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STOCHASTIC CARBON DIOXIDE FORECASTING
MODEL FOR CONCRETE DURABILITY
APPLICATIONS
Bassel Habeeb, Emilio BastidasArteaga, Helena Gervásio, Maria Nogal
To cite this version:
Bassel Habeeb, Emilio BastidasArteaga, Helena Gervásio, Maria Nogal. STOCHASTIC CAR
BON DIOXIDE FORECASTING MODEL FOR CONCRETE DURABILITY APPLICATIONS.
18th International Probabilistic Workshop, Lecture Notes in Civil Engineering 153, pp.753765, 2021,
�10.1007/9783030736163_58�. �hal03227811�
18th International Probabilistic Workshop
May 1214, 2021, Guimarães, Portugal
STOCHASTIC CARBON DIOXIDE FORECASTING MODEL FOR
CONCRETE DURABILITY APPLICATIONS
Bassel Habeeb1, Emilio BastidasArteaga1,2, Helena Gervásio3 and Maria Nogal4
1 Institute for Research in Civil and Mechanical Engineering UMR CNRS 6183, University of Nantes,
France.
2 Laboratory of Engineering Sciences for Environment UMR CNRS 7356, La Rochelle University, France.
3 Institute for Sustainability and Innovation in Structural Engineering, University of Coimbra, Portugal.
4 Faculty of Civil Engineering and Geosciences, TU Delft, Netherlands.
Keywords: Seasonal Stochastic Markovian Autoregressive Integrated Moving Average model, In
frastructure reliability, Carbon dioxide forecasting, Concrete carbonation, Climate change.
Abstract
Over the Earth’s history, the climate has changed considerably due to natural processes affecting
directly the earth. In the last century, these changes have perpetrated global warming. Carbon
dioxide is the main trigger for climate change as it represents approximately up to 80 percent of
the total greenhouse gas emissions. Climate change and concrete carbonation accelerate the cor
rosion process increasing the infrastructure maintenance and repair costs of hundreds of billions
of dollars annually. The concrete carbonation process is based on the presence of carbon dioxide
and moisture, which lowers the pH value to around 9, in which the protective oxide layer surround
ing the reinforcing steel bars is penetrated and corrosion takes place. Predicting the effective re
tained service life and the need for repairs of the concrete structure subjected to carbonation
requires carbon dioxide forecasting in order to increase the lifespan of the bridge. In this paper,
short term memory process models were used to analyze a historical carbon dioxide database, and
specifically to fill in the missing database values and perform predictions. Various models were
used and the accuracy of the models was compared. We found that the proposed Stochastic Mar
kovian Seasonal Autoregressive Integrated Moving Average (MSARIMA) model provides
!!
value
of 98.8%, accuracy in forecasting value of 89.7% and a variance in the value of the individual
errors of 0.12. When compared with the CO2 database values, the proposed MSARIMA model pro
vides a variance value of 0.1 and a coefficient of variation value of 8.0
""#
.
Habeeb et al.
IPW2020 –18th International Probabilistic Workshop
2
1 INTRODUCTION
Civil infrastructure investment in the European Union has been in a steady decline since the
outbreak of the economic and financial crisis. Although the decrease appears to gradually level
off from 2015 onwards with an increase of 5% [1]. The increase in the infrastructure investment
from 2015 onwards was illustrated as an action by the European Union for the sake of designing
and maintaining these systems for a certain service lifetime, which was recognized as critical
issues worldwide.
Decision making in the civil infrastructure investment in the European Union utilizing the qual
ity control plan is involved in the case of repairing or demolition of the reinforced concrete
bridges, depending on the recent key performance indicators (KPI). The KPI are specified by
engineering consultants regarding the current condition of the bridge and the strategies to be
followed (Reference strategy / Representative strategy) taking into consideration the reliability,
the cost and the availability of the bridge.
Reinforced concrete bridges are characterized by high durability, despite that, they are also
vulnerable to natural hazards, as well as extreme events that affect their performance and ser
viceability. Statistics on bridge collapses worldwide reveal that natural hazards are the predom
inant cause of failure. French government revealed that among the 12,000 maintained bridges
after the collapse of the motorway bridge located in Genoa, 840 are at risk of collapsing. This
issue is common across Europe [2].
