Heliyon 9 (2023) e22544

Contents lists available at ScienceDirect

Heliyon

journal homepage: www.cell.com/heliyon

Research article

Time series based road traffic accidents forecasting via SARIMA 

and Facebook Prophet model with potential changepoints

Edmund F. Agyemang a,b,c,∗, Joseph A. Mensah c, Eric Ocran a, Enock Opoku a,c, 
Ezekiel N.N. Nortey a

a Department of Statistics and Actuarial Science, College of Basic and Applied Sciences, University of Ghana, Ghana
b School of Mathematical and Statistical Science, College of Sciences, University of Texas Rio Grande Valley, USA
c Department of Computer Science, Ashesi University, No. 1 University Avenue, Berekuso, Accra, Eastern Region, Ghana

A R T I C L E I N F O A B S T R A C T

Keywords:

Road traffic accident

SARIMA

Facebook Prophet

Potential changepoints

Ghana

Road traffic accident (RTA) is a critical global public health concern, particularly in developing 
countries. Analyzing past fatalities and predicting future trends is vital for the development 
of road safety policies and regulations. The main objective of this study is to assess the 
effectiveness of univariate Seasonal Autoregressive Integrated Moving Average (SARIMA) and 
Facebook (FB) Prophet models, with potential change points, in handling time-series road 
accident data involving seasonal patterns in contrast to other statistical methods employed by 
key governmental agencies such as Ghana’s Motor Transport and Traffic Unit (MTTU). The 
aforementioned models underwent training with monthly RTA data spanning from 2013 to 
2018. Their predictive accuracies were then evaluated using the test set, comprising monthly 
RTA data from 2019. The study employed the Box-Jenkins method on the training set, yielding 
the development of various tentative time series models to effectively capture the patterns in 
the monthly RTA data. 𝑆𝐴𝑅𝐼𝑀𝐴 (0,1,1) × (1,0,0)12 was found to be the suitable model for 
forecasting RTAs with a log-likelihood value of −266.28, AIC value of 538.56, AICc value of 
538.92, BIC value of 545.35. The findings disclosed that the 𝑆𝐴𝑅𝐼𝑀𝐴 (0,1,1) × (1,0,0)12 model 
developed outperforms FB-Prophet with a forecast accuracy of 93.1025% as clearly depicted by 
the model’s MAPE of 6.8975% and a Theil U1 statistic of 0.0376 compared to the FB-Prophet 
model’s respective forecasted accuracy and Theil U1 statistic of 84.3569% and 0.1071. A Ljung-

Box test on the residuals of the estimated 𝑆𝐴𝑅𝐼𝑀𝐴 (0,1,1) × (1,0,0)12 model revealed that 
they are independent and free from auto/serial correlation. A Box-Pierce test for larger lags also 
revealed that the proposed model is adequate for forecasting. Due to the high forecast accuracy 
of the proposed SARIMA model, the study recommends the use of the proposed SARIMA model 
in the analysis of road traffic accidents in Ghana.

* Corresponding author at: Department of Statistics and Actuarial Science, College of Basic and Applied Sciences, University of Ghana, Ghana.
Available online 20 November 2023
2405-8440/© 2023 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/).

E-mail address: efagyemang@st.ug.edu.gh (E.F. Agyemang).

https://doi.org/10.1016/j.heliyon.2023.e22544

Received 14 June 2023; Received in revised form 5 November 2023; Accepted 14 November 2023

http://www.ScienceDirect.com/
http://www.cell.com/heliyon
mailto:efagyemang@st.ug.edu.gh
https://doi.org/10.1016/j.heliyon.2023.e22544
http://crossmark.crossref.org/dialog/?doi=10.1016/j.heliyon.2023.e22544&domain=pdf
https://doi.org/10.1016/j.heliyon.2023.e22544
http://creativecommons.org/licenses/by/4.0/


Heliyon 9 (2023) e22544E.F. Agyemang, J.A. Mensah, E. Ocran et al.

1. Introduction

The invention of automobiles brought great relief to humanity; people had little complaints due to fatigue from walking long dis-

tances, lateness to work and goods and services could be procurred within the shortest possible time [1–3]. The use of automobiles 
have increased access to remote places and enhanced livelihood due to related research, promoted social and economic interactions 
geographically and created jobs [4–6]. This not withstanding the continuous use and misuse of automobiles have negatively impacted 
lives as well as property [7]. The rampant occurrence of road accidents seem to have no cure especially in developing countries un-

like many studies pertaining to the control and elimination of diseases as reported in the scientific literature [8]. Therefore, many 
reseachers have given considerable amount of time and space to studying models that could predict the occurrence of road accidents 
over the years. A study by [9] measured the effects of randomness, exposure, weather, and daylight to variations of road accidents 
by using a generalized Poisson regression model based on data from four countries (Denmark, Norway, Finland, and Sweden). Their 
study concluded that randomness and exposure account for eighty to ninety per cent of the variation in road traffic accidents. On 
the contrary, road accidents are viewed as deterministic occurrences according to a study by [10]. However, inadequate information 
makes it uncertain how accidents happen. As a result, in this work, we uphold the notion that road accidents are more random than 
deterministic, as argued by [9], though the vehicle’s driver may have a hand in road traffic accidents.

A road traffic accident is a significant cause of death, injury and a disadvantage or handicap worldwide, both in high-income, low-

middle income and low-income countries [11]. [12] asserted that “Road Traffic Accidents (RTAs) manifest when a motor vehicle 
collides with another vehicle, pedestrian, animal, geographical features, or architectural barriers, potentially leading to injuries, 
property damage, and fatalities”. A host of researchers have commented on the causes of road traffic accidents. [13], among a ton 
of researchers, attributed the causes to overspeeding, drunk driving, wrong overtaking, poor road network and poor worthiness of 
many vehicles in the country. Notable causes of RTAs include but not limited to unnecessary speeding, reckless driving, fatigue [14], 
inadequate experience, traffic rules violation, road surface defects, wrong overtaking, machine failure and defective light [15,16], 
overloading, poor vision [17,18] among others. It is a robust superstitious belief in Africa and Ghana that witches also cause road 
accidents, as many converted witches and wizards confess and attest to this fact [19,20]. However, studies have shown that road 
traffic accidents result from drivers’ unethical behaviours [21]. The force behind this could be strongly linked to the inability of the 
drunk driver to control the vehicle because of sleeping [22]. In addition to drunk drivers, passengers and other drunk road users may 
not know what happens before, during, and after a road accident. When passengers and other road users behave in such a way, they 
are prevented from taking action to avoid serious injuries or death. Drivers do what they want and cause accidents that cost lives. 
The effects of RTAs, apart from causing injuries and death, have also brought about other consequences. Road traffic accidents have 
been known to cause traumas [23], reduce family or persons involved in the accident’s financial position [24], disabilities to people 
[25,26] and psychological effects [27] among others. The impact of road accidents has gone far to the extent that some people refuse 
to drive a vehicle again. RTAs are classified as fatal, serious, or minor based on the damage they inflict on human lives and property 
[28,29].

Ghana, classified as a low-middle income earning country, suffers the most regarding road traffic accidents. RTAs are rising each year 
on Ghana’s roads and have become a significant concern for all and sundry in recent times. Even though the road system network in 
Ghana is terrible, some drivers have also considerably contributed to the number of fatalities registered on Ghana’s roadways each 
year [30]. Although numerous African nations have made strides in reducing road fatalities, some have encountered challenges in 
this effort. Ghana, like several others, faces difficulties in effectively addressing this issue. Over the last three decades, traffic injuries 
and deaths in Africa have increased [31]. In 2005, South Africa, one of the most industrialized countries on the African continent, 
had seventeen (17) allowed automobiles per 100 citizens and no sign of a decrease in road traffic accident deaths as of now [32]. 
Forecasting future RTA-related deaths worldwide is difficult, although past patterns might be thought to give a realistic picture of 
what may occur later. However, a few nations thoroughly veer off from these expectations. Moreover, drifts in numerous parts of the 
world are inconsistent, and there is a confirmation of an increment in deaths in Africa and Asia/Pacific. Statistics from developing 
nations are consistent with changes in the total number of road traffic accidents recorded year after year. Annually, an estimated 
1.2 million individuals succumb to fatalities resulting from RTAs, and another fifty (50) million are injured. Statistical analysts have 
forecasted that these descriptive statistics will increase by about 65% over the next 20 years if care is not taken [33–35]. 75% of 
road traffic deaths came about because vehicles collided with each other in low-earning income countries despite owning only 32% 
of yearly fatalities for every 10,000 cars around the world, making this claim unbelievable. Globally, an estimated cost of US$ 518 
billion is spent on RTAs [36]. The share of the developing countries is about US $100 billion, representing 1 to 3 per cent of their 
gross national product [37]. These stunning numbers indicate that road traffic accidents happen on all landmasses and in every 
nation. Numerical and computational approaches such as those suggested by [38–42] can be employed as baseline mathematical 
models in conjunction with optimization algorithms to help reduce the alarming rate of the occurrence RTAs.

Recognizing the limitations of traditional regression techniques, particularly in handling road traffic accident-related cases due to 
their reliance on independence assumptions, numerous studies have shifted toward time series methodologies. Approaches such as 
ARMA, ARIMA, DRAG, state space models, and structural models are favoured for their ability to enhance the forecasting of fac-

tors related to RTAs. Models have additionally been utilized to examine injuries and deaths caused by RTAs. Various models have 
been used to model road accident data. [43] estimated the influence of speed limit modifications on the number of road crashes in 
metropolitan and provincial interstate thruways in the United States using a structural equation of stochastic modelling technique. 
[44] analyzed RTAs in Kuwait using an Autoregressive Integrated Moving Averages (Box Jenkins) model and compared it to Artificial 
Neural Networks (ANN) to predict RTA deaths in Kuwait. The study found that ANN was superior if there should arise an occurrence 
2

of long-term series without regular variations of accidents. Several researchers have used collision prediction models to model RTAs 



Heliyon 9 (2023) e22544E.F. Agyemang, J.A. Mensah, E. Ocran et al.

in various regions of the world. However, due to differences in numerous parameters in different geographical locations, it becomes 
challenging to apply models that have worked elsewhere in the globe to data gathered from other parts of the world [45]. In Ghana, 
there has been minimal statistical modelling of RTAs. This problem stems from the unavailability of data acquisition on road accident 
cases by the authorities in charge of road accident data. Considering the rate at which RTAs are increasing annually in Ghana, there is 
a need for this study. As a result, statistical analysis of the Madina-Adenta highway RTAs is required to determine the validity or fal-

sity of current literature on RTAs in Ghana. When seasonal patterns in the road traffic accident data are validated, statistical models 
such as SARIMA and FB Prophet would be employed to fit a model to the RTAs data for improved prediction and decision-making. 
The education and research department of the National Motor Transport and Traffic Unit (MTTU) has used descriptive statistical 
techniques and charts for reporting road traffic accidents in Ghana over the past few years. This method’s notable drawback lies in 
its failure to provide essential estimates of road accident occurrences, injuries, and fatalities in Ghana, hampering the ability of the 
National Road Safety Commission (NRSC) and Motor Transport and Traffic Unit (MTTU) stakeholders to make informed projections. 
Utilizing time series models like SARIMA and FB Prophet is, therefore, deemed crucial in addressing this knowledge gap.

