PreprintPDF Available

Discussing anthropogenic global warming from an econometric perspective: a change scenario based on the Arima paleoclimate time series model

Preprints and early-stage research may not have been peer reviewed yet.

Abstract and Figures

Global warming has divided the scientific community worldwide with predominance for anthropogenic alarmism. This article aims to project a climate change scenario using a stochastic model of paleotemperature time series and compare it with the dominant thesis. The ARIMA model, an integrated autoregressive process of moving averages, popularly known as Box-Jenkins, was used for this purpose. The results showed that the estimates of the model parameters were below 1 degree Celsius for a scenario of 100 years which suggests a period of temperature reduction and a probable cooling, contrary to the prediction of the IPCC and the anthropogenic current of an increase in 1,50 degree to 2,0 degree Celsius by the end of this century. Thus, we hope with this study to contribute to the discussion by adding a statistical element of paleoclimate in counterpoint to the current consensus and to placing the debate in a long term historical dimension, in line with other research already present in the scientific literature.
Content may be subject to copyright.
Discussing anthropogenic global warming from an econometric
perspective: a change scenario based on the Arima paleoclimate time
series model
Gilmar Veriato Fluzer Santos¹†*, Lucas Gamalel Cordeiro¹†, Claudio Antonio Rojo¹†,
and Edison Luiz Leismann¹†.
¹Western Paraná State University, Graduate Program in Management/Professional Master’s Degree, Cascavel,
85819-110, Brazil.
*Gilmar Santos, mail to:
Global warming has divided the scientific community worldwide with predominance for
anthropogenic alarmism. This article aims to project a climate change scenario using a stochastic
model of paleotemperature time series and compare it with the dominant thesis. The ARIMA model –
an integrated autoregressive process of moving averages, popularly known as Box-Jenkins - was used
for this purpose. The results showed that the estimates of the model parameters were below 1°C for a
scenario of 100 years which suggests a period of temperature reduction and a probable cooling,
contrary to the prediction of the IPCC and the anthropogenic current of an increase in 1.50° C to 2.0°
C by the end of this century. Thus, we hope with this study to contribute to the discussion by adding a
statistical element of paleoclimate in counterpoint to the current consensus and to placing the debate
in a long-term historical dimension, in line with other research already present in the scientific
Global warming. Paleoclimatology. Time series. Arima model. Climate scenarios.
The controversies over global warming and its effects on the economy and the environment are
the subject of discussion and debate around the world and in some ways determine how governments
and companies develop their policies and conduct their business.
The human action according to the followers of anthropogeny and other international bodies
such as the IPCC (Intergovernmental Panel on Climate Change - UN), has been responsible for
climate change and global warming (greenhouse effect). This is endorsed by most scientific
publications by showing that more than 90% of studies on the subject say that the cause of global
warming is anthropogenic, established as the "official version" by IPCC advocates (Salzer, Neske &
Rojo, 2019; Cook et al. 2013; Bray, 2010; Anderegg et al., 2010; Oreskes, 2004).
Shwed and Bearman (2010) bring an important contribution in the strategy of assessing
the state of scientific contestations on certain issues when the scientific community considers a
proposition a fact and how the importance of internal dissent in the face of consensus
The IPCC Working Group Chair, Jim Skea, states: "Limiting warming to 1.5°C is possible
within the laws of chemistry and physics but doing so requires unprecedented changes" (IPCC
Special Report, 2019). Supporters of the naturalistic cause, on the other hand, present arguments that
challenge these studies by claiming that anthropogenic global warming is theoretically fragile with
calculated misinformation, and its historical sample of only 150 years would be insufficient to
establish a consensus often supported by agnotology and metric uncertainties (Molion, 2008; Legates
et al. 2015; Legates, Soon & Briggs, 2013; Reinsinger et al. 2010).
What is noticeable is that the more research explores the past, the more the anthropogenic
thesis is weakened, as demonstrated by Davis (2017) and Harde (2019)
by finding that changes in
the atmospheric CO2 concentration did not cause changes in ancient climate temperature and climate
change is not related to the carbon cycle, but rather to native impacts. Easterbrook (2016), in his
evidence-based book brought data opposing CO2 emissions as the primary source of global warming,
the thesis of which has been captured by politics and dubious computer modeling.
Other pro-anthropogenic studies ignore paleoclimatology as a relevant factor in research or
have it as a factor of uncertainty, such as that of Haustein et al. (2017), Cook et al. (2013), Mitchell
et al. (2017), Medhaug et al. (2017), in addition to those that underpinned the IPCC report (Solomon
et al. 2007). However, increasingly scientists are pointing to data which suggests that climate changes
are a result of natural cycles, which have been occurring for thousands of years, says Easterbrook
Thus, it is possible to identify a gap in this debate which is a broader time horizon research
and give statistical predictability to climate change. This is the objective of this study, whose essence
is to establish a climate prediction scenario based on a series of paleotemperatures of 12 thousand
years (Holocene)
until nowadays, plus the uncertainties that the data used predict. For this, the
integrated autoregressive moving average econometric model (ARIMA - BJ) was adopted as a
method whose database was extracted from the article by Kaufman et al. (2020), who applied five
statistical methods of thermal reconstruction to ascertain the global average surface temperature
to the present day, which served as the basis for this research.