Carbonation of concrete is one of the main causes of corrosion and occurs by the reaction given
in Equation (1) between atmospheric CO2 and the hydrated phases of concrete. This reaction
generates calcium carbonate, leading to a drop in the pH value, in which the protective oxide
layer of the reinforcing steel bars is broken and corrosion starts. Therefore, the life span of the
concrete infrastructure is affected by the enhanced risk of carbonation induced corrosion [3].
#$%&'()!*#$'!+$%$'$*(!'
(1)
The temperature significantly affects the diffusion coefficient of CO2 into concrete, the rate of
reaction between CO2 and Calcium Hydroxide (Ca(OH)2), and their rate of dissolution in pore
water [4]. The optimum relative humidity condition for the carbonation process is between 50%
and 70%, including wetting and drying cycles that enhance the reaction [5].
The carbonation process is very sensitive to the local climate depending on the environmental
conditions [6]. Climate change impacts the infrastructure as the increase in CO2 levels associ
ated with global warming will increase the carbonationinduced corrosion. Moreover, changes
in humidity and temperature significantly affect the initiation time of corrosion [7]. Since stud
ies on global warming have predicted several changes in climate, the impact of climate change
on structural reliability should be considered. For example, BastidasArteaga has calculated
numerically in the oceanic environment a reduction in the lifetime of failure that ranges between
1.4% and 2.3% and up to 7% when cyclic loading is considered [8].
A carbon dioxide database is essential to study the influence of realistic exposure conditions on
concrete carbonation. Databases could be also used to establish probabilistic prediction models.
Therefore, this study proposes a prediction model that is established based on the timedomain
analysis of the database and evaluated with a short memory process. The model is also com
pared with other autoregressive models. The proposed Stochastic Markovian Seasonal Auto
regressive Integrated Moving Average model (MSARIMA) is also used to fill the missing
Stochastic carbon dioxide forecasting model for concrete durability applications
IPW2020 –18th International Probabilistic Workshop
3
database and to perform predictions, taking into account the statistical analysis on the previ
ously existing historical database and seasonality.
Climate models are based on wellnotarized physical processes that simulate the transfer of
energy and materials through the climate system. Climate models, also known as general cir
culation models, use mathematical equations to characterize how energy and matter interact in
different parts of the ocean, atmosphere and land [9]. Climate models are operated using varia
bility that is driving the climate and predicting the climate change in the future. External factors
are the main inputs into the climate models that affect the amount of the solar energy absorbed
by the Earth or the amount trapped by the atmosphere, these external factors are called “forcing”.
They include variations in the sun’s output, greenhouse gases and tiny particles called aerosols
that are emitted from burning fossil fuels, forest fires and volcanic eruptions. The aerosols re
flect incoming sunlight and influence cloud formation except the black carbon.
Climate models provide results that vary with respect to the actual historical database; those
variations are at the expense of each model differences in: (ensemble, data source, forcing, the
initial state of run, driving model, aerosols influence and jet stream impact). However, the pro
posed model is based on stochastic time series analysis that avoids the climate models variations
and provides database that is statistically related to the existing historical database.
2 CARBON DIOXIDE FORECASTING
Time series forecasting is a quantitative approach that uses information based on historical val
ues and associated patterns to predict future observations. Time series analysis comprises meth
ods for analyzing timeseries data to extract meaningful statistics and other characteristics of
the data. The analysis includes trend, seasonality and irregular components. A timeseries anal
ysis quantifies the main features in data and random variation. These reasons, combined with
improved computing power, have made time series methods widely applicable.
2.1 Methodology
2.1.1 Time series analysis
Time series analysis for carbon dioxide database is based on the timedomain analysis (auto
correlation analysis and crosscorrelation analysis), in which the type of the process deduced is
a shortterm memory process with shortrange dependence that is characterized by an exponen
tial decay of the autocorrelation function (Acf) for the historical database.
2.1.2 Decomposition
Time series consists of two systematic components: trend and seasonality, and a nonsystematic
component called noise. A multiplicative nonlinear model is used as the seasonality increases
with the increase in the trend. The autocorrelation function of the nonsystematic component
demonstrates the characteristics of the autoregressive model in terms of damaged cosine shape.
2.1.3 Stationarity
Stationarity of the database is essential to maintain the statistical properties of the time series,
a stationarized series is relatively easy to predict, the stationarity is achieved through differenc
ing and log transformation. The basic idea of stationarity is that the probability laws that govern
the behavior of the process do not change over time. In a sense, the process is in statistical
equilibrium. Specifically, a process is strictly stationary if the distribution of existed state is the
same as the distribution of the previous state for all choices of time points and all choices of
Habeeb et al.
