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Abstract

The review comments that resulted in rejection of the paper “Hen-or-egg causality: Atmospheric CO₂ and temperature” by the Science of the Total Environment, along with the submission letter and the rejection letter, are seen in https://www.researchgate.net/publication/343879018. Here, as a rebuttal to the review comments, we post the relevant parts of our final published paper, the full version of which can be found here: https://www.researchgate.net/publication/346356891.
Rebuttal to review comments on:
Hen-or-egg causality: Atmospheric CO₂ and temperature
by D. Koutsoyiannis, and Z. W. Kundzewicz
As a rebuttal to the review comments of the manuscript with the above title, rejected by the Science
of the Total Environment, which we made available on ResearchGate, here we post parts of our final
published paper whose beginning is shown below.
Specifically, the rebuttals are contained in the following sections of the final paper:
4.2. Complications in Seeking Causality
4.3. Additional Clarifications of Our Approach
Appendix A.4. Some Notes on the Alternative Procedures on Causality
Appendix A.5. Additional Graphical Depictions
These are contained in the following pages (along with the References). Comments are welcome.
Links of related material:
Rejected manuscript: https://www.researchgate.net/publication/343878781
Review comments: https://www.researchgate.net/publication/343879018
Final published paper: http://dx.doi.org/10.3390/sci2040083
Sci 2020,2, 83 11 of 33
If η1>0, then the dominant direction is xτyτ.
If η1<0, then the dominant direction is yτ
xτ.
Justification and further explanations of these conditions are provided in Appendix A.3.
4.2. Complications in Seeking Causality
It must be stressed that the above conditions are considered as necessary and not sucient
conditions for a causative relationship between the processes
xτand yτ
. Following Koutsoyiannis [
30
]
(where additional necessary conditions are discussed), we avoid seeking sucient conditions, a task
that would be too dicult or impossible due to its deep philosophical complications as well as the
logical and technical ones.
Specifically, it is widely known that correlation is not causation. As Granger [62] puts it,
when discussing the interpretation of a correlation coecient or a regression, most textbooks warn that
an observed relationship does not allow one to say anything about causation between the variables.
Perhaps that is the reason why Suppes [
61
] uses the term “prima facie cause” in his definition
given above which, however, he does not explain, apart for attributing “prima facie” to Jaakko Hintikka.
Furthermore, Suppes discusses spurious causes and eventually defines the genuine cause as a “prima
facie cause that is not spurious”; he also discusses the very existence of genuine causes which under
certain conditions (e.g., in a Laplacean universe) seems doubtful.
Granger himself also uses the term “prima facie cause”, while Granger and Newbold [
63
] note
that a cause satisfying a causality test still remains prima facie because it is always possible that,
if a dierent information set were used, it would fail the new test. Despite the caution issued by its
pioneers, including Granger, through the years, the term “Granger causality” has become popular
(particularly in the so-called “Granger causality test”, e.g., [
64
]). Probably because of that misleading
term, the technique is sometimes thought of as one that establishes causality, thus resolving or
overcoming the “correlation is not causation” problem. In general, it has rarely been understood that
identifying genuine causality is not a problem of choosing the best algorithm to establish a statistical
relationship (including its directionality) between two variables. As an example of misrepresentation
of the actual problems, see [65], which contains the statement:
Determining true causality requires not only the establishment of a relationship between two variables,
but also the far more dicult task of determining a direction of causality.
In essence, the “Granger causality test” studies the improvement of prediction of a process
yτ
by
considering the influence of a “causing” process xτthrough the Granger regression model:
yτ=
η
X
j=1
ajyτj+
η
X
j=1
bjxτj+ετ(14)
where
aj
and
bj
are the regression coecients and
ετ
is an error term. The test is based on the null
hypothesis that the process xτis not actually causing yτ, formally expressed as
H0:b1=b2=. . . =bη=0 (15)
Algorithmic details of the test are given in [
64
], among others. The rejection of the null hypothesis
is commonly interpreted in the literature with a statement that xτ“Granger-causes” yτ.
This is clearly a misstatement and, in fact, the entire test is based on correlation matrices. Thus,
it again reflects correlation rather than causation. The rejection of the null hypothesis signifies
improvement of prediction and this does not mean causation. To make this clearer, let us consider the
following example: people sweat when the atmospheric temperature is high and also wear light clothes.
