INTRODUCTION

Nowcasting refers to the projection of information about the present, the near future, and even the recent past. The importance of nowcasting has shifted from weather forecasting to economics, where economists use it to track the economy status through real-time GDP forecasts, as it is the main low-frequency (quarterly – annually) indicator reflecting the state of the country’s economy. This is like satellites that reflect the weather on earth. It does this by relying on high-frequency measured variables (daily – monthly) that are reported in real-time. The importance of using nowcasting in the economy stems from the fact that data issuers, statistical offices and central banks release the main variables of the economy, such as gross domestic product and its components, with a long lag. In some countries, it may take up to five months. In other countries, it may take two years depending on the capabilities each country has, leading to a state of uncertainty about the economic situation among economic policy makers and state followers business. Real-time indicators related to the economy (e.g consumer prices and exchange rates) are used here in order to obtain timely information about variables published with a delay. The first use of nowcasting technology in economics dates back to Giannone et al 2008, by developing a formal forecasting model that addresses some of the key issues that arise when using a large number of data series released at varying times and with varying delays ^[1]. They combine the idea of “bridging” the monthly information with the nowcast of quarterly GDP and the idea of using a large number of data releases in a single statistical framework. Banbura et al proposed a statistical model that produces a series of Nowcasting forecasts on real-time releases of various economic data ^[2]. The methodology enables the processing of a large amount of information from nowcasting’s Eurozone GDP Q4 2008 study. Since that time, the models that can be used to create nowcasting have expanded. Kuzin et al compared the mixed-frequency data sampling (MIDAS) approach proposed by Ghysels et al ^[4,5] with the mixed-frequency VAR (MF-VAR) proposed by Zadrozny and Mittnik et al [6,7], with model specification in the presence of mixed-frequency data in a policy-making situation, i.e. nowcasting and forecasting quarterly GDP growth in Eurozone on a monthly basis. After that time, many econometric models were developed to allow the use of nowcasting and solve many data problems. Ferrara et al ^[8] proposed an innovative approach using nonparametric methods, based on nearest neighbor’s approaches and on radial basis function, to forecast the monthly variables involved in the parametric modeling of GDP using bridge equations. Schumacher et al ^[9] compare two approaches from nowcasting GDP: Mixed Data Sampling (MIDAS) regressions and bridge equations. Macroeconomics relies on increasingly non-standard data extracted using machine learning (text analysis) methods, with the analysis covering hundreds of time series. Some studies examined US GDP growth forecasts using standard high-frequency time series and non-standard data generated by text analysis of financial press articles and proposed a systematic approach to high-dimensional time regression problems ^[10-11]. Another team of researchers worked on dynamic factor analysis models for nowcasting GDP ^[12], using a Dynamic Factor Model (DFM) to forecast Canadian GDP in real-time. The model was estimated using a mix of soft and hard indices and the authors showed that the dynamic factor model outperformed univariate criteria as well as other commonly used nowcasting models such as MIDAS and bridge regressions. Anesti et al ^[13] proposed a release-enhanced dynamic factor model (RA-DFM) that allowed quantifying the role of a country’s data flow in the nowcasting of both early (GDP) releases and later revisions of official estimates. A new mixed-frequency dynamic factor model with time-varying parameters and random fluctuations was also designed for macroeconomic nowcasting, and a fast estimation algorithm was developed ^[14]. Deep learning models also entered the field of GDP nowcasting, as in many previous reports ^[15-17]. In Syria, there are very few attempts to nowcasting GDP, among which we mention a recent report that uses the MIDAS Almon Polynomial Weighting model to nowcasting Syria’s annual GDP based on the monthly inflation rate data ^[18].  Our research aims to solve a problem that exists in the Arab and developing countries in general, and Syria in particular, namely the inability to collect data in real-time on the main variable in the economy due to the weakness of material and technical skills. Therefore, this study uses Bayesian mixed-frequency VAR models to nowcast GDP in Syria based on a set of high-frequency data. The rationale for choosing these models is that they enable the research goal to be achieved within a structural economic framework that reduces statistical uncertainty in the domain of high-dimensional data in a way that distinguishes them from the nowcasting family of models, according to a work by Cimadomo et al and Crump et al ^[19-20]. The first section of this research includes the introduction and an overview of the previous literature. The second section contains the research econometric framework, through which the architecture of the research model is developed within a mathematical framework. The third section is represented by the data used in the research including the exploratory phase. The fourth section contains the discussion and interpretation of the results of the model after the evaluation. The fifth section presents the results of the research and proposals that can consider a realistic application by the authorities concerned.

MATERIALS AND METHODS

The working methodology in this research is divided into two main parts. The first in which the low-frequency variable (Gross Domestic Product) is converted from annual to quarterly with the aim of reducing the forecast gap and tracking the changes in the GDP in Syria in more real-time., by reducing the gap with high-frequency data that we want to predict their usage. To achieve this, we used Chow-Lin’s Litterman: random walk variant method.

Chow-Lin’s Litterman Method

This method is a mix and optimization of two methods. The Chow-Lin method is a regression-based interpolation technique that finds values ​​of a series by relating one or more higher-frequency indicator series to a lower-frequency benchmark series via the equation:

                                        x(t)=βZ(t)+a(t)                     (1)

Where  is a vector of coefficients and a random variable with mean zero and covariance matrix  Chow and Lin ^[21] used generalized least squares to estimate the covariance matrix, assuming that the errors follow an AR (1) process, from a state space model solver with the following time series model:

                          a(t)=ρa(t-1)+ϵ(t)                 (2)

Where ϵ(t)~N(0,σ^2) and |ρ|<1. The parameters ρ and βare estimated using maximum likelihood and Kalman filters, and the interpolated series is then calculated using Kalman smoothing. In the Chow-Lin method, the calculation of the interpolated series requires knowledge of the covariance matrix, which is usually not known. Different techniques use different assumptions about the structure beyond the simplest (and most unrealistic) case of homoscedastic uncorrelated residuals. A common variant of Chow Lin is Litterman interpolation ^[22], in which the covariance matrix is ​​computed from the following residuals:

                                 a(t)=a(t-1)+ϵ(t)                 (3)

Where ϵ(t)~N(0,V)

                                  ϵ(t)=ρϵ(t-1)+e(t)               (4)

and the initial state a(0)=0. . This is essentially an ARIMA (1,1,0) model.

In the second part, an econometric model suitable for this study is constructed by imposing constraints on the theoretical VAR model to address a number of statistical issues related to the model’s estimation, represented by the curse of dimensions.

Curse of Dimensions

The dimensional curse basically means that the error increases with the number of features (variables). A higher number of dimensions theoretically allows more information to be stored, but rarely helps as there is greater potential for noise and redundancy in real data. Collecting a large number of data can lead to a dimensioning problem where very noisy dimensions can be obtained with less information and without much benefit due to the large data ^[23]. The explosive nature of spatial scale is at the forefront of the Curse of Dimensions cause. The difficulty of analyzing high-dimensional data arises from the combination of two effects: 1- Data analysis tools are often designed with known properties and examples in low-dimensional spaces in mind, and data analysis tools are usually best represented in two- or three-dimensional spaces. The difficulty is that these tools are also used when the data is larger and more complex, and therefore there is a possibility of losing the intuition of the tool’s behavior and thus making wrong conclusions. 2- The curse of dimensionality occurs when complexity increases rapidly and is caused by the increasing number of possible combinations of inputs. That is, the number of unknowns (parameters) is higher than the number of observations. Assuming that m denotes dimension, the corresponding covariance matrix has m (m+1)/2 degrees of freedom, which is a quadratic term in m that leads to a high dimensionality problem. Accordingly, by imposing a skeletal structure through the initial information of the Bayesian analysis, we aimed to reduce the dimensions and transform the high-dimensional variables into variables with lower dimensions and without changing the specific information of the variables. With this, the dimensions were reduced in order to reduce the feature space considering a number of key features.

