a. Memory in the data and Markovian representation

Let the observed data be given in the form of a time-discrete sequence {z_t}_{t=1, . . . , T} of c dimensional data vectors that describe the observation or measurement of a process at T subsequent instances. We will say that the process underlying the observations has a memory depth d ≥ 0 if the conditional probability distribution P of future states of the process, given the present state and all past states, depends only on the present state and d previous states but not on all past states. Mathematically, this property can be expressed as

View Expanded

We will call a process Markovian if d = 0. For D ≥ d > 0 it is obvious that the extended stochastic process x^(D)_t = (z_t, z_t−1, . . . , z_t−D) (which we will call a d-frame recasting of the original process) is Markovian; i.e.,

View Expanded

We will further omit the upper index D to simplify the notation.

This means that any observed process with finite memory can be cast into the Dc-dimensional extended space and become Markovian (allowing us to apply Markovian techniques of time series analysis, such as HMMs).

There are two major problems associated with this strategy: (i) reliable estimation of the memory depth d is not a trivial task if the dimension c of the observation data is high and (ii) the numerical cost of the time series analysis increases significantly for large D because the dimension of the extended space is D times larger than the dimension of the original space.

The first of the abovementioned problems becomes even more serious if the physics of the underlying process is unknown, that is, if it is not a priori clear what kind of stochastic dynamics should be expected (linear or nonlinear, additive or multiplicative noise, etc.). Linear approaches, such as multivariate autoregressive processes (MVARs; Brockwell and Davis 2002), can be used for estimation of d in multiple dimensions. However, such analyses do not guarantee reliability because there are examples of systems with finite nonlinear memory (e.g., the time series of stock returns in finance) where linear analysis methods do not reveal any significant memory effects (Tsay 2005). Another problem of such methods is their high numerical cost: the MVAR method, for example, scales as O(c⁶). This prohibits the application of these methods to very high-dimensional systems without making additional assumptions about the analyzed data (the single dimensions are statistically independent, etc.).

On the other hand, the reported examples of application of nonlinear memory estimation methods, like conditional heteroscedastic models [such as ARCH (Tsay 2005) or its generalizations], are limited to specific application areas (like econometrics and financial data analysis) and low-dimensional cases; in general they do not allow a robust estimation for very large datasets.

b. State-specific dimension reduction

All of the above arguments underline the importance of dimension reduction methods in time series analysis. To be able to find hidden metastable states in very high-dimensional data, one should be able to couple the problem of the identification of those states to an appropriate dimension reduction strategy. We will now briefly outline the main idea of one such approach, the topological dimension reduction (Horenko et al. 2006; HoSc; Horenko et al. 2008).

Let us assume that with the help of one of the methods described above, we were able to estimate the upper bound D of the memory depth for the given time series {z_t}_{t=1, . . . , T}. It is worthwhile to mention that we do not need to determine the memory depth exactly because all we are interested in later on is to cast the process into Markovian framework, as explained above. Therefore we need a lower bound on D. To account for memory effects in the analyzed data, we can extend the vector space of observables z_t at each time t with D previous observations {z_t−1, . . . , z_t−D}. The resulting vector x_t = {z_t, z_t−1, . . . , z_t−d} is a component in n = Dc-dimensional space. The idea of the method is to identify the m principal directions with the highest variance in n-dimensional data x_t(m ≪ n). In contrast to standard PCA, in which these principal directions are supposed to be global (i.e., valid for the whole time series x_t), the idea of state-specific topological dimension reduction consists of the assumption that the principal directions can vary in time and are defined with the help of a sequence of K linear projectors 𝗧_i ∈ ℝ^n×m, i = 1, . . . , K; that is, 𝗧_i is understood to project onto the subspace spanned by the local principal directions. Mathematically the problem of identifying 𝗧_i can be stated as a minimization problem w.r.t. the residuum functional, describing the least squares difference between the original observation and its reconstruction by means of the m-dimensional projection

View Expanded

where γ_i(t) (i.e., the hidden path) denotes the probability to optimally describe the n-dimensional vector x_t at time t with the local projector 𝗧_i and Σ^K_i=1γ_i(t) = 1 for all t. The quantity γ_i(t) provides a relative weight to the statement that an observation x_t belongs to the ith hidden state. For the moment we assume the sequence of probabilities γ_i(t) to be known and fixed; in the next section we will present a way to estimate this sequence from a given observation x_t. The functional L depends on the projector matrices 𝗧_i and center vectors μ_i ∈ ℝⁿ. Moreover, the projectors 𝗧_i are subject to the orthogonality condition

View Expanded

The solution of the optimization problem (3) subjected to orthogonality constraints (4) is possible in three cases (HoSc).

