Traffic Intensity Detection in Lagos State Using Bayesian Estimation Model

Traffic congestion is a vast trouble in city areas, impacting financial boom, productivity, and the environment. Various studies have targeted on estimating and quantifying the socioeconomical impact of congestion. The study [10] conducted a systematic literature review on the causes, effects, and techniques to minimize traffic congestion. The study [11] proposed a complex algorithm that considers factors which includes time cost, gasoline consumption, and CO₂ emissions to evaluate the socioeconomic impact of congestion. A comprehensive evaluation of site traffic prediction literature, highlighting studies procedures and developments in traffic glide prediction was conducted in the study [12] with the objective of identifying key trends and methodologies in the field. Consequently, traffic intensity detection is a key component in traffic monitoring, and intersection traffic state calculation. Different techniques are used for detecting site traffic depth, including using movement detection sensors in clever avenue lighting fixtures systems [13], laptop imaginative and prescient-based structures the use of traffic surveillance cameras [14], and traffic parameter mixtures in intersection detection gadgets [15]. These techniques’ purpose to correctly decide the variety of cars or passengers present in a given place throughout a specific term.

In addition, previous researchers have proven the usage of system learning algorithms to acquire excessive accuracy in predicting specific degrees of traffic congestion. And beautify the accuracy and performance of traffic intensity detection systems. Traffic depth detection the use of system mastering has been a popular research area, where accurate and timely data approximately cutting-edge and forecast parameters of traffic flows can improve the performance of using delivery infrastructure [16]. Various system getting to know strategies and fashions have been proposed to come across and predict traffic depth, such as using convolutional neural networks (CNN) for powerful detection of site traffic congestion [17]. Additionally, deep studying fashions have been developed for traffic car detection, which could expect the varieties of motors the use of strategies together with body difference and contour mapping [18, 19]. These models aim to help in shooting and reading traffic data from CCTV surveillance in highways. Overall, system learning and deep getting to know strategies have shown promise in detecting and studying traffic intensity for numerous packages, consisting of traffic congestion manipulate and traffic surveillance.

2.1 Overview of Bayesian estimation

Bayesian estimation of traffic intensity is a statistical technique for estimating the ratio of the advent rate to the service fee in a queueing system. The Bayesian method uses prior statistics approximately the site traffic depth to update the posterior distribution of the traffic depth given the located information. This can be used to reap greater accurate estimates of the site traffic depth than conventional methods, consisting of maximum chance estimation. Bayesian estimation is a way of revising ideals considering observed statistics. It does not produce a single parameter estimate, however instead a distribution over the possible parameter values [9]. Bayesian estimation is based totally at the Bayesian interpretation of chance, in which probability expresses a degree of perception in an occasion [2]. The degree of belief may be primarily based on previous knowledge or personal ideals about the event.

To perform Bayesian estimation, we want to specify a prior distribution that displays our initial beliefs approximately the parameter of hobby. Then, we use Bayes' theorem to update our ideals based on the information we observe. The posterior distribution, or the conditional distribution of the parameter given the records, can be computed the usage of the Bayes theorem [20]. The facts from the statistics and our preceding ideals are combined to create the posterior distribution.

Bayesian estimation can be applied to various types of data and models. For example, we can use Bayesian estimation to estimate the proportion of right-handed people in a population, or the mean height of a certain species of plants, or the effect of a treatment on a disease outcome. In each case, we need to choose an appropriate prior distribution and a likelihood function that models how the data are generated given the parameter value [6].

Bayesian estimation has some advantages over other methods of inference, such as frequents statistics where by Bayesian approach offers advantages in traffic intensity detection by allowing for the incorporation of prior knowledge, handling uncertainty, integrating multiple data sources, adapting to changing conditions, and providing robust estimates in the presence of outliers or limited data. These benefits make Bayesian methods to be well-suited for applications where accurate and reliable estimation of traffic intensity is highly critical.