A significant limitation to researchers in RTA research is the inability to obtain data on people who suffer from road accidents. In 
most parts of the world, people at an accident scene may fail to report the incident to the police for records to be taken, or if it is 
reported to the police, they fail to keep the records. Additionally, the emergency unit of various hospitals refuses to keep records 
of RTA victims once they are admitted [46]. In Ghana, accurate data on RTA cases are usually hard to come by. The information is 
inadequate even if acquired, mainly because not all accidents are reported to the police for records to be kept [47]. Furthermore, the 
police may have neglected to complete some of the accident report forms on RTAs submitted to them. However, other researchers 
who have utilized data on RTAs from MTTU in Ghana have provided adequate proof that their data is credible. This research makes 
several key contributions. Firstly, it showcases the application of the SARIMA model for capturing temporal patterns in accident 
data. Additionally, it introduces the implementation of the Facebook Prophet model, which adeptly handles holidays, special events, 
and outliers. As a result of incorporating potential changepoints into the study, the models are more capable of adapting to shifts in 
accident patterns than traditional methods. A rigorous comparative analysis of SARIMA and Facebook Prophet models evaluates their 
predictive capabilities, collectively providing an innovative and practical framework for accurate road traffic accident forecasting. 
Due to the increasing rate of RTAs in Ghana, undertaking this research is helpful. The results and findings of this study would be 
beneficial for road safety planning to help minimize road traffic accidents and fatalities in Ghana. The time series model developed 
in this study is recommended for use by the MTTU, NRSC, and relevant stakeholders to help monitor the efficacy of diverse road 
safety policies. Additionally, the study’s findings will contribute to the body of academic literature concerning RTAs

The remainder of the paper is organized as follows: Section 2 discusses the data and methods used for the study, including ARMA, 
ARIMA, SARIMA, FB Prophet models, SARIMA model building process, model identification tools, model diagnostics, and model 
accuracy. Section 3 presents and discusses the results of the investigation. Section 4 concludes the research and provides recommen-

dations.

2. Data and methods

Secondary RTAs data on the Madina-Adenta Highway were retrieved from police reports from 2013 to 2019 and analyzed using 
univariate SARIMA and FB Prophet time series model with potential changepoints. The recorded number of RTAs data used in the 
study include rear-end collisions, head-on collisions, side impact collisions, rollovers, pedestrian or cyclist Accidents, multi-vehicle 
pileups and run-off accidents. Monthly RTA data from 2013 to 2018 (72 months) were used in building the two models, while the 
monthly RTAs for 2019 (12 months) were used in testing the accuracy of the two models under consideration. Data from 2020 to 
the first half of 2023 of the number of Road Traffic Accidents (RTAs) per month were regrettably excluded from this study due to 
unforeseen circumstances resulting from the COVID-19 pandemic. These circumstances led to non-representative and incomplete RTA 
data specific to the Madina-Adenta Highway. The study focused on analyzing RTA data from this highway, revealing a conspicuous 
seasonal pattern. This observation prompted the application of SARIMA and FB Prophet models. The study’s analysis was conducted 
using the R programming language. The study’s data and codes are publicly accessible on GitHub via the repository located at 
github .com /Agyemang1z /Road -Accidents.

2.1. AutoRegressive moving average (ARMA) model

The autoregressive moving average ARMA (𝑝, 𝑞) model is formulated by the combination of autoregressive AR (𝑝) and moving 
average MA (𝑞) model, which is a suitable model for univariate time series data. The AR (𝑝) model is given mathematically by (1):

𝑥𝑡 = 𝜗0 + 𝜗1𝑥𝑡−1 + 𝜗2𝑥𝑡−2 + . . . + 𝜗𝑝𝑥𝑡−𝑝 + 𝜀𝑡 = 𝜗0 +
𝑝∑

𝑖=1
𝜗𝑖𝑥𝑡−𝑖 + 𝜀𝑡 (1)

where 𝑥𝑡 are the observed values, 𝜀𝑡 is random shocks at time 𝑡, 𝜗𝑖 (𝑖 = 1 , 2 , . . . , 𝑘) are the parameters of the 𝐴𝑅 (𝑝) model, 
3

𝜗0 is the constant term, and 𝑝 is the order of the time series model.

http://github.com/Agyemang1z/Road-Accidents


Heliyon 9 (2023) e22544E.F. Agyemang, J.A. Mensah, E. Ocran et al.

The 𝑀𝐴 (𝑞) model is likewise given by (2):

𝑥𝑡 = 𝜇 + 𝜀𝑡 − 𝛼1𝜀𝑡−1 − 𝛼𝜀𝑡−2 − . . . − 𝛼𝑞𝜀𝑡−𝑞 = 𝜇 + 𝜀𝑡 −
𝑞∑
𝑖=1

𝛼𝑖𝜀𝑡−𝑖 (2)

where 𝜇 is the mean of the series, 𝛼𝑖 (𝑖 = 1 , 2 , . . . , 𝑞) represents parameters of the model with order 𝑞, with random errors 𝜀𝑡
are assumed as a white noise process.

The mixed autoregressive moving average 𝐴𝑅𝑀𝐴 (𝑝, 𝑞) model is also expressed mathematically in (3) by:

𝑦𝑡 = 𝜇 +
𝑝∑

𝑖=1
𝜗𝑖𝑥𝑡−𝑖 + 𝜀𝑡 −

𝑞∑
𝑖=1

𝛼1𝑖𝜀𝑡−𝑖 (3)

where the order (𝑝, 𝑞) represents 𝑝 order for autoregressive 𝐴𝑅 (𝑝) and 𝑞 for the moving average 𝑀𝐴 (𝑞) terms.

2.2. AutoRegressive integrated moving average (ARIMA) model

The 𝐴𝑅𝐼𝑀𝐴 (𝑝, 𝑑, 𝑞) model using the lag operator is mathematically expressed in (4) as:

𝜗 (𝐿) (1 −𝐿)𝑑 𝑥𝑡 = 𝜇 + 𝛼 (𝐿)𝜀𝑡(
1 −

𝑝∑
𝑖=1

𝜗𝑖𝐿
𝑖

)
(1 −𝐿)𝑑 𝑥𝑡 = 𝜇 +

(
1 −

𝑝∑
𝑖=1

𝛼𝑖𝐿
𝑖

)
𝜖𝑡 (4)

The order of autoregressive, integrated, and moving average terms of the model are given respectively by 𝑝, 𝑑 and 𝑞; 𝑑 is the 
differencing required to achieve series stationarity.

2.3. Box-Jenkins seasonal ARIMA (SARIMA) model

Many real-world time series datasets feature a seasonal component that repeats after every 𝑆 observation. For example, consider 
utilizing a monthly observation time series dataset, where 𝑆 = 12. We can generally anticipate that 𝑋𝑡 to a large extent rely on 
𝑋𝑡−12 and probably 𝑋𝑡−24 in addition to terms such as 𝑋𝑡−1, 𝑋𝑡−2, …. The Box and Jenkins generalization 𝐴𝑅𝐼𝑀𝐴 (𝑝, 𝑑, 𝑞) include 
seasonal components and are often characterized as a general multiplicative Seasonal ARIMA model abbreviated as SARIMA (𝑝, 𝑑, 𝑞)×
(𝑃 ,𝐷,𝑄)𝑆 model and expressed mathematically in the study in (5) by:

𝜙𝑝 (𝐵)Φ𝑃 (𝐵𝑠)∇𝑑∇𝑆
𝐷𝑋𝑡 = 𝜃𝑞 (𝐵)Θ𝑄

(
𝐵𝑆

)
𝑍𝑡 (5)

Where 𝑆 denotes the seasonal lag, 𝐵 denotes the backshift operator and 𝑍𝑡 is the random error component. 𝜙𝑝 and Φ𝑃 are the 
non-seasonal and autoregressive seasonal parameters. Additionally, 𝜃𝑞 and Θ𝑄 are the non-seasonal and moving average seasonal 
parameters. 𝑝 and 𝑞 are, respectively, the orders of non-seasonal autoregressive and moving average parameters, whilst 𝑃 and 
𝑄 are orders of autoregressive and moving average seasonal parameters, respectively. Lastly, 𝑑 and 𝐷 respectively represent the 
non-seasonal and seasonal differences. ∇𝑑 means we apply the ∇ operator 𝑑 times and similarly for ∇𝑆

𝐷 .

2.4. ARIMA model building process

The ARIMA model uses a three-stage approach to get a suitable model for a forecast. These include:

1. Model Identification: The model identification involves determining if the time series data is stationary or non-stationary [48]. If 
it is non-stationary, determine the degree of differencing needed to make it stationary. The acquisition of the AR order 𝑝 and the 
MA order 𝑞 follows afterwards. Typically, the non-stationary time series data is frequently shown by an autocorrelation graph 
with slow decay. For this study, the Augmented Dickey-Fuller Test (ADF) test was used to test for series stationarity.

2. Model Estimation: This entails determining the best feasible estimates for the Box-Jenkins model parameters [49]. Nonlinear 
least squares and maximum likelihood estimation are the primary methodologies for fitting Box-Jenkins models. The parameters 
in this study were estimated using Maximum Likelihood Estimation (MLE).

3. Model Diagnostic: This stage checks if the model is adequate or not. If the model is inadequate, it is essential to return to stage 
one and choose a better model. Once the model has been selected, estimated, validated, and determined to be acceptable, it is 
utilized to generate forecasts.

2.5. Test of significance of model coefficients

For each coefficient, the estimated t-value is given by (6) as:

𝑒𝑠𝑡𝑖𝑚𝑎𝑡𝑒𝑑 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡
4

𝑡 =
𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑒𝑟𝑟𝑜𝑟

(6)



Heliyon 9 (2023) e22544E.F. Agyemang, J.A. Mensah, E. Ocran et al.

If |𝑡| ≥ 2, the estimated coefficient is significantly different from zero (0), and the model coefficient is statistically significant. Also, if 
the p-value of a model coefficient is less than the 5% significance level, the estimated coefficient is adjudged statistically substantial 
and otherwise.

2.6. Model identification tools

They evaluate the balance between model adequacy and model complexity. Various indicators measuring the quality of fit applied 
in this study’s model identification process include Akaike Information Criterion (AIC), corrected Akaike Information Criterion (AICc), 
Bayesian Information Criterion (BIC), Mean Absolute Percentage Error (MAPE), Mean Square Error (MSE), Root Mean Square Error 
(RMSE), and Mean Absolute Error (MAE). The AIC uses the MLE approach. The MLE technique is utilized to estimate a variety of 
feasible SARIMA models for this approach, and each AIC computed using (7)

𝐴𝐼𝐶 = −2 ln𝐿+ 2𝑊 (7)

where ln𝐿 is the model’s log-likelihood, and 𝑊 is the number of model parameters. In the case of two or more competing models, 
the one with the lower AIC is superior. The AIC exhibits bias, particularly evident when the ratio of parameters to available data is 
high. [50] demonstrated that the tendency might be approximated by introducing an additional non-static penalty factor to the AIC, 
resulting in the development of the corrected AIC, denoted by AICc, and mathematically given by (8) as

𝐴𝐼𝐶𝑐 =𝐴𝐼𝐶 + 2𝑊 (𝑊 + 1)
𝑁 −𝑊 − 1

(8)

where 𝑁 is the sample size or the number of time series observations.