1) Case 1: Known hidden path

If the hidden path γ_i(t) is known, then the minimum of the functional (3) can be found, analytically resulting in a state-specific version of the PCA:

View Expanded

View Expanded

where Λ_i is a matrix with m dominant eigenvalues of the weighted covariance matrix Σ^T_t=1γ_i(t)(x_t − μ_i)(x_t − μ_i)^T on the diagonal (nondiagonal elements are zero); that is, each of the K hidden states is characterized by a specific set of essential dimensions 𝗧_i (which can be defined as corresponding dominant eigenvectors) and center vectors μ_i ∈ ℝⁿ calculated from the conditional averaging of the time series w.r.t. corresponding occupation probabilities γ_i(t) (Horenko et al. 2006).

3) Case 3: Wavelets–PCA

Because both of the HMM–PCA model assumptions are very difficult to check (especially for high-dimensional data), we need to construct a method that is free of those assumptions and which we can use for a posteriori verification of the HMM–PCA results. Therefore, we assume that the unknown function γ_i(t) can be represented as a finite linear combination of (few) discrete Haar-wavelet functions ϕ(x):

View Expanded

With their help any hidden occupation probability function γ_i(t) ∈ L₂(R) for a fixed scaling factor J ∈ Z can be represented by a linear combination of the scaled wavelet-functions:

View Expanded

If the number of ansatz functions involved in expansion (8) can be assumed to be small, it allows us to project the original high-dimensional optimization problem to the low-dimensional space of the wavelet coefficients cⁱ_r. The integral transformation between the wavelet representation and the occupation probabilities γ_i(t) can be efficiently implemented using the fast Haar-wavelet transformation (FWT; Strang and Nguyen 1997).

In our specific implementation of the wavelet-based optimization procedure (HoSc), we made two simplifying assumptions: (i) we assumed that the occupation probability functions γ_i(t) can take only discrete values 0 and 1 (i.e., the occupation probabilities are assumed to be discrete step functions) and (ii) we fixed the upper limit of the Galerkin subspace dimension for each of the optimization runs (i.e., together with the first assumption, it means that we set the upper limit of transitions between K hidden states).

The main advantage of the resulting Wavelet–PCA approach is that it is independent of the model assumptions (Markovianity and Gaussianity) of the HMM–PCA method. However, our specific implementation of the method scales quadratically with the number of involved Haar-wavelet functions; that is, the method is not applicable for very long time series with large numbers of transitions between the hidden states. But it can be used for validation of the model assumptions of the HMM–PCA by comparison of the γ_i(t) values identified by both methods for relatively short segments of the analyzed time series.

a. The expectation step

In this step, the state occupation probability γ^l_t(i) = P(X_t = i|x_t, λ) and the transition probability η^l_t(i, j) = P(X_t = i, X_t+1 = j|x_t, λ) are calculated for each time t ∈ [t^l₁, . . . , T^l], given the observation x_t and the current model λ. To calculate the two conditional probabilities of the expectation step (or E-step), we first define two additional variables:

View Expanded

View Expanded

where α^l_t(i) and β^l_t(i) are forward and backward variables, respectively. The interpretation of α^l_t(i) is as follows: it denotes the probability of the observation subsequence l up to time t together with the information that the system is in hidden state i at time t conditioned w.r.t. the given model parameters λ. The following formulas show that the computation of the sequence α^l_t(i) for the whole sequence is possible with K² T operations:

View Expanded

View Expanded

View Expanded

where ρ_j(x_t+1) = ρ(x_t+1|X_t+1 = j) defines the conditional observation probability of the data at time t + 1 in the hidden state j. The backward variable β^l_t(i) can be computed with analogous formulas:

View Expanded

View Expanded

View Expanded

Using the definitions (11) and (12), one can finally write for the probability P(x_t^l1, . . . , x_T^l|λ) (Bilmes 1998)

View Expanded

or equivalently

View Expanded

where l is chosen in such a way that t ∈ [t^l₁, . . . , T^l]. The two conditional probabilities of the E-step can be calculated efficiently by using the forward–backward variables:

View Expanded

With these values, the probability of being in state i at time t can be expressed as

View Expanded

Note that the expected number of transitions from i to any other state (including itself) within the whole observation is , and the expected number of transitions from i to j is .

b. The maximization step

This step finds a new model λ̂ via a set of re-estimation formulas. The maximization guarantees that the likelihood does not decrease in each iteration.

