice, we found that 30 iterations were long enough for most models to converge. We therefore obtained the estimated model parameters Ã,B̃,π̃=A30,B30,π30, as well as the posterior probability TS of P(st=Sl|o1,o2,…,oT,Ã,B̃,π̃) for both HSs and all t values, which we write in short form as P1=Ps1=S1,Ps2=S1,…,PsT=S1 and P2=Ps1=S2,Ps2=S2,…,PsT=S2. Here, we noted that BWA assigns the indexing of HSs randomly; therefore, the S1 of one station is not guaranteed to be equivalent to the S1 of another station.

We cannot simply do the above BWA estimation once to obtain Ã,B̃,π̃ 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 Po1,o2,…,oT|Ã,B̃,π̃ 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 P̃1=(Ps1=S1, Ps2=S1,…,PsT=S1|o1,o2,…,oT, Ã,B̃,π̃).

For each initial condition, the BWA randomly assigns one HS to be S1 and the other to be S2. To show all 15 HMMs simultaneously in Fig. 4a, we need to standardize S1 and S2 across all HMMs. For this purpose, we set P̃1 as the “standard”. For the remaining 14 posterior probabilities P1ii=2,…,15, we checked their expected absolute difference, EAD = mean|P̃1-P1i}, from P̃1, whose value ranges from 0 to 1. If EAD > 0.5, P1i is more similar to P̃2 than to P̃1, and we proceed to swap the HS indexing for the ith HMM by assigning P1i(new)≡P2i and P2i(new)≡P1i. Otherwise, P1i corresponds to the same HS as P̃1, and we leave its HS indexing unchanged. In this way, we standardized all 15 models so that their P1 can be visualized together in Fig. 4a, with the P̃1 TSs sorted by their model scores Po1,o2,…,oT|Ã,B̃,π̃ 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 P̃1, 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 Lw. 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.

Up to this point, we did not incorporate any EQ catalog information into P̃1 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 S1 and S2 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 S1 and the latter S2 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 S1 and S2 for this station and use S1 and S2 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 P̃1 and classified time periods across the 7 years as belonging to two sets T1 and T2. The time point ti was assigned to T1 if P̃tiS1>0.5 and to T2 if P̃tiS2>0.5. After this is done, we checked how EQs are distributed between T1 and T2 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 ix,iy, for ix and iy in 0,1,…,15, are grouped together (see Fig. 6).

For each grid cell ix,iy, we defined the EQ frequencies for HSs S1 and S2 as 6FEQ,1=N1T1,FEQ,2=N2T2, where N1 is the number of EQs occurring within T1, N2 is the number of EQs occurring within T2, T1 is the total duration of T1 time periods, and T2 is the total duration of T2 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 7RF=FEQ,1FEQ,1+FEQ,2. For any cell containing at least one EQ, the range of its RF is [0,1]. Intuitively, any cell with RF<0.5 is a region having lower EQ frequency in S1 compared to S2; and any cell with RF>0.5 is a region having a higher EQ frequency in S1 compared to S2. For example, for a cell with RF=0.2, FEQ,1 is only one-quarter of FEQ,2. The RF 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 (P̃1) and EQ activities for all 20 stations, starting from 20 grid maps of RF values.

Once we obtained the P̃1 TS for each station, the natural first step of our analysis was to examine the RF 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 N1 and N2 in Fig. 7a and b respectively to clearly show how many EQs occurred during T1 and T2. The resulting RF 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 S1 and S2 are not very different in terms of EQ frequency; whereas red/green cells are regions with significantly lower/higher EQ frequencies in S1.

As can be seen in Fig. 7c, for different regions the HS with higher EQ activities can be either S1 or S2. This is true not only for the CHCH station but also for all 20 stations, whose RF 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 RF 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 S1 or S2 is the proxy for increased EQ probabilities because neither can be associated consistently with the active or the passive state. Instead, we should understand S1 and S2 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 S1 and S2 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.

We defined the discrimination power D for each cell as 8D=RF-0.5. 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 S1 is very different from that associated with S2. 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.

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 Ã,B̃,π̃ 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 RD, defined for each cell in the grid map as 9RD=(no. simulatedD<empiricalD)400. In the language of statistical hypothesis testing, the p value for the test is given by p=1-RD. The value of RD ranges from 0 to 1. If RD 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 RD values (as percentages) for all 20 stations. Dark-red cells are regions with RD close to 1, and white and pink cells are regions with RD close to 0. From these grid maps, we can better appreciate the utility of HS TSs across the grid map since the RD 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 RD=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 RD≥0.9 (dark-red color), while RD≥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 RD 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 (RD 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 RD, 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 RD, 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 RD 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 RD 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 RD≥RD_min at the given grid cell. The user-defined threshold RD_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 RD≥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.

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 RD 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 RD 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 RD 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 RD by chance. A truly informative station should have significantly more cells with high RD 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 RD≥RD_min as satisfactory cells and the rest as unsatisfactory cells, where RD_min is the user-defined threshold that determines how high the RD 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 RD_min. Here for simplicity let us consider a scheme with a uniform RD_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 REQS, as 10REQS=∑satisfactorycellsNEQ∑allcellsNEQ, where NEQ is the number of EQs in each cell. This ratio of EQs in satisfactory cells takes on values 0≤REQS≤1. Intuitively, if REQS=0.4, it means that given the RD_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., RD_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 REQS as the empirical HS TS or higher.

We carried out this hypothesis test station by station by first computing the REQS 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 11GCL=(no. simulatedREQS<empiricalREQS)400. Similarly to the p value for the cellular-level hypothesis test, the p value for this global-level hypothesis test is given by p=1-GCL, where the GCL range is [0,1], and gives the probability that the empirical HS TSs have higher REQS values than their simulated counterparts. For example, if a station has GCL =0.99, we can say that given the specified RD_min, we are 99 % confident that the empirical HS TS yields a higher REQS than its simulated counterparts.

In Fig. 12, we show the results of our global-level significance tests, for a choice of RD_min=0.95, in the form of histograms of the 400 simulated REQS values compared against the empirical REQS 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 REQS values of the 18 stations are statistically significant. We also observed that for RD_min=0.95, the simulated REQS 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 REQS 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 Lw (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.

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: Lw 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 [Lw,Q], for each station. Due to the high computational cost to test each combination of Lw 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 Lw values: {0.02,0.03,0.04,0.05,0.1,0.2,0.25} d (or {28.8, 43.2, 57.6, 72, 144, 288, 360} min) and four different Q values {30,40,60,80}. 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 REQS values, as well as the resulting GCL value. In Figs. 13 and 14, we show the 20 heatmaps of REQS and GCL across the hyperparameter space respectively for RD_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 REQS 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 Lw=0.03 and Q=80 as optimal since it has the highest REQS 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.