The BIC, like AIC, also uses the MLE. It is expressed by (9) as:

𝐵𝐼𝐶 = −2 ln𝐿+𝑊 ln (𝑁) (9)

The BIC penalizes the number of estimated model parameters more severely than the AIC. Applying minimal BIC for model selection 
results in a model with fewer parameters than that chosen for AIC. According to the concept of parsimony, BIC is considerably 
superior in model selection over AIC. A lower BIC value indicates that the model fits better.

2.7. Test of model diagnostics for SARIMA model

The Box-Pierce and Ljung-Box tests is employed to check the adequacy of the study’s estimated model. The Ljung-Box test fits 
residual (error term) randomness based on several lags. If the autocorrelations of the residuals are small, the model does not exhibit 
a significant lack of fit and is thus assumed adequate. The Lilliefors (KS) test is also used in this study to check for the normality 
of the model’s residuals, and it must have a p-value more significant than 0.05; otherwise, the model’s residuals are considered not 
to be normally distributed. The Ljung-Box statistics is a function of the accumulated sample autocorrelation, 𝜌ℎ, up to any specified 
time lag 𝑘. It is obtained as a function of ℎ given by (10) as

𝑄 = 𝑛 (𝑛+ 2)
𝑘∑

ℎ=1

𝜌2
ℎ

𝑛− ℎ
(10)

where 𝑄 ∼ 𝜒2
𝛼,𝑛−𝑝−𝑞 , and 𝑛 is the number of data points that can be used after any differencing processes. When the calculated value 

of 𝑄 is obtained, the critical region for rejection of the hypothesis of randomness is 𝑄 > 𝜒2
𝛼,𝑛−𝑝−𝑞 . This means that the model under 

consideration is inadequate but adequate if otherwise. When the model is insufficient, there arises a need to fit an appropriate model. 
That is, going back to the model identification and developing a better model.

2.8. Test of model accuracy

Detecting the best-fit model based on accuracy ensures that the chosen model is not over fitted. It is important to note that a high 
error rate indicates that the model is built poorly, whereas a low error rate indicates that it is built well. The accuracy of the two 
competitive models were computed using (11), (12), (13), (14) and (15) respectively: 𝑀𝐴𝑃𝐸, 𝑀𝐴𝐸(𝑀𝐴𝐷), 𝑀𝑆𝐸, 𝑅𝑀𝑆𝐸 and 
Theil U1 Statistic (𝜏).

𝑀𝐴𝑃𝐸 = 1
𝑛

𝑛∑
𝑡=1

|𝑦𝑡 − �̂�𝑡|
𝑦𝑡

× 100%. (11)

𝑀𝐴𝐸(𝑀𝐴𝐷) = 1
𝑛

𝑛∑
𝑡=1

|𝑦𝑡 − �̂�𝑡|. (12)

∑𝑛 (
𝑦 − �̂�

)2

5

𝑀𝑆𝐸 = 𝑡=1 𝑡 𝑡

𝑛
. (13)



Heliyon 9 (2023) e22544E.F. Agyemang, J.A. Mensah, E. Ocran et al.

𝑅𝑀𝑆𝐸 =

√∑𝑛

𝑡=1(𝑦𝑡 − �̂�𝑡)2

𝑛
. (14)

𝜏 =

√√√√1
𝑛

𝑛∑
𝑡=1

𝜖2
𝑡
÷
⎛⎜⎜⎝
√√√√1

𝑛

𝑛∑
𝑡=1

𝑦2
𝑡
+

√√√√1
𝑛

𝑛∑
𝑡=1

�̂�2
𝑡

⎞⎟⎟⎠ ,. (15)

where 𝑦𝑡 are the actual values, �̂�𝑡 are the forecast values and 𝑦𝑡 − �̂�𝑡 = 𝜖𝑡 are the forecast errors. 0 ≤ 𝜏 ≤ 1, for 𝜏 ≊ 0 implies good fit 
of model to data and 𝜏 ≊ 0 implies poor fit of model to data.

2.9. Forecasting

After successfully identifying, estimating, diagnosing, and deciding on the appropriate time series model to use, forecasting can 
be done. If the current time is denoted by 𝑡, the forecast for 𝑌𝑡+𝑟 is the 𝑟− period ahead forecast and denoted by 𝑌𝑡+𝑟. The infinite 
MA representation of the forecast is given in (16) by;

𝑌𝑡+𝑟 = 𝜇 +
∞∑
𝑖=1

𝜓𝑖𝜀𝑡+𝑟−𝑖 (16)

and an 𝐴𝑅𝐼𝑀𝐴(𝑝, 𝑑, 𝑞) process at time 𝑡 + 𝑟 (that is, a period in the future) is given in (17):

𝑌𝑡+𝑟 =
𝑝+𝑑∑
𝑖=1

∅𝑖𝑦𝑡+𝑟−𝑖 + 𝜀𝑡+𝑟 −
𝑞∑
𝑖=1

𝜃𝑖𝜀𝑡+𝑟−𝑖 (17)

where, 𝜓𝑖 is the weight (a constant). Once a forecast is obtained for 𝑌𝑡+1, it can be used to obtain a forecast for 𝑌𝑡+2 and then, these 
two generate a forecast for 𝑌𝑡+3. This can be used to acquire forecasts for any point in time.

2.10. Facebook Prophet forecasting model

Developed by Facebook, the FB Prophet, an additive regression model is in high demand for forecasting purposes due to its three 
main features: trend, seasonality, and holiday. The model is expressed in (18) as

𝑦 (𝑡) = 𝛼 (𝑡) + 𝛽 (𝑡) + 𝜂 (𝑡) + 𝜀 (𝑡) (18)

where 𝑦 (𝑡) is the forecast; the model parameters 𝛼 (𝑡), 𝛽 (𝑡) and 𝜂 (𝑡) are respectively the trend (non-periodic changes), seasonal 
(periodic changes) and holidays effects, which gives irregular schedules. 𝜀 (𝑡) is the error term of the forecast 𝑦 (𝑡) which represent 
any unusual changes. The FB Prophet model adopts a Fourier series to fit models with seasonality effects 𝑠 (𝑡) represented in (19) as

𝑠 (𝑡) =
𝑁∑
𝑘=1

𝛼𝑘 cos
(
2𝜋𝑘𝑡
𝑝

)
+ 𝛽𝑘 sin

(
2𝜋𝑘𝑡
𝑝

)
(19)

where 𝑝 is the period of the seasonal pattern, 𝛼𝑘 and 𝛽𝑘 are the Fourier coefficients. Employing the data’s rising points as a reference, 
the Prophet model adopts a logistic growth curve trend to discern trends. FB Prophet is adept at managing time series data charac-

terized by significant seasonal fluctuations and a substantial historical data span. Notably, the Prophet model effectively manages 
outliers, even in scenarios involving missing data or shifts in trends [51,52]. The effective application of a Prophet model necessi-

tates the variables y (target) and ds (Date Time) in the time series. It demonstrates optimal performance when applied to datasets 
encompassing multiple seasons and featuring notable seasonal impacts [53]. For the purpose of this study, the potential change 
points were chosen as the months with the major holidays in Ghana. This study chose January, March, April, May and December as 
our potential changepoints and the number of changepoints was set as 30. We operationalized the trend model using a saturating 
growth approach, establishing the logistic growth model’s carrying capacity at 10. We set the interval width and change point prior 
Scale to 0.8 and 0.05 respectively.

The introduction and comparison of the FB Prophet model (with potential changepoints) with other competitive time series 
models (such as SARIMA) to forecast RTAs in Ghana has not been explored and to the best of our knowledge, this is the first study 
to explore the FB Prophet model in this domain. Figs. 1 and 2 presents the working model of the SARIMA and FB Prophet adopted 
for the study.

2.11. Comparison between Facebook Prophet and ARIMA models

1. Model Complexity: Facebook Prophet has been designed to be a user-friendly forecasting tool with minimal configuration re-

quirements [54]. It automates several steps involved in time series forecasting, such as handling seasonality, trend detection, and 
outliers. ARIMA is a more traditional and widely used time series forecasting model. A key aspect of ARIMA models is tuning 
model parameters, such as order (𝑝, 𝑑, 𝑞) values for autoregressive, differencing, and moving average components.

2. Seasonality Handling: As part of its integrated functionality, FB Prophet is able to handle a variety of seasonalities, such as daily, 
6

weekly, and yearly patterns [55]. It can handle multiple seasonal components and also handle irregular holidays and events. 



Heliyon 9 (2023) e22544E.F. Agyemang, J.A. Mensah, E. Ocran et al.

Fig. 1. Working model of ARIMA adopted from [53].

Fig. 2. Working model of Facebook Prophet adopted from [53].

Models that use ARIMA can also handle seasonal data through seasonal differencing or by manually incorporating seasonal 
components.

3. Trend Detection and Outlier handling: FB Prophet automatically detects and models both linear and non-linear trends and can 
handle outliers in the data. It can identify and adjust for these outliers, preventing them from overly influencing the forecast 
and can handle situations where the trend changes over time. ARIMA models can capture linear trends but may not handle 
non-linear trends effectively.

4. Interpretability: The FB Prophet provides a more interpretable forecast due to its breakdown into the trend, seasonality, and 
holiday components. In addition, it provides visualizations and diagnostics for evaluating the model’s performance. The ARIMA 
model is less interpretable since it focuses mostly on the statistical properties of the time series. An understanding of the 
underlying mathematics is necessary to interpret the model parameters and diagnostics of an ARIMA model.

2.12. Examples of solving optimization problems in road traffic accident research

It is worth knowing that solving optimization problems in road accident research is key to reducing toad traffic accidents. Below 
are three concrete examples of solving optimization problems in road accident related research.

1. Traffic Signal Timing Optimization

• Problem: Enhancing the synchronization of traffic signals at intersections to alleviate congestion and lower the probability of 
accidents.

• Solution: By optimizing the timing of traffic signals, researchers and traffic engineers can work to minimize the likelihood 
of accidents occurring. An objective function is defined, which could include minimizing the total number of conflict points 
(locations where accidents are more likely to occur) or maximizing the throughput of vehicles. Constraints are established 
to ensure that traffic signal timings adhere to safety and operational standards. These constraints may include minimum 
green time, maximum cycle length, and pedestrian crossing times. Various optimization algorithms, such as those established 
by [56,57] may be applied to find the optimal signal timing plan that minimizes the objective function while satisfying the 
constraints. The optimized signal timing plan is then simulated to assess its impact on traffic flow and safety. Once an optimized 
signal timing plan is validated, it can be implemented at the intersection.
7

2. Route Planning for Emergency Services



Heliyon 9 (2023) e22544E.F. Agyemang, J.A. Mensah, E. Ocran et al.

Fig. 3. Box-plot with data on traffic accidents by months (2013–2019).

• Problem: Finding the optimal routes for emergency vehicles (e.g., ambulances, fire trucks) to reach accident scenes quickly 
while avoiding traffic congestion.

• Solution: Applying optimization algorithms such as that established by [58–60] to take into account real-time traffic data, ac-

cident locations, and the urgency of the situation. These algorithms can recommend the fastest and safest routes for emergency 
responders, potentially saving lives by reducing response times.

3. Vehicle Fleet Optimization for Safety Inspections

• Problem: Optimizing the scheduling and routing of safety inspection teams to inspect a large number of vehicles efficiently, 
ensuring compliance with safety regulations and reducing the risk of accidents due to faulty vehicles.

• Solution: Use vehicle routing optimization algorithms to determine the best inspection routes for a fleet of inspectors, con-

sidering factors like the locations of inspection sites, inspection durations, and traffic conditions. The goal is to maximize the 
number of inspections performed within a given time frame while minimizing travel distance and time.