Over Parameterizations

This problem, which is an integral part of a high dimensionality problem, is statistically defined as adding redundant parameters and the effect is an estimate of a single, irreversible singular matrix ^[24]. This problem is critical for statistical estimation and calibration methods that require matrix inversion. When the model starts fitting the noise to the data and the estimation parameters, i.e. H. a high degree of correlation existed in the co-correlation matrix of the residuals, thus producing predictions with large out-of-sample errors. In other words, the uncertainty in the estimates of the parameters and errors increases and becomes uninterpretable or far from removed from the realistic estimate. This problem is addressed by imposing a skeletal structure on the model, thereby transforming it into a thrift. Hence, it imposes constraints that allow a correct economic interpretation of the variables, reducing the number of unknown parameters of the structural model, and causing a modification of the co – correlation matrix of the residuals so that they become uncorrelated with some of them, in other words, become a diagonal matrix.

Overfitting and Underfitting

Overfitting and Underfitting are major contributors to poor performance in models. When Overfitting the model (which works perfectly on the training set while ineffectively fitting on the test set), it begins with matching the noise to the estimation data and parameters, producing predictions with large out-of-sample errors that adversely affect the model’s ability to generalize. An overfit model shows low bias and high variance ^[25]. Underfitting refers to the model’s inability to capture all data features and characteristics, resulting in poor performance on the training data and the inability to generalize the model results ^[26]. To avoid and detect overfitting and underfitting, we tested the validity of the data by training the model on 80% of the data subset and testing the other 20% on the set of performance indicators.

Theoretical VAR model

VAR model developed by Sims ^[27] has become an essential element in empirical macroeconomic research. Autoregressive models are used as tools to study economic shocks because they are based on the concept of dynamic behavior between different lag values for all variables in the model, and these models are considered to be generalizations of autoregressive (AR) models. The p-rank VAR is called the VAR_p model and can be expressed as:

y_t=C+β_1 y_(t-1)+⋯+β_P y_(t-P)+ϵ_t ϵ_t~(0,Σ)               (5)

Where y_t is a K×1  vector of endogenous variables.  Is a matrix of coefficients corresponding to a finite lag in, y_t, ϵ_t: random error term with mean 0 representing external shocks, Σ: matrix (variance–covariance). The number of parameters to be estimated is K+K^2 p, which increases quadratic with the number of variables to be included and linearly in order of lag. These dense parameters often lead to inaccuracies regarding out-of-sample prediction, and structural inference, especially for high-dimensional models.

Bayesian Inference for VAR model

The Bayesian approach to estimate VAR model addresses these issues by imposing additional structure on the model, the associated prior information, enabling Bayesian inference to solve these issues [28-29], and enabling the estimation of large models [2]. It moves the model parameters towards the parsimony criterion and improves the out-of-sample prediction accuracy [30]. This type of contraction is associated with frequentist regularization approaches [31]. Bayesian analysis allows us to understand a wide range of economic problems by adding prior information in a normal way, with layers of uncertainty that can be explained through hierarchical modelling [32].

Prior Information

The basic premise for starting a Bayesian analysis process must have prior information, and identifying it correctly is very important. Studies that try not to impose prior information result in unacceptable estimates and weak conclusions. Economic theory is a detailed source of prior information, but it lacks many settings, especially in high-dimensional models. For this reason, Villani [33] reformulates the model and places the information at a steady state, which is often the focus of economic theory and which economists better understand. It has been proposed to determine the initial parameters of the model in a data-driven manner, by treating them as additional parameters to be estimated. According to the hierarchical approach, the prior parameters are assigned to hyperpriors. This can be expressed by the following Bayes law:         

Wherey= (y_(p+1)+⋯+y_T)^T,θ:  is coefficients AR and variance for VAR model, γ: is hyperparameters. Due to the coupling of the two equations above, the ML of the model can be efficiently calculated as a function of γ. Giannone et al ^[34] introduced three primary information designs, called the Minnesota Litterman Prior, which serves as the basis, the sum of coefficients ^[35] and single unit root prior ^[36].

Minnesota Litterman Prior

Working on Bayesian VAR priors was conducted by researchers at the University of Minnesota and the Federal Reserve Bank of Minneapolis ^[37] and these early priors are often referred to as Litterman prior or Minnesota Prior. This family of priors is based on assumption that Σ is known; replacing Σwith an estimate Σ. This assumption leads to simplifications in the prior survey and calculation of the posterior.

The prior information basically assumes that the economic variables all follow the random walk process. These specifications lead to good performance in forecasting the economic time series. Often used as a measure of accuracy, it follows the following torques:

The key parameter ⋋ is which controls the magnitude influence of the prior distribution, i.e. it weighs the relative importance of the primary data. When  the prior distribution is completely superimposed, and in the case of  the estimate of the posterior distributions will approximate the OLS estimates.  is used to control the break attenuation, and  is used to control the prior standard deviation when lags are used. Prior Minnesota distributions are implemented with the goal of de-emphasizing the deterministic component implied by the estimated VAR models in order to fit them to previous observations. It is a random analysis-based methodology for evaluating systems under non-deterministic (stochastic) scenarios when the analytical solutions are complex, since this method is based on sampling independent variables by generating phantom random numbers.

Dummy Observations

Starting from the idea of Sims and Zha ^[38]​​ to complement the prior by adding dummy observations to the data matrices to improve the predictive power of Bayesian VAR. These dummy observations consist of two components; the sum-of-coefficient component and the dummy-initial-observation component. The sum-of-coefficients component of a prior was introduced by Doan et al ^[37] and demonstrates the belief that the average of lagged values ​​of a variable can serve as a good predictor of that variable. It also expresses that knowing the average of lagged values ​​of a second variable does not improve the prediction of the first variable. The prior is constructed by adding additional observations to the top (pre-sample) of the data matrices. Specifically, the following observations are constructed: 

Where  is the vector of the means over the first  observed by each variable, the key parameter  is used to control for variance and hence the impact of prior information. When  the prior information becomes uninformative and when  the model is formed into a formula consisting of a large number of unit roots as variables and without co-integration. The dummy initial observation component ^[36] creates a single dummy observation that corresponds to the scaled average of the initial conditions, reflecting the belief that the average of the initial values ​​of a variable is likely to be a good prediction for that variable. The observation is formed as:

Where  is the vector of the means over the first  observed by each variable, the key parameter  is used to control on variance and hence the impact of prior information. As , all endogenous variables in VAR are set to their unconditional mean, the VAR exhibits unit roots without drift, and the observation agrees with co-integration.

Structural Analysis

The nowcasting technique within the VAR model differs from all nowcasting models in the possibility of an economic interpretation of the effect of high-frequency variables on the low-frequency variable by measuring the reflection of nonlinear changes in it, which are defined as (impulse and response). A shock to the variable not only affects the variable directly, but it is also transmitted to all other endogenous variables through the dynamic (lag) structure of VAR. An impulse response function tracks the impact of a one-off shock to one of the innovations on current and future values ​​of the endogenous variable.

BVAR is estimated reductively, i.e., without a contemporary relationship between endogenous variables in the system. While the model summarizes the data, we cannot determine how the variables affect each other because the reduced residuals are not orthogonal. Recovering the structural parameters and determining the precise propagation of the shocks requires constraints that allow a correct economic interpretation of the model, identification constraints that reduce the number of unknown parameters of the structural model, and lead to a modification of the co-correlation matrix of the residuals so that they become uncorrelated, i.e., they become a diagonal matrix. This problem is solved by recursive identification and achieved by Cholesky’s analysis ^[39] of the variance and covariance matrix of the residuals , Where Cholesky uses the inverse of the Cholesky factor of the residual covariance matrix to orthogonalize the impulses. This option enforces an ordering of the variables in the VAR and attributes the entire effect of any common component to the variable that comes first in the VAR system. For Bayesian frame, Bayesian sampling will use a Gibbs or Metropolis-Hasting sampling algorithm to generate draws from the posterior distributions for the impulse.  The solution to this problem can be explained mathematically from the VAR model:

Where  show us contemporaneous correlation.  Coefficient matrix at lag1,  error term where:

In order to estimate the previous equation, we must abandon the contemporaneous relations between the endogenous variables since we transfer them to the other side:

Now parsimony VAR model can be estimated:

Where A_1 are reduced coefficient. u_t Represent the weighted averages of the structural coefficientsβ_1,ϵ_t. Where:

Mixed Frequency VAR

Complementing the constraints imposed on the theoretical VAR model, as a prelude to achieving the appropriate form for the research model and developing the model presented by AL-Akkari and Ali ^[40] to predict macroeconomic data in Syria, we formulate the appropriate mathematical framework for the possibility to benefit from high-frequency data emitted in real-time in nowcasting of GDP in Syria.