To apply the EM algorithm, we need to re-estimate parameters λ describing the hidden Markov model and essential linear manifolds via the maximum likelihood estimator. Thus, the observation x_t at time t ∈ [t^l₁, . . . , T^l] has to be weighted with the probability for the hidden state i γ^l_t(i) for the respective subsequence l. To calculate this re-estimation, we fix the sequence X_t of hidden states [this means also keeping the sequence of γ^l_t(i) fixed] and calculate the derivatives of the functional (10) w.r.t. the parameter set λ. By setting all of the partial derivatives to zero for some fixed reduced dimensionality m we get a coupled system of nonlinear algebraic equations for the parameters that can be solved analytically [analogous to the derivation shown in Horenko et al. (2006) and HoSc]. We will skip the derivation here and just present the final re-estimation formulas:

View Expanded

View Expanded

where max_m[spec(Cov_i)] denotes m dominant eigenvalues of the covariance matrix Cov_i;

View Expanded

The E and M steps are iteratively repeated until a predetermined maximal number of iterations is reached or the improvement of the likelihood becomes smaller than a given limit. The entire EM algorithm has the nice property that the likelihood function is nondecreasing in each step (i.e., we iteratively approximate local maxima). We will call the presented method ensemble HMM–PCA to refer to the ability of the new method to deal with an ensemble of statistically independent subsequences and to stress the difference from the standard HMM–PCA. As for the scaling of numerical effort, the resulting ensemble HMM–PCA method is linear in the length of the observation series x_t, quadratic in the number K of hidden Markov states (essentially because the transition matrix elements of the hidden Markov chain should be estimated), and scales as O(mn²) in the reduced dimensionality m (because only m dominant eigenvectors of Cov_i matrix are required, they can be obtained with numerically efficient subspace methods such as the Raley–Ritz iteration or Lanczos method). Therefore the ensemble HMM–PCA approach is applicable to systems with very high dimensionality and very long observation data sequences. This feature is demonstrated in section 5 where the method is used for analysis of a multidimensional meteorological dataset.

a. Hidden Markov process

To estimate the confidence intervals of the hidden transition probabilities A_ij we first make use of the second derivatives (∂²L/∂A²_ij)(A) (also called Fisher information) of the log-likelihood function (10) subject to the constraint

View Expanded

(A is the hidden transition matrix of the Markov chain estimated by the HMM–PCA algorithm). We denote the number of the transitions in the identified Markovian sequence X_t between the states i and j as N_ij. The most probable sequence X_t of the hidden states can be directly computed from the hidden probabilities γ^l_t(i) by applying, for example, the Viterbi algorithm (Viterbi 1967). Then it is easy to verify that the explicit expression for the Fisher information of the identified Markov chain X_t is

View Expanded

The confidence intervals of the hidden Markov process are then given by [A_ij − δ(A_ij), A_ij + δ(A_ij)], where

View Expanded

the multiplier 1.96 comes from the definition of 95% confidence interval in Gaussian statistics.

b. Extended EOFs

The Gaussianity assumption for the observation process in the HMM–PCA method gives an opportunity to estimate the confidence intervals of the manifold parameters (μ_i, 𝗧_i) straightforwardly. This can be done in a standard way of multivariate statistical analysis because the variability of the weighted covariance matrices (25) involved in the calculation of the optimal projectors 𝗧_i is given by the Wishart distribution (Mardia et al. 1979). The confidence intervals of 𝗧_i can be estimated by sampling from this distribution and calculating the m dominant eigenvectors of the sampled matrices, whereas the confidence intervals of μ_i can be acquired from the respective standard deviations (Mardia et al. 1979).

c. Optimal choice of K

If there exist two states with confidence intervals overlapping for each of the respective reduced model parameters, then those are statistically indistinguishable and K should be reduced and the HMM–PCA calculation repeated. In other words, confidence intervals implicitly give a natural upper bound for the number of hidden states. On the other hand, the spectral theory of the Markov processes connects the number K of metastable states with the number of the dominant eigenvalues in the so-called Perron cluster (Schütte and Huisinga 2003). This allows us to apply the Perron cluster–cluster analysis (PCCA; Deuflhard and Weber 2005) to find the lower bound of K. Both these criteria in combination can help to find the optimal number K of the hidden states in each specific application.