Thus, it is reasonably expected that in the prediction of sweat quantity, temperature matters. In the
Sci 2020,2, 83 12 of 33
absence of temperature measurements (e.g., when we have only visual information, like when watching
a video), the weight of the clothes algorithmically improves the prediction of sweat quantity. However,
we could not say that the decrease in clothes weight causes an increase in sweat (the opposite is more
reasonable and becomes evident in a three-variable regression, temperature–clothes weight–sweat,
as further detailed in Appendix A.4).
Cohen [
66
] suggested replacing the term “Granger causality” with “Granger prediction” after
correctly pointing out this:
Results from Granger causality analyses neither establish nor require causality. Granger causality
results do not reveal causal interactions, although they can provide evidence in support of a hypothesis
about causal interactions.
To avoid such philosophical and logical complications, here, we replace the “prima facie” or
“Granger” characterization of a cause and, as we already explained, we abandon seeking for genuine
causes by using the notion of necessary conditions for causality. One could say that if two processes
satisfy the necessary conditions, then they define a prima facie causality, but we avoid stressing
that as we deem it unnecessary. Furthermore, we drop “causality” from “Granger causality test”,
thus hereinafter calling it “Granger test”.
Some have thought they can approach genuine causes and get rid of the caution “correlation is not
causation” by replacing the correlation with other statistics in the mathematical description of causality.
For example, Liang [
44
] uses the concept of information (or entropy) flow (or transfer) between two
processes; this method has been called “Liang causality” in the already cited work he co-authors [
43
].
The usefulness of such endeavours is not questioned, yet their vanity to determine genuine causality is
easy to infer: It suces to consider the case where the two processes, for which causality is studied,
are jointly Gaussian. It is well known that in any multivariate Gaussian process, the covariance matrix
(or the correlation matrix along with the variances) fully determines all properties of the multivariate
distribution of any order. For example, the mutual information in a bivariate Gaussian process is
(Papoulis, [67])
H[y|x] = ln σyq2πe(1r2)(16)
where
σ
and rdenote standard deviation and correlation, respectively. Thus, using any quantity
related to entropy (equivalently, information), is virtually identical to using correlation. Furthermore,
in Gaussian processes, whatever statistic is used in describing causality, it is readily reduced to
correlation. This is evident even in Liang [
44
], where, e.g., in his Equation (102), the information flow
turns out to be the correlation coecient multiplied by a constant. In other words, the big philosophical
problem of causality cannot be resolved by technical tricks.
From what was exposed above (Section 4.1), the time irreversibility (or directionality) is most
important in seeking causality. In this respect, we certainly embrace Suppes’s condition (i) and
Granger’s first axiom, as stated above. Furthermore, we believe there is no meaning in refusing that
axiom and continuing to speak about causality. We note though that there have been recent attempts
to show that
coupled chaotic dynamical systems violate the first principle of Granger causality that the cause
precedes the eect. [68]
Apparently, however, the particular simulation experiment performed in the latter work which,
notably, is not even accompanied by any attempt for deduction based on stochastics, cannot show any
violation. In our view, such a violation, if it indeed happened, would be violation of logic and perhaps
of common sense.
Additional notes for other procedures detecting causality, which are not included in the focus of
our study, are given in Appendix A.4.
Sci 2020,2, 83 13 of 33
4.3. Additional Clarifications of Our Approach
After the above theoretical and methodological discourse, we can clarify our methodological
approach by emphasizing the following points.
1.
To make our assertions and, in particular, to use the “hen-or-egg” metaphor, we do not rely on
merely statistical arguments. If we did that, based on our results presented in the next section,
we would conclude that only the causality direction T
[CO
2
] exists. However, one may
perform a thought experiment of instantly adding a big quantity of CO
2
to the atmosphere.
Would the temperature not increase? We believe it would, as CO
2
is known to be a greenhouse gas.
The causation in the opposite direction is also valid, as will be discussed in Section 6, “Physical
Interpretation”. Therefore, we assert that both causality directions exist, and we are looking for
the dominant one under the current climate conditions (those manifest in the datasets we use)
instead of trying to make assertions of an exclusive causality direction.
2.
While we occasionally use statistical tests (namely, the Granger test, Equations (14) and (15)), we opt
to use, as the central point of our analyses, Equation (13) (and the conditions below it) because it
is more intuitive and robust, fully reflects the basic causality axiom of time precedence, and is
more straightforward, transparent (free of algorithmic manipulations), and easily reproducible
(without the need for specialized software).
3.