We estimate a mixed frequency VAR model with no constant or exogenous variables with only two data frequencies, low and high (quarterly-monthly), and that there are high frequency periods per low frequency period. Our model consists of  variable  observed at low frequency and  variables  observed at high frequency. Where:

: represent the  low-frequency variable observed during the low-frequency period .

: represent the  high-frequency variable observed during the low-frequency period .

By stacking the  and  variables into the matrices  and  ignoring the intersections and exogenous variables for brevity, respectively, we can write the VAR:

Performance indicators

We use indicators to assess the performance of the models which are used to determine their ability to explain the characteristics and information contained in the data. It does this by examining how closely the values ​​estimated from the model correspond to the actual values, taking into account the avoidance of an underfitting problem that may arise from the training data and an overfitting problem that arises from the test data. The following performance indicators include:

Theil coefficient (U):

Mean Absolute Error (MAE):

Root Mean Square Error (RMSE):                                   

Where  the forecast is value;  is the actual value; and  is the number of fitted observed. The smaller the values of these indicators, the better the performance of the model. 

RESULTS & DISCUSSION

As mentioned, the data consists of high and low frequency variables collected from the websites of official Syrian organizations and the World Bank ^[41-46], summarized in Table 1, which shows basic information about data. The data of the variable GDP were collected annually and converted using the combination of Chow-Lin’s Litterman methods, (Fig 1), which shows that the volatility in annual GDP was explained by the quarter according to the hybrid method used, and gives us high reliability for the possibility of using this indicator in the model. For high-frequency data, the monthly closing price was collected from the sources listed in Table (1) and its most recent version was included in the model to provide real-time forecasts.

Figure (2) shows the evolution of these variables. We presented in Figure (2), the data in its raw form has a general stochastic trend and differs in its units of measurement. To unify the characteristics of this data, growth rates were used instead of prices in this case, or called log, because of their statistical properties. These features are called stylized facts; first, there is no normal distribution. In most cases, the distribution deviates to the left and has a high kurtosis. It has a high peak and heavy tails ^[47]. Second, it has the property of stationary and there is almost no correlation between the different observations ^[48]. The log of this series is calculated.

Figure 2. Evolution of high frequency study variables

Figure (3) shows the magnitude growth of the macroeconomic and financial variables in Syria during the indicated period. We note that the data is characterized by the lack of a general trend and volatility around a constant. We found that the fluctuations change over time and follow the characteristic of stochastic volatility that decreases, increases and then decreases. We found that the most volatile variable is the exchange rate (EXR) and the period between 2019 and 2020 is the one when the variables fluctuated the most due to uncertainty and US economic sanctions. We also noted that the periods of high volatility are offset by negative growth in Syria’s GDP. Using the following Table (2), we show the most important descriptive statistics for the study variables.

                                                 *** denotes the significance of the statistical value at 1%. ** at 5%. * at 10%.

Table (2) shows that the probability value of the normal distribution test statistic is significant at 1%, and we conclude that the data on the growth rates of the Syrian economy is not distributed according to the normal distribution, and both the mean and the standard deviation were not useful for the prediction in this case since they have become a breakdown. We also note from Table (2) the positive value of the skewness coefficient for all variables, and therefore the growth rates of economic variables are affected by shocks in a way that leads to an increase in the same direction, except for the economic growth rate, in which the shocks are negative and the distribution is skewed to the left. We also found that the value of the kurtosis coefficient is high (greater than 3 for all variables), indicating a tapered leptokurtic-type peak. Additionally, we noted that the largest difference between the maximum and minimum value of the variable is the exchange rate, which indicates the high volatility due to the devaluation of the currency with the high state of uncertainty after the start of the war in Syria. Also, one of the key characteristics of growth rates is that they are stable (i.e. they do not contain a unit root). Since structural changes affect expectations, as we have seen in Figure (1), there is a shift in the path of the variable due to political and economic events. We hence used the breakpoint unit root test proposed by Perron et al ^[49-50], where we assume that the structural change follows the course of innovation events and we test the largest structural breakpoint for each variable and get the following results:

Where  is break structural coefficients,  is dummy variable indicated on structural change, : intercept, : lag degree for AR model. Figure (4) shows the structural break estimate for the study variables. Structural break refers to a change in the behavior of a variable over time, shifting relationships between variables. Figure (4) also demonstrates that all macroeconomic variables have suffered from a structural break, but in different forms and at different times, although they show the impact of the same event, namely the war in Syria and the resulting economic blockade by Arab and Western countries, as all structural breaks that have occurred after 2011. We found that in terms of the rate of economic growth, it has been quickly influenced by many components and patterns. For EXR, CPI, GOP, the point of structural change came after 2019, the imposition of the Caesar Act and the Corona Pandemic in early 2020, which resulted in significant increases in these variables. We noted that the structural break in the Damascus Stock Exchange Index has occurred in late 2016 as the security situation in Syria improved, resulting in restored investor confidence and realizing returns and gains for the market.

The step involves imposing the initial information on the structure of the model according to the properties of the data. Based on the results of the exploratory analysis, the prior Minnesota distributions were considered a baseline, and the main parameter is included in the hierarchical modeling:

Rho H = 0.2 is high frequency AR(1), Rho L = 0 is low frequency AR(1), Lambda = 5 is overall tightness, Upsilon HL = 0.5 is high-low frequency scale, Upsilon LH = 0.5 is low-high frequency scale, Kappa = 1 is exogenous tightness. C1 = 1 is residual covariance scale. The number of observations in the frequency conversion specifies the number of high frequency observations to use for each low frequency period. When dealing with monthly and quarterly data, we can specify that only two months from each quarter should be used. Last observations indicate that the last set of high-frequency observations from each low-frequency period should be used. Initial covariance gives the initial estimate of the residual covariance matrix used to formulate the prior covariance matrix from an identity matrix, specifying the number of Gibbs sampler draws, the percentage of draws to discard as burn-in, and the random number seed. The sample is divided into 90% training (in–of –sample) and 10% testing data (out-of–sample).

Table (3) provides us with the results of estimating the model to predict quarterly GDP in Syria. The results show the basic information to estimate the prior and posterior parameters, and the last section shows the statistics of the models such as the coefficient of determination and the F-statistic. We note that every two months were determined from a monthly variable to forecast each quarter of GDP in the BMFVAR model. Although there is no explanation due to the imposition of constraints, the model shows good prediction results with a standard error of less than 1 for each parameter and a high coefficient of determination of the GDP prediction equation explaining 93.6% of the variance in GDP.

Figure (5) also shows us the reliability of the prediction results. It is clear that the roots of the estimation of the parameters follow the inverse AR polynomial process. The inverted AR roots have modules very close to one which is typical of many macro time series models.

 Table (4) shows the evolution of GDP Forecasting in Syria in and out the sample based on a number of indicators. When the value of these indicators is close to 0, the estimated values ​​are close to the actual values, which is shown by Table 4, since the values ​​of these indicators are all less than 1. The out-of-sample predicted values ​​show a better performance of the model, and hence it can be adopted to track changes in quarterly GDP over time. Up-to-date with current versions.

Through the data visualization technique, Figure (6) shows us the closeness of the expected values of quarterly GDP in Syria in of sample, which leads to the exclusion of the presence of an undue problem in the estimate (overfitting – underfitting). The median is used in the prediction because it is immune to the values of structural breaks and the data are not distributed according to the normal distribution. The out-of-sample forecast results (Fig 7) also indicate negative rates of quarterly GDP growth in Syria with the negative impact of internal and external shocks on the Syrian economy with the accumulation of the impact of sanctions, the lack of long-term production plans and the ineffectiveness of internal monetary policy tools, as the latest forecasts for GDP growth this quarter are (-3.69%).