a. Relative statistical weights

The vector π of relative statistical weights of the hidden states can be calculated as the fixed point of the Markovian transition operator; that is,

View Expanded

Note that we use multiplication from the left because 𝗔 is a stochastic matrix with row sums all equal to 1.0.

b. Mean exit times

The mean exit time τ^ex_i is the expected time for the process X_t to stay in the hidden state i until it switches to any other state. Thus, it is one of the basic quantities and can be used to compare different hidden states w.r.t. their metastability. It can be directly computed from the diagonal elements of the transition matrix 𝗔:

View Expanded

where δt is the time step between the observations.

c. Mean first passage times

For any pair of two different hidden states i and j, the mean first passage time τ^pas_ij represents the expected time for the process X_t to start in the state i and to reach the state j for the first time. It can be calculated from the solution of the following linear system of equations:

View Expanded

This quantity describes the dynamical properties of the process X_t and can be used to analyze and compare different transition pathways between metastable states.

a. Description of the data

Using the method presented in the previous sections, we analyze daily mean values of the 500-hPa geopotential height field from the ERA-40 data (Simmons and Gibson 2000). We consider a region with the coordinates 32.5°–75.0°N, 27.5°W–47.5°E, which includes Europe and a part of the eastern North Atlantic. The combination of land and sea makes the selected region preferable for the appearance of dynamically relevant phenomena; it also captures the area of maximum Atlantic block formation (Wiedenmann et al. 2002). The resolution of the data is 2.5°, which implies a grid with 31 points in the zonal and 18 in the meridional direction. We have also tested the sensitivity of the results presented here by reducing the resolution by a factor of 2, taking only 16 × 9 grid points.

For the analysis we have considered geopotential height values only for winter and for the period 1958/59 to 2001/02, where a winter includes the months December to February; thus, we end with a nonequidistant time series of 3960 days. The reasons for considering winter months only were (i) because of the increased equator-to-pole temperature gradient, the synoptic eddies and the quasi-stationary Rossby waves in the atmosphere are much more intense during winter, which suggests much more pronounced regime behavior, and (ii) if we focus on blocking events only, representing a kind of metastability in the circulation, there is a pronounced maximum in the block formation for the considered region during winter (Lupo et al. 1997).

We have mentioned already in the introduction the problem with the seasonal cycle when analyzing atmospheric data w.r.t. metastable behavior. To remove the seasonal trend we apply a standard procedure, where from each value in the time series we subtract a mean build over all values corresponding to the same day and month (e.g., from the data on 1 January 1959 we subtract the mean value over all days which are the first of January, and so on).

b. The blocking index

For the purpose of interpreting the results of the presented method w.r.t. metastability of blocking events, we compute the Lejenas–Okland index from the data. It indicates the appearance of a blocking anticyclone and the duration of the event. We have a blocking if the geopotential height difference at 500 hP between 40°N and 60°N is negative over a region with 20° zonal extent. The exact formula is given in Lupo et al. (1997); for the purpose of representation we have computed a zonally averaged value of the index, rescaled it, and reversed its sign. (A part of the time series of the index is shown in Fig. 7.)

c. Discussion of the results

To choose the lower bound of the frame length in the algorithm, the memory depth of the data was estimated from the autocorrelation and partial autocorrelation function. The dominant eigenvalues of the autocorrelation matrix and of the autoregressive (AR) coefficients computed at different time lags are presented in Fig. 1. From the spectrum of the AR coefficients one can see that the data has an internal memory of about 5 days and it can be approximately modeled by an autoregressive process of the order 5; the oscillations after the fifth day are interpreted as noise. We conclude that a frame length of 5 days will be sufficient to make the data Markovian.

To choose the optimal number of hidden states K, we first start the HMM–PCA algorithm with K = 8 for different values of d = 1, 5, 10, 20, and 40 and m = 1. As mentioned above, because only a relatively short time series is available we need first to estimate the upper bound for K by comparing the confidence intervals of HMM–PCA parameters. To avoid the inherent problem of the EM algorithm—namely, that it only converges to the local maximum of the likelihood functional (dependent on the initial parameter values)—we perform the optimization with different randomly chosen sets of initial parameters 100 times and take the result with maximal likelihood. One of the transition matrix spectra is shown in Fig. 2. If the confidence intervals for a pair of states are overlapping it means that the corresponding states are statistically undistinguishable and the whole optimization procedure should be repeated for K = K − 1. It comes out that only for K = 4 are all of the hidden states statistically distinguishable; therefore, we proceed further with 4 hidden states.