IPW2020 –18th International Probabilistic Workshop
4
time step lag. The stationarity of the time series is checked using Kwiatkowski–Phillips–
Schmidt–Shin (KPSS) test and augmented DickeyFuller (ADF) test [10].
2.1.4 Models
The statistical technique utilized for forecasting the carbon dioxide is Seasonal Stochastic Mar
kovian Autoregressive Integrated Moving Average (MSARIMA) which provides high accuracy
and precise results. Moreover, other statistical techniques that include moving average based
methods, such as Autoregressive Moving Average (ARMA), Autoregressive Integrated Mov
ing Average (ARIMA), HoltWinters’ Triple Exponential Smoothing and Seasonal Autoregres
sive Integrated Moving Average (SARIMA) were performed in order to compare the variations
in the accuracy of the models.
Lately, Autoregressive Integrated Moving Average (ARIMA) model has been used to study the
short timevarying processes. However, one limitation of ARIMA is its natural tendency to
concentrate on the mean values of the past series data. Therefore, it remains challenging to
capture a rapidly changing process, in which the proposed model (MSARIMA) solves this issue
by triggering a Markovian step when the value of the integration part is >1 and the probability
of occurrence is related to the previous seasonal events.
2.2 Models description
Models presented are divided into two categories: auto regression (AR) moving average (MA)
parameters and exponential smoothing parameters. The proposed MSARIMA model is based
on the AR and MA parameters. In addition, it accounts for seasonality and Markovian step
technique.
The autoregressive model of order p, which is denoted as AR(p), writes:
,%+*./&,%"& *0%
'
&() #; 0%1#23&456*
!#)
(2)
where
7+
is the state,
8
is a parameter of the model,
9
is constant,
:+
is a random white noise
WN
and
𝜎!
"
is the variance of the random white noise.
In this case, we denote by {
7+
} 1 AR (p). In the same way, we can rewrite a process AR(p)
with a polynomial
φ
(B).
8&;)#7+#+ :+#; 8&;)#+#<#=#8);#=8!;!#=#>>>=#8,;,
(3)
The moving average model of order q, which is denoted as MA(q), writes:
7++?+.@:+" *:+
.
()
(4)
where
@
is a parameter of the model and
?
is the expectation of
7+
, often equals to zero.
Use the backshift operator B to rewrite Equation (4).
7++@&;)#:+#; @&;)#+ #<#*#@);*@!;!*#>>>*@.;.##
(5)
2.2.1 Autoregressive moving average model (ARMA)
The general ARMA model was described in the 1951 by Peter Whittle [11].
7++#9*?*.@:+"
.
() *.87+" *:+/
,
()
(6)
The model could be written using the polynomials φ (B) and θ (B) in which the constant
𝑐
and
?
&
are zero values:
Stochastic carbon dioxide forecasting model for concrete durability applications
IPW2020 –18th International Probabilistic Workshop
5
7+=.87+"
,
() +.@0:+"0
.
0() *:+
(7)
8&;)#7++#@&;)#:+#A#&<=.8;)
,
() 7++&<*.#@0;0)
.
0() :+#
(8)
The ARMA model omits the integration part of its calculation leading to a nonstationary time
series model in which statistical parameters will vary with time. On the contrary, embedding
the integration part in the time series will control the stationarity in which the statistical prop
erties such as mean, variance, autocorrelation, etc. are all constant over time.
2.2.2 Autoregressive integrated moving average model (ARIMA)
The ARIMA is an advanced ARMA model that solves the stationarity of the time series by
using difference operation, this value is up to the secondorder of integration (
𝑑!"# = 2)
based
on the backshift operator Equation (9). Otherwise, it is solved using log transformation.
;&7+)#+#7+")#B#;1&7+)+#7+"1##
(9)
The general equation taking into account the constant
𝑐
and
?
&
as a nonzero value, in which
𝑐
=
&?&<=82=C=83)#
and
?
is the mean of
&<=;)17+
, is as follows:
(
<=8);#=>>>=#8,;,)&<=;)1&7+

?D1EFG)+
&<*@);*>>>*@.;.):+
(10)
2.2.3 Seasonal Autoregressive integrated moving average model (SARIMA)
The seasonality of a model is detected using an autocorrelation function in which the peaks
evolve over the lag values of a defined time series with a scale value >24. The monthly seasonal
stationarity of a model is based on a lag value of
H
= 12 and is known as the seasonal monthly
differencing operator in Equation (11).
