## 2 Region, Data, Method
### Spectral Measure of Coherent Behavior of Seismic Noise Parameters  
To diagnose synchronization effects in changes to the 4-dimensional time series of noise parameters, we used the value of the spectral measure of their coherent behavior \(\
u(\tau,\,\omega)\), estimated in a sliding time window. Previously, the spectral measure of coherent behavior was used in [5; 11; 12] for diagnosing synchronization effects in changes in geophysical, geochemical, hydrological, meteorological and other multidimensional time series. The spectral measure of coherence of a 4-dimensional series of seismic noise parameters is constructed as the modulus of the product of the by-component canonical coherences of a multidimensional series \(\
u(\tau,\,\omega)=\prod_{i=1}^{m}\left|\mu_{j}(\tau,\,\,\omega)\right|\), where \(m=4\) is the total number of analyzed time series that make up a multidimensional series (dimension of a multivariate time series); \(\omega-\)frequency, day\({}^{-1}\); \(\tau\) is the time coordinate of the right end of the sliding time window, consisting of a given number of samples of the time series; \(\mu_{j}(\tau,\,\omega)\) is the canonical coherence of the \(j\)-th scalar time series, which characterizes the degree of connection of this series with all other series that make up the multidimensional series. The value \(\mid\mu_{j}(\tau,\,\omega)\mid^{\,2}\) is a generalization of the quadratic coherence spectrum between two signals, where the first signal represents the \(j\)-th scalar time series, and the second signal is a vector that reflects the overall changes of the remaining three series. The quantity \(\mu_{j}(\tau,\,\omega)\) satisfies the inequalities \(0\leq\mid\mu_{j}(\tau,\,\omega)\mid\,\leq 1\), from which it follows that the closer the value \(\mid\mu_{j}(\tau,\,\omega)\mid\) to unity, the more linearly related are the variations at the frequency \(\omega\) in the time window with the coordinate \(\tau\) of the \(j\)-th time series, with similar variations in the other three time series. Thus, the value \(0\leq\
u(\tau,\,\omega)\leq 1\), by virtue of its construction, describes the effect of the cumulative coherent (synchronous, collective) behavior in the time-frequency domain of all seismic noise parameters' time series that form a 4-dimensional time series. Since the values of \(\
u(\tau,\,\omega)\) vary in the interval [0, 1], the closer the corresponding value is to unity, the stronger relationship between the variations of the 4-dimensional time series components at the frequency \(\omega\) for the time window with the coordinate \(\tau\). It should be noted that comparison of the \(\
u(\tau,\,\omega)\) absolute values is possible only for the same number \(m\) of simultaneously processed time series, because, by virtue of the formula for \(\
u(\tau,\,\omega)\), as \(m\) grows, the value of \(\
u\) decreases, as the product \(m\) values less than one. In this paper, the number of simultaneously analyzed time series is \(m=4\). To estimate the spectral matrix of 4-dimensional time series, a 5 th order vector autoregressive model was used (AR = 5) [9]. Taking into account the problem of identifying prognostic effects preceding earthquakes, the \(\
u(\tau,\,\omega)\)-calculated values in all diagrams are referred to the right edge of the current time window.