## 3 Methodologies
### Time Series Count Data Model  
The practice of counting discontinuous occurrences over a certain length of time yields counting information. Most of the temporal approaches presumptively presume that the mechanism that produces the occurrences is time varying (\(t\)). They lack recollection as a result. Suppose \(\{Y_{t}:t\in N\}\) is a count time series and \(\{X_{t}:t\in N\}\) is a time-varying r-dimensional covariate vector, where \(X_{t}=\{X_{t1},\ldots,\ X_{t,r}\}^{T}\). The multivariate time series' standard statistical system has the following basic principle:  
\[g(\lambda_{t})=\beta_{0}+\sum_{k=1}^{p}\beta_{k}\widetilde{g}\big{(}Y_{t-i_{k }}\big{)}+\sum_{l=1}^{q}\alpha_{l}\Big{(}g\big{(}\lambda_{t-j_{l}}\big{)}- \eta^{T}diag(e)X_{t-j_{l}}\Big{)}+\eta^{T}X_{t} \tag{10}\]  
where \(g:\mathbb{R}^{+}\rightarrow\mathbb{R}\) is the connection function, and \(\widetilde{g}:\mathbb{N}_{0}\rightarrow\mathbb{R}\) is the conversion function. The parameter \(\eta\) alludes to the consequence of covariates, and \(\{Y_{t-i_{1}},\ Y_{t-i_{2}\ldots},\ Y_{t-i_{p}}\}\) is the set of lagged observations to be regressed on, and \(\{\lambda_{t-j_{l}},\ \lambda_{t-j_{2}\ldots},\ \lambda_{t-j_{l}}\}\) are the lagged conditional means. \(P=\{1,2,\ldots,p\}\) and \(Q=\{1,2,\ldots,q\}\) are the ordering variables for the observable information's correlation processes [32]. Log-linear projections suggest how the samples rely on the degrees of explanatory data, going beyond simple statistical measures. They simulate the relationships and interactions between descriptive analysis. The log linear model of the count data time series which allows for negative covariate effects can be written as:  
\[v_{t}=log(\lambda_{t})=\beta_{0}+\sum_{k=1}^{p}\beta_{k}log(Y_{t-k}+1)+\sum_{l= 1}^{q}\alpha_{l}v_{t-l} \tag{11}\]Using the Poisson presumption in combination, \(Y_{t}=y|\mathcal{F}_{t}-1\ \sim\ Poisson\ (\lambda_{t})\), with the multivariate time series standard statistical model's basic context, it follows that:  
\[P(Y_{t}=y|\mathcal{F}_{t}-1)=\frac{\lambda_{t}^{y}exp(-\lambda_{t})}{y!},\ y=0,1,\ldots \tag{12}\]  
with the supposition \(var(Y_{t}|\mathcal{F}_{t}-1)=E(Y_{t}|\mathcal{F}_{t}-1)=\lambda_{t}\). The Poisson distribution can be replaced by the inverse probability density, particularly under the above-described variant description [32]. It is particularly helpful for univariate values with a standard deviation greater than the sample mean and an infinite positivity spectrum. With a significantly larger variability in the predictor variables due to heteroscedasticity in other data points, the impact of many variables that cause the relative frequency to rely on earlier experiences, the appearance of exceptions, and the appearance of surplus nulls in the regression analysis are all causes of overdispersion [32]. To tackle potential over-dispersion issues in some scenarios, in this case crash-frequency, the Negative Binomial distribution, instead of Poisson distribution, \(Y_{t}=y|\mathcal{F}_{t}-1\ \sim\ Poisson\ (\lambda_{t},\ \varnothing)\), is considered with the dispersion parameter \(\varnothing\), satisfying:  
\[P(Y_{t}=y|\mathcal{F}_{t}-1)=\frac{\Gamma(\phi+y)}{\Gamma(y+1)\Gamma(\phi)} \bigg{(}\frac{\phi}{\phi+\lambda_{t}}\bigg{)}^{\phi}\bigg{(}\frac{\lambda_{t }}{\phi+\lambda_{t}}\bigg{)}^{y},y=0,1,\ldots. \tag{13}\]  
where the variance refers to \(var(Y_{t}|\mathcal{F}_{t}-1)=\lambda_{t}+\lambda_{t}^{2}/\sigma\).