For example, Bayesian estimation allows us to incorporate prior knowledge and uncertainty into our analysis, this is particularly useful because in real-world scenarios, we often have some idea about the typical patterns of traffic intensity based on historical data or domain expertise. By integrating this prior know-how into the Bayesian version, we are able to improve the accuracy of traffic depth detection, it additionally inference can offer dependable estimates even if information is limited or noisy. This is beneficial in traffic intensity detection in which records may not continually be abundant or may be suffering from various factors such as climate, events, or road conditions. The Bayesian framework permits for updating ideals as new information becomes to be had, that can help in converting visitors’ conditions. Bayesian models also can adapt and research over time as extra information is amassed. This adaptability is vital in traffic depth detection where styles can also trade through the years [16]. By continuously updating the version with new statistics, Bayesian methods can offer greater accurate and updated estimates of traffic depth. Even observed you however you and to make probabilistic statements about the parameter without counting on repeated sampling or self-belief durations. Bayesian estimation additionally offers an herbal manner to evaluate different fashions or hypotheses using the Bayes issue, that's the ratio of posterior chances for 2 models [4].

However, Bayesian estimation also has a few challenges and barriers. For instance, selecting a previous distribution may be subjective and controversial, and might have an effect on the outcomes of the analysis. Also, computing the posterior distribution may be computationally extensive and require specialized algorithms, which includes Markov chain Monte Carlo (MCMC) strategies [2]. In summary, taking pictures spatial and temporal variations in Bayesian models for traffic depth detection entails structuring the version to explicitly account for geographical modifications over time. By utilizing suitable priors, hierarchical systems, and dynamic modeling techniques, Bayesian strategies can efficaciously and correctly capture the complicated dynamics of site visitors’ depth across each space and time in Lagos State.

Furthermore, decoding and speaking Bayesian consequences may be tough for some audiences who aren't acquainted with the Bayesian framework.

2.2 Bayesian estimation model

A direct and powerful class approach based totally at the Bayes theorem is referred to as Naive Bayes. It makes the idea that, given the magnificence name, the facts’ capabilities are conditionally impartial. Stated in another way, it presupposes that a function's effect at the magnificence is ignorant of other functions. In data technological know-how and gadget learning, the Naive Bayes set of rules is a famous, honest technique. It's an algorithm for classifiers that makes use of possibility.

It also the inspiration of this version is Bayes’ theorem:

$P(A \mid B)=\frac{P(B \mid A) P(A)}{P(B)}$                    （1）

where, A and B are any events, $P(A \mid B)$ is the likelihood of A conditionally given B.

To implement Naive Bayes in Python, we can use the scikit-learn library [21], which provides various implementations of Naive Bayes for different types of features. In this project, we consider the Gaussian Naive Bayes for numerical features with Gaussian distribution, and multinomial Naive Bayes for discrete features with multinomial distribution.

2.2.1 Gaussian Naive Bayes’ classification model

In the context of classification, we can use Bayes' theorem to calculate the probability of a class label $C_k$.

For each class $C_k$, estimate $\mu_{i k}$ and variance of $\sigma_{i k}^2$ of each feature $x_i$, assuming it follows a Gaussian distribution given a feature vector $x=\left(x_1, x_2, \ldots, x_n\right)$ as follows:

$P\left(C_k \mid x\right)=\frac{P\left(x \mid C_k\right) P\left(C_k\right)}{P(x)}$                   (2)

where, $P\left(x \mid C_k\right)$ is the probability of the characteristics given the class, $P\left(C_k\right)$ is the class's prior probability, and $P(x)$ provides proof of the characteristics' marginal probability.

To make a prediction for a new instance $x$, we use the following rule:

$\hat{y}=\arg \max _k P\left(C_k \mid x\right)$                   (3)

Step 1: Calculate the prior probability $\left(C_k\right)$ for each class $\left(C_k\right)$

Step 2: Use Bayes' theorem to calculate the posterior probability $P\left(x \mid C_k\right)$

That is, we choose the class that has the highest probability given the features. However, calculating $P\left(x \mid C_k\right)$ and $P(x)$ can be difficult, particularly if there are a lot of features. The Gaussian assumption is useful in this situation [22]. We can simplify the likelihood as follows by assuming that the features are conditionally independent given the class:

$\begin{gathered}P\left(x \mid C_k\right)=P\left(x_1 \mid C_k\right) P\left(x_2 \mid C_k\right) \ldots P\left(x_n \mid C_k\right)= \prod_{i=1}^n P\left(x_i \mid C_k\right)\end{gathered}$                 (4)

where, $\mathrm{P}\left(C_k\right)$ is the prior probability of $C_k, P\left(x \mid C_k\right)$ is the likelihood of feature $x_i$ given class $C_k$ for each $P\left(x_i \mid C_k\right)$ is a Gaussian probability density function with $\mu_{i k}$ and standard deviation $\sigma_{i k}$:

$P\left(x_i \mid C_k\right)=\frac{1}{\sqrt{2 \pi \sigma_{i k}^2}} e^{-\frac{\left(x_i-\mu_{i k}\right)^2}{2 \sigma_{i k}^2}}$                (5)

This means that we only need to estimate the probability of each feature given the class, which can be done using various methods depending on the type of the feature (e.g., categorical, or numerical). Similarly, we can ignore P(x) in the denominator since it is constant for all classes. As a result, the posterior probability may be rewritten as:

$P\left(C_k \mid x\right) \propto P\left(C_k\right) \prod_{i=1}^n P\left(x_i \mid C_k\right)$                 (6)

Step 3: Assign the class label $\hat{y}$ for a new instance x by setting the class $C_k$ with the highest posterior probability $P\left(x \mid C_k\right)$

2.2.2 Multinomial Naive Bayes classification model

When it comes to classification, the Bayes theorem allows us to determine the likelihood of a class label $C_k$.

For each class $C_k$, estimate the probability $P\left(C_k \mid x\right)$ of each term $x_i$ occurring, given a feature vector $x=\left(x_1, x_2, \ldots, x_n\right)$ as follows:

$P\left(C_k \mid x\right)=\frac{P\left(x \mid C_k\right) P\left(C_k\right)}{P(x)}$               (7)

where, $P\left(x \mid C_k\right)$ is the likelihood of the features given the class, $P\left(C_k\right)$ is the prior probability of the class, and $P(x)$ is the evidence of marginal probability of the features [23].

To make a prediction for a new instance x, we use the following rule:

$\hat{y}=\arg \max _k P\left(C_k \mid x\right)$                (8)

Count ( $x_i$ in $C_k$, ) is the number of terms $x_i$ appears in documents of class $C_k$.

$\propto$ is a smoothing parameter (often set to 1, Laplace smoothing to handle terms not present in training data.

Step 1: calculate the prior probability $\mathrm{P}\left(C_k\right)$ for each class $C_k$

Step 2: use Bayes' theorem to calculate the posterior probability $P\left(C_k \mid x\right)$

That is, we choose the class that has the highest probability given the features. However, calculating $P\left(x \mid C_k\right)$ and $P(x)$ can be challenging, especially when the number of features is large. This is where the multinomial assumption comes in handy. By assuming that the features are conditionally independent given the class, we can simplify the likelihood as follows:

$P\left(x \mid C_k\right)=P\left(x_1 \mid C_k\right) P\left(x_2 \mid C_k\right) \ldots P\left(x_n \mid C_k\right)=\prod_{i=1}^n P\left(x_i \mid C_k\right)$                        (9)

where, each $P\left(x_i \mid C_k\right)$ is a multinomial probability function with parameter $\theta_{i k}$ :

$P\left(x_i \mid C_k\right)=\binom{n}{x_i} \theta_{i k}^x\left(1-\theta_{i k}\right)^{n-x_i}$                   (10)

where, $n$ is the total number of trials (e.g., words or pixels), and $x_i$ is the number of successes (e.g., occurrences or intensities) for feature $i$. The parameter $\theta_{i k}$ is the probability of success for feature $i$ given class $k$, and it can be estimated by:

$\widehat{\theta}_{i k}=\frac{N_{i k}+\alpha}{N_k+n \alpha}$                  (11)

where, $N_{i k}$ is the number of times feature $i$ appears in class $k$, $N k$ is the total count of all features in class $k$, and $\alpha$ is a smoothing parameter that prevents zero probabilities.

Similarly, we can ignore $P(x)$ in the denominator since it is constant for all classes. Therefore, we can rewrite the posterior probability as:

$P\left(C_k \mid x\right) \propto P\left(C_k\right) \prod_{i=1}^n P\left(x_i \mid C_k\right)$                   (12)

Step 3: assign the class label $\hat{y}$ for a new document $x$ by selecting the class $C_k$ with the highest posterior probability $P\left(x_i \mid C_k\right)$.

