Unveiling non-stationary coupling between Amazon and ocean during recent extreme events

Abstract

The interplay between extreme events in the Amazon’s precipitation and the anomaly in the temperature of the surrounding oceans is not fully understood, especially its causal relations. In this paper, we investigate the climatic interaction between these regions from 1999 until 2012 using modern tools of complex system science. We identify the time scale of the coupling quantitatively and unveil the non-stationary influence of the ocean’s temperature. The findings show consistently the distinctions between the coupling in the recent major extreme events in Amazonia, such as the two droughts that happened in 2005 and 2010 and the three floods during 1999, 2009 and 2012. Interestingly, the results also reveal the influence over the anomalous precipitation of Southwest Amazon has become increasingly lagged. The analysis can shed light on the underlying dynamics of the climate network system and consequently can improve predictions of extreme rainfall events.

Appendix

We estimate the probability via ordinal pattern symbolization (Bandt and Pompe 2002) of the time series measurements. The procedure transforms the experimental signal into a series of n-tuples that captures the dynamical patterns in the variable evolution. Each n-tuples contains n chosen indexes of the original time series ordering (usually ascending) according to its respective variable value, the n-tuple is called order pattern $\pi$ of size n. The order patterns are generated by sampling n values with delay $d$, so that the i-sample is defined as $(x_{i},x_{i+ d },x_{i+2 d },\ldots ,x_{i+(n-1) d })$. In the context of phase-space reconstruction of dynamical systems, n is also known as the embedding dimension and $d$ as the embedding delay, the setting of this two parameters are discussed in the following two paragraph.

First, let us define the choice of the embedding delay parameter. Considering Eq. (1) as the difference between the information required to explain Y given just $Y^{\tau }$ and the information required to explain Y knowing both $X^{\tau }$, $Y^{\tau }$, that is $I(X^{\tau };Y|Y^{\tau }) = H(Y|Y^{\tau }) - H(Y|Y^{\tau },X^{\tau })$ (McGill 1954), we propose an improvement in the probability estimation by choosing appropriately the embedding delay $d^{*}$ that maximize Eq. (1).

$$\begin{aligned} d^{*} = \mathrm{arg\,max}_{d \in [1, 50] \subset \mathbb {N}} \{ I(X^{\tau };Y|Y^{\tau }) \}. \qquad \qquad \end{aligned}$$

(2)

With this procedure, we are able to sample the variable through a symbolic reconstruction that is closest to the intrinsic time scale of the flux of information between the X and Y variables (Zunino et al. 2010). This approach can be applied to other coupling problems whenever the time scale of individual dynamics is different from the coupling time scales, like heart beat and blood pressure response, stock price and currency relation, etc.

Secondly, we treat the number of ordinal pattern n. The probability of each pattern $\pi$ is defined as $p_{\pi } = \lim _{N \rightarrow \infty } N_{\pi }/N$ where $N_{\pi }$ is the number of occurrences of the pattern $\pi$ and $N=\sum _{\pi } N_{\pi }$. In the case, we are dealing with randomly independent variable then $p_{\pi }\rightarrow 1/n!$. This give us the upper bound of the entropy $H \rightarrow \log _{2} n!$. We adopt a frequentist approach where the probability of each pattern $p_{\pi }$ is approximated by $p_{\pi } \approx N_{\pi }/N$. Notice that the last term of Eq. (1) depends on the evaluation of the joint probability $p(X^{\tau },Y,Y^{\tau })$. In the scenario where all variable are randomly independent, the joint probability is $1/\left( n!\right) ^{3}$ and it would be required to sample a number of points much greater than $\left( n!\right) ^{3}$ to capture the independence between variables and avoiding artifacts due to small sample sizes. For this reason, it is prudent to use two order pattern ($n=2$), so that the size of the sample points in the time series N should be $N \gg 8$.