For simplicity, we do not use any statistic other than correlation here. We stress that the system we
are examining is indeed classified as Gaussian and, thus, it is totally unnecessary to examine any
statistic in addition to correlation. The evidence of Gaussianity is provided by Figures A1 and A2
in Appendix A.5, in terms of marginal distributions of the processes examined and in terms of
their relationship. In particular, Figure A2 suggests a typical linear relationship for the bivariate
process. We note that the linearity here is not a simplifying assumption or a coincidence as there
are theoretical reasons implying it, which are related to the principle of maximum entropy [
67
,
69
].
4.
All in all, we adhere to simplicity and transparency and, in this respect, we illustrate our results
graphically, so they are easily understandable, intuitive, and persuasive. Indeed, our findings
are easily verifiable even from simple synchronous plots of time series, yet we also include plots
of autocorrelations and lagged cross-correlation, which are also most informative in terms of
time directionality.
5. Results
5.1. Original Time Series
Here, we examine the relationship of atmospheric temperature and carbon dioxide concentration
using the available modern data (observations rather than proxies) in monthly time steps, as described in
Section 3. To apply our stochastic framework, we must first make the two time series linearly compatible.
Specifically, based on Arrhenius’s rule (Equation (1)), we take the logarithms of CO
2
concentration
while we keep Tuntransformed. Such a transformation has also been performed in previous studies,
which consider the logarithm of CO
2
concentration as a proxy of total radiative forcing (e.g., [
41
]).
However, by calling this quantity “forcing”, we indirectly give it, a priori (i.e., before investigating
causation), the role of being the cause. Therefore, here, we avoid such interpretations; we simply call
this variable the logarithm of carbon dioxide concentration and denote it as ln[CO2].
A synchronous plot of the two processes (specifically, UAH temperature and
ln[CO2]
at Mauna
Loa) is depicted in Figure 8. Very little can be inferred from this figure alone. Both processes show
increasing trends and thus appear as positively correlated. On the other hand, the two processes
appear to have dierent behaviours. Temperature shows an erratic behaviour while
ln[CO2]
has a
smooth evolution marked by the annual periodicity. It looks impossible to infer causality from that
graph alone.
Sci 2020,2, 83 24 of 33
For η > 0, using the property that cx[η]is an even function (cx[η]=cx[η]), we get
cxy [η]=X
j=0αjcx[jη]=Xη1
j=0αjcx[ηj]+X
j=ηαjcx[jη], (A9)
and for the negative part
cxy [η]=X
j=0αjcx[j+η]. (A10)
With intuitive reasoning, assuming that the autocovariance function is decreasing (
cx[j]<cx[j]
for
j>j
), as usually happens in natural processes, we may see that the rightmost term of Equations
(A9) and (A10) should be decreasing functions of
η
(as for
j>j
it will be
cx[jη]<cx[jη]
and
cx[j+η]<cx[j+η]
). However, the term
Pη1
j=0αjcx[ηj]
of Equation (A9) is not decreasing. Therefore,
it should attain a maximum value at some positive lag
η=η1
. Thus, a positive maximizing lag,
η=η1>
0, is a necessary condition for causality direction from
xτto yτ
. Conversely, the condition that
the maximizing lag is negative is a sucient condition to exclude the causality direction exclusively
from xτto yτ.
All above arguments remain valid if we standardize (divide) by the product of standard deviations
of the processes
xτ
and
yτ
and, thus, we can replace cross-covariances
cxy [η]
with cross-correlations
rxy [η](or, in the case of dierenced processes, r˜
x˜
y[ν,η]).
Appendix A.4. Some Notes on the Alternative Procedures on Causality
Reviewer Yog Aryal [
85
] opined that we missed referring to the recent relevant works by
Hannart et al. [
92
] and Verbitsky et al. [
93
]. In response to this comment, we include this Appendix
(not contained in Version 1 of our paper) explaining, in brief, why we do not compare our results with
the ones of those studies, also noting that only the latter study contains material that is prima facie
comparable to ours. The former study, focusing on the so-called causal counterfactual theory, is more
theoretical and also much more interesting. While we, too, are preparing a theoretical study, in which
we will discuss some theories in detail, in this Appendix, we give some key elements of our theoretical
disagreements and a counterexample that illustrates the disagreements.
We first note that in order to define causality, Hannart et al. [
92
] refer to the work on the 18th
century philosopher David Hume and, in particular, his famous book Enquiry concerning Human
Understanding [
94
] first published in 1748. From this book, we wish to quote the following important
passage, which emphasizes the diculties even in defining causality:
Our thoughts and enquiries are, therefore, every moment, employed about this relation: Yet so imperfect
are the ideas which we form concerning it, that it is impossible to give any just definition of cause,
except what is drawn from something extraneous and foreign to it.