2.2.3 Simple statement of Bayes’ theorem

In its most basic setting, which involves only discrete distributions, the Bayes theorem links the conditional and marginal probabilities of event A and B as a non-vanishing probability. A specific scenario involving discrete probability distribution of data and continuous prior and posterior probability distributions was presented by Thomas Bayes [12]:

$P(A \mid B)=\frac{P(B \mid A) P(A)}{P(B)}$                  (13)

Every term in the Bayes theorem has a common name:

1. $P(A)$ is either A's marginal probability or prior probability. It is "prior" in the sense that it disregards any knowledge that B may have.
2. $P(A \mid B)$ is the likelihood of A given B, contingent on. Since it is based on or derives from the given value of B, it is also known as the posterior probability.
3. $P(B \mid A)$ is the probability of B given A, conditionally. Another name for it is the chance.
4. $P(B)$ is used as a normalization constant and represents the marginal or prior probability of B.

The mathematical illustration provided by the Bayes theorem illustrates how the conditional probability of B given A [24].

2.2.4 Bayes'  extension for two or more events

Bayes-like theorems encompass more than two events. As an instance:

$P(A \mid B \cap C)=\frac{P(A) P(B \mid A) P(C \mid A \cap B)}{P(B) P(C \mid B)}$                  (14)

A few simple steps will get you there from the definition of conditional probability and the Bayes theorem:

$\begin{gathered}P(A \mid B \cap C)=\frac{P(A \cap B \cap C)}{P(B \cap C)}=\frac{P(C \mid A \cap B) P(A \cap B)}{P(B) P(C \mid B)}=\frac{P(A) P(B \mid A) P(C \mid A \cap B)}{P(B) P(C \mid B)}\end{gathered}$                  (15)

Similarly,

$P(A \mid B \cap C)=\frac{P(B \mid A \cap C) P(A \mid C)}{P(B \mid C)}$                     (16)

can be derived in the following way and is thought of as a conditional Bayes theorem:

$\begin{aligned} & P(A \mid B \cap C)=\frac{P(A \cap B \cap C)}{P(B \cap C)} & =\frac{P(B \mid A \cap C) P(A \mid C) P(C)}{P(C) P(B \mid C)}=\frac{P(B \mid A \cap C) P(A \mid C)}{P(B \mid C)}\end{aligned}$                          (17)

A common method is to work with a decomposition of the joint possibility and marginalizing (integrating or summarizing) over the variables that are not of interest. Making use of this function would possibly drastically reduce the calculations. Depending at the form of decomposition, it can be smooth to show that a few integrals need to be 1 so as for the decomposition to terminate [25]. For example, a Bayesian community specifies the factorization of a joint distribution of several variables, wherein the conditional possibility of every variable given the opposite variables takes a fairly easy form.

The methodology of this research consists of four main steps: data collection, data preprocessing, data analysis, and data evaluation. Figure 2 shows the model architecture for study from the input data, the processing stage, to the output.

2.png

Figure 2. Model architecture

The following is a brief overview of each step:

Data collection: This step involves obtaining the secondary data from Kaggle repository, which contains 48120 observations of the number of vehicles intersecting at four junctions in Lagos State at various times of the day and on different days of the week [7]. The data was collected by observing the number of vehicles passing through each junction using cameras or sensors. The data was obtained from https://www.kaggle.com/ahmedlahlou/accidents-in-france-from-2015-to-2017.

Data preprocessing: This step involves ensuring the quality and suitability of the data for further analysis by handling missing values, outliers, and inconsistencies. An Exploratory Data Analysis was also carried out to identify any data quality issues, missing values, outliers, or patterns within the dataset [20]. Additionally, data transformation was used, which included turning unstructured data into characteristics that made sense and enhanced the functionality of machine learning models. The following were extracted: the hours, the day of the week, month, and year, generating cyclic features for time of day, or calculating time differences between events. Data visualization provided intuitive insights into traffic patterns, data quality, and aid in decision-making for subsequent analysis steps. All these were subject to Feature selection, Data scaling, cross-validation and data splitting [8].

Data analysis: This step involves applying the Bayesian Estimation Model to estimate the traffic intensity in Lagos State. The Gaussian Naive Bayes classifier and the multinomial Naive Bayes classifier are the models chosen for this project. These are certain kinds of Bayesian Estimation Models that use the Bayes theorem under the mistaken presumption that each pair of features provided the class variable will always be conditionally independent [29]. The features are assumed to be regularly distributed by the Gaussian Naive Bayes classifier, whereas they are assumed to be multinomially distributed by the multinomial Naive Bayes classifier. These models were chosen because they are simple, scalable, and suitable for high-dimensional data [25].