&<=;)4#7+=7+7+"4
(11)
I&;4)8&;)+=?)+ J&;4)@&;):+
(12)
The SARIMA model without the differencing operations is mentioned in Equation (12) and the
terms are illustrated below:
8&;)+#<=8);#=8!;!#=#>>>=#8,;,
(13)
I&;4) + <=I);4=I!;!4#=#>>>=#I,;,4
(14)
@&;)+<#*#@);*@!;!*#>>>*@.;.##
(15)
K&;4)+<*K);4*K!;!4*#>>>*#K.;.4
(16)
where
I
is the seasonal AR parameter,
8
is the AR parameter,
#K
is the seasonal MA parameter
and
#@
is the MA parameter.
2.2.4 Markovian Seasonal Autoregressive Integrated Moving Average model (MSARIMA)
The proposed model is based on the SARIMA model. The MSARIMA solves the SARIMA
only limitation with its tendency to concentrate on the mean values of the past series data by
working on a sequence of time intervals changing their mean value in each time and by trigger
ing a Markovian step Equation (17).
L + MN#O5PO5")! +#Q5")! R#
#
+ST&:6")!)U <######57+V5" *W#####
TX:6")!YZ <#####57+V5" *#<#####
(17)
Habeeb et al.
IPW2020 –18th International Probabilistic Workshop
6
where
O
is the state,
W
is the mean value of the monthly seasonal errors of the value
7
and
L
is
the Markovian step value.
The MSARIMA model is developed based on the SARIMA model with a triggering condition
when the integration value >1, the model works on increasing the accuracy of the prediction
regarding the seasonal errors for the current state.
The step process depends on the most recent past event and the Markovian step is a renewable
process because it presents only positive values. This model neglects the
?
values in the pre
vious equations and presents the Markovian step process value
𝛿
for more accurate results. The
equation is as follows:
I&;4)8&;)+)+J&;4)@&;):+*L
(18)
2.2.5 HoltWinters’ multiplicative seasonal model
Winters (1960) extended Holt’s method to capture seasonality [12]. The HoltWinters’ seasonal
method comprises the forecast equation and three smoothing equations. The multiplicative
method is used when the seasonal variations are changing proportionally to the trend of the
series. The seasonal component is expressed in relative terms and the series is seasonally ad
justed by dividing through by the seasonal component. Within each year, the seasonal compo
nent will sum up to approximately the seasonal frequency value.
7
[
+789+ +#&\+*]>^+)_+7:";<=7)>
(19)
\++` ?!
@!"# *&<=`)#&\+") *^+") )
(20)
^++!a&\+=\+"))*&<=a)^+")
(21)
_b#+c A%
B!"$"C!"$ *&<=c)_+"D
(22)
where
\+
is the level,
^b
is the trend,
_b
is the seasonal component, m is the seasonal frequency,
and
`
,
a
and
c
are the model smoothing parameters.
3 RESULTS AND DISCUSSION
The main objective of this section is to estimate the ability of the proposed approach in fore
casting carbon dioxide concentration using an incomplete database. The forecasting and pre
diction of the missing values are performed using the following mathematical and stochastic
models: ARMA, ARIMA, SARIMA, HoltWinters’ and MSARIMA.
3.1 Database description
The concentration of greenhouse gases in Portugal is measured on the island of Terceira, which
is one of the nine islands in the archipelago of the Azores, located in the middle of the Atlantic
Ocean. The database is available since 1979 for different greenhouse gases. In particular, for
the three main gases, carbon monoxide (CO) since 1990, carbon dioxide (CO2) since 1979, and
methane (CH4) since 1983. However, the carbon dioxide database includes missing values. The
samples are collected on the island of Terceira and the analysis is carried out in NOAA lab,
Hawaii, in the scope of the Cooperative Global Air Sampling Network.
Stochastic carbon dioxide forecasting model for concrete durability applications
IPW2020 –18th International Probabilistic Workshop
7
3.2 MSARIMA CO2 database prediction
The database offered by NOAA lab, Hawaii, in the scope of the Cooperative Global Air Sam
pling Network includes missing values. Therefore, a stochastic MSARIMA model presents ac
curate results in filling the database shown in Figure 1 and can be used in for forecasting
purposes.
