Causally coupled autoregressive processes and local information-theoretic signatures of the response of a system with three types of regime shifts: the change in (a) strength of a causal driver, (b) strength of a hidden driver, and (c) intrinsic uncertainty. Figures report mean at each time step and the standard error in the mean, obtained from 100 independent simulations. The yellow vertical line indicates t C , purple line shows causal coupling strength C(t), and the gray dashed line shows the strength K(t) of a hidden driver. The parameters α 1 = α 2 = 0.2, d = 0.5 in all figures. In (a) and (b), β 1 = β 2 = 1, while in (c), β 1 = β 2 are functions of time, defined using (17) and depicted in the figure in gray dashed and magenta dash-dot lines, while K(t) and C(t) are constants, equal to 0.01 and 0.5, respectively.

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... s relates to the maximum value of the function and b to the sharpness of the transition. For example, a regime shift of a causal driver may be modeled as a change of a coupling strength term C(t). As Fig. 5(a) shows, in case of a regime shift in a causal driver, we observe a significant change in the absolute values of local information dynamical measures. 38 This situation is characterized by large TE and large AIS in the high-coupling regime and no significant change in MI. In Appendix D, we also show that a change in MI is also possible, ...

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... the case of a regime shift of a hidden (common, simultaneous) driver can be modeled via variation in the coupling strength K(t). The result, shown in Fig. 5(b) suggests that the change in a hidden driver's strength is signified by high MI when coupling is strong, and vice versa when coupling is weak. Note that changes in AIS and TE are possible: the former is possible when K(t) is sufficiently high-valued, while the latter is possible when C(t) is ...

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... (17) i.e., the system's overall uncertainty is a time functional, while C(t) = K(t) = 0.3 = const. In this case, we find that all three information dynamic measures can be mutually low-valued on the side of transition that models high-uncertainty regime, and mutually lowvalued on the other, where the system's intrinsic uncertainty is low, see Fig. ...

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... that the results shown in Fig. 5 are not coincidental, further statistical proof is needed to prove that signatures are persistent in a general spectrum of nonlinearly coupled autoregressive systems. We also note that it is more than possible that a combination of these effects comes into play in real data, and such convolution and its signature is not studied any ...

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... time is thought to be characterized by a peak in MI among agents that constitute the system. [42][43][44] The authors of Ref. 20 claim that stock market crashes exhibit a peak in MI at the point in time when one would expect a significant regime change to take place. The peak in MI observed in Fig. 7 for spread is similar to the case simulated in Fig. 5(b), where a mutual hidden driver increased in strength, making the system more ...

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... spread variables, we see a large amount of total apparent transfer entropy, as well as large average AIS, persisting for weeks preceding the price peak, similar to a situation simulated in Fig. 5(a), where the importance of information sharing increased to a larger value after t C in contrast with before. Therefore, we suggest that spread system variables transit from a strong-coupling regime before the price peak to a weak-coupling regime after the crash. Our intuition is further supported by similar results observed by ...

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... peak of the bubble is characterized by a clear, although not dramatic, reduction in both TE and MI, and to a lesser extent, reduction in AIS. We observe that for O (and to a smaller extent, for r), all three measures-AIS, TE, MI-follow the same pattern, that is, when TE is large, AIS, MI are also large. We were able to simulate this scenario in Fig. 5(c) by changing the autoregressive parameters responsible for the amount of uncertainty in the signals. When the amount of noise, unique for each system's constituents, is varied, the three information dynamics measures either increase or decrease in magnitude simultaneously as a response, being high-valued when the uncertainty is low. The ...

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... we perform a parameter sensitivity analysis for the results, shown in Fig. 5. Note that in Figs. 13 and 14 we do not consider the parameters α 1 , α 2 , β 1 , β 2 as variables and set them to α 1 = α 2 = 0.2 and β 1 = β 2 = 1. In all figures, we also chose d = 0.5. We note that for a definite conclusion regarding these information dynamical signatures, a thorough analysis of the relation to these parameters ...

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... Fig. 13, we show that a signature of high TE and AIS with no change in MI indicates strong causal coupling [illustrated in Fig. 5(a) of the main text], and this statement is robust for a wide range of parameters. It is worth pointing out, however, that at high levels of K, a significant amount of MI is also detectable in a weak-coupling regime. Fig. 14, we show that for a wide range of parameter values that characterize the strength and steepness of the regime ...

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... Fig. 15 shows that a change in overall system's uncertainty [the amount of uncorrelated random noise, that was illustrated in Fig. 5(c) of the main text] is detectable with a signature of having low TE, MI, and AIS in a high-uncertainty regime and vice versa in a low-uncertainty regime. Parameter b quantifies the abruptness of the transition, ...

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... Fig. 15 shows that a change in overall system's uncertainty [the amount of uncorrelated random noise, that was illustrated in Fig. 5(c) of the main text] is detectable with a signature of having low TE, MI, and AIS in a high-uncertainty regime and vice versa in a low-uncertainty regime. Parameter b quantifies the abruptness of the transition, whereas s quantifies the magnitude of change in the two regimes centered at t C . MI, TE, AIS are high in the high-uncertainty ...