Results generated indicated the fragility of the anthropo-warming thesis, which showed
significant divergence from the scenario projected by the IPCC, which in its latest report predicted
an increase of more than 1.5°C in the planet's temperature by 2050 (IPCC, 2019).
Therefore, we sought to establish one more variable for the global warming issue, in order to
innovate the discussion and enable a technically critical approach, with the intention of comparing it
with the consensus that prevails today.
The parameters used to reach at the results were the median and the 5th and 95th percentiles
representing the estimate of uncertainties with 90% confidence, as the authors themselves indicate
by recommending that
“future users of this reconstruction use the full ensemble when considering
the plausible Holocene GMST evolution. By representing the multi-method
reconstruction as a single time series, the median of the ensemble may be best
along with the 90% range of the ensemble to represent uncertainty."
(Kaufman et al., 2020, p.04).
For building the results, the data were represented graphically and fed the software IBM -
SPSS Statistics, v. 22, for processing the ARIMA methodology - Box Jenkins methodology and the
corresponding outputs according to each step of the calculation. Figure 1 shows the evolution of the
12k median of the data set extracted from Kaufmann et al. (2020) on a 100-year scale, with milestone
"0" being the year 2019 (p. 8) calculated from the different reconstruction methods.
Figure 1. Evolution of the Global Median 12k years temperature.
Source: Author elaboration (adapted from Kaufman et al. (2020 p. 06) from CSV file data at
Figure 2 represents the 5th and 95th percentile range of the set bringing together the
various sources of uncertainty, including proxy temperature, chronology, and methodological
choices, as per Kaufman et al. (2020 p. 03).
Figure 2. Evolution of the parameters 5th and 95th global percentiles (uncertainties).
Source: Author elaboration (adapted from CSV file data - temp 12K all methods percentiles
The average temperature of the 1800-1900 period for each composite was used as the
pre-industrial reference period defined by the authors as an anomaly of 0° C and which served
as the reference for the IPCC (1850-1900). For this reason, it was removed from each member
of the ensemble to avoid issuing individual records and different reconstructions (Kaufman et
al, 2020).
Box-Jenkins ARIMA model's objective is to provide a valid basis for forecasting, after
Evolu t ion of t he m edian glob a l t e mpera t ure 12,000 ye ars
∆ global temperature 5th e 95th
Global 12 K GMS T t e mpera t ure 5th a n d 9 5 t h p ercen t is
global_5 global_95
all tests, parameters, and diagnostics have been performed. The forecasts of the two-time
series, median and uncertainties, were generated in the IBM - SPSS Statistics software, version
22, in a specific session for ARIMA modeling.
According to the model parameters, predictions for the median were expressed in the
form of temperature estimates, for the next 100 years, represented by AR and MA. For statistical
reliability purposes, the degree of significance (Marôco, 2018) of the parameters must be
measured, being extremely significant in AR and very significant in MA, as described in figure
Arima model parameters
Estimate SE t Sig.
MdTempGlob No transformation Constant ,191
AR Lag 1 ,932
MA Lag 1 -,266
Figure 3: 100-year scale temperature estimates of AR and MA parameters.
Source: Author elaboration with Software SPSS - Statistics v. 22.
An important condition for model reliability is the residuals of the ACF and PACF
correlations, the white noise. For the model to be validated as the most adequate, they should be
concentrated around the mean, and the degree of significance is absolute (0 or close), thus
represented in figure 4. (Note: Retardo means Lag; “de resíduo” means of waste)
Figure 4: Residuals of the ACF and PACF correlograms (White noise).
Source: prepared by the author (SPSS - Statistics v. 22).
Thus, once stationarity is achieved (see p. 7-9), we can model it with an autoregressive
process (AR), which we will represent by Y the Median (Md) at period t (Holocene) as:
    
   
where δ is the mean of Y and ut is an uncorrelated random error with zero mean and constant
variance ² (this is white noise), then we will say that Yt follows a first-order stochastic
autoregressive or AR process (1).
The AR process we have just discussed is not just a mechanism that may have generated Y.
In this case, Y may evolve into a first-order moving average process, or an MA (1). If we model Y in
this way:
where µ is a constant and u, as before, is a white noise stochastic error term. Here Y at period t is
equal to a constant plus a moving average of the current and past error terms. More generally, we can
represent it like this
               (3)
which is an MA(q) process. In short, a moving average process is just a linear combination of white
noise error terms. In this case, most likely Y has characteristics of both AR and MA and is therefore
ARMA. Then Yt follows an ARMA (1,1) process, and can be written as
         (4)
because there is an autoregressive term and a moving average term. In the Equation, ɵ represents a
constant term. In general, in an ARMA (p, q) process, there will be p autoregressive terms and q
moving average terms.