Next, we have to verify the assumptions needed to apply the HMM–PCA method. The first possibility is to a posteriori check the Gaussianity of the data in the hidden states and the Markovianity of the hidden process. However, it will not guarantee that these assumptions will also be fulfilled in any of the EM iterations. Another possibility is to compare the results of the HMM–PCA optimization with, for example, some fragment of Wavelets–PCA results (because Wavelets–PCA is much slower but does not imply any assumptions about the analyzed data). This will give us a possibility to estimate the robustness of optimization w.r.t. the model assumptions. As we see from Fig. 3, the respective Viterbi paths are almost identical for both methods; therefore, it verifies the usage of the HMM–PCA analysis.

Next, we have studied the sensitivity of the results w.r.t. different frame lengths. The calculated Viterbi paths, showing the most probable sequence of hidden states, are displayed in Fig. 4. When the frame length increases, the transitions between the hidden states reduce and the occupation duration increases. The discrepancy of the Viterbi paths for different frame lengths can happen because data with smaller frame lengths are non-Markovian but the algorithm can still find some metastable regime behavior, which is filtered out if the larger frame length is applied.

We have tested the dependence of the results on the resolution, using data on a 16 × 9 and on a 32 × 18 grid for the analysis. The Viterbi paths for both grids are shown in Fig. 4; they are nearly identical. Figures 5 and 6 display the center vectors μ_i for the two different resolutions and d = 1. In both cases, the large-scale structure of the pattern is captured by the algorithm.

From Figs. 5 and 6, we see that the hidden states describe two different regimes: μ₁ and μ₃ are characterized by a negative geopotential anomaly at higher latitudes and a positive anomaly at lower latitudes, whereas the other two states, μ₂ and μ₄, have anomalies reversed in sign. Thus, the states in the first regime are associated with an intensification of the zonal flow and those in the second regime with a weakening of it. Each regime can be then subdivided into states with stronger anomalies (μ₃ and μ₄) and weaker anomalies (μ₁ and μ₂).

We expect that blocking events will be captured mostly by hidden state 4; this is confirmed if we plot the probability γ₄ and the blocking index (see Fig. 7). From the Viterbi paths and the blocking index, we calculated that state 4 and state 2 respectively capture 46% and 36% of all blocking events. If we consider as blocking situations cases in which the blocking index is negative over a period larger than 6 days (filtered index), the numbers above change to 58% and 29%, respectively. Looking at individual events, we found that the two states also represent other weather patterns with an anomalous geopotential gradient (e.g., cut-off lows). Nevertheless, about 73% of all days in state 4 are associated with blockings; for state 2 this number is 47%. If we consider the filtered blocking index, the numbers change to 52% and 21%, respectively.

Calculating the projection matrix 𝗧_i, we find the leading m EOFs within each of the hidden states and compare the variance patterns computed in this way with those from a standard EOF analysis of the dataset. A particular difference is the absence of the first EOF pattern from the standard method in the local EOFs 𝗧_i from the HMM–PCA algorithm. This mode describes the variability of the meridional geopotential gradient and, as discussed above, such a dynamics is already captured by the time evolution of the functions γ_i, i = 1, . . . , 4. The leading three variance patterns produced by the HMM–PCA algorithm looked very similar for the four hidden states; other EOFs differed if the corresponding states had positive/negative or weak/strong geopotential anomalies.

But how do the results change when we make the data Markovian, considering an extended space with the dimension n = d * c? We can split the center vector μ_i into d parts with the original dimension c, representing the mean state of the system at different time lags. The resulting sequence can be interpreted as the “mean time evolution” of the mean state in i. Fig. 8 displays such a sequence for μ₄, showing the growth in time of the meridional geopotential gradient anomaly.

To represent the results for larger frame lengths and different states, we have computed the geopotential height difference between 40°N and 60°N from the vector μ_i at different time lags, using exactly the same criteria as for the calculation of the blocking index (see section 2b), but now we consider all values, not only the negative one. The results are displayed in Fig. 9. We see that the overall time evolution is characterized by a growth or a decay of the meridional geopotential gradient, which for q = 5 reaches at the end its values from the analysis with q = 1. For larger frame lengths, the amplitude of the gradient is strongly reduced but the time evolution shows more complex character with changing phases of decay and growth (e.g., state 4 in the case of q = 40). This can probably be explained by the fact that because in those cases the duration of the blocking is smaller than the dynamical frame length d, many creations and/or destructions of the blocking situations are getting averaged out.