Hannart et al. [
92
], while studying the probability of occurrence of an event Y, introduced the
two-valued variable Xfto indicate whether or not a forcing fis present, and continue as follows:
The probability
p1=P(Y=1Xf=1)
of the event occurring in the real world, with f present,
is referred to as factual, while
p0=P(Y=1Xf=0)
is referred to as counterfactual. Both terms
will become clear in the light of what immediately follows. The so-called fraction of attributable risk
(FAR) is then defined as
FAR =1p0
p1
(A11)
The FAR is interpreted as the fraction of the likelihood of an event that is attributable to the
external forcing.
They also show that under some conditions, FAR is a probability which they denote PN and call
probability of necessary causality. They stress that it “is important to distinguish between necessary
and sucient causality” and they associate PN (or FAR) “with the first facet of causality, that of
Sci 2020,2, 83 25 of 33
necessity”. They claim to have “introduced its second facet, that of suciency, which is associated
with the symmetric quantity 1
(1p1)/(1p0)
”; they denote it as PS, standing for probability of
sucient causality.
Central to the logical framework of Hannart et al. [
92
] is the notion of intervention of an experimenter,
which is equivalent to experimentation with the ability to set the value of the assumed cause to a
desired value. Clearly, this is feasible in laboratory experiments and infeasible in natural processes.
The authors resort to the “so-called in silico experimentation” which, despite the impressive name chosen,
is intervention in a mathematical model that represents the process. Hence, objectively, they examine
the “causality” that is embedded in the model rather than the natural causality. One may argue that this
it totally unnecessary. It would be better to inspect the model’s equations or code to investigate what
causality has been embedded in the model instead of running simulations and calculating probabilities.
In particular, if the models used are climate models as in [
92
], their inability to eectively describe
(perform in “prime time”) the real-world processes [
50
,
95
100
] makes the entire endeavour futile.
Another notion these authors use is exogeneity, which is related to the so-called causal graph, reflecting
the assumed dependencies among the studied variables. Specifically, they state “a sucient condition
for Xto be exogenous wrt any variable is to be a top node of a causal graph”.
Here, we will use the simple example of Section 4.2, temperature–clothes weight–sweat, to show
that using the quantities FAR (or PN) and PS may give spurious results that do not correspond to
necessary or sucient conditions for causality, at least with their meaning in our paper.
We use the two-valued random variables
x
,
y
,
z
to model the states of temperature, clothes weight,
and sweat, respectively. We designate the following states:
x=1: being hot above a threshold;
y=1: wearing clothes with weight below a threshold;
z=1: sweat quantity above a threshold;
and the opposite states with
x=
0,
y=
0,
z=
0, respectively. We choose the threshold of temperature
so that
Pnx=0o=Pnx=1o=
0.5 and that of clothes weight so that
Pny=0o=Pny=1o=
0.5.
We choose a small probability, 0.05, of wearing light clothes when cold, or heavy clothes when hot, i.e.,
Pny=1x=0o=Pny=0x=1o=
0.05 (generally, we avoid choosing zero probabilities; rather the
minimum value we choose is 0.05).
Using the definition of conditional probability,
Pny=yx=xo=Pny=y,x=xo
Pnx=xo, (A12)
we find the probability matrix Awith elements aij =Pnx=i,y=joas follows:
A="0.475 0.025
0.025 0.475 #x=0
x=1
y=0y=1
. (A13)
Now, we assign plausible values to the conditional probabilities of high sweat,
Pnz=1x=x,y=yo, as follows:
Cold, heavy clothes: Pnz=1x=0, y=0o=0.2
Cold, light clothes: Pnz=1x=0, y=1o=0.1
Hot, heavy clothes: Pnz=1x=1, y=0o=0.95
Hot, light clothes: Pnz=1x=1, y=1o=0.80
Sci 2020,2, 83 26 of 33
Again, we have avoided setting any of the conditional probabilities to 0 (or 1), and we have used
multiples of 0.05 for all of them.
Using the definition of conditional probability in the form
Pnz=zx=x,y=yo=Pnz=z,y=y,x=xo
Pny=y,x=xo, (A14)
we find the joint probabilities for each of the triplets x,y,zthat are shown in Table A1.