In the fit chart, shown in figure 5, it is observed that the two lines coincide, almost
overlapping, indicating that this is the best of the models tested. The outliers present between 1 and
5 dates were kept in the setup since if we were to remove them, the series would not be robust. This
guarantees its impartiality and uncertainty for future events (Stockinger & Dutter, 1987). Note:
observado means observed; ajuste means adjust; UCL: the upper control limit; LCL: the lower
control limit.
Figure 5. Graph of the adjusted 12k median series.
Source: elaborated by the author (SPSS- Statistics).
Regarding the uncertainty results 5th and 95th percentiles, the process follows the same
model as the median, whose configuration is described in a supplementary file. The following
parameters were generated, according to figure 6:
ARIMA model parameters
Estimate SE t Sig.
GLOBAL5-Model_1 GLOBAL5 No transformation Constant -2,403
AR Lag 1 ,999
Lag 1 -,700
GLOBAL95-Model_2 GLOBAL95 No transformation Constant ,149
AR Lag 1 ,996
Lag 1 -,382
Figure 6. Parameters of the 5th and 95th percentile temperatures (model uncertainty).
Source: Author elaboration (SPSS - Statistics v. 22).
We then have a set of six different extremely significant temperature results for the estimates
of the two models, namely 0.932; -0.266 (fig. 3) and 0.999; -0.70; 0.996; -0.382 (fig.
To fulfill the objective of this study, it is necessary that a standard measure be calculated and
adopted as a reference. The median, extracted from the set of estimates of both models is the most
appropriate statistical measure in this case, whose result was 0.333 (calculated from Microsoft
Excel). It is evident, thus, a temperature well below the 1.50% to 2.00% projected by the IPCC by
the end of this millennium. The results generated here indicate that as opposed to warming, the
scenario being drawn is that the world may go through a period of decreasing temperatures in the
next hundred years, which could imply a cooling of global scope.
Given the results presented, one must ask why there is so much consensus around a scenario
that as the evidence shows here, leaves much room for doubt? Another question that arises is why
there is so much scientific unanimity around anthropogenic warming (97.2% according to Cook et
al., 2013), now called "climate change"?
It is understanding that if we compare recent temperatures to the distribution of global
maximum temperatures during the Holocene, there was on average a 1 increment over the pre-
industrial period (1850-1900) and for most members of the ensemble, no 200-year interval during
the series exceeded the warmth of the most recent decade (Kaufman et al., p. 5). We see, therefore,
that the time horizon of the anthropogenic thesis is recent to the time of man's existence on earth
(Holocene) and when compared to the results of this research, lacks substantiation if analyzed in the
light of statistical science.
On the other hand, Kaufman et al., (2020), when relying on the IPCC projections, admit that
temperatures for the rest of this century are likely to exceed 1 if compared to those of the pre-
industrial era (1800-1900), which they considered as an anomaly of 0. Although the authors claim
that the Holocene GMST reconstruction is comparable with the IPCC long-term projections and those
seen in the last decade, the results presented here show a different and antagonistic scenario if one
considers a hundred-year scale and the historical temporality present in the statistical series.
It is known that one of the villains of anthropogenic genesis, the greenhouse effect, was
already unveiled in 1896 by Arrhenius as a natural phenomenon beneficial to the development of
biological life on the earth's surface (troposphere) whose subsequent studies were duly confirmed
(Miller & Spoolman, 2016). Therefore, reinventing this evidence is something that does not hold up
in light of the intuitive and deductive capacity of science, as the proponents of anthropogeny claim
in establishing a rationale for global warming.
It does not exclude the impact that human action has brought to recent climate change, which
might be important and timely, but seems to be insignificant in the face of the millennial variability
of the climate, the size and complexity of the universe, and all the natural and astronomical
phenomena that interact with the earth in the planetary system.
Lastly, it should be argued that the climate scenario predicted here is not enough to determine
which are the true causes of recent climate change, whether natural or anthropogenic, since the two
may be complementary, not divergent. For this, new studies on paleoclimate and its variability are
needed to corroborate the estimates resulting from this research and to bring more evidence in the
search for scientific truth.
It is not credible to submit the world and organizations to the intention of a doubtful thesis
with all the consequences that this brings to the strategic planning of political and economic agents.
It would condemn humanity to an environmentalist dogmatism and a catastrophism without any
support to justify them.
Data and methods
The data for this paper were collected from Kaufmann et al., (2020) unprecedented multi-
method reconstruction research of mean land surface temperature (GMST) during the Holocene era
(12,000 years) to the present day, "whose database is the most comprehensive global compilation
of previously available published Holocene proxy temperature time series" (Kaufman et al., 2020,
p. 01).