These examples demonstrate how optimization techniques can be applied in road traffic accident research to enhance safety, improve 
traffic flow, and allocate resources effectively. They leverage data and modelling to make informed decisions and reduce the risk of 
accidents on the road.

3. Results and discussion

This section presents the outcomes of the forecast generated by the SARIMA and FB Prophet models.

Over the seven-year span under study, January demonstrated the lowest occurrence of road accidents, with an average of 29 incidents. 
The first and third quarters each witnessed an average of 33 road accidents per month. Notably, the fourth quarter experienced the 
highest monthly average of reported RTAs, with an average of 37 incidents, marking this period as the most perilous for drivers, 
passengers, pedestrians, and other road users. Unexpectedly, the most perilous month of the year is November, despite the festive 
activities typically associated with December in Ghana. November falls within the fourth quarter and records an average of 41 traffic 
accidents per month. The ratio between the highest average value (November) and the lowest average value (January) stands at 
41.38%. The standard deviations of the monthly road accidents cases depicted in Fig. 3 by year are: 10 (2013), 11 (2014), 8 (2015), 
4 (2016), 11 (2017), 12 (2018), and 4 (2019).

3.1. Development of the SARIMA model

Fig. 4 depicts the frequency of RTAs on the Madina-Adenta highway as spikes and troughs increase and decrease. The average 
number of traffic accidents per year are 26 (2013), 35 (2014), 27 (2015), 29 (2016), 39 (2017), 41 (2018) and 42 (2019). From the 
reference year 2013, it is evident that the average number of road accidents has risen by 34.62% (2014), 3.85% (2015), 11.54% 
(2016), 50.00% (2017), 57.69% (2018) and 61.54% (2019). From Fig. 4, we observe some upward and downward surges in the 
series. Also, Fig. 4 shows that the monthly RTAs time series dataset under examination has a solid and persistent seasonal tendency. 
This indicates that seasonal components exist in the monthly RTAs. This further suggests that the monthly RTAs data on the Madina-

Adenta highway is non-stationary. To effectively apply an appropriate time series model to the RTA data, it’s imperative to eliminate 
both the seasonality and trend present within the dataset. The single exponential smoothing method was employed to deal with the 
seasonality and trend components as suggested by the ndiffs() function in R.

3.2. Single exponential smoothing method

The road traffic accident data underwent a first-order differencing method to effectively eliminate both the trend and seasonality 
from the original dataset. Fig. 5 helps us to see that the differenced data exhibits stationarity with constant mean and variance. The 
8

Augmented Dickey-Fuller Test (ADF) test was employed to confirm series stationarity test.



Heliyon 9 (2023) e22544E.F. Agyemang, J.A. Mensah, E. Ocran et al.

Fig. 4. Time series plot of Road Traffic Accident cases from 2013 to 2019.

Fig. 5. Single Exponential plot of Road Traffic Accident cases.

Table 1

ADF test for Level Stationarity.

Test Statistic Lag Order P-value

ADF −5.8053 4 0.0100

We reject the null hypothesis of non-stationarity and conclude that the first differenced RTAs data is stationary since 0.0100 <
0.05 as evident in Table 1.

3.3. Autocorrelation and partial autocorrelation function plots of the first difference

From Fig. 6(a), looking at the autocorrelation function with the error limits, autocorrelation at lag one is significant. This depicts 
a possible behaviour of MA (1). For the plot of ACF, we notice sinusoidal waves shown in the autocorrelation plot.

Fig. 6(b) depicts the partial autocorrelation function plot, which demonstrates that the partial autocorrelation coefficient exceeds 
the error bounds at lags 1, 2, and 4, which are significant. The error limit at lag 3 is insignificant, and it is the first to do so. 
Since the PACF cuts off after lags 1, 2 and lag 4, this suggests possible behaviour of SMA (1), SMA (2) or SMA (4). The ACF and 
PACF plots suggest that 𝑄 = 1, 2 𝑜𝑟 4, 𝑝 = 1, 𝑃 = 1, 𝑑 = 1, 𝐷 = 0 and 𝑞 = 1 would be needed to model the road traffic accident 
data. The following tentative models; 𝑆𝐴𝑅𝐼𝑀𝐴 (1,1,1) × (1,0,1)12, 𝑆𝐴𝑅𝐼𝑀𝐴 (1,1,1) × (1,0,0)12, 𝑆𝐴𝑅𝐼𝑀𝐴 (1,1,0) × (0,0,1)12, 
𝑆𝐴𝑅𝐼𝑀𝐴 (0,1,1) × (1,0,0)12, 𝑆𝐴𝑅𝐼𝑀𝐴 (0,1,2) × (1,0,0)12, 𝑆𝐴𝑅𝐼𝑀𝐴 (1,1,2) × (1,0,0)12 and 𝑆𝐴𝑅𝐼𝑀𝐴 (0,1,3) × (1,0,2)12
9

can be suggested after careful examination of the ACF and PACF plots in Fig. 6.



Heliyon 9 (2023) e22544E.F. Agyemang, J.A. Mensah, E. Ocran et al.

Fig. 6. Plot of the ACF and PACF of First Difference road traffic accident cases.

Table 2

Final parameter estimates of 𝑆𝐴𝑅𝐼𝑀𝐴 (1,1,1) × (1,0,1)12 .

Parameter Coefficient Standard Error Z-value P-value

AR (1) −0.0057 0.0007 −7.8014 0.000∗∗∗

MA (1) −0.7831 0.0069 −1111.2357 0.000∗∗∗

SAR (1) −0.5513 NaN NaN NaN

SMA (1) 0.3654 0.0028 132.0225 0.000∗∗∗

Table 3

Final parameter estimates of 𝑆𝐴𝑅𝐼𝑀𝐴 (1,1,1) × (1,0,0)12 .

Parameter Coefficient Standard Error Z-value P-value

AR (1) 0.0422 0.1543 0.2733 0.7846

MA (1) −0.8080 0.0975 −8.2895 0.000∗∗∗

SAR (1) −0.2176 0.1363 −1.5965 0.1104

Table 4

Final parameter estimates of 𝑆𝐴𝑅𝐼𝑀𝐴 (1,1,0) × (0,0,1)12 .

Parameter Coefficient Standard Error Z-value P-value

AR (1) −0.4929 0.1075 −4.5863 0.000∗∗∗

SMA (1) −0.1678 0.1325 −1.2654 0.2057

Table 5

Final parameter estimates of 𝑆𝐴𝑅𝐼𝑀𝐴 (0,1,1) × (1,0,0)12 .

Parameter Coefficient Standard Error Z-value P-value

MA (1) −0.7905 0.0807 −9.7978 0.000∗∗∗

SAR (1) −0.2170 0.1364 −1.5911 0.1116

3.4. Possible seasonal ARIMA models parameter estimation

The tables below display the final parameter estimates of the possible seasonal ARIMA models under consideration with their 
respective coefficients, standard error, and p-values.

From Table 2, AR(1), MA (1) and SMA (1) are statistically significant at the 5% significance level.

From Table 3, MA (1) with a coefficient of -0.8080 and a Z-value of -8.2895 is statistically significant.

AR (1) with a coefficient of -0.4929 and a standard error of 0.1075 statistically significant as seen in Table 4.

From Table 5, MA (1) is statistically significant with p-value < 0.05.

From Table 6, MA (1) is statistically significant with p-value < 0.05.

Table 6

Final parameter estimates of 𝑆𝐴𝑅𝐼𝑀𝐴 (0,1,2) × (1,0,0)12 .

Parameter Coefficient Standard Error Z-value P-value

MA (1) −0.7665 0.1186 −6.4636 0.000∗∗∗

MA (2) −0.0325 0.1209 −0.2686 0.7883

SAR (1) −0.2179 0.1363 −1.5983 0.1100
10



Heliyon 9 (2023) e22544E.F. Agyemang, J.A. Mensah, E. Ocran et al.

Table 7

Final parameter estimates of 𝑆𝐴𝑅𝐼𝑀𝐴 (1,1,1) × (1,0,0)12 .

Parameter Coefficient Standard Error Z-value P-value

AR (1) 0.0422 0.1543 0.2733 0.7846

MA (1) −0.8080 0.0975 −8.2895 0.000∗∗∗

SAR (1) −0.2176 0.1363 −1.5965 0.1104

Table 8

Final parameter estimates of 𝑆𝐴𝑅𝐼𝑀𝐴 (0,1,3) × (1,0,2)12 .

Parameter Coefficient Standard Error Z-value P-value

MA (1) −0.7654 0.1204 −6.3592 0.000∗∗∗

MA (2) −0.0072 0.1710 −0.0420 0.9665

MA (3) −0.0269 0.1639 −0.1645 0.8693

SAR (1) −0.3745 0.4348 −0.8615 0.3890

SMA (1) 0.1952 0.4297 0.4543 0.6496

SMA (2) 0.1129 0.2598 0.4346 0.6638

Table 9

Potential SARIMA models with their 𝐴𝐼𝐶 , 𝐴𝐼𝐶𝐶 and 𝐵𝐼𝐶 val-

ues.

SARIMA Models 𝐴𝐼𝐶 𝐴𝐼𝐶𝐶 𝐵𝐼𝐶

SARIMA (1,1,1) × (1,0,1)12 542.08 543.01 553.4

SARIMA (1,1,1) × (1,0,0)12 540.49 541.10 549.54

SARIMA(1,1,0) × (0,0,1)12 552.94 553.30 559.73

SARIMA (0,1,1) × (1,0,0)12 538.56 538.92 545.35

SARIMA (0,1,2) × (1,0,0)12 540.49 541.10 549.54

SARIMA (1,1,2) × (1,0,0)12 542.51 543.43 553.82

SARIMA(0,1,3) × (1,0,2)12 546.10 547.88 561.94

Table 10

Ljung-Box test of SARIMA (0,1,1) ×
(1,0,0)12 .

Test Statistic P-value

Ljung-Box 6.5417 0.8864

From Table 7, MA (1) is also statistically significant at alpha level of < 0.05.

MA(1) is also statistically significant in the 𝑆𝐴𝑅𝐼𝑀𝐴 (0,1,3) × (1,0,2)12 model as seen in Table 8.

3.5. Fitting the suitable SARIMA model to road accident time series data

Both the initial differenced ACF and PACF plots were rigorously examined when modelling the road traffic accident series data 
to produce the suitable-fitted model based on their respective 𝐴𝐼𝐶 , 𝐴𝐼𝐶𝐶 and 𝐵𝐼𝐶 values.

From Table 9, comparing the 𝐴𝐼𝐶 , 𝐴𝐼𝐶𝐶 and 𝐵𝐼𝐶 values of the candidate models, it can be deduced that the SARIMA (1,1,0)×
(0,0,1)12 (highlighted in red) has an 𝐴𝐼𝐶 value of 538.56, an 𝐴𝐼𝐶𝐶 value of 538.92 and a 𝐵𝐼𝐶 value of 545.35 which are the lowest 
accuracies of all the tentative SARIMA models constructed. Therefore, per the model selection criteria, SARIMA (1,1,0) × (0,0,1)12
is the suitable SARIMA time series model for modelling the Madina-Adenta highway road traffic accident.