The uncertainty at each point in time is included in the Syrian GDP projections, thereby achieving two objectives: incorporating the range of GDP change at each point in time and knowing the amount of error in the forecasts. We found that uncertainty increases as the forecast period lengthens (Figures 8-10).

The model also provides us with important results through scenario analysis. Figure 11 shows that the quarterly GDP in Syria is affected by the shocks of high-frequency variables, since this Figure shows that the shocks have negative impacts. This recommends a better activation of the instruments of The Central Bank and those responsible for monetary policy in Syria. These results are considered important tools for them to know and evaluate the effectiveness of their tools.

Figure 6. Forecasting quarterly GDP in Syria in of sample with 35%, 10%, 5% distribution quantities.

Figure 7. Forecasting quarterly GDP in Syria out of sample with 35%, 10%, 5% distribution quantities.

Figure 8. Uncertainty for forecasting quarterly GDP in Syria in-of –sample

Figure 10. Uncertainty for forecasting quarterly GDP in Syria out-of –sample

Figure 11. Shocks of high-frequency variables in the quarterly GDP of Syria

CONCLUSIONS AND RECOMMENDATIONS

This paper showed that BMFVAR could be successfully used to handle Parsimony Environment Data, i.e., a small set of macroeconomic time series with different frequencies, staggered release dates, and various other irregularities – for real-time nowcasting. BMFVAR are more tractable and have several other advantages compared to competing nowcasting methods, most notably Dynamic Factor Models. For example, they have general structures and do not assume that shocks affect all variables in the model at the same time. They require less modeling choices (e.g., related to the number of lags, the block-structure, etc.), and they do not require data to be made stationary. The research main finding was presenting three strategies for dealing with mixed-frequency in the context of VAR; First, a model – labelled “Chow-Lin’s Litterman Method” – in which the low-frequency variable (Gross Domestic Product) is converted from annual to quarterly with the aim of reducing the forecast gap and tracking the changes in the GDP in Syria in more real-time and reducing the gap of high-frequency data that we want to predict usage. Second, the research adopts a methodology known as “blocking”, which allows to treat higher frequency data as multiple lower-frequency variables. Third, the research uses the estimates of a standard low-frequency VAR to update a higher-frequency model. Our report refers to this latter approach as “Polynomial-Root BVAR”. Based on a sample of real-time data from the beginning of 2010 to the end of the first quarter of 2023, the research shows how these models will have nowcasted Syria GDP growth. Our results suggests that these models have a good nowcasting performance. Finally, the research shows that mixed-frequency BVARs are also powerful tools for policy analysis, and can be used to evaluate the dynamic impact of shocks and construct scenarios. This increases the attractiveness of using them as tools to track economic activity for both the Central Bank of Syria and the Central Bureau of Statistics.

Introduction

The Internet of Things (IoT) makes all smart devices connect to the Internet anywhere at any time to form the Internet of Things ^[1]. Smart devices are described to be low power and limited capability of processor and memory. These devices can be linked together to form a network without an infrastructure in which nodes act as a router. Smart devices use IEEE 802.15.4 ^[2] and routing protocol RPL ^[3], but RPL did not support mobile nodes, so many researchers worked to improve it, but their results showed more overhead because of sending more control messages in the networks. other researches results were not accurate because of depending on RSSI value (Received signal strength indication) to detect the mobility of nodes, which is affected by obstacles and interference. Our contribution is to improve RPL protocol by making it able to ensure the nodes aware of the mobile node movement, so the node can reconnect to the network as soon as possible and reduce the disconnection time without increasing the transmission rate of control messages. The proposed protocol is lightweight and suitable for devices with limited and restricted specifications because it uses new control message which is not periodically sent.

The rest of this paper is organized as following: The first section introduces the RPL routing protocols, then related works are mentioned with discussions about the challenges of the research to design an efficient LLNs routing protocol. After that the proposed control message is explained, followed by the extensive simulation, performance evaluation and interpretations. Finally, the paper is concluded.

RPL (Routing Protocol for Low Power and Lossy Networks(LLN)) ^[3]:

In the network layer of the IoT protocol stack using 6LoWPAN technology, The RPL routing protocol was developed by (ROLL) group and was described in RFC (6550). RPL is suitable for fixed devices not the mobile devices. The root node serves as a gateway to the Internet for devices in the network. RPL organizes a topology as a Directed Acyclic Graph (DAG) that is partitioned into one or more Destination Oriented DAGs (DODAGs), one DODAG per root. It sends periodic DIO (DODAG Information Object) messages to inform nodes of its existence. It invites the neighbour nodes to call it, and in turn, each node that hears the DIO message and wants to communicate with root node will send a DAO (Destination Advertisement Object) message, then each node that communicates with parent node sends DIO messages to make the rest of the nodes connect and this process is repeated until all nodes are connected together, which is shown in Figure 1.

Fig. 1: Control messages used in RPL

The RPL protocol depends on the Trickle algorithm to manage the timers used to send periodic messages, to reduce the overhead on these networks. When the state of instability is detected, the transmission rate of control messages increases to spread updates quickly and reduces when the network is stable ^[4]. The node selects the parent node using Objective Function (OF) that depends on the number of hops, then it was developed to MARHOF (The Minimum Rank with Hysteresis Objective Function) [5] in which the parent node is selected based on the value of the Expected Transmission Count (ETX) that determine the quality of the link. The RPL protocol did not specify any mechanism for detecting routing adjacency failures of mobile node because such a mechanism causes overhead in bandwidth and power consumption. It is not suitable for Battery-powered devices to send periodic messages, so it requires external mechanisms to detect that the neighbour node is no longer reachable. This mechanism should preferably focus on links that were already used . ^[3]