Extraction of the primary data from this study is available as individual CSV files and
merged as a netCDF file at figshare 35 and at NOAA Palaeoclimatology 36
( A CSV file with the multi-method joint median
and 5th and 95th percentiles is also available in both data repositories. All were used as input data
to compose the 12k time series of paleotemperatures in the two variables and fed into IBM SPSS-
Statistics software (v. 22) for the calculation of parameters and estimates. The data generated for
the development of this research are
available in supplementary file.
Stochastic Processes and the Stationarity Test
To introduce the development of the forecast, we justify graphically and mathematically the
results that the SPSS software generated with their respective outputs, in the two variables of this
study, the median and the uncertainty set. To better understand, we will use the graphs in this section
and the mathematical formulation of their results as well as the structuring of the uncertainty set (same
pattern) in a supplementary file.
First of all, it is necessary to apply two tests to verify the stationarity of the time series: (1)
graphical analysis and (2) the correlogram test, since it is a condition for using the ARIMA (BJ)
By analyzing Figures 1, 2 and 5, we verify that the series are not stationary, that is, by
establishing a mean line for the 12K global temperature median series (Figure 7) we verify that the
data do not circulate around it and express a trend. Note: número de sequência means sequence
Figure 7. Graphical test for stationarity.
Next, we apply the correlation tests, also called "F" correlation function: ACF (automatic)
and ACFP (partial), the next step to make the series stationary, as shown in figures 8 and 10, and
their respective reports, represented by figure 9.
Note: coeficiente means coefficient. Número de retardo
means Lag numbers.
Figure 8. Graphical test of autocorrelation (automatic).
Automatic correlations
Series: MdTempGlob
Lag Autocorrelation Standard
Box-Ljung Statistics
Value df Sig.
1 ,956 ,090 113,376 1 ,000
2 ,887 ,089 211,775 2 ,000
3 ,821 ,089 296,777 3 ,000
4 ,759 ,089 370,121 4 ,000
5 ,702 ,088 433,357 5 ,000
- - - - - -
- - - - - -
16 ,246 0,84 723,810 16 ,000
a. The underlying process considered is independence (white noise).
b. Based on the asymptotic chi-square approximation.
Figure 9. LJung Box statistical report (Ho and H hypotheses).
Figure 10. Graphical test of partial autocorrelation - PCA. Graphical and correlation analysis
indicates that we have to normalize the series making it stationary. The process occurs with the choice
of the first lag (lag), which exceeded the confidence interval in both tests and whose degree of
graphical significance is higher, i.e., has the highest correlation and the lowest value according to the
Ljung-Box statistic. The lag that meets these criteria, therefore, is number 1, highlighted in fig. 9.
From these results, we can graphically represent (figure 11) the stationarity adjusted, as a
function of the first differentiation (lag 1):
Figure 11. Adjusted stationarity as a function of lag 1. So, we can then replicate this modeling for
the probabilistic analysis of the uncertainties, represented by the 5th and 95th percentiles, at a 90%
confidence level, since it assumed the same stationarity criteria and tests (graph and correlogram)
of the median. The graphical representation of the uncertainty set is described in the supplementary
Applying the Box- Jenkins model
Box-Jenkins’s method aims (figure 12) is to estimate a statistical model and interpret it
according to the sample data. If this estimated model is used for forecasting, we should assume that
its characteristics are constant over the period and particularly in future periods. A simple reason for
requiring the stationary data is that any model that is inferred based on that data can be interpreted
as stationary or stable and therefore provides a valid basis for prediction (Pokorny, 1987, Gujarati
and Porter, 2011).
1. Identification o
f the model
(Choosing tentative p, d, q)
2. Parameter estimation of
the chosen model
3. Diagnostic checking:
Are the estimated residuals white noise?
Yes (Go to Step 4) No (Return to Step 1)
4. Forecasting
Figure 12. The Box–Jenkins methodology. About step 4, Forecasting: One of the reasons for the
popularity of the ARIMA modeling is its success in forecasting. In many cases, the forecasts obtained
by this method are more reliable than those obtained from the traditional econometric modeling,
particularly for short-term forecasts. Of course, each case must be checked (Gujarati and Porter,
2011, p. 778).
We concluded that the MedTempGlobal (as described in the data/figures) time series model
was not stationary and we had to normalize it, making it stationary with constant mean and variance
and its covariance invariant over time. Therefore, it is an integrated time series, i.e., it combines the
two autoregressive processes (AR and MA) in the same set.
An important point to note is that when using the Box- Jenkins methodology, we must have
both a stationary time series and a time series that is stationary after one or more differentiations
(Gujarati and Porter, 2011).
Then, we can state that if a time series is integrated of order 1, therefore, it is I (1), after
differentiating it becomes I (0), that is, stationary. In general, if a time series is I (d), after
differentiating it d times, we get an I (0) series.
If one has to differentiate a time series d times to make it stationary and apply the ARMA (p,
q) model to it, one will say that the original time series is ARIMA (p, d, q), that is, it is a moving
average integrated autoregressive time series, where p denotes the numbers of the autoregressive
terms, d the number of times the series must be differentiated before it becomes stationary, and q the
number of moving average terms.