Because the proposed technique for frame lengths ≥2 is a special type of time-lagged statistics, it can be used to study onsets and withdrawal of diagnosed features. In this, given a time lag q, we have computed conditional composites for diagnosed events. For onsets, we have selected time slices t_j, j = j₁, j₂, . . . , j_Ne (N_e is the number of diagnosed events) when the occupation probability for the state 4(γ^l₄) reaches unity. This state was selected because it corresponds most closely to blockings as diagnosed by the employed blocking index. An additional condition is imposed that γ^l₄ remains unity at least for five consequent days (a condition of persistence). For these time slices, a conditional average is computed:

View Expanded

An analogous conditional average is computed for withdrawals by selecting the time slices t_k, k = k₁, k₂, . . . , k_Ne as the last days of the diagnosed events when γ^l₄=1. After that, a conditional average for withdrawals is computed:

View Expanded

In both cases q = 0, . . . , d − 1, where d is the frame length.

We note different interpretations of x^o(q) and x^w(q). In the former case, q covers time intervals before the block onset. As a result, the composite x^o(q) corresponds to typical synoptic conditions before the block onset. In contrast, for x^w(q), q covers time moments when a block exists and, generally, is well developed. As a result, x^w(q) has to be interpreted as a typical pattern of mature blocking state.

For onsets, the composite pattern exhibits developing meridional wavy structure (Fig. 10). This feature first appears in the southwestern part of the studied domain as a positive anomaly of geopotential height (q = 4 − 2). Afterward, at q = 1 − 0 this anomaly spreads to the east and becomes more pronounced, forming a ridge (a trough) in the southern (northern) part of the domain. Eventually, this trough–ridge system evolves to the blocked state. These features are common for the development of typical Atlantic blocking (Berggren et al. 1949; Rex 1950a; Diao et al. 2006).

For withdrawals (Fig. 11), we see a very marked positive anomaly of geopotential height in the southern part of the domain and a negative one in the northern part. Neither anomaly moves for different values of q within this composite. This emphasizes a stationarity of blockings within their life cycles. However, it becomes more marked if one travels from q = 4 to q = 0. The reason for this is the chosen length of frame, 5 days, which is comparable to the typical duration of blockings (e.g., Rex 1950b; Wiedenmann et al. 2002; Lupo et al. 1997; Diao et al. 2006; Croci-Maspoli et al. 2007). The fully developed anomaly spreads above the greater part of the northern Atlantic and attains a large magnitude.

Next we analyze the hidden transition matrix identified by the HMM–PCA in the Markovian case (K = 4, m = 1, d = 5). The transition graph correspondent to the identified matrix A is shown in Fig. 12. Each of the hidden states corresponds to a dynamical pattern of 5 days. As we have seen above in Fig. 9, each of the patterns is associated with specific blocking formation or destruction events. Therefore, by analyzing the transition graph from Fig. 12 we can gain some insight into the kinetics of such events. We start with the calculation of relative statistic weights of the respective hidden states. The solution of (29) yields π₁ = 0.2363, π₂ = 0.1836, π₃ = 0.4234, and π₄ = 0.1567; that is, the dynamical pattern corresponding to the blocking formation in hidden state 4 is the most infrequent one. To compare the metastability of the hidden states, we can calculate the mean exit times τ^ex_i from (30). We get the following values: τ^ex₁ = 4.3, τ^ex₂ = 5.3, τ^ex₃ = 14, and τ^ex₄ = 16 days. Together with Fig. 12, these can be interpreted to mean that both 3 and 4 are metastable states, whereas 1 and 2 correspond to a transition pathway between them. Blocking events associated with the hidden state 4 represent a metastable event in the Markovian model: its typical duration is 16 days and two typical transition pathways in the system are 3 → 1 → 2 → 4 and 4 → 2 → 1 → 3. To characterize and compare these two pathways, we calculate the mean first passage times. As results from (31), τ^pas₃₄ = 131 and τ^pas₄₃ = 49 days; that is, it takes much longer to “create” a blocking situation then to “destroy” it. This is also in a good agreement with the respective statistical weights π of the corresponding states; the “unblocked” metastable state 3 is visited almost 3 times more frequently then the “blocked” state 4.