Related work

Fotouhi (2015) proposed the MRPL protocol by integrating the RPL protocol with the smart hop using beacons. The protocol has two phases: the discovery phase and the data transmission phase. In the route discovery phase, the mobile node sends a broadcast of n DIS messages. The node that receives DIS messages calculates the ARSSI and adds this value within a DIO message. The mobile node selects its parent node that sent the DIO message with the highest ARSSI .^[6] Gara (2016) suggested to use adaptive timer algorithm to regulate the transmission of DIO and DIS messages by the mobile nodes. The proposed algorithm computes (d) the remaining distance for a node to leave the parent node’s radio range by subtracting the preferred parent node’s radio range from the value of the distance between the two nodes. When (d) distance becomes shorter, the node discovers to find a new parent node. The researcher suggested using ETX and RSSI values to determine the best parent node .^[7]Cobârzan (2016) proposed Mobility-Triggered RPL (MT-RPL) is a cross-layer protocol operating between the MAC and routing layers. It enables X-Machiavel operations to use information from layer 2 to trigger actions at layer 3 to ensure the node connects to the network. It reduces the disconnection time, increases the packet delivery ratio, and reduces overhead. A restriction of MT-RPL is that it relies on a fixed node that acts as an opportunistic forwarder for packets sent by a mobile node .^[8]Wang (2017) proposed an RRD (RSSI, Rank, and Dynamic) method to develop the RPL protocol based on a combination of Received Signal Strength Indicator (RSSI) monitoring, Rank updating, and dynamic control message.  It proposes a new DIO interval by modifying it dynamically according to Rank updates. RRD increased the packet delivery ratio and decreased the end-to-end delay and overhead, but it did not consider energy consumption .^[9] Fotouhi (2017) proposed mRPL+ ^[10], when the link quality decreases during the transfer process with the parent node, the node will start sending periodic DIO messages to search for a better parent node. It also relied on the principle of overhearing to allow the parent node’s neighbour nodes listen for all messages exchanged in their radio range. When a neighbour node detects good link quality with the mobile node, it sends a DIO message to link with it. mRPL+ achieved good results in terms of the packet delivery rate, reaching 100%, but it led to more power consumption. Bouaziz and Rachedi (2019) ^[11] focused on two phases in the protocol, the first one is motion detection for the nodes, and the second is predicting the new connection before the current connection is lost. It relied on the principle of excluding the mobile nodes from the path to avoid interruptions caused by them and considered the mobile nodes as a leaf in the network that isn’t a router. EMA-RPL assumed that the node is moving away when the value of ARSSI decreases, but this is not always true because this value is affected by obstacles. Bouaziz (2019) ^[12] used the Kalman filter algorithm to predict the movement of mobile nodes and choose the parent node based on the predicted path. EKF-MRPL assumes that the mobile nodes will not participate in any routing process, but this is not always possible, especially in applications with many mobile nodes. Predicting movement and calculating distances are based on RSSI value which is inaccurate because it is affected by obstacles. Sanshi (2019) ^[13] modified the RPL protocol using fuzzy logic with several parameters (residual power, expected transfer count ETX, RSSI, mobility timer). FL-RPL uses the mobility timer parameter, which is the expected time to remain within the radio range depending on the location obtained from the RSSI value. This method isn’t accurate because it is affected by barriers and interference. Mobile nodes are considered leaf nodes and cannot participate in the routing process, This concept is not correct in the case of a network with more mobile nodes than fixed ones. Manikannan (2020) ^[14] used the firefly algorithm (FA) inspired by the firefly that produces light to communicate or attract prey or as a protective warning mechanism. When the distance increases, the light becomes weaker. It depends on choosing the parent node with a high RSSI value and re-implementing the FA algorithm until it reaches an optimal solution. In the simulation, only 12 nodes were used, and FA-RPL improved the packet delivery rate by 2.31%. The firefly algorithm (FA) is a good and powerful optimization algorithm, but its computational complexity is high .Safaei (2022) ^[15]. The research proposed ARMOR protocol by using a new parameter TTR to select the best parent node that will stay the longest within the radio range. TTR is calculated based on the node’s speed and position, and it is added to the DIO message. A new timer was added to increase the rate of sending DIO messages by the fixed node in order to introduce itself and to be selected as the parent node by the mobile nodes. The mobile nodes did not modify their timer, but this is not suitable for its neighbor nodes to be aware of their current speed in case it changes .The related works showed there was a need for a protocol that supports mobility, they worked on that but this led to an increase in delay and overhead in the network. In this research, new control message was proposed to make the nodes aware of parent node movement to take into account the appropriate time for changing the parent node without waiting for the timer specified in standard RPL protocol.

                                    Table 1: Related works

Fig. 2: ICMPv6 message format [16]

HERE Base object

HERE Base object is proposed to contain the following fields shown in Figure 3:

1- Flags: 8 bits. only 2 bits are reserved for S and H. the 6 bits remaining unused in the Flags field and reserved for flags. The field must be set to zero by the sender and must be ignored by the receiver.

(STOP) S: the ‘S’ flag indicates that the mobile node has stopped and sends a HERE message to its child nodes.

(LISTEN) L: the ‘H’ flag indicates that the node has heard Here message that is sent by mobile nodes and it is still within its radio range even after its movement, so the mobile node does not need to find a new parent node. However, if it moves and no such message arrives, the mobile node needs to find a new parent node as soon as possible to reduce the delay caused by separation resulted by moving the mobile modes when they change their place.

The values of all flag fields remain zero when a message is sent by the mobile node to its parent and child nodes if it moves to let them know.

(0,0) I MOVE TO HERE.

(0,1) LISTEN I’M HERE

(1,0) I STOP HERE.

(1,1) Invalid state.

Fig. 3: HERE Base object

2-   Control Message Option Format ^[16]:

No type of options was proposed, so padding will be used in proposed message. The general form of option contains three parts (Type, Option Length, Option Data). For padding option, the fields will be as following:

• Option Type: 8-bit, the value (0x00,0x01) is for padding type.
• Option Length: 8-bit, measured in octets, not including the Type and Length fields.
• Option Data: consists of N-2 zero octets.

The Pad1: option is used to add an octet of zeros for alignment. it has neither option length nor option data. The value of this type is 0x00. Pad1 is shown in Figure 4.

Fig. 4: Pad1

The PadN: option is used to add two or more octets of zeros. The value of this type is 0x01. PadN is shown in Figure 5.

Fig. 5: PadN

Proposed protocol rules

A mobile node must reconnect with a parent node within 5 seconds after it stops moving ^[17], so the following rules is suggested for proposed protocol:

• After sending a HERE message, a timer (HERE INTERVAL: 2.5 seconds) is triggered. If the LISTEN message does not arrive, then the mobile node will search for a new parent node.
• A node that has received a HERE message from a parent node has a timer run for 2.5 seconds. If it does not receive another HERE message indicating that the node is moving within its domain, then it will need to search for a new parent node.
• A node that received a HERE message from a parent node and did not receive another message after 2.5 seconds then has to wait for 0.5 seconds. After this period (STOP INTERVAL: 3 seconds), if it does not receive a STOP message, indicating that the mobile node has stopped and is still within its range, then this detects that this node is out of its range and will need to search for a new parent node.
• After the node stops moving for 3 seconds, it sends a STOP message to tell the child node that it has stopped. If the child node hears this message, it will not search for a new parent node. Otherwise, it will search for a new parent node. The mechanism of the proposed protocol is described in Figure 6. When the mobile node moves, it sends the proposed control message ‘HERE’ to children and parent node. As the parent node receives this control message, it sends a ‘STOP’ control message indicating that it is still within its domain. Otherwise, it will search for a new parent node. The child node which receives ‘HERE’ control message waits for a time “HERE INTERVAL”. After this period, if it does not receive another message, it has to wait for an additional time. If it receives a ‘STOP’ control message, this indicates that the node has stopped and is still in the same radio range. If the child node does not receive the STOP message, it will search for a new parent node.

The main purpose of this paper is to maintain the networks from disconnection for long time although of moving nodes, and to less changing parent node, so we make the node stay connected with the parent node although of its movement while it still in its radio range. The proposed control message is sent to inform the node of the change in the location of the parent node, which maintains communication between nodes by searching for the parent node immediately if it moves. The advantages of the proposed message compared to previous works that it is more suitable for LLN networks because it is sent only when node changes his location so it does not increase the overhead in the network.

Fig. 6: The mechanism of the proposed protocol

Fig. 7: Case study1

Figure 7 shows case study where all messages are received correctly despite of the mobile node movement. The parent and child nodes are still in the mobile node’s radio range.

Figure 8 shows another case study where the mobile node was moving, which led the child node to exit outside the mobile node radio range, so it did not receive a HERE or STOP message. After that, it will search for a new parent node. When the mobile node stops, the parent node is out of range, so it does not receive a LISTEN message and it will search for a new parent node by sending DIS messages to its neighbours.

Fig. 8: Case study2

Protocol Performance Evaluation

In this paper, the proposed protocol was evaluated using the Cooja emulator ^[18] that supports IoT and all its protocols. Cooja is used because it is an emulator, not a simulator, meaning that its performance is closer to reality because it is running real devices in the network, which makes the results we get more

accurate and simulating reality. This emulator runs on Contiki OS which is open source, multitasking and designed specifically for constrained memory. It supports a wide range of low-power wireless devices, like a Z1 chip or sky mote, etc.

         Performance metrics [13]

The proposed protocol was evaluated in terms of PDR, power consumption, overhead, and end-to-end delay. The calculations are as follows:

1. 1. Packet Delivery Ratio (PDR): Represents the ratio between the number of received data packets and sent data packets.
2. 2. overhead: The ratio between the number of routing packets and the number of successfully received data packets.
3. 3. Average End-to-End Delay: The average time is taken to propagate the packet from the source to the destination.
4. 4. Average power consumption: is the average amount of power used by nodes during the working time of devices in the network.