We, therefore, have in this time series an ARIMA (1,0,1) model, as it was differentiated once
(d = 1) before becoming stationary (of first difference), and can be modeled as an ARMA (1,1)
process, as it has an AR term and an MA post stationarity.
Finally, it is important to emphasize that to optimize the results, it was necessary to run in the
software SPSS - Statistics all the possible combinations of the ARIMA model (p,d,q) in the two
parameters, to arrive at the statistically optimal model after the decomposition of the data and meeting
the criteria of analysis and execution.
1. Salzer, E. Neske, D. A. L. & Rojo, C. A. Global warming: bias analysis in divergent strategic
scenarios. Journal Multi-Science Research (Msr), Vitoria, v. 2, n. 2, p. 144-158. Semestral.
Disponível em: (2019).
2. IBM – SPSS Statistics v. 22.
download (2020).
3. Cook J. et al 2013 Environ. Res. Lett. 8 024024.
9326/8/2/024024 (2013).
4. Bray, D. The scientific consensus of climate change revisited. Ciência e política ambiental , 13
(5), 340-350, (2010).
5. Anderegg W. R. L., Prall J.W., Harold J., Schneider S. H. Expert Credibility in Climate Change.
Proceedings of the National Academy of Sciences Jul 2010, 107 (27) 12107-12109; DOI: 10.1073
/ pnas.1003187107 (2010).
6. Oreskes, N. Science.Vol. 306, Issue 5702, p.1686. DOI: 10.1126/science.1103618 (2004).
7. Shwed, U., & Bearman, P. The Temporal Structure of Scientific Consensus Formation. American
Sociological Review, 75(6), 817-840. doi:10.2307/25782168 (2010).
8. IPCC - Intergovernmental Panel on Climate Change. United Nations. N.Y. (2019).
9. Molion, L. C. B. Aquecimento global: Uma visão crítica. Revista brasileira de climatologia, 3. .
DOI: (2008).
10. Legates, D. R., Soon, W, &. Briggs, W. M. & C. Monckton of Brenchley. Consenso Climático e
'Desinformação': Uma Tréplica à Agnotologia, Consenso Científico e o Ensino e Aprendizagem
das Mudanças Climáticas . Sci & Educ 24, 299-318. https://doi- (2015).
11. Legates, D. R., Soon, W. & Briggs, WM Learning and Teaching Climate Science: The Perils of
Consensus Knowledge Using Agnotology. Sci & Educ 22, 2007–2017 (2013). https://doi- (2013).
12. Reisinger, A. , Meinshausen, M. , Manning, M. & Bodeker, G. Uncertainties of global warming
metrics: CO2 and CH4, Geophys. Res. Lett. , 37 , L14707, doi: 10.1029 / 2010GL043803 (2010).
13. Davis, W.J. The Relationship between Atmospheric Carbon Dioxide Concentration and Global
Temperature for the Last 425 Million Years. Climate 2017, 5, 76. (2017).
14. Harde, H. What Humans Contribute to Atmospheric CO2: Comparison of Carbon Cycle Models
with Observations. Earth Sciences. Vol. 8, No. 3, 2019, pp. 139-158. doi:
10.11648/ (2019).
15. Easterbrook, D. (Ed.). (2016). Evidence-based climate science: data opposing CO2 emissions as
the primary source of global warming. Elsevier (2016).
16. Haustein, K., Allen, MR, Forster, PM et al. A real-time Global Warming Index. Sci
Rep 7, 15417. (2017).
17. Mitchell, D., James, R., Forster, P. et al. Realizing the impacts of a 1.5 °C warmer world. Nature
Clim Change 6, 735–737 (2016). (2016).
18. Medhaug, I., Stolpe, M., Fischer, E. et al. Reconciling controversies about the 'global warming
hiatus'. Nature 545, 41–47 (2017).
19. Solomon, S. et al. Climate Change 2007: The Physical Science Basis. Working Group I
Contribution to the Fourth Assessment Report of the IPCC (Cambridge University Press, 2007).
20. Kaufman, D., McKay, N., Routson, C. et al. Holocene global mean surface temperature, a multi-
method reconstruction approach. Sci Data 7, 201 (2020).
0530-7 (2020).
21. Maroco, J. Análise estatística com o SPSS Statistics. 7ª. ed. 54-60 (Pero Pinheiro, 2018).
22. Stockinger N., Dutter R. Robust time series analysis: a survey Kybernetika, Vol. 23 (1987), No.
Suppl, (1), 3-88 Persistent URL:
23. Miller, G. T., & Spoolman, S. E. Environmental Science. 22-24 (Cengage Learning, 2016).
24. Gujarati, D. N. & Porter, D. C. Basic Econometrics, 5th edition. 767-778. AMGH Editora,
25. Box, G. P. & Jenkins, G. M. Time series analysis: forecasting and control. Ed. rev. holden day,
são francisco: holden, (1978). Document shared on
26. Box, G. E. P., Jenkins, G. M. & Reinsel, G. C. (2008) Time Series Analysis: Forecasting and
Control. 4ª Edição, Wiley, Oxford.