The general multiplicative Seasonal ARIMA model is thus given in the study by equation (5) as:

𝜙𝑝 (𝐵)Φ𝑃 (𝐵𝑠)∇𝑑∇𝑆
𝐷𝑋𝑡 = 𝜃𝑞 (𝐵)Θ𝑄

(
𝐵𝑆

)
𝑍𝑡

Hence, the seasonal ARIMA (0,1,1) × (1,0,0) can be represented in (20) as

𝜙0 (𝐵)Φ1
(
𝐵12)∇1∇12

0𝑋𝑡 = 𝜃0 (𝐵)Θ0
(
𝐵12)𝑍𝑡 (20)

3.6. Diagnostic checking of estimated model

The suitable-fitted model for modelling the road accident data is then tested further to make theoretical conclusions about the 
11

model as a good fit and an optimal model for both estimation and forecasting.



Heliyon 9 (2023) e22544E.F. Agyemang, J.A. Mensah, E. Ocran et al.

Table 11

Lilliefors (KS) Normality test of SARIMA 
(0,1,1) × (1,0,0)12 .

Test Statistic P-value

Lilliefors (KS) 0.0841 0.2364

Fig. 7. Plot of ACF and PACF plot of residuals of SARIMA (0,1,1) × (1,0,0)12 model.

From Table 10, Given the p-value of 0.8864, which exceeds the significance threshold of 0.05, we conclude that the residuals 
of the estimated model demonstrate independence and follow an identically distributed or white noise process. Hence, the model 
displays no substantial lack of fit.

At a 5% significance level, since the p-value is greater than the alpha level (thus, 0.2364 > 0.05), the residuals of the estimated 
SARIMA (0,1,1) × (1,0,0)12 model are normally distributed as evident in Table 11.

3.7. Test of model adequacy of SARIMA

The limits of the 2𝜎 are given by ±𝑍𝛼

2
× 1√

𝑛
.

−1.96

(
1√
72

)
≤ 𝑟 ≤ 1.96

(
1√
72

)
This is simplified as −0.2310 ≤ 𝑟 ≤ 0.2310. Hence, the confidence interval of the random error spans across two (2) standard devia-

tions. Fig. 7 shows two blue horizontal lines indicating this claim. It could be seen that the errors are within ±0.2310 as evident in 
the ACF/PACF plots. It should also be noted that the plot of these autocorrelations shows no systematic structure, indicating that the 
residuals are purely random.

3.8. Box-Pierce test for larger lags

The modified Ljung-Box test (Box-Pierce test) for larger lags was employed to ascertain whether the proposed model is adequate 
for forecasting.

Observing Table 12, it is apparent that the p-values at different lags considerably surpass the 0.05 significance level. As a result, 
we lack substantial evidence to reject our model. Given its adequacy for lags 12, 24, 36, 48, and 60, it is plausible to assume its 
adequacy for larger lags as well. Hence, the SARIMA (0,1,1) × (1,0,0)12 model is adequate at a 0.05 level of significance and can be 
used to forecast future road accident cases optimally.

3.9. Evaluation of SARIMA model

It is clear from Table 13 that the SARIMA (0,1,1) × (1,0,0)12 model developed for RTAs on the Madina-Adenta Highway has a 
forecast accuracy of 93.1025%, displayed by the model’s MAPE of 6.8975% which is indicative of highly accurate forecasting as 
12

suggested by [61]. A 𝜏 statistic of 0.0376 further indicates good model fit.



Heliyon 9 (2023) e22544E.F. Agyemang, J.A. Mensah, E. Ocran et al.

Table 12

Modified Ljung-Box (Box-Pierce test) for 
larger lags of SARIMA (0,1,1) × (1,0,0)12 .

Lags Chi-Square Statistic P-value

12 5.7045 0.9302

24 8.1798 0.9989

36 14.956 0.9992

48 19.566 0.9999

60 25.717 1.0000

Table 13

Out of sample Validation for 𝑆𝐴𝑅𝐼𝑀𝐴 (0,1,1) × (1,0,0)12 with Forecast Performance Statistics.

Month Year 𝑦𝑡 �̂�𝑡 𝑦2
𝑡

�̂�2
𝑡

𝜖𝑡 |𝜖𝑡| 𝜖2
𝑡

| 𝜖𝑡
𝑦𝑡
| |% 𝐸𝑟𝑟𝑜𝑟|

Jan 2019 38 43 1444 1849 5 5 25 0.1316 13.16

Feb 2019 43 47 1849 2209 4 4 16 0.0930 9.30

Mar 2019 36 40 1296 1600 4 4 16 0.1111 11.11

Apr 2019 45 44 2025 1936 −1 1 1 0.0227 2.27

May 2019 40 42 1600 1764 2 2 4 0.0500 5.00

Jun 2019 39 40 1521 1600 1 1 1 0.0256 2.56

Jul 2019 50 46 2500 2116 −4 4 16 0.0800 8.00

Aug 2019 42 43 1764 1849 1 1 1 0.0233 2.33

Sep 2019 39 44 1521 1936 5 5 25 0.1282 12.82

Oct 2019 44 42 1936 1764 −2 2 4 0.0454 4.54

Nov 2019 41 38 1681 1444 −3 3 9 0.0732 7.32

Dec 2019 46 44 2116 1936 −2 2 4 0.0435 4.35

Total 503 513 21253 22003 34 122 0.8277

MAE 2.8333

MAPE 6.8975

MSE 10.1667

RMSE 3.1885

Theil’s U1 Statistic (𝜏) 0.0376

Fig. 8. Plot of Forecasted values of Facebook Prophet model with 95% confidence limits.

From Table 14, it is evident that the best model by the auto.arima function in R programming software confirms that our 
selected model (ARIMA(0,1,1)(1,0,0)[12]) using the diagnostic of the ACF and PACF plots is the ideal seasonal model to forecast the 
underlying RTAs data.

3.10. Evaluation of Facebook Prophet model

The performance of the FB Prophet model, with change points being the months with the major holidays in Ghana is assessed. 
January, March, April, May and December were chosen as potential change points.

It is evident from Table 15 and Fig. 8 that the Facebook Prophet model developed for RTAs on the Madina-Adenta Highway has 
13

a forecast accuracy of 84.3569%, displayed by the model’s MAPE of 15.6431%.



Heliyon 9 (2023) e22544E.F. Agyemang, J.A. Mensah, E. Ocran et al.

Table 14

Possible tentative ARIMA models extracted by auto.arima func-

tion.

Model Evaluation Statistic

ARIMA(0,1,0)(0,0,1)[12] 569.2443

ARIMA(0,1,0)(0,0,1)[12] with drift 571.4203

ARIMA(0,1,0)(0,0,2)[12] 570.8983

ARIMA(0,1,0)(0,0,2)[12] with drift 573.145

ARIMA(0,1,0)(1,0,0)[12] 568.8583

ARIMA(0,1,0)(1,0,0)[12] with drift 571.0352

ARIMA(0,1,0)(1,0,1)[12] 570.9207

ARIMA(0,1,0)(1,0,1)[12] with drift 573.1644

ARIMA(0,1,0)(1,0,2)[12] 572.3704

ARIMA(0,1,0)(2,0,0)[12] with drift 573.0677

ARIMA(0,1,0)(2,0,1)[12] 573.0674

ARIMA(0,1,0)(2,0,1)[12] with drift 575.3814

ARIMA(0,1,0)(2,0,2)[12] Inf

ARIMA(0,1,0)(2,0,2)[12] with drift Inf

ARIMA(0,1,1)(0,0,1)[12] with drift 540.1289

ARIMA(0,1,1)(0,0,2)[12] 541.2295

ARIMA(0,1,1)(0,0,2)[12] with drift 542.4327

ARIMA(0,1,1)(1,0,0)[12] 538.9229

ARIMA(0,1,1)(1,0,0)[12] with drift 539.9962

ARIMA(0,1,1)(1,0,1)[12] 540.9437

ARIMA(0,1,3)(0,0,1)[12] 543.619

ARIMA(0,1,3)(0,0,1)[12] with drift 544.4773

ARIMA(0,1,3)(0,0,2)[12] 545.7991

ARIMA(1,1,1)(0,0,1)[12] 541.3203

ARIMA(1,1,1)(0,0,1)[12] with drift 542.2117

ARIMA(1,1,1)(0,0,2)[12] 543.4696

ARIMA(1,1,1)(0,0,2)[12] with drift 544.597

ARIMA(1,1,1)(1,0,0)[12] 541.0963

ARIMA(1,1,1)(1,0,0)[12] with drift 542.1079

ARIMA(1,1,1)(1,0,1)[12] Inf

ARIMA(1,1,1)(1,0,1)[12] with drift Inf

ARIMA(1,1,1)(1,0,2)[12] 545.4409

ARIMA(1,1,1)(1,0,2)[12] with drift 546.8489

ARIMA(1,1,1)(2,0,0)[12] Inf

ARIMA(1,1,1)(2,0,0)[12] with drift Inf

ARIMA(2,1,1)(0,0,1)[12] 543.6271

ARIMA(2,1,1)(0,0,1)[12] with drift 544.5017

ARIMA(2,1,1)(0,0,2)[12] 545.8267

ARIMA(2,1,1)(0,0,2)[12] with drift 546.9558

ARIMA(2,1,1)(1,0,0)[12] 543.4051

ARIMA(3,1,0)(0,0,1)[12] 549.7264

ARIMA(3,1,0)(0,0,1)[12] with drift 551.9101

ARIMA(3,1,0)(0,0,2)[12] 552.022

ARIMA(3,1,0)(0,0,2)[12] with drift 554.3071

ARIMA(3,1,0)(1,0,0)[12] 549.4729

ARIMA(4,1,0)(0,0,1)[12] 543.9409

ARIMA(4,1,0)(0,0,1)[12] with drift 546.0397

ARIMA(4,1,0)(1,0,0)[12] 543.4893

ARIMA(4,1,0)(1,0,0)[12] with drift 545.5827

ARIMA(4,1,1) with drift 548.1061

ARIMA(5,1,0) with drift 548.3584

Best model: ARIMA(0,1,1)(1,0,0)[12].

The SARIMA (0,1,1) × (1,0,0)12 model developed in this study outperforms the Facebook Prophet model per their respective 
forecast performance statistics. It is then ideal to use the SARIMA model to make forecasts.

It is observed that road accidents on the Madina-Adenta Highway will show decreasing and increasing spikes from the start of 
January 2019 to the last quarter of 2019. There will be an anticipated steady pattern in RTA cases from the first quarter of 2020 and 
afterwards, as depicted by Fig. 9.

4. Conclusion and recommendation

Introducing and producing several vehicles has led to increased traffic accidents and negative consequences. Therefore, constantly 
reviewing, analyzing, and evaluating the existing situation is necessary. This will help identify the main causes of accidents and the 
most effective prevention strategies. Creating awareness campaigns to educate people on safe driving practices is also imperative. 
14

An integral facet of traffic management within a jurisdiction involves forecasting the frequency of RTAs during specific periods of 



Heliyon 9 (2023) e22544E.F. Agyemang, J.A. Mensah, E. Ocran et al.

Table 15

Out of sample Validation for Facebook Prophet model with Forecast Performance Statistics.