Simulation Results and Analysis

This section presents a performance analysis of proposed protocols compared to protocols MARPL, FL-RPL, and ARMOR. The networks in the simulation were built using Cooja program, where the simulation parameters were adopted according to previous studies that were compared with. The research ^[19] suggested the MARPL protocol, which adopted the idea of detecting node movement through the value of RSSI and determining the availability of the neighboring node. If the node receives a DIO message, it updates the Neighbor Variability metric, otherwise, if it receives DAO or DIS control messages, it reduces the time interval between each transmission of DIO messages, which increases their transmission rate and thus more overhead. In the simulation, a 300*300 m2 area was considered with 50 mobile nodes at a maximum speed of 3 m/s with a different number of root nodes (1,2,3). The results in Figure 9 show the value of the packet delivery rate which increases when the number of roots nodes increases.

Fig. 9: Packet Delivery Ratio vs num of roots

By comparing the MARPL simulation results with the proposed protocol, we notice the superiority of the proposed protocol due to its ability to support mobile nodes by making the parent and child nodes of the mobile node aware of its movement via the proposed control message (HERE). MARPL increases the ratio of sending control messages, which causes more overhead on the network and increases collisions because it sends DIO messages to all neighbors, while the HERE control messages are sent only to the parent and child nodes that need this information. Average end-to-end delay: The result shown in Figure 10. Considering that MARPL relied on calculating the value of node availability through the RSSI, this leads to recalculating the value for all nodes every time. In the proposed protocol, even if the node moves and changes its location, it makes sure that the parent node is still within its radio range in order to reduce repeating the search process for a new parent node, and that reduces the power consumption, collision, and delay.

Fig.10: Average End to End delay vs num of roots

FL-RPL ^[13] is modified the RPL using fuzzy logic with several parameters. The simulation was implemented with an area of 10000^m2, 9 mobile nodes and (15,20,25,30) fixed nodes, the simulation results showed that the value of the packet delivery rate is close when comparing the proposed protocol with the FL-RPL protocol, where the proposed protocol exceeded 3% (Figure 11). The proposed protocol outperforms FL-RPL through reducing the delay by about half because the FL-RPL protocol performs many operations every time a node receives a DIO message, (see Figure 12). The routing metrics are given as input of fuzzy inference system to obtain a confidence score of the node and recalculate the mobility time. Therefore, these steps cause more delay and an increase in power consumption. Figure 13 shows that the proposed protocol reduced energy consumption, especially when the number of mobile nodes was more than fixed ones.

Fig. 11: Packet Delivery Ratio vs num of fixed nodes

Fig. 12: Delay vs num of fixed nodes

ARMOR ^[15], the research proposed a new parameter TTR to select the best parent node that will stay the longest within the radio range. TTR is calculated based on the node’s speed and position, and it is added to the DIO message. In this paper, a new timer was added to increase the rate of sending DIO messages by the fixed node in order to introduce itself and to be selected as the parent node by the mobile nodes. The mobile nodes did not modify their timer, but this is not suitable for its neighbor nodes to be aware of their current speed in case it changes. The simulation was implemented with an area of 10000^m2, 20 nodes(10 static nodes and 10 mobile nodes) at a speed of 0.5 to 1.5m/s, and one root node. Another scenario was with 40 nodes (20 static nodes and 20 mobile nodes).

Fig. 13: Power Consumption vs num of fixed nodes

The simulation results showed that the packet delivery rate of the proposed protocol is 10% higher than ARMOR (Figure 14) because it supports mobile nodes by making them directly aware of the state of the parent node connected to them. If it becomes out of radio range, it will search for a new parent node.

Fig. 14: Packet Delivery Ratio vs num of all nodes (mobile and fixed)

The routing load of the ARMOR protocol increased because it modified the timer algorithm for static nodes which made them send more control messages, so the mobile nodes are aware and communicate with them.

Fig. 15: overhead vs num of all nodes (mobile and fixed)

The proposed protocol did not increase the rate of sending control messages (Figure 15), so it was less routing load. It relied only on a suggested control message sent by the mobile node to its parent and child nodes when it moves. The power consumption of the ARMOR protocol is higher than the proposed protocol (Figure 16)  because it sends more control messages.

Fig. 16: Power consumption vs num of all nodes (mobile and fixed)

Discussion

The research shows the need to support mobile nodes in IoT networks. The proposed work helped to achieve this and reduce the impact caused by the nodes when they move within the network. From the simulation results, we observed that the proposed protocol improved the performance of the RPL protocol. It increased the packet delivery ratio because it made the parent and child nodes of the mobile node aware of its state. So they search for a new parent node immediately when the node moves away without waiting to expire the timer of the trickle algorithm, so it decreased the delay .The proposed protocol characterized that it doesn’t send control messages periodically because this is not suitable for the nature of LLN networks. It minimized overhead because it maintained routing adjacency, focused on links that are used by the mobile node, and did not depend on broadcasting the proposed control message to all neighboring nodes to be more suitable for this type of network because this decreased the power consumption. Therefore, the proposed protocol helps to spread the devices that support mobility (smart watches, smart vacuum cleaners) in IoT networks without impact on the network.

Conclusions:

In this research, a new mechanism is proposed to discover disconnection in the network and makes a node reconnect as soon as possible. This disconnection results from movement of the nodes. Our goal is to ensure that the protocol is lightweight while working to support mobility, because many related studies led to increased the overhead, and others depended on discovering movement of nodes through the value of the signal strength RSSI, which is affected by interference and barriers. The new control message ‘HERE’ is proposed to be sent by the node when it moves to both the parent node and its children. If the parent node receives this message, it will respond by LISTEN message, but if it does not receive LISTEN message, it will search for a parent node. If a node stops moving, it sends a STOP message to notify its child nodes. These operations were set with a timer under the standard times for this type of network. The results showed the superiority of the proposed protocol over previous studies because it helped to reconnect the nodes in the network quickly, which increases the packet delivery rate and reduces the delay caused by disconnections in the network when nodes move. Also, it did not depend on increasing the rate of sending control messages which causes an increase in the network overhead .As this paper focuses on supporting the mobility in LLN networks, our future work is to propose a method to determine the type of device (fixed / mobile) and suggest a mechanism for detecting the movement of nodes in real-time within the network and changing its parent node to more appropriate node, so that it reduces the number of changes the parent node .In addition, we will work on enhancing network stability depending on additional parameters to choose the parent node in the network. This research contributes to increase IoT spreading in many applications in several areas, such as parking system, smart home, health care, etc. Which contain mobile nodes without affecting network performance.

Introduction

Many researchers are concerned with studying the behavior of concrete under the influence of compression loads. Then giving proposals for mathematical models through a mathematical extrapolation that explains the material behavior, which is determined by the relation between stress and strain (σ,ε). However, most of the proposed mathematical equations depend mainly on the results of the experimental tests which could be similar in the general shape with variation in the terms of its application, while depending on the mechanical, chemical, and physical properties of the various particles of concrete. In comparison with the studies that were carried out on concrete containing chemical and filler additives, such as hydraulic, pozzolanic materials, and chemical additives such as plasticizers ^[1], the topic of SCC self-compacting concrete containing improved materials and the effect of these materials on the molecular structure of the material is considered an important one for research. This is due to the effectiveness and wide spread of this type of modern concrete SCC in many engineering applications. Mohammed H M presented an experimental investigation on the stress-strain behavior of normal ^[2] and high-strength self-compacting concrete, with two different maximum aggregate sizes. The results show that the ascending parts of the stress-strain curves become steeper as the compressive strength increases, and maximum aggregate size decreases. Jianjie Yu studied the changing regularity of rubber particles’ impact on self-compacting concrete deformation performance ^[3]. The results show that the rubber particles in self-compacting concrete is more uniform distribution, compared with the reference group. In addition, Selvi K presented an experimental investigation on the modulus of elasticity of self-compacting concrete, involving various flying ash proportions ^[4], where the stress-strain relationship was studied for the M20 concrete mix. All previous research has studied self-compacting concrete containing fine filler additives, such as flying ash ^[5]. That gives the concrete distinctive fresh and hardened properties.

This paper will present the production of self-compacting concrete SCC clear of fine fillers by using cement as a fine material that is available and representative of those additives’ fillers.