27. Pokorny, M. An introduction to econometrics. (ed. Blackwell, B.) p. 343 (Basil Blackwell, 1987).
28. Routson, CC, McKay, NP, Kaufman, D. S. et al. Mid-latitude net precipitation decreased with
Arctic warming during the Holocene. Nature 568, 83–87.
1060-3 (2019).
29. PAGES 2k Consortium. Consistent multi-decadal variability in global temperature
reconstructions and simulations over the Common Era. Nat. Geosci. 12, 643–649 (2019).
30. Marcott, S. A., Shakun, J. D., Clark, P. U. & Mix, A. C. A reconstruction of regional and global
temperature for the past 11,300 years. Science 339, 1198 (2013).
Supplementary Materials
Mathematical rationale and statistical guidelines used in the methodology (Ref. fig. 7 to10, pg. 8
- 9).
When translating this analysis into a mathematical expression, the FAC correlation denoted by
k as a function of k is defined as:
 
!"#$%#& '(%)*+#,
-#$%#& '
where the value of k is the chosen delay (1), and the system considers covariance with lag k and
variance already calculated according to the data. In practice, as we have a stochastic uniequational
series, we can compute the function of sample correlation, . For such, we first need to calculate
the covariance of the sample with lag k, , and the variance of the sample, defined as:
where n is the sample size and Ȳ is the sample mean.
Thus, the function of sample correlation, with lag k is:
which is simply the ratio of sample covariance (with klag) and sample variance. The k-versus-k
graph is known as a sample correlogram.
Modeling the time series according to the methods
(AR) of moving averages (MA) and ARIMA (p. 07 to 10).
Once the parking period is conquered, we can model it with an autorregressive process
(AR),which we will represent by Yt, the Median (Md) in the t period (Holocene) as:
in which δ is the mean of Y and u is an error that’s not correlated with m is day zero and constant
variance ² (this is a white noise),so we will say that Yt follows a first-order stochastic
autorregressive process or AR (1). Here the value of Y in period t depends on its value in the previous
period and on a random term; y values are expressed as deviations based on an average value. In other
words, this model states that the predicted value of Y in period t is simply some proportion (= ) plus
a random shock or disturbance in the t-period; again, the Y values are expressed around their mean
The AR process we just discussed is not just a mechanism that may have generated Y. In this
case, Y can evolve into a first-order moving average process, or an MA (1). If we model Y this way:
in which μ is a constant and u,as before, is a stochastic error term of white noise. Here Y in period t
is equal to a constant plus a moving average of the current and past terms of error. In a more general
way, we can represent
which is an MA process(q). In short, a moving average process is just a linear combination of white
noise error terms.
Self-regressive process of moving averages (ARMA)
It is most likely that Y has both AR and MA characteristics and is therefore ARMA. So, Yt
follows an ARMA process (1.1), and can be written as
because there is an autoregressive term and a moving average term. In the Equation, ɵ represents a
constant term. In general, in an ARMA process (p, q), there will be autoregressive terms p and moving
average terms q.
Graphical configuration of the uncertainty set (p. 5 to 10, v. Data and methods).
Figure 1. Graphic description ofthe 5th and 95th percentiles decomposed (uncertainties)
Next, we represent the graphs of the correlations of both models, the ACF and ACFp, and
their subsequent differentiation for parking.
Figure 2. 5th percentile Autocorrelation Graphic Test (Automatic ACF)
Source: Prepared by the authors (SPSS - Statistics v. 22)
Figure 3. Partial autocorrelation graphic test 5th percentile - ACFP.
Source: prepared by the authors (SPSS- Statistics v. 22).
Figure 4. 95th percentile autocorrelation graph test (automatic ACF)
Source: Prepared by the authors (SPSS - Statistics v. 22)
Figure 5. 95th percentilepartial autocorrelation graph test - ACFP.
Source: prepared by the authors (SPSS- Statistics v. 22).
Figure 6. Residues of the ACF and PACF correlograms – 5th and 95th percentiles (White
Figure 7. Adjustment chart of the 5th and 95th percentiles parameters.
Source: SPSS output - Statistics powered by the authors.
Data Records
The data that led to this research were reused by Kaufman et al., 2020, as already referenced
in the text. After treatment, the data fed the IBM-SPSS Statistics v. 22, at, for the development of this
research and the generation of results. They can be found in the figshare
Technical Validation
All validations aiming to verify the technical quality and accuracy of the results were done in
the ARIMA platform of SPSS-Statistics, and are described in the body of the text and in the data
repository. For space reasons, only the data from the model that satisfied the research methodology
was sent, according to the foundations found in the specific literature.
scientific foundation on which the results of this paper are based, without ideological or political
bias, and the timing of the research data are important to consider. No less important is the legacy
left by the authors cited and researched in this work, besides Kaufman et al. (2020): Routson et al.