Month Year 𝑦𝑡 �̂�𝑡 𝑦2
𝑡

�̂�2
𝑡

𝜖𝑡 |𝜖𝑡| 𝜖2
𝑡

| 𝜖𝑡
𝑦𝑡
| |% 𝐸𝑟𝑟𝑜𝑟|

Jan 2019 38 38 1444 1444 0 0 0 0.0000 0.00

Feb 2019 43 43 0 0 0 0 0 0.0000 0.00

Mar 2019 36 55 1296 3025 −19 19 361 0.5278 52.78

Apr 2019 45 35 2025 1225 10 10 100 0.2222 22.22

May 2019 40 33 1600 1089 7 7 49 0.1750 17.50

Jun 2019 39 41 1521 1681 −2 2 4 0.0513 5.13

Jul 2019 50 38 2500 1444 12 12 144 0.2400 24.00

Aug 2019 42 42 0 0 0 0 0 0.0000 0.00

Sep 2019 39 37 1521 1369 2 2 4 0.0513 5.13

Oct 2019 44 46 1936 2116 −2 2 4 0.0455 4.54

Nov 2019 41 57 1681 3249 −16 16 256 0.3902 39.02

Dec 2019 46 38 2116 1444 8 8 64 0.1739 17.39

Total 503 503 21253 21699 78 986 1.8772

MAE 6.5000

MAPE 15.6431

MSE 82.1667

RMSE 9.0646

Theil’s U1 Statistic 0.1071

Fig. 9. Plot of Forecasted values of (0,1,1) × (1,0,0)12 model with 95% confidence limits.

the year [61]. This is mainly to get citizens informed about road safety status to understand the issue better, improve their attitudes, 
and improve their driving habits. Our primary objective for this study is to compare how well SARIMA and Facebook Prophet 
models with potential change points handle time series data with seasonal components. The study aimed to identify a suitable model 
that fits the Madina-Adenta RTA data and use it to forecast. The time series plot of the RTA data showed both increasing and 
decreasing spikes with sharp upward and downward surges, indicative of seasonality in the data. This gives a suspicion of the data 
being non-stationary. The RTA data was then differenced once to make it stationary. An estimated seasonal ARIMA and FB Prophet 
models were developed using the monthly RTA time series data from 2013 to 2018 (72 months) as the training set. The monthly 
RTA time series data for 2019 (12 months) were used to test the accuracy of candidate models. A comparison of all the possible 
model accuracy metrics (𝐴𝐼𝐶 , 𝐴𝐼𝐶𝐶 and 𝐵𝐼𝐶) of the suggested tentative models for the Madina-Adenta monthly RTAs time series 
data revealed that 𝑆𝐴𝑅𝐼𝑀𝐴 (0,1,1) × (1,0,0)12 which has an 𝐴𝐼𝐶 value of 538.56, 𝐴𝐼𝐶𝐶 value of 538.92 and a 𝐵𝐼𝐶 value of 
545.35, which were respectively the lowest among all the possible SARIMA models formulated was chosen as the suitable model in 
modelling the RTAs on the Madina-Adenta highway, Ghana per the selection criteria approaches. 𝑆𝐴𝑅𝐼𝑀𝐴 (0,1,1) × (1,0,0)12 was 
then subjected to model diagnostic tests. The diagnostic test was conducted on the normality of the residuals, independence of the 
residuals and the test for model accuracy. The Lilliefors test for normality proved that the residuals were normally distributed. In 
contrast, the Ljung-Box test shows that the residuals were free of serial or autocorrelation and followed a white noise process. The 
modified Ljung-Box, also known as the Box-Pierce test for larger lags, proved significant at the 5% significance level for lags 12, 
24, 36, 48 and 60, which indicates that it will be significant for larger lags, too, making the estimated model considered ideal for 
forecasting. A comparative analysis was then made between 𝑆𝐴𝑅𝐼𝑀𝐴 (0,1,1) × (1,0,0)12 and FB Prophet model, where the former 
provided a high forecast accuracy of 93.1025% relative to the latter’s forecast accuracy of 84.3569%. A Theil U1 statistic of 0.0376 
15

for the SARIMA model compared to 0.1071 for the FB Prophet model further indicates a good model fit of the SARIMA model to 



Heliyon 9 (2023) e22544E.F. Agyemang, J.A. Mensah, E. Ocran et al.

the RTA data. Even though the FB Prophet model has outperformed SARIMA models in most domains regarding RTA modelling, 
its applicability to the Ghanaian setting is missing in action. For example, forecasting daily time series of passenger demand for 
urban rail transit [62], road traffic injury prediction in Northeast China [63], road traffic forecasting in Bangladesh [64]. The study, 
therefore, recommended that the MTTU and National Safety Road Commission (NRSC) should adopt 𝑆𝐴𝑅𝐼𝑀𝐴 (0,1,1) × (1,0,0)12
model in their RTAs safety intervention planning activities due to its high forecast accuracy. Several avenues for further research can 
be pursued. First, use model comparison and ensemble techniques to compare SARIMA and Prophet against advanced counterparts 
such as machine learning techniques as in [65–67]. Researchers can also explore more intricate time series models, such as state 
space models, VARs, and Bayesian structural models, for enhanced predictive accuracy. Anomaly detection within accident data 
can be explored, identifying unusual patterns and sudden spikes, which can help inform the development of early warning systems 
and targeted intervention strategies for more proactive accident prevention. The government of Ghana is also urged to support 
institutions like the MTTU and NRSC in terms of recruiting qualified personnel and providing logistics and quality education on 
road traffic accident prevention for the citizenry. The fight against RTA is the responsibility of every citizen. This research should 
be used to develop effective road safety strategies and policies, evaluate and monitor road safety initiatives’ progress, and identify 
improvement areas. To enhance road safety and mitigate accidents, multifaceted approaches must be employed in road accident-

related research in Ghana. These interventions span diverse domains, including infrastructure enhancement through improved road 
design and pedestrian/cyclist facilities; stringent enforcement of traffic regulations encompassing speed limits, seat belt and helmet 
laws, and measures against impaired driving; fostering public awareness through road safety campaigns and educational programs 
integrated into school curricula; elevation of vehicle safety standards through mandatory incorporation of advanced safety features 
and rigorous crash testing; establishment of comprehensive accident databases for data-driven decision-making and identification of 
accident patterns; legal and regulatory reforms about liability, insurance, and penalty structures, to bolster road safety. By integrating 
these measures, we aim to reduce the burden of road accidents on society and create a safer road environment in a low-middle income 
country such as Ghana.

Funding statement

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Additional information

No additional information is available for this paper.

CRediT authorship contribution statement

Edmund F. Agyemang: Writing – review & editing, Writing – original draft, Validation, Software, Methodology, Investigation, 
Formal analysis, Data curation, Conceptualization. Joseph A. Mensah: Writing – original draft, Supervision, Project administration, 
Methodology, Investigation, Formal analysis. Eric Ocran: Writing – original draft, Visualization, Software, Methodology, Formal 
analysis. Enock Opoku: Writing – original draft, Software, Project administration, Investigation, Formal analysis. Ezekiel N.N. 
Nortey: Writing – review & editing, Methodology, Investigation, Formal analysis, Conceptualization.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to 
influence the work reported in this paper.

Data availability

The data used to support the findings of this study are available from the corresponding author upon request and can also be 
assessed from the R codes available on GitHub repository at github .com /Agyemang1z /Road -Accidents.

Acknowledgement

The authors express their gratitude to the Motor Traffic and Transport Unit for granting access to the dataset utilized in this 
research. Additionally, they acknowledge the invaluable contributions of the anonymous reviewers and editors, whose insightful 
comments greatly enriched this work. The first and corresponding author also acknowledges the enormous support of the University 
of Texas Rio Grande Valley Presidential Research Fellowship fund.

References

[1] Ryan Collin, Yu Miao, Alex Yokochi, Prasad Enjeti, Annette Von Jouanne, Advanced electric vehicle fast-charging technologies, Energies 12 (10) (2019) 1839.

[2] Brian Caulfield, James Kehoe, Usage patterns and preference for car sharing: a case study of Dublin, Case Stud. Transp. Policy 9 (1) (2021) 253–259.

[3] Wenqi Zhou, Lisheng Fan, Fasheng Zhou, Feng Li, Xianfu Lei, Wei Xu, Arumugam Nallanathan, Priority-aware resource scheduling for uav-mounted mobile edge 
16

computing networks, IEEE Trans. Veh. Technol. (2023).

http://github.com/Agyemang1z/Road-Accidents
http://refhub.elsevier.com/S2405-8440(23)09752-9/bibBD6400421E7DD162F3C21CD614B94FB8s1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bib65D593BE4550C72A9A3B01BDE821C524s1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bibEC5B2558E80ACC1AE6CBB47F8971A138s1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bibEC5B2558E80ACC1AE6CBB47F8971A138s1


Heliyon 9 (2023) e22544E.F. Agyemang, J.A. Mensah, E. Ocran et al.

[4] Rodolfo I. Meneguette, R. De Grande, A.A. Loureiro, Intelligent Transport System in Smart Cities, Springer International Publishing, Cham, 2018.

[5] R. Kanthavel, S.K.B. Sangeetha, K.P. Keerthana, Design of smart public transport assist system for metropolitan city Chennai, Int. J. Intell. Netw. 2 (2021) 57–63.

[6] Wenqi Zhou, Junjuan Xia, Fasheng Zhou, Lisheng Fan, Xianfu Lei, Arumugam Nallanathan, George K. Karagiannidis, Profit maximization for cache-enabled 
vehicular mobile edge computing networks, IEEE Trans. Veh. Technol. (2023).

[7] Sampurna Mandal, Swagatam Biswas, Valentina E. Balas, Rabindra Nath Shaw, Ankush Ghosh, Motion prediction for autonomous vehicles from lyft dataset 
using deep learning, in: 2020 IEEE 5th International Conference on Computing Communication and Automation (ICCCA), IEEE, 2020, pp. 768–773.

[8] Ahmad M. Khalil, The genome editing revolution, J. Genet. Eng. Biotechnol. 18 (1) (2020) 1–16.

[9] Lasse Fridstrøm, Jan Ifver, Siv Ingebrigtsen, Risto Kulmala, Lars Krogsgård Thomsen, Measuring the contribution of randomness, exposure, weather, and daylight 
to the variation in road accident counts, Accid. Anal. Prev. 27 (1) (1995) 1–20.

[10] Jingru Gao, Gary A. Davis, Using naturalistic driving study data to investigate the impact of driver distraction on driver’s brake reaction time in freeway rear-end 
events in car-following situation, J. Saf. Res. 63 (2017) 195–204.

[11] Amrit Banstola, Julie Mytton, Cost-effectiveness of interventions to prevent road traffic injuries in low- and middle-income countries: a literature review, Traffic 
Inj. Prev. 18 (4) (2017) 357–362.

[12] Wubetie Habtamu Tilaye, Human Injury Causing Road Traffic Accident at Debre Markos Town, 2020.

[13] Thiri Ko, Analysis and forecasting of road traffic accidents in Yangon municipal area (2014-2018), PhD thesis, MERAL Portal, 2019.

[14] Debela Deme, Review on factors causes road traffic accident in Africa, J. Civ. Eng. Res. Technol. 1 (1) (2019) 1–8.

[15] P. Shunmuga Perumal, M. Sujasree, Suresh Chavhan, Deepak Gupta, Venkat Mukthineni, Soorya Ram Shimgekar, Ashish Khanna, Giancarlo Fortino, An insight 
into crash avoidance and overtaking advice systems for autonomous vehicles: a review, challenges and solutions, Eng. Appl. Artif. Intell. 104 (2021) 104406.

[16] Muhammad Hussain, Jing Shi, Zahara Batool, An investigation of the effects of motorcycle-riding experience on aberrant driving behaviors and road traffic 
accidents-a case study of Pakistan, Int. J. Crashworthiness 27 (1) (2022) 70–79.