Materials and Methods

Many previous studies and research have discussed the performance of concrete and described its behavior using the stress-strain curves (σ,ε). This expresses the mechanical behavior of the material ^[6], throughout deducing a formula that enables us to analyze the behavior of concrete mathematically. In 1951, HOGNESTAD proposed mathematical formula (equations 1 and 2), this enables us to describe the relationship between stress and strain in the ascending, and descending parts of curves ^[7] for traditional concrete. Figure /1/. So, stress f_c is calculated as a function of the relative strain $ (\frac{\varepsilon_c}{\varepsilon_{co}}) $.

$ f_{c,1}=0.85{f\prime}_c\left[2\frac{\varepsilon_c}{\varepsilon_{co}}-{(\frac{\varepsilon_c}{\varepsilon_{co}})}^2\right]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0>\varepsilon_c>\varepsilon_{co} $          (1)

$ f_{c,2}=0.85{f\prime}_c\left[1-{0.15(\frac{\varepsilon_c-\varepsilon_{co}}{\varepsilon_{cu}-\varepsilon_{co}})}^2\right]\ \ \ \ \ \ \ \ \ \ \ \ \varepsilon_{co}>\varepsilon_c>\varepsilon_{cu} $          (2)

$ f_{c,1} $: the stress of the concrete in the ascending part MPa.

$ f_{c,2} $: the stress of the concrete in the descending part MPa.

$ {f\prime}_c $: the maximum compression strength of the concrete MPa.

$ \varepsilon_{co} $: the strain corresponding to the maximum compression strength of the concrete.

$ \varepsilon_{cu} $: the critical strain of the concrete.

$ \varepsilon_c $: the strain of the concrete.

Fig. 1. HOGNESTAD stress-strain curve

KENT and PARK also presented, in 1971, a proposal for a stress-strain curve model Equation 3 for the confined and unconfined concrete with the descending part only ^[8]. Figure /2/. So, the stress fc is calculated as a function of the strain corresponding to the maximum strain εco, and the critical strain

$ f_c={f\prime}_c\left[1-\frac{0.5}{\varepsilon_{50u}-\varepsilon_{co}}(\varepsilon_c-\varepsilon_{co})\right]\ \ \ \ \ \ \ \ \ \ \ \ \ \varepsilon_{co}>\varepsilon_c>\varepsilon_{cu} $   (3)

$ \varepsilon_{50u}=\frac{3+0.29{f^\prime}_c}{145{f^\prime}_c-1000}{f^\prime}_c(Mpa) $

Fig. 2. KENT and PARK stress-strain curve

Also, in 1973 POPOVICS presented mathematical formulas. Equation /4/ describing the stress-strain relation of concrete [9]. It was considered that the relative stress  as a function of the relative strain  according to the following:

$ \frac{f_c}{{f\prime}_c}=\left[\frac{n\frac{\varepsilon_c}{\varepsilon_{co}}}{\left(n-1\right)+{(\frac{\varepsilon_c}{\varepsilon_{co}})}^n}\right] $          (4)

$ n=0.058.{f\prime}_c+1 $

He also presented a mathematical formula. Equation 5. in order to calculate the strain of concrete as a function of the maximum compression strength in the concrete:

$ \varepsilon_{co}=\frac{2{f\prime}_c}{12500+450{f\prime}_c} $          (5)

As for CARREIRA, he followed what POPOVICS had reached and developed a mathematical formula. Equation /6/ for calculating stress as a function of strain [3], where he took the effect of the concrete elasticity factor into account. As follows:

$ \frac{f_c}{{f\prime}_c}=\left[\frac{R(\frac{\varepsilon_c}{\varepsilon_{co}})}{\left(R-1\right)+{(\frac{\varepsilon_c}{\varepsilon_{co}})}^R}\right] $           (6)
$ R=\frac{E_c}{E_c-E_O} $

Where:
$ E_c=5000.\sqrt{{f\prime}_c}\ Mpa $
$ E_O=\frac{{f\prime}_c}{\varepsilon_{co}} $

$ ???: is the strain at the maximum compressive strength of confined concrete ?′?. $
The European Code EN 1992-1-1 also presented a mathematical formula. Equation /7/describes the stress-strain behavior of concrete [4] as a function of the relative strain $ \frac{\varepsilon_c}{\varepsilon_{co}}, $, and related to the elasticity factor of the material. According to the following:
$ \frac{f_c}{{f\prime}_c}=\left[\frac{k\eta-\eta^2}{1+(k-2)\eta}\right] $

$ \eta=\frac{\varepsilon_c}{\varepsilon_{co}} $

$ k={1.05.E}_c\frac{\varepsilon_{co}}{{f\prime}_c} $

SCOPE OF WORK

The Fresh Properties of Self-Compacting Concrete

Three mixtures of self-compacting concrete SCC were produced, in addition to the reference mixture, in different cement quantities [11]. The proportions of materials for the mixes are given in Table 1.

Table 1. The proportions of materials

The tests of the fresh properties of SCC were conducted [14]. They include J-ring Test, Slump Flow (SF) and  Test, Visual Stability Index (VSI), and Segregation Test (SR). As shown in Figure 3.

Fig. 3. SCC-HRW Fresh Properties Tests

The rules for adopting the ratio of superplasticizer depend on the technical specification and guidelines for using this type of material, where the recommended ratio range (from 0.5 to 2.5) % of cement weight. In the lab tests, the initial ratio was 1% and then increased till it achieved the required operability in each one of the mixes.

It was found that the mixture with a proportion of cement quantity (550 kg/m³) containing plasticizer coded as HRW gives the preferred properties the operability, Table /2/. The mixture with (450 kg/m³) cement quantity does not give any of the required properties for this type of concrete.

Table 2. SCC fresh properties:

Strength of SCC Samples on the Axial Compression

The laboratory tests result of SCC cylindrical samples produced locally at uni-axial compression [13] show the contribution of the plasticizer by reducing the ratio W/C up to 13%. Thus, there is an improvement in the SCC concrete in terms of strength and strain behavior. Figure 4.

Fig. 4. Fracture Test of SCC Cylindrical Samples

According to laboratory tests, an increase in the strength of concrete containing plasticizer HRW (high reduction water) by 2% of the cement weight up to 12% has been noted, compared to the reference sample, which has the same proportions of materials but without a plasticizer.

The plasticizer in all concrete mixtures contributed to an increase in the operability and a decrease in the W/C ratio. The percentage up in the strength was due to the quantity of cement used, the type of plasticizer, the percentage of plasticizer, and the W/C ratio [12]. The decrease in the compression strength was observed by using a lower quantity of cement with a higher percentage of plasticizer. This can be explained by the change in the molecular structure of SCC mixtures when the ratio W/C is stable. So, it is better to use a higher quantity of cement for a lower plasticizer percentage, and stability by a ratio of W/C.

Since the concrete molecular structure is made up of two components, aggregates and paste. Aggregates are generally classified into two groups, fine and coarse, and occupy about 70% of the mix volume. The paste is composed of cement, water, and entrained air and ordinarily constitutes 30% of the remaining volume.

The major change in the molecular structure of the mixture under the influence of the superplasticizer is on the cement paste, which affects one component or more of the three mentioned.

Table 3. below shows the increase in the cylindrical compression strength depending on the cement quantity, the percentage of plasticizer, and W/C ratio.

       Table 3. Compressive strength of SCC mixtures

In the mixes with cement quantities 450 kg/m³, it is noted that the compression strength takes the same result with or without a plasticizer. This can explain the excess dose of the plasticizer causes additional dispersion and scattering of the cement granules, due to the repulsion of the negative charges that envelop them. Thus, there was a lack of bonding of the cement granules. Another reason that could occur was an increase in the additional air to the mixture.

Figure 5. shows the relation between the strength and the plasticizer percentage used, according to the quantity of cement in the SCC samples.

Fig. 5. SCC Strength and Plasticizer Percentage Relation

According to the previous data of the tests, and by searching for the mathematical equation that is closest and most representative of the relation between the strength of SCC and cement quantity and by using Curve Expert1.4 software program for processing curves, many mathematical curves were obtained. Figure 6. shows the closest curves to the study case with correlation coefficient R.