A Gaussian distribution is assumed to easily estimate the probabilities for input variables using the Gaussian Probability Density Function. The GaussianNB function of the sklearn Naive_Bayes class in the scikit-learn library of Python was used to implement the model. The data will be divided into 80% for the training and the remaining 20% for testing and validation.

Data evaluation: This step entails evaluating the performance and accuracy of the Bayesian Estimation Model for visitors’ intensity detection in Lagos State. An Evaluation Metrics was completed in understanding the version’s accuracy, precision, recall, and standard overall performance, allowing effective model assessment and contrast [30]. Accuracy gauges how frequently the version makes accurate predictions, precision determines the percentage of wonderful predictions that the model as it should be predicting, recall evaluates the frequency with which the model properly predicts among fine cases, and F1-rating calculates the precision and consider harmonic suggest. These metrics have been calculated the use of confusion matrices and type reviews. The fashions were additionally compared the usage of the Confusion Matrices, which measure how well the version can distinguish between exclusive lessons [20].

4.1 Evaluations metrics

These metrics collectively help in knowledge the version's accuracy, precision, recollect, and ordinary overall performance, allowing powerful model assessment and contrast [31].

(1) Accuracy score: The accuracy rating is a typically used assessment metric for category fashions. It calculates the proportion of the model's total wide variety of predictions that are correct. It gives a large image of the version's effectiveness, but while running with unbalanced datasets, it couldn't be sincere.

(2) Classification report: The class record gives an intensive assessment of the effectiveness of a classification model. Metrics like remember, accuracy, F1 rating, and assistance are included for every class. The F1 rating is the harmonic mean of precision and remember. Precision gauges the accuracy of fantastic predictions, consider gauges the capability to understand nice cases. The quantity of occurrences of every magnificence within the dataset is called the aid.

(3) Confusion matrix: A confusion matrix is a table that visualizes the performance of a category version by way of summarizing the results of predictions. Its presentations the quantity of true positives, proper negatives, fake positives, and false negatives by evaluating the projected training with the real instructions. It enables to perceive the version's overall performance in phrases of efficiently and incorrectly labelled times for each magnificence. The Gaussian Naive Bayes classifier's anticipated elegance labels (y_pred) and real class labels (y_test) had been used to calculate the confusion matrix. The confusion matrix became then visualized the use of a confusion matrix Display item, imparting a graphical representation of the distribution of predicted and real elegance labels. The plot displayed the confusion matrix, and the name "Confusion Matrix Distribution of Gaussian Naive Bayes" became assigned to the plot.

As Nigeria's economic center, Lagos attracts humans from all over the country in search of employment possibilities, further growing the population and the variety of automobiles on the road. Lagos State, faces sizeable challenges with visitors’ congestion because of several factors, along with fast populace growth, infrastructure boundaries, and socioeconomic situations. Lagos, with a populace estimated to be over 20 million human beings and awareness of industrial activities in areas like Victoria Island, Ikeja, and Lekki. This fast urbanization places titanic stress on existing transportation infrastructure specially at some point of peak hours as human beings go back and forth to and from paintings. The street community in Lagos is insufficient to deal with the extent of visitors as roads are narrow, poorly maintained, lack proper drainage structures, and restrained alternative routes leading to common site visitors’ jams. The city is predicated heavily on casual transport structures like minibuses (danfos) and bikes (okadas), which can be chaotic and make contributions to congestion. Hence, because of insufficient public shipping, many citizens rely on non-public motors, increasing the number of motors on the street.

Many traffic alerts used for traffic control are old or not synchronized, leading to inefficient traffic glide. Traffic laws aren't continuously enforced, main to troubles like unlawful parking, avenue buying and selling, and disregard for site visitors’ guidelines. Lack of Real-Time Traffic Data hinders powerful site visitors’ management and well-timed response to incidents. Delays and inefficiencies in implementing transportation policies and projects further exacerbate congestion problems. Addressing these challenges requires a comprehensive approach, including significant investments in infrastructure, improved public transportation systems, efficient traffic management, and better urban planning practices.