(2019), Cook et al. (2013), PAGES 2k Consortium, Marcott et al. (2013), Harde (2019), Box &
Jenkins (1978), Gujarati & Porter (2011). Special gratitude and acknowledgement to professors
Claudio Antonio Rojo and Edison Luiz Leismann, who gave all the trust and support necessary to
carry out this research.
Author contributions statement
G.V.F.S directed the project, wrote the manuscript and developed the methodology, with
input from all authors. E.L.L provided version 22 of the IBM Statistics software to conduct the
research and guide the methodology. L.G.R participated in the elaboration of the theoretical
framework and in the structuring of the texts. C.A.R oriented the line of research and the projection
of scenarios. All authors have reviewed the manuscript.
Competing interests and funding sources
The author declare no competing interests and have not received any specific grants from
funding agencies in the public, commercial, or non-profit sectors to do this research.
Additional information .
Correspondence and requests for materials should be addressed to G.V. F. S.
Author’s information
G.V.F.S: mail to:
C. A. R:
E. L. L:
ResearchGate has not been able to resolve any citations for this publication.
Full-text available
An extensive new multi-proxy database of paleo-temperature time series (Temperature 12k) enables a more robust analysis of global mean surface temperature (GMST) and associated uncertainties than was previously available. We applied five different statistical methods to reconstruct the GMST of the past 12,000 years (Holocene). Each method used different approaches to averaging the globally distributed time series and to characterizing various sources of uncertainty, including proxy temperature, chronology and methodological choices. The results were aggregated to generate a multi-method ensemble of plausible GMST and latitudinal-zone temperature reconstructions with a realistic range of uncertainties. The warmest 200-year-long interval took place around 6500 years ago when GMST was 0.7 °C (0.3, 1.8) warmer than the 19th Century (median, 5th, 95th percentiles). Following the Holocene global thermal maximum, GMST cooled at an average rate −0.08 °C per 1000 years (−0.24, −0.05). The multi-method ensembles and the code used to generate them highlight the utility of the Temperature 12k database, and they are now available for future use by studies aimed at understanding Holocene evolution of the Earth system.
Full-text available
The Intergovernmental Panel on Climate Change assumes that the inclining atmospheric CO2 concentration over recent years was almost exclusively determined by anthropogenic emissions, and this increase is made responsible for the rising temperature over the Industrial Era. Due to the far reaching consequences of this assertion, in this contribution we critically scrutinize different carbon cycle models and compare them with observations. We further contrast them with an alternative concept, which also includes temperature dependent natural emission and absorption with an uptake rate scaling proportional with the CO2 concentration. We show that this approach is in agreement with all observations, and under this premise not really human activities are responsible for the observed CO2 increase and the expected temperature rise in the atmosphere, but just opposite the temperature itself dominantly controls the CO2 increase. Therefore, not CO2 but primarily native impacts are responsible for any observed climate changes.
Full-text available
The latitudinal temperature gradient between the Equator and the poles influences atmospheric stability, the strength of the jet stream and extratropical cyclones1–3. Recent global warming is weakening the annual surface gradient in the Northern Hemisphere by preferentially warming the high latitudes4; however, the implications of these changes for mid-latitude climate remain uncertain5,6. Here we show that a weaker latitudinal temperature gradient—that is, warming of the Arctic with respect to the Equator—during the early to middle part of the Holocene coincided with substantial decreases in mid-latitude net precipitation (precipitation minus evapotranspiration, at 30° N to 50° N). We quantify the evolution of the gradient and of mid-latitude moisture both in a new compilation of Holocene palaeoclimate records spanning from 10° S to 90° N and in an ensemble of mid-Holocene climate model simulations. The observed pattern is consistent with the hypothesis that a weaker temperature gradient led to weaker mid-latitude westerly flow, weaker cyclones and decreased net terrestrial mid-latitude precipitation. Currently, the northern high latitudes are warming at rates nearly double the global average4, decreasing the Equator-to-pole temperature gradient to values comparable with those in the early to middle Holocene. If the patterns observed during the Holocene hold for current anthropogenically forced warming, the weaker latitudinal temperature gradient will lead to considerable reductions in mid-latitude water resources. A reduced gradient in temperatures between low and high latitudes during the Holocene led to drier mid-latitudes.
Full-text available
We propose a simple real-time index of global human-induced warming and assess its robustness to uncertainties in climate forcing and short-term climate fluctuations. This index provides improved scientific context for temperature stabilisation targets and has the potential to decrease the volatility of climate policy. We quantify uncertainties arising from temperature observations, climate radiative forcings, internal variability and the model response. Our index and the associated rate of human-induced warming is compatible with a range of other more sophisticated methods to estimate the human contribution to observed global temperature change.