[17] Huiying Wen, Yingxin Du, Zheng Chen, Sheng Zhao, Analysis of factors contributing to the injury severity of overloaded-truck-related crashes on mountainous 
highways in China, Int. J. Environ. Res. Public Health 19 (7) (2022) 4244.

[18] Williams Ackaah, Benjamin Aprimah Apuseyine, Francis K. Afukaar, Road traffic crashes at night-time: characteristics and risk factors, Int. J. Inj. Control Saf. 
Promot. 27 (3) (2020) 392–399.

[19] Gabriel Klaeger, Stories of the road: perceptions of power, progress and perils on the Accra-Kumasi road, Ghana, in: The Making of the African Road, Brill, 2017, 
pp. 86–115.

[20] Irenius Konkor, Moses Kansanga, Yujiro Sano, Roger Antabe, Isaac Luginaah, Community perceptions and misconceptions of motorcycle accident risks in the 
upper West region of Ghana, Travel Behav. Soc. 15 (2019) 157–165.

[21] Hilde Iversen, Risk-taking attitudes and risky driving behaviour, Transp. Res., Part F Traffic Psychol. Behav. 7 (3) (2004) 135–150.

[22] Hai-peng Shao, Juan Yin, Wen-hao Yu, Qiu-ling Wang, Aberrant driving behaviours on risk involvement among drivers in China, J. Adv. Transp. 2020 (2020) 
1–8.

[23] Bekir Nihat Dogrul, Ibrahim Kiliccalan, Ekrem Samet Asci, Selim Can Peker, Blunt trauma related chest wall and pulmonary injuries: an overview, Chin. J. 
Traumatol. 23 (03) (2020) 125–138.

[24] T.L. Gunaruwan, P.C.J. Nayanalochana, Economic cost of human fatalities due to road traffic accidents in Sri Lanka: an estimation based on the human capital 
approach, J. South Asian Logist. Transp. 3 (1) (2023).

[25] Rannveig Svendby, Becoming the ‘Other’a Qualitative Study of Power, Masculinities and Disabilities in the Lives of Young Drivers After Road Traffic Accidents, 
2019.

[26] Kudzai Mwapaura, Witness Chikoko, Kudzai Nyabeze, Kwashirai Zvokuomba, Socio-economic challenges faced by persons with disabilities induced by road 
traffic accidents in Zimbabwe: the case of st giles rehabilitation centre, Harare, AfriFuture Res. Bull. (2021) 117.

[27] Geleta Mussa Yimer, Yonas Fissha Adem, Yosef Haile, Determinants of post-traumatic stress disorder among survivors of road traffic accidents in dessie compre-

hensive specialized hospital North-East Ethiopia, BMC Psychiatry 23 (1) (2023) 1–11.

[28] Sharaf Alkheder, Madhar Taamneh, Salah Taamneh, Severity prediction of traffic accident using an artificial neural network, J. Forecast. 36 (1) (2017) 100–108.

[29] C. Cabrera-Arnau, R. Prieto Curiel, S.R. Bishop, Uncovering the behaviour of road accidents in urban areas, R. Soc. Open Sci. 7 (4) (2020) 191739.

[30] Frances Agyapong, Thomas Kolawole Ojo, Managing traffic congestion in the Accra Central market, Ghana, J. Urban Manag. 7 (2) (2018) 85–96.

[31] Davies Adeloye, Jacqueline Y. Thompson, Moses A. Akanbi, Dominic Azuh, Victoria Samuel, Nicholas Omoregbe, Charles K. Ayo, The burden of road traffic 
crashes, injuries and deaths in Africa: a systematic review and meta-analysis, Bull. World Health Organ. 94 (7) (2016) 510.

[32] Pali Lehohla, Road Traffic Accident Deaths in South Africa, 2001-2006: Evidence from Death Notification, Statistics South Africa Pretoria, 2009.

[33] Paul J. Moroz, David A. Spiegel, The world health organization’s action plan on the road traffic injury pandemic: is there any action for orthopaedic trauma 
surgeons?, J. Orthop. Trauma 28 (2014) S11–S14.

[34] Adnan A. Hyder, Nino Paichadze, Tamitza Toroyan, Margaret M. Peden, Monitoring the decade of action for global road safety 2011–2020: an update, Global 
Public Health 12 (12) (2017) 1492–1505.

[35] Margaret M. Peden, Prasanthi Puvanachandra, Looking back on 10 years of global road safety, Int. Health 11 (5) (2019) 327–330.

[36] Ahmad Salisu Aliyu, Nuru Yakubu Umar, Funmilayo Abubakar Sani, Hussaini Mohammed, Epidemiological study on the prevalence of road traffic accident and 
associated risk factors among drivers in bauchi state, Nigeria, Am. J. Surg. Clin. Case Rep. 3 (8) (2021) 1–8.

[37] Charles G. Manyara, Combating road traffic accidents in Kenya: a challenge for an emerging economy, in: Kenya After 50: Reconfiguring Education, Gender, 
and Policy, 2016, pp. 101–122.

[38] Muhammad Abdul Basit, Umar Farooq, Muhammad Imran, Nahid Fatima, Abdullah Alhushaybari, Sobia Noreen, Sayed M. Eldin, Ali Akgül, Comprehensive 
investigations of (Au-Ag/Blood and Cu-Fe3O4/Blood) hybrid nanofluid over two rotating disks: numerical and computational approach, Alex. Eng. J. 72 (2023) 
19–36.

[39] Umar Farooq, Muhammad Abdul Basit, Sobia Noreen, Nahid Fatima, Abdullah Alhushaybari, Sayed M. El Din, Muhammad Imran, Ali Akgül, Recent progress 
in Cattaneo-Christov heat and mass fluxes for bioconvectional carreau nanofluid with motile microorganisms and activation energy passing through a nonlinear 
stretching cylinder, Ain Shams Eng. J. (2023) 102316.

[40] Muhammad Abdul Basit, Muhammad Imran, Shan Ali Khan, Abdullah Alhushaybari, R. Sadat, Mohamed R. Ali, Partial differential equations modeling of 
bio-convective sutterby nanofluid flow through paraboloid surface, Sci. Rep. 13 (1) (2023) 6152.

[41] M.A. Basit, Madeeha Tahir, Ayesha Riasat, S.A. Khan, Muhammad Imran, Ali Akgül, Numerical simulation of bioconvective Casson nanofluid through an 
exponentially permeable stretching surface, Int. J. Mod. Phys. B (2023) 2450128.

[42] Muhammad Imran, Sumeira Yasmin, Hassan Waqas, Shan Ali Khan, Taseer Muhammad, Nawa Alshammari, Nawaf N. Hamadneh, Ilyas Khan, Computational 
analysis of nanoparticle shapes on hybrid nanofluid flow due to flat horizontal plate via solar collector, Nanomaterials 12 (4) (2022) 663.

[43] Sandy Balkin, J. Keith Ord, Assessing the impact of speed-limit increases on fatal interstate crashes, J. Transp. Stat. 4 (1) (2001) 1–26.

[44] Abdul Hameed, Al Abbasi, Comparison Between Neural Networks and Traditional Statistical Methods to Predict the Number of Deaths Due to Traffic Accidents 
in Kuwait, 2005.

[45] Luke Fletcher, Alexander Zelinsky, Driver inattention detection based on eye gaze—road event correlation, Int. J. Robot. Res. 28 (6) (2009) 774–801.

[46] Angela Watson, Barry Watson, Kirsten Vallmuur, Estimating under-reporting of road crash injuries to police using multiple linked data collections, Accid. Anal. 
17

Prev. 83 (2015) 18–25.

http://refhub.elsevier.com/S2405-8440(23)09752-9/bibCE351C50F1882E227B75FFAD6D218958s1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bibA9F28936C448F93B50CB7A72218A11A1s1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bib5D0F2ED16F971997161CF0A3F44EBCC2s1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bib5D0F2ED16F971997161CF0A3F44EBCC2s1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bib7869D8CB19DF8AAE6B1E35C8AF960A58s1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bib7869D8CB19DF8AAE6B1E35C8AF960A58s1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bib3346934386339EE9692B26155605333Cs1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bibE346A5026924B231E5D455239CC61A7Cs1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bibE346A5026924B231E5D455239CC61A7Cs1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bib1C4629AE4DD5E44824433C8A495C4B3As1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bib1C4629AE4DD5E44824433C8A495C4B3As1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bibF804495B72FEF19DEC4EDC7B8F85FF09s1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bibF804495B72FEF19DEC4EDC7B8F85FF09s1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bibD82C64E33E0CC6B93D1A2ABB70CD9B84s1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bibFC132D0318227604840D8006D123D087s1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bib7E592BE636079D794AA1EF1CB9721AD4s1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bib1F4504808CDBBCDAC7E5248F0AB5D829s1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bib1F4504808CDBBCDAC7E5248F0AB5D829s1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bibC4DC33627C18317EFB4BB56F811A197Cs1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bibC4DC33627C18317EFB4BB56F811A197Cs1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bibE8E3EC9029F0C1B94BA2ADFEEC421D39s1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bibE8E3EC9029F0C1B94BA2ADFEEC421D39s1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bib15157D1AD7C37587A88E62DB4DD7A2CFs1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bib15157D1AD7C37587A88E62DB4DD7A2CFs1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bibAB22F49704759EF84FE76A7B08AAD39Es1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bibAB22F49704759EF84FE76A7B08AAD39Es1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bib13045F802CE41718C4ABA51D1304E3FFs1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bib13045F802CE41718C4ABA51D1304E3FFs1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bib283BF63542AD5FA0CD8366E8E9B50827s1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bib8A2A6BD6FDA8F4CDB1C9D236C358FF8As1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bib8A2A6BD6FDA8F4CDB1C9D236C358FF8As1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bibFF609AF8507F978B0A601E51DCAA5F48s1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bibFF609AF8507F978B0A601E51DCAA5F48s1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bib7B0DAED36657839B6A4EE3200479D49Fs1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bib7B0DAED36657839B6A4EE3200479D49Fs1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bib2501366EC5AB12BEFBD9CB083B3817B2s1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bib2501366EC5AB12BEFBD9CB083B3817B2s1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bib7D8FFFE765AC37B4F36C563A1C46E6D0s1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bib7D8FFFE765AC37B4F36C563A1C46E6D0s1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bib8283F2DDC15BA3ACD8D5E9CFEB8DDCD2s1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bib8283F2DDC15BA3ACD8D5E9CFEB8DDCD2s1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bibEA5E0E4DB6B19E902DC12F1EFD9F6156s1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bibBC03B4F0B6A5B5CBDFD16FAACA0BB7F9s1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bibC46F15B606C1EA8C42760356A260DDC9s1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bibB0166F80C6D35428D114D11207368CF0s1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bibB0166F80C6D35428D114D11207368CF0s1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bibDE4B6040E67297C2B2460E6188AD9038s1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bibFC94AACA0A2400758B13AAA8BE32B311s1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bibFC94AACA0A2400758B13AAA8BE32B311s1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bib744D18A953FC9F92911DE75CE7FC8628s1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bib744D18A953FC9F92911DE75CE7FC8628s1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bib87F381560961E99185AF5EC91848486Cs1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bib1AD2C715133D0F09117E57B03A071DD2s1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bib1AD2C715133D0F09117E57B03A071DD2s1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bib6C2FAD22D89ABA5D55CDD2A4620BF2CAs1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bib6C2FAD22D89ABA5D55CDD2A4620BF2CAs1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bibAFA2C2948DABFE7E944C490AB859D125s1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bibAFA2C2948DABFE7E944C490AB859D125s1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bibAFA2C2948DABFE7E944C490AB859D125s1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bibDFFEEB1C8099BFE1D168E4702288037Bs1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bibDFFEEB1C8099BFE1D168E4702288037Bs1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bibDFFEEB1C8099BFE1D168E4702288037Bs1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bib7E4E612DF652EB06AA1F6061BD4555F2s1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bib7E4E612DF652EB06AA1F6061BD4555F2s1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bib75E7D8AE3575ED20461ED6B899BD8F4Fs1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bib75E7D8AE3575ED20461ED6B899BD8F4Fs1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bib3E46B4A6F6B0ED5B0047839A574FEDC7s1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bib3E46B4A6F6B0ED5B0047839A574FEDC7s1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bib2D3F5563057FF36BDD35C6FEA07FF3C6s1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bib22B0B778D20EEE526B3A442E260A0AB9s1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bib22B0B778D20EEE526B3A442E260A0AB9s1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bib9DD1B39EF9866640AFD94C556C3A1C53s1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bib5376CBC31EB924177FD63F89E19E67E9s1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bib5376CBC31EB924177FD63F89E19E67E9s1


Heliyon 9 (2023) e22544E.F. Agyemang, J.A. Mensah, E. Ocran et al.