Table A1. Joint probabilities Pnx=x,y=y,z=zofor all triplets x,y,z
x y z =0z=1
0 0 0.38 0.095
0 1 0.0225 0.0025
1 0 0.00125 0.02375
1 1 0.095 0.38
Pnz=zo=0.49875 0.50125
Now, assume that we let an “artificial intelligence entity” (AIE) decide on causality based on the
probability rules of the Hannart et al. [
92
] framework. Our AIE has access to numerous videos of
people and is “trained” to assign accurate values of yand z, referring to clothes and sweat, based on the
images in videos. In the video images, no thermometers are shown and, thus, our AIE cannot assign
values of x, nor can it be aware of the notion of temperature. Our AIE tries to construct a causal graph
putting, say,
y
as a top node and
z
as an end node; hence, it assumes that
y
is exogenous. Based on
the huge information it can access, our AIE can (a) claim that it has constructed a prediction model
based on one part of the data (e.g., using the so-called deep-learning technique) and, hence, is able to
perform “in silico experimentation” (even though this is not absolutely necessary) and (b) accurately
estimate the joint and conditional probabilities related to
y,z
using either the model, the data, or both.
Provided that the dataset is large enough, it will come up with the true values for the conditional
probabilities, which are
bij =Pny=i,z=jo
and
cij =Pnz=jy=io
, and form the matrices Band C,
respectively, with values as follows:
B="0.38125 0.11875
0.1175 0.3825 #y=0
y=1
z=0z=1
,C="0.7625 0.2375
0.235 0.765 #y=0
y=1
z=0z=1
. (A15)
Here, the true values bij have been determined from the values of Table A1 noting that
bij =Pny=i,z=jo=Pnz=j,y=i,x=0o+Pnz=j,y=i,x=1o(A16)
and the true values cij have been determined from the definition of conditional probability:
Pnz=zy=yo=Pnz=z,y=yo
Pny=yo. (A17)
Our AIE will then implement the causality conditions of sweat on clothes weight, assigning
p0=Pnz=1y=0o=
0.2375 and
p1=Pnz=1y=1o=
0.765. It will further calculate the probability
of necessary causality as PN =0.690, and the probability of sucient causality even higher, PS =0.692.
Hence, our AIE will inform us that there is all necessary and sucient evidence that light clothes cause
high sweat.
Sci 2020,2, 83 27 of 33
Now, coming to the study by Verbitsky et al. [
93
], we notice that it assumes that “each time
series is a variable produced by its hypothetical low dimensional system of dynamical equations”
and uses the technique of distances of multivariate vectors for reconstructing the system dynamics.
As demonstrated in Koutsoyiannis [
101
], such assumptions and techniques are good for simple toy
models but, when real-world systems are examined, low dimensionality appears as a statistical artifact
because the reconstruction actually needs an incredibly high number of observations to work, which are
hardly available. The fact that the sums of multivariate vectors of distances is a statistical estimator
with huge uncertainty is often missed in studies of this type, which treat data as deterministic quantities
to obtain unreliable results. We do not believe that the Earth system and Earth processes (including
global temperature and CO
2
) are of low dimensionality, and we deem it unnecessary to discuss the
issue further. We only note the fact that global temperature and CO
2
virtually behave as Gaussian,
which enables reliable estimation of standard correlations and dismiss the need to use the overly
complex and uncertain correlation sums.
Appendix A.5. Additional Graphical Depictions
Δ𝑇 Δln
[CO]
Δ𝑇 Δ ln
[CO]₂ ₂
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
-3-2-10123
ΔT
Standard normal variate, z
Empirical
Theoretical
95% confidence limits
-0.002
0
0.002
0.004
0.006
0.008
0.01
0.012
-3 -2 -1 0 1 2 3
Δln[CO₂]
Standard normal variate, z
Empirical
Theoretical
95% confidence limits
0
0.002
0.004
0.006
0.008
0.01
0.012
-1 -0.5 0 0.5 1
y= Δln[CO₂]
x= ΔΤ
Figure A1.
Normal probability plots of
Tand
ln[CO
2
] where Tis the UAH temperature and [CO
2
]
is the CO2concentration at Mauna Loa at monthly scale.
Figure A2.
Scatter plot of
T
and
ln[CO2]
where Tis the UAH temperature and [CO
2
] is the CO
2
concentration at Mauna Loa at monthly scale; the two quantities are lagged in time using the optimal
lag of 5 months (Table 1). The two linear regression lines are also shown in the figure.
