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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 xτ→yτ.

•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 suﬃcient

conditions for a causative relationship between the processes

xτand yτ

. Following Koutsoyiannis [

30

]

(where additional necessary conditions are discussed), we avoid seeking suﬃcient conditions, a task

that would be too diﬃcult 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 coeﬃcient 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 diﬀerent 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 diﬃcult 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

ajyτ−j+

η

X

j=1

bjxτ−j+ετ(14)

where

aj

and

bj

are the regression coeﬃcients 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 suﬃces 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(1−r2)(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 coeﬃcient 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 eﬀect. [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 diﬀerent 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[j−η]=Xη−1

j=0αjcx[η−j]+X∞

j=ηαjcx[j−η], (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[j′−η]<cx[j−η]

and

cx[j′+η]<cx[j+η]

). However, the term

Pη−1

j=0αjcx[η−j]

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 suﬃcient 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 diﬀerenced 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 diﬃculties 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 =1−p0

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 suﬃcient 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 suﬃciency, which is associated

with the symmetric quantity 1

−(1−p1)/(1−p0)

”; they denote it as PS, standing for probability of

suﬃcient 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 eﬀectively 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 suﬃcient 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 suﬃcient 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 suﬃcient causality even higher, PS =0.692.

Hence, our AIE will inform us that there is all necessary and suﬃcient 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]₂

₂

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.

Δ𝑇 Δln

[CO]₂

₂

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 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

₂

₂

₂

-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 diﬀerenced time series of UAH temperature and

Barrow CO2concentration.

₂

₂

₂

-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 diﬀerenced time series of UAH temperature and South

Pole CO2concentration.

₂

₂

₂

-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 diﬀerenced time series of UAH temperature and global

CO2concentration.

Sci 2020,2, 83 29 of 33

₂

₂

Πλούταρχος Συμποσιακά ΒΒικιθήκη

Συμποσιακά Β

₂

-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 diﬀerenced time series of CRUTEM4 temperature and

Mauna Loa CO2concentration.

₂

₂

Πλούταρχος Συμποσιακά ΒΒικιθήκη

Συμποσιακά Β

₂

-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 diﬀerenced time series of CRUTEM4 temperature and

global CO2concentration.

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