[47] Mohammed Salifu, Williams Ackaah, Under-reporting of road traffic crash data in Ghana, Int. J. Inj. Control Saf. Promot. 19 (4) (2012) 331–339.

[48] Guy P. Nason, Stationary and non-stationary time series, Stat. Volcanol. 60 (2006).

[49] Keith William Hipel, Angus Ian McLeod, William C. Lennox, Advances in Box-Jenkins modeling: 1. Model construction, Water Resour. Res. 13 (3) (1977) 
567–575.

[50] Clifford M. Hurvich, Chih-Ling Tsai, Regression and time series model selection in small samples, Biometrika 76 (2) (1989) 297–307.

[51] Mohan Mahanty, K. Swathi, K. Sasi Teja, P. Hemanth Kumar, A. Sravani, Forecasting the spread of Covid-19 pandemic with prophet, Rev. Intell. Artif. 35 (2) 
(2021) 115–122.

[52] Rajat Kumar Rathore, Deepti Mishra, Pawan Singh Mehra, Om Pal, Ahmad Sobri Hashim, Azrulhizam Shapi’i, T. Ciano, Meshal Shutaywi, Real-world model for 
bitcoin price prediction, Inf. Process. Manag. 59 (4) (2022) 102968.

[53] Sweeti Sah, B. Surendiran, R. Dhanalakshmi, Sachi Nandan Mohanty, Fayadh Alenezi, Kemal Polat, Forecasting Covid-19 pandemic using prophet, arima, and 
hybrid stacked LSTM-GRU models in India, Comput. Math. Methods Med. (2022) 2022.

[54] Karuna Gull, Suvarna Kanakaraddi, Ashok Chikaraddi, Covid-19 outbreak prediction using additive time series forecasting model, Trends Sci. 19 (22) (2022) 
1919.

[55] Pousali Chakraborty, Marius Corici, Thomas Magedanz, A comparative study for time series forecasting within software 5g networks, in: 2020 14th International 
Conference on Signal Processing and Communication Systems (ICSPCS), IEEE, 2020, pp. 1–7.

[56] V.A. Turchenko, S.V. Trukhanov, V.G. Kostishin, F. Damay, F. Porcher, D.S. Klygach, M.G. Vakhitov, Dmitry Lyakhov, D. Michels, Bernat Bozzo, et al., Features 
of structure, magnetic state and electrodynamic performance of SrFe12−𝑥ln𝑥O19 , Sci. Rep. 11 (1) (2021) 18342.

[57] Munirah A. Almessiere, Yassine Slimani, Norah A. Algarou, Maksim G. Vakhitov, Denis S. Klygach, Abdulhadi Baykal, Tatyana I. Zubar, Sergei V. Trukhanov, 
Alex V. Trukhanov, Hussein Attia, et al., Tuning the structure, magnetic, and high frequency properties of Sc-doped Sr0.5Ba0.5Sc𝑥Fe12−𝑥O19/NiFe2O4 hard/soft 
nanocomposites, Adv. Electron. Mater. 8 (2) (2022) 2101124.

[58] Lunyuan Chen, Lisheng Fan, Xianfu Lei, Trung Q. Duong, Arumugam Nallanathan, George K. Karagiannidis, Relay-assisted federated edge learning: performance 
analysis and system optimization, IEEE Trans. Commun. (2023).

[59] Sihui Zheng, Cong Shen, Xiang Chen, Design and analysis of uplink and downlink communications for federated learning, IEEE J. Sel. Areas Commun. 39 (7) 
(2020) 2150–2167.

[60] Junjuan Xia, Lisheng Fan, Wei Xu, Xianfu Lei, Xiang Chen, George K. Karagiannidis, Arumugam Nallanathan, Secure cache-aided multi-relay networks in the 
presence of multiple eavesdroppers, IEEE Trans. Commun. 67 (11) (2019) 7672–7685.

[61] Nemanja Deretić, Dragan Stanimirović, Mohammed Al Awadh, Nikola Vujanović, Aleksandar Djukić, Sarima modelling approach for forecasting of traffic 
accidents, Sustainability 14 (8) (2022) 4403.

[62] Dung David Chuwang, Weiya Chen, Forecasting daily and weekly passenger demand for urban rail transit stations based on a time series model approach, 
Forecasting 4 (4) (2022) 904–924.

[63] Tianyu Feng, Zhou Zheng, Jiaying Xu, Minghui Liu, Ming Li, Huanhuan Jia, Xihe Yu, The comparative analysis of sarima, Facebook prophet, and lstm for road 
traffic injury prediction in northeast China, Front. Public Health 10 (2022).

[64] Mohammmad Asifur Rahman Shuvo, Muhtadi Zubair, Afsara Tahsin Purnota, Sarowar Hossain, Muhammad Iqbal Hossain, Traffic forecasting using time-series 
analysis, in: 2021 6th International Conference on Inventive Computation Technologies (ICICT), IEEE, 2021, pp. 269–274.

[65] Yinghao Guo, Rui Zhao, Shiwei Lai, Lisheng Fan, Xianfu Lei, George K. Karagiannidis, Distributed machine learning for multiuser mobile edge computing 
systems, IEEE J. Sel. Top. Signal Process. 16 (3) (2022) 460–473.

[66] Le He, Lisheng Fan, Xianfu Lei, Xiaohu Tang, Pingzhi Fan, Arumugam Nallanathan, Learning-based mimo detection with dynamic spatial modulation, IEEE 
Trans. Cogn. Commun. Netw. (2023).

[67] Shunpu Tang, Lunyuan Chen, Ke He, Junjuan Xia, Lisheng Fan, Arumugam Nallanathan, Computational intelligence and deep learning for next-generation 
18

edge-enabled industrial IoT, IEEE Trans. Netw. Sci. Eng. (2022).

http://refhub.elsevier.com/S2405-8440(23)09752-9/bibEE1236C4B78CAF9DADA4B52AA01E3B0Es1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bib9B2CD29C98188C679F618EEBFA4D352Fs1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bibDE9E84596F8DE9C993CE4028D7E79F05s1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bibDE9E84596F8DE9C993CE4028D7E79F05s1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bib183A486D8E68477DD55DBEB93EB29034s1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bib6D5970852E2633539F7569A824830661s1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bib6D5970852E2633539F7569A824830661s1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bib7361FAFD36B013A25DD8159D1F775E0Es1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bib7361FAFD36B013A25DD8159D1F775E0Es1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bib0BF9A43921539C6C928CF7327749EB1Bs1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bib0BF9A43921539C6C928CF7327749EB1Bs1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bib91E2AB577BFAC2A7D9929271EFB19CFDs1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bib91E2AB577BFAC2A7D9929271EFB19CFDs1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bib09730577DA4B0F3BEC02C8D4A6ED28B6s1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bib09730577DA4B0F3BEC02C8D4A6ED28B6s1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bib092F433BE40F417EAE9544EA86B93534s1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bib092F433BE40F417EAE9544EA86B93534s1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bib2E809188985E2E50242B9FB8C9707F3Bs1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bib2E809188985E2E50242B9FB8C9707F3Bs1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bib2E809188985E2E50242B9FB8C9707F3Bs1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bib8D6E911CC6738F83DDA5326CD7021EE8s1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bib8D6E911CC6738F83DDA5326CD7021EE8s1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bib2D51818F5A718A4AF29A702DA92F1F3As1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bib2D51818F5A718A4AF29A702DA92F1F3As1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bibF6284CCA11A449C4E3673F997471704Ds1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bibF6284CCA11A449C4E3673F997471704Ds1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bibA60E22CD3D2F9F6494EE6C5737206EECs1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bibA60E22CD3D2F9F6494EE6C5737206EECs1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bib90097DD17AE080D5E652C2D60CC547ACs1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bib90097DD17AE080D5E652C2D60CC547ACs1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bib7B6F1F4D74A51DD41B7260142D218748s1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bib7B6F1F4D74A51DD41B7260142D218748s1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bibD17773546F2541C80C68F194AB5123BAs1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bibD17773546F2541C80C68F194AB5123BAs1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bib6380AE86A2C53480FC3BB0D842D3EFF2s1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bib6380AE86A2C53480FC3BB0D842D3EFF2s1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bib95B3FA4CCB87EED7BD94B6342DA6A04Fs1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bib95B3FA4CCB87EED7BD94B6342DA6A04Fs1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bibDBA18E00C853F4E757C64B0B4BF48E41s1
http://refhub.elsevier.com/S2405-8440(23)09752-9/bibDBA18E00C853F4E757C64B0B4BF48E41s1

	Time series based road traffic accidents forecasting via SARIMA and Facebook Prophet model with potential changepoints
	1 Introduction
	2 Data and methods
	2.1 AutoRegressive moving average (ARMA) model
	2.2 AutoRegressive integrated moving average (ARIMA) model
	2.3 Box-Jenkins seasonal ARIMA (SARIMA) model
	2.4 ARIMA model building process
	2.5 Test of significance of model coefficients
	2.6 Model identification tools
	2.7 Test of model diagnostics for SARIMA model
	2.8 Test of model accuracy
	2.9 Forecasting
	2.10 Facebook Prophet forecasting model
	2.11 Comparison between Facebook Prophet and ARIMA models
	2.12 Examples of solving optimization problems in road traffic accident research

	3 Results and discussion
	3.1 Development of the SARIMA model
	3.2 Single exponential smoothing method
	3.3 Autocorrelation and partial autocorrelation function plots of the first difference
	3.4 Possible seasonal ARIMA models parameter estimation
	3.5 Fitting the suitable SARIMA model to road accident time series data
	3.6 Diagnostic checking of estimated model
	3.7 Test of model adequacy of SARIMA
	3.8 Box-Pierce test for larger lags
	3.9 Evaluation of SARIMA model
	3.10 Evaluation of Facebook Prophet model

	4 Conclusion and recommendation
	Funding statement
	Additional information
	CRediT authorship contribution statement
	Declaration of competing interest
	Data availability
	Acknowledgement
	References