Figure 1: Example of CO2 assessment based on previous database
3.3 Stochastic models analysis
3.3.1 Stochastic models predictions
In this section, the prediction of the MSARIMA model is compared with SARIMA and Holt
Winters’ models as both include seasonal components. This is implemented by forecasting a
historical CO2 starting from 2010 through 2018 Figure 2. The prediction of the MSARIMA
model seems to provide the best fitting results to the original database compared to the other
models. The errors associated with the predictions will be further studied in the next section.
Figure 2: Stochastic models comparison
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IPW2020 –18th International Probabilistic Workshop
8
3.3.2 Stochastic models accuracy
A statistical study was performed to derive the variations between the mathematical stochastic
models and the meteorological station’s database. The difference of the relative frequencies for
CO2 presented in Figure 3 was performed for a time series starting from 01/2010 to 01/2018 to
describe the variations in the models. The proposed MSARIMA model presents the lowest var
iations. Moreover, ARIMA and ARMA models show higher variations than the other models
as seasonality is not considered.
Figure 3: Difference in relative frequency
The stochastic models' statistical study in Table 1 illustrates the variation of the models with
the original database in terms of mean value, variance and coefficient of variation, in which the
MSARIMA model shows the lowest variation with the meteorological station’s CO2 database.
On the contrary, the others present higher variations in the results.
Table 1: Statistical differences with meteorological station's CO2 database
Model
Mean value
Variance
Coefficient of variation (%)
MSARIMA
0.32
0.1
0.0008
SARIMA
1.567
11.94
0.20
HoltWinters’
1.562
7.07
0.11
ARIMA
1.27
5.48
0.08
ARMA
4.82
3.05
0.07
The accuracy of the stochastic models is finally demonstrated by comparing SARIMA and
MSARIMA models with the original database for the data given in Figure 2. This study will be
carried out in terms of the error indicators in Table 2. In this table ME is the mean error, RMSE
is the square root of the average of the square errors, MAE is the mean absolute error, MAPE
is the mean absolute percentage error and
!!#
is the proportion of the fitted model variation with
the original database.
Table 2: Accuracy between MSARIMA and SARIMA models
Model
ME
RMSEMAE
1MAPE[%]
𝑹𝟐[%]
MSARIMA
9.78
0.12
89.7
98.8
SARIMA
24.0
0.16
85.3
97.8
Stochastic carbon dioxide forecasting model for concrete durability applications
IPW2020 –18th International Probabilistic Workshop
9
The MSARIMA model presents the highest
!!
value in which 98.8% of the CO2 database var
iation is explained by the fitted model. The mean error refers to the average of all errors, it is
also described as the uncertainty in measurements, the proposed MSARIMA model provides
the lowest value in errors. The variation in the errors in the set of forecasts is diagnosed by the
difference between RMSE and MAE, in which lower values in RMSEMAE show lower vari
ance in the individual errors, as shown in Table 2 the MSARIMA model has the lowest RMSE
MAE values. The accuracy of a model prediction is presented by the 1MAPE value, as it cal
culates the relation between forecasted values and original values, in which the MSARIMA
model has the highest accuracy in forecasting.
CONCLUSIONS
The prediction of the proposed Stochastic Markovian Seasonal Autoregressive Integrated Mov
ing Average (MSARIMA) model seems to provide the best fitting results to the original CO2
database compared to the other models.
The proposed MSARIMA model provides
!!#
value of 98.8%, accuracy in forecasting value of
89.7% higher than all the other models and variance in the individual errors value of 0.12. When
compared with the CO2 database values, the proposed MSARIMA model provides a mean value
of 0.32, a variance value of 0.1 and a coefficient of variation value of 8.0
""#
.
The provided results demonstrate that there is no overestimation in the predictions using the
proposed MSARIMA model, which might be an obstacle due to the proposed step methodology.
On the contrary, the MSARIMA model provided the best fit in predictions when compared with
the original CO2 database.
ACKNOWLEDGMENTS
This paper was carried out in the framework of the Strengthening the Territory’s Resilience to
Risks of Natural, Climate and Human Origin (SIRMA) project, which is cofinanced by the
European Regional Development Fund (ERDF) through INTERREG Atlantic Area Program
with application code: EAPA_826/2018. The sole responsibility for the content of this publica
tion lies with the author. It does not necessarily reflect the opinion of the European Union.
Neither the INTERREG Europe program authorities are responsible for any use that may be
made of the information contained therein.
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