Fig. 6. The Closest Mathematical Curves

The above three mathematical graphic curves are closest to our experimental tests. It is clear from Figure 6 that the model known as the Rational Function gives the closest model with the best correlation factor R closer to 1. Equation 8, the realistic behaviour of the SCC, and the most expressive of the relation between the cement quantity and the strength of the self-compacting concrete, on the axial compression for different plasticizer ratios. While the other curves show a continuous sharp decline in the strength between the defined points, and an infinitely increasing of strength. This contradicts the relation between the concrete strength and the cement grade.

The Equation of Rational Function is given as:

$ y=\frac{a+bx}{1+cx+dx^2} $              (8)
y: Compression Strength of SCC (MPa).
x: Cement Grade (kg/m3).
a,b,c,: Equation constants, substituting the values of the constants into Equation (1). Founded that the equation takes the following form (9):

$ y=\frac{-45572.93+48.36x}{1-7.74x+0.0124x^2} $

Proposed Model of the Stress-Strain Formula for the Self-Compacting Concrete

The chemical materials in the self-compacting concrete led to a change in the molecular composition of the material. Thus, a change in its mechanical properties affects the behaviour of the concrete in fresh and hardened cases.

Through the application and analysis of previous models of the relation between stress-strain on SCC prepared in the lab, it was found that models can express the behaviour of concrete SCC in varying proportions. As for, the convergence was clear with the lab model, in the ascending part of the curve until reaching the ultimate stress Figure 7. But the difference was in the descending part of the curve. Therefore, it was interesting to search for a mathematical model that describing and expressing the closest behaviour for SCC in ascending and descending parts of the curve.

To understand the behaviour of hardened concrete, and through accurate processing of the numerical results of the fracture tests on the uni-axial compression of cylindrical samples. Models were achieved to describe the stress-strain occurred under the influence of the uni-axial compressive force.

To obtain a general unconditional formula, unrestricted in terms of the quantity of cement and the proportion of plasticizer used in the mixture. Data were processed and converted into non-dimensional relative values allowing the conversion of the data from the specific case into the general state of the tested concrete.

Fig. 7. SCC and Reference Stress-Strain Curves

It was started from the same premise of previous studies and models in mathematical treatment by converting stress f_c into relative stress by dividing it by $ {f\prime}_c $, and the measured strain \varepsilon_c into relative strain by dividing it by $ \varepsilon_{co} $. So, the values of the mathematical treatment for stress and strain, are according to the following:
$ \frac{f_c}{{f\prime}_c} $: nominal relative stress, where: $ {f\prime}_c $: the maximum cylindrical strength of the concrete.
$ \frac{\varepsilon_c}{\varepsilon_{co}} $: nominal relative strain, where:$ \varepsilon_{co} $: strain corresponding to the maximum stress of the concrete.
To find a mathematical model that expresses the behavior of this type of concrete, the results of lab experiments of samples produced of SCC were entered at several cement quantities, and by defining it as nominal relative values $ \frac{f_c}{{f\prime}_c} $, $ \frac{\varepsilon_c}{\varepsilon_{co}} $ to obtain the optimal mathematical curve for this case, which gives the best correlation coefficient R and the smallest standard error.
The treatment showed that the most appropriate mathematical formula for the curve, in this case, is the Rational Function form, Equation 10. which is the general form:
$ y=\frac{a+bx}{1+cx+dx^2} $                          (10)

y: the compressive strength in the concrete.
x: the strain in the concrete.
a, b, c, d equation constants.
The equation shown above and in its general form does not achieve the conditions of the model we are looking for, in terms of stress and relative strain. Figure 8 below shows the experimental and the equation curves form:

Fig. 8. SCC and Rational Function Equation Stress-Strain Curves

Starting from the performance form of the SCC expressed by the stress-strain curve. The equation was reformulated of the closed model in its general form, using the values of relative stress and strain. So that form of Equation 11. and Equation 12. becomes according to the following:

$ \frac{f_c}{{f\prime}_c}=\frac{a+b(\frac{\varepsilon_c}{\varepsilon_{co}})}{1+c(\frac{\varepsilon_c}{\varepsilon_{co}})+d{(\frac{\varepsilon_c}{\varepsilon_{co}})}^2} $  (11)

$ f_{cnorm}=\frac{a+b(\varepsilon_{cnorm})}{1+c(\varepsilon_{cnorm})+d{(\varepsilon_{cnorm})}^2} $   (12)

$ f_{cnorm}=\frac{f_c}{{f\prime}_c} $    nominal relative stress

$ \varepsilon_{cnorm}=\frac{\varepsilon_c}{\varepsilon_{co}} $      nominal relative strain.

Therefore, the proposed general mathematical formula could be expressed with the values of constant b in the ascending and descending parts of the curve.

Results and Discussion

The Proposed General Mathematical Formulas for the Stress-Strain Curve of SCC:

The final mathematical formula using the overall values for the b constant could be expressed for the two parts of the stress-strain curve. Figure 9. As the Equation 18 and Equation 19 below:

Ascending part of the curve:

$ f_{cnorm}=\frac{2.14\varepsilon_{cnorm}}{1+0.14\varepsilon_{cnorm}+{\varepsilon_{cnorm}}^2} $

                                                          $ \frac{f_c}{{f\prime}_c}=\frac{2.14(\frac{\varepsilon_c}{\varepsilon_{co}})}{1+0.14(\frac{\varepsilon_c}{\varepsilon_{co}})+{(\frac{\varepsilon_c}{\varepsilon_{co}})}^2} $                            (18)

Descending part of the curve:

$ f_{cnorm}=\frac{0.57\varepsilon_{cnorm}}{1-1.43\varepsilon_{cnorm}+{\varepsilon_{cnorm}}^2} $

                                                            $ \frac{f_c}{{f\prime}_c}=\frac{0.57(\frac{\varepsilon_c}{\varepsilon_{co}})}{1-1.43(\frac{\varepsilon_c}{\varepsilon_{co}})+{(\frac{\varepsilon_c}{\varepsilon_{co}})}^2} $                            (19)

Fig. 9. Stress-Strain Curves for SCC and Proposed Eq. at all Cement Grade

The behavior using the derived equation is very close to the laboratory samples. It also improves the stress values and the fluidity in the strain values.In comparing the experimental results of the (σ,ε) with the previous curves, the results for the two ascending and descending parts were different. As close as possible to the POPOVICS curve in the ascending part and the EURO-CEB in the descending one through comparing the proposed formula that describes the behavior of SCC as mentioned in formula .17. with that of POPOVICS, which takes the form 20. below:
$ \frac{f_c}{{f\prime}_c}=\left[\frac{n\frac{\varepsilon_c}{\varepsilon_{co}}}{\left(n-1\right)+{(\frac{\varepsilon_c}{\varepsilon_{co}})}^n}\right] $           (20)

Through the observation of the two formulas, there is an incomplete similarity in general. However. this similarity can be complete when the value of the constants as below:
n=b=2
The proposed and POPOVICS formulas take the following form 21:
$ \frac{f_c}{{f\prime}_c}=\frac{2\frac{\varepsilon_c}{\varepsilon_{co}}}{1+{(\frac{\varepsilon_c}{\varepsilon_{co}})}^2} $                         (21)
It can also verify the conformity of the behaviour proposed in Eq.21 with the actual samples graphically. Below are the stress-strain curves of the proposed formula and the experimental sample. Figure .10:

Fig. 10. Stress-Strain Curves for The SCC Proposed Mathematical Formula

It has been found that the mathematical equation 21 in which the POPOVICS formula and the proposed formula are similar, gives graphic curves describing the behavior of local SCC with a convergence of up to 90%.

CONCLUSION

This paper aimed to understand the mechanical behavior of SCC samples. Through laboratory verification and mathematical analysis. The key findings of this study are presented below:

The possibility to produce an acceptable Self-Compacting Concrete SCC from Syrian raw materials.

Outputting a mathematical model that enables us to directly calculate the behavior of all types of SCC samples, and to represent it graphically.

The proposed mathematical model describes the behavior of concrete up to 90% approximate, whatever the quantity of cement.