Questionable dynamical evidence for causality between galactic cosmic rays and interannual variation in global temperature.

Tsonis et al. (1) claim a significant causality between cosmic rays (CRs) and interannual variation in global temperature (ΔGT) by using convergent cross mapping (CCM) (2). Their results are potentially helpful to model the climate system. However, we are unable to reproduce their results and suspect that this dynamical evidence uncovered by CCM is questionable. CCM is based on the theory of state space reconstruction, and it was originally introduced by Rulkov et al. (3). If variable Y unidirectionally causes X, then the values of Y can be reconstructed from the state information of X via cross mapping (denoting by X xmap Y) based on a library (=training dataset) MX,Y containing pairs of ð~x, yÞ recorded simultaneously, but not vice versa. For causality, the prediction performance of X xmap Y improves with increasing library size L, which is what convergence means. Choosing libraries suitably is crucial. Different methods may lead to different results. However, there is no evidence in refs. 1, 2, and 4 that the authors are aware of this problem. No relevant information can be found in ref. 1. Different methods are used in refs. 2 and 4. According to the shared code in ref. 4, they first form M ~ X,Y containing all pairs ð~x, yÞ and then, for a given L, construct ML,X,Y by sampling pairs from M ~ X,Y using the bootstrap method (method 1). In ref. 2, ~x constructed from a segment of X and the corresponding y are used to form ML,X,Y (method 2). For unbiased results, it is essential that, for each predictee yi, the pair ð~xi, yiÞ must be excluded from ML,X,Y. Thus, the library size after exclusion may shrink for methods 1 and 2. Therefore, we propose method 3 which constructs libraries by bootstrapping from M ~ X,Ynfð~xi, yiÞg for each predictee yi. Advantages of this method over methods 1 and 2 are that it keeps the library size fixed and removes any nonstationarity of libraries reconstructed on the basis of a segment of X by uniformly sampling from the whole embedding space. Therefore, method 3 is recommended when implementing CCM. Using methods 1–3, results in Fig. 1 show no evidence for causality between CR and ΔGT. Studying the prediction performance for many L values is a version of multiple testing (5). A statistically significant rejection of the null hypothesis requires that significantly more than 5% of all trials lie outside the 95% confidence interval, which is not given if a single value is outside (as in ref. 1). Alternatively, one could enlarge the confidence interval using, e.g., a Bonferroni correction (which is not necessary in Fig. 1, because all curves are located inside the confidence intervals of null hypotheses). In summary, our analysis of the same datasets does not yield any statistically significant indication. Considering the possible effect of multiple testing, it is more appropriate to conclude that no significant causality between CR and ΔGT can be uncovered by CCM.