Peer review of: Partial cross mapping eliminates indirect causal influences (2020)


Reviewers' comments: 

Reviewer #3 (Remarks to the Author): 

The manuscript discusses a very important question of detecting causality from data and addresses 
the problem of misidentifying indirect causal inﬂuences as direct ones. This is a very important 
question relevant for many applications, and is of interest to those who are developing such methods 
as well as wider – almost in any area of research where cause-consequence relations in dynamical 
systems are considered. 

The authors propose an interesting and novel method that could help advance the filed. While this 
referee agrees with the authors that "the methods based on prediction frameworks, such as the 
Granger causality, the transfer entropy, and all their extensions, mathematically are not suitable for 
dealing with the time series data produced by nonlinear dynamical systems...", their claims of 
universality of their method may be slightly stretched. Here are the reasons why this may be so – 

1. The method is based on the embedding theorem, proposed independently by Takens and Mane
(authors may consider adding Mane’s reference as well: R. Mañé (1981). "On the dimension of the
compact invariant sets of certain nonlinear maps". In D. A. Rand and L.-S. Young (ed.). Dynamical
Systems and Turbulence, Lecture Notes in Mathematics, vol. 898. Springer-Verlag. pp. 230–242). The
theorem is applicable for autonomous systems, while it has been shown in several studies that it has
drawbacks when applied to data from nonautonomous systems (see e.g. Clemson and Stefanovska,
Physics Reports, 2014, DOI: 10.1016/j.physrep.2014.04.001 for discussion). Because most of the
real-world systems are nonautonomous, authors could discuss the limitations of their method, or how
it can be applicable to nonautonomous systems.
2. In their comparison authors focus on Granger causality and entropy methods, but there are also
other methods that have been proposed to study couplings, such as those based on dynamical
Bayesian inference (see for example Stankovski et al, Coupling functions in networks of oscillators,
New Journal of Physics, 17: 035002, 2015, or the recent review Stankovski et al Rev. Mod. Phys. 89,
2017). To make the claim of universality stronger authors could include comparison/discussion of this
type of methods as well.
3. The effect of noise and the number of points. This is a very important issue, but seems insufficiently
elaborated in the current version of the manuscript, and again makes the claim of universality weaker.
Here, it would be helpful if authors use units of time for their time series (both numerically generated
or obtained from measurements), otherwise they can use the term points series instead. Namely, both
the effect of noise and the effect of length of the recordings are relative to the dynamical properties of
the system in question, hence the usage of absolute time units would help evaluate the effect of noise
and the length of recording much more realistically. This referee suggests that all time series are
presented in time-units and the discussion about the effect of noise and length of recording is made
relative to the time-scales of the system.

Additional comments 

1. Authors may consider bringing parts of the supplementary material to the main text. Namely, one
of the important part of the work is the concept of non-separability and it is explained in the
supplementary material, while it naturally belongs to the main text. The procedure is also central part
of the work, and again, is provided in the supplementary material. The examples could also be added
to the main text, as, once sufficiently elaborated, they will strengthen the claim of the universality of
the method.
2. “Takens’ embedding theory” is more specifically the Takens-Mane theorem.

3. The references are relatively well formatted, however many journal titles are written with small 
letters and abbreviations are not consistently used. 
4. Authors may consider making the code publicly available, as well as the data from examples that 
will be included, to increase the transparency of their work and make possible for others to reproduce 
it. 
 
Aneta Stefanovska