Full-text available
Assessing human impacts on climate and biodiversity requires an understanding of the relationship between the concentration of carbon dioxide (CO2) in the Earth’s atmosphere and global temperature (T). Here I explore this relationship empirically using comprehensive, recently-compiled databases of stable-isotope proxies from the Phanerozoic Eon (~540 to 0 years before the present) and through complementary modeling using the atmospheric absorption/transmittance code MODTRAN. Atmospheric CO2 concentration is correlated weakly but negatively with linearly-detrended T proxies over the last 425 million years. Of 68 correlation coefficients (half non-parametric) between CO2 and T proxies encompassing all known major Phanerozoic climate transitions, 77.9% are non-discernible (p > 0.05) and 60.0% of discernible correlations are negative. Marginal radiative forcing (ΔRFCO2), the change in forcing at the top of the troposphere associated with a unit increase in atmospheric CO2 concentration, was computed using MODTRAN. The correlation between ΔRFCO2 and linearly-detrended T across the Phanerozoic Eon is positive and discernible, but only 2.6% of variance in T is attributable to variance in ΔRFCO2. Of 68 correlation coefficients (half non-parametric) between ΔRFCO2 and T proxies encompassing all known major Phanerozoic climate transitions, 75.0% are non-discernible and 41.2% of discernible correlations are negative. Spectral analysis, auto- and cross-correlation show that proxies for T, atmospheric CO2 concentration and ΔRFCO2 oscillate across the Phanerozoic, and cycles of CO2 and ΔRFCO2 are antiphasic. A prominent 15 million-year CO2 cycle coincides closely with identified mass extinctions of the past, suggesting a pressing need for research on the relationship between CO2, biodiversity extinction, and related carbon policies. This study demonstrates that changes in atmospheric CO2 concentration did not cause temperature change in the ancient climate.
Evidence-Based Climate Science: Data Opposing CO2 Emissions as the Primary Source of Global Warming, Second Edition, includes updated data related to the causes of global climate change from experts in meteorology, geology, atmospheric physics, solar physics, geophysics, climatology, and computer modeling. This book objectively gathers and analyzes scientific data concerning patterns of past climate changes, influences of changes in ocean temperatures, the effect of solar variation on global climate, and the effect of CO2 on global climate. This analysis is then presented as counter-evidence to the theory that CO2 is the primary cause behind global warming. Increasingly, scientists are pointing to data which suggests that climate changes are a result of natural cycles, which have been occurring for thousands of years. Unfortunately, global warming has moved into the political realm without enough peer-reviewed research to fully validate and exclude other, more natural, causes of climate change. For example, there is an absence of any physical evidence that CO2 causes global warming, so the only argument for CO2 as the cause of warming rests entirely in computer modeling. Thus, the question becomes, how accurate are the computer models in predicting climate? What other variables could be missing from the models? In order to understand modern climate changes, we need to look at the past history of climate changes. Vast amounts of physical evidence of climate change over the past centuries and millennia have been gathered by scientists. Significant climate changes have clearly been going on for many thousands of years, long before the recent rise in atmospheric CO2 Evidence-Based Climate Science, Data Opposing CO2 Emissions as the Primary Source of Global Warming, Second Edition, documents past climate changes and presents physical evidence for possible causes. Provides scientific evidence for issues related to global climate change that is not readily available elsewhere. Offers detailed analysis of temperature measurements with the goal of helping readers to understand conflicting claims about global warming heard every day in the news media. Presents real-time data on polar ice. Presents the real-time effect of CO2 on global warming, rather than forecasts based on computer models.
Between about 1998 and 2012, a time that coincided with political negotiations for preventing climate change, the surface of Earth seemed hardly to warm. This phenomenon, often termed the ‘global warming hiatus’, caused doubt in the public mind about how well anthropogenic climate change and natural variability are understood. Here we show that apparently contradictory conclusions stem from different definitions of ‘hiatus’ and from different datasets. A combination of changes in forcing, uptake of heat by the oceans, natural variability and incomplete observational coverage reconciles models and data. Combined with stronger recent warming trends in newer datasets, we are now more confident than ever that human influence is dominant in long-term warming.
The academic community could make rapid progress on quantifying the impacts of limiting global warming to 1.5 [deg]C, but a refocusing of research priorities is needed in order to provide reliable advice.
Agnotology has been defined in a variety of ways including “the study of ignorance and its cultural production” and “the study of how and why ignorance or misunderstanding exists.” More recently, however, it has been posited that agnotology should be used in the teaching of climate change science. But rather than use agnotology to enhance an understanding of the complicated nature of the complex Earth’s climate, the particular aim is to dispel alternative viewpoints to the so-called consensus science. One-sided presentations of controversial topics have little place in the classroom as they serve only to stifle debate and do not further knowledge and enhance critical thinking. Students must understand not just what is known and why it is known to be true but also what remains unknown and where the limitations on scientific understanding lie. Fact recitation coupled with demonizing any position or person who disagrees with a singularly-derived conclusion has no place in education. Instead, all sides must be covered in highly debatable and important topics such as climate change, because authoritarian science never will have all the answers to such complex problems.