Sci 2020,2, 83 28 of 33
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
-48-36-24-12 0 12243648
Correlation coefficient
Lag (months)
ΔΤ
Δln[CO₂]
ΔΤ - Δln[CO₂], monthly
ΔΤ - Δln[CO₂], annual
ΔΤ - Δln[CO₂], fixed year
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
-48-36-24-120 12243648
Correlation coefficient
Lag (months)
ΔΤ
Δln[CO₂]
ΔΤ - Δln[CO₂], monthly
ΔΤ - Δln[CO₂], annual
ΔΤ - Δln[CO₂], fixed year
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
-48-36-24-120 12243648
Correlation coefficient
Lag (months)
ΔΤ
Δln[CO₂]
ΔΤ - Δln[CO₂], monthly
ΔΤ - Δln[CO₂], annual
ΔΤ - Δln[CO₂], fixed year
Figure A3.
Auto- and cross-correlograms of the dierenced time series of UAH temperature and
Barrow CO2concentration.
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
-48-36-24-12 0 12243648
Correlation coefficient
Lag (months)
ΔΤ
Δln[CO₂]
ΔΤ - Δln[CO₂], monthly
ΔΤ - Δln[CO₂], annual
ΔΤ - Δln[CO₂], fixed year
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
-48-36-24-120 12243648
Correlation coefficient
Lag (months)
ΔΤ
Δln[CO₂]
ΔΤ - Δln[CO₂], monthly
ΔΤ - Δln[CO₂], annual
ΔΤ - Δln[CO₂], fixed year
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
-48-36-24-120 12243648
Correlation coefficient
Lag (months)
ΔΤ
Δln[CO₂]
ΔΤ - Δln[CO₂], monthly
ΔΤ - Δln[CO₂], annual
ΔΤ - Δln[CO₂], fixed year
Figure A4.
Auto- and cross-correlograms of the dierenced time series of UAH temperature and South
Pole CO2concentration.
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
-48-36-24-12 0 12243648
Correlation coefficient
Lag (months)
ΔΤ
Δln[CO₂]
ΔΤ - Δln[CO₂], monthly
ΔΤ - Δln[CO₂], annual
ΔΤ - Δln[CO₂], fixed year
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
-48-36-24-120 12243648
Correlation coefficient
Lag (months)
ΔΤ
Δln[CO₂]
ΔΤ - Δln[CO₂], monthly
ΔΤ - Δln[CO₂], annual
ΔΤ - Δln[CO₂], fixed year
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
-48-36-24-120 12243648
Correlation coefficient
Lag (months)
ΔΤ
Δln[CO₂]
ΔΤ - Δln[CO₂], monthly
ΔΤ - Δln[CO₂], annual
ΔΤ - Δln[CO₂], fixed year
Figure A5.
Auto- and cross-correlograms of the dierenced time series of UAH temperature and global
CO2concentration.
Sci 2020,2, 83 29 of 33
Πλούταρχος Συμποσιακά ΒΒικιθήκη
Συμποσιακά Β
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
-48 -36 -24 -12 0 12 24 36 48
Correlation coefficient
Lag (months)
ΔΤ
Δln[CO₂]
ΔΤ - Δln[CO₂], monthly
ΔΤ - Δln[CO₂], annual
ΔΤ - Δln[CO₂], fixed year
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
-48-36-24-12 0 12243648
Correlation coefficient
Lag (months)
ΔΤ
Δln[CO₂]
ΔΤ - Δln[CO₂], monthly
ΔΤ - Δln[CO₂], annual
ΔΤ - Δln[CO₂], fixed year
Figure A6.
Auto- and cross-correlograms of the dierenced time series of CRUTEM4 temperature and
Mauna Loa CO2concentration.
Πλούταρχος Συμποσιακά ΒΒικιθήκη
Συμποσιακά Β
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
-48 -36 -24 -12 0 12 24 36 48
Correlation coefficient
Lag (months)
ΔΤ
Δln[CO₂]
ΔΤ - Δln[CO₂], monthly
ΔΤ - Δln[CO₂], annual
ΔΤ - Δln[CO₂], fixed year
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
-48-36-24-12 0 12243648
Correlation coefficient
Lag (months)
ΔΤ
Δln[CO₂]
ΔΤ - Δln[CO₂], monthly
ΔΤ - Δln[CO₂], annual
ΔΤ - Δln[CO₂], fixed year
Figure A7.
Auto- and cross-correlograms of the dierenced time series of CRUTEM4 temperature and
global CO2concentration.
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