## 3 Theory and Methods
### Time-Series and Automated Statistical Downscaling (ADS)  
ADS refers to a collection of data points arranged in chronological order, and it finds application in various fields that involve temporal measurements. It can be either univariate, relying on a single parameter, or multivariate, depending on multiple parameters.  
In statistical downscaling, the Autoregressive-Scaling Model with covariate selection and prediction (ASD) is used for downscaling precipitation, and two critical steps are involved.  
Researchers can obtain insights into the geographical distribution of road surface characteristics and how they vary over time by discussing the spatial and temporal aspects separately. Analyzing spatial characteristics can assist in identifying regional patterns and potential hotspots of specific State categories, whereas analyzing temporal elements can reveal seasonal trends and daily variations in road surface conditions. This understanding is critical for developing effective road maintenance methods and adjusting to changing weather conditions, ultimately improving the resilience and safety of the road network.  
Firstly, the classification of rainy and non-wet days (precipitation exceeding 1 mm) is performed, followed by the prediction of the total amount of precipitation specifically for rainy days. Subsequently, the anticipated precipitation can be expressed as follows:  
\[E(Y)=R\times E(Y|R)\text{ }\text{where }R=\left\{\begin{aligned} & 0,&\text{if }P(Rainy)<0.5\\ & 1,&\text{Otherwise}\end{aligned}\right. \tag{1}\]  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**No. of Ref.** & **Research** & **Method** & **Main Category** \\ \hline
[44] & & CNN & Building \\
[45] & Identifying damage to & CNN & Road \\
[46] & infrastructure assets & CNN & Water \\
[37] & & KNN & Bridges \\
[47] & & CNN & Power \\ \hline
[48] & & ML & Bridges \\
[41] & Timing of & RL & Road \\
[32] & Maintenance and & RL & Road \\
[43] & Rehabilitation & RL & Road \\
[29] & & GA & Road \\ \hline
[22] & & KNN & Road \\
[37] & & ANN & Dam \\
[49] & Performance Forecast & ANN & Sewer \\
[50] & & RNN & Power \\ \hline \hline \end{tabular}
\end{table}
Table 1: Earlier research on the use of AI for infrastructure management.  
Rainy and non-rainy days are represented as a binary variable in the framework, denoted as \(R\). The proposed framework [55] employs five pairs of classification and regression approaches to test its effectiveness.  
The discrepancy between the actual and expected results is then quantified as the loss. A loss function is employed to measure this difference. This study utilized two loss functions: Mean Square Error (MSE) and Categorical Cross-Entropy.  
\[MSE=\frac{1}{n}\sum_{i=1}^{n}(y_{i}-\widehat{y}_{i})^{2} \tag{2}\]  
\[CategoricalCrossEntropy=-\sum_{i=1}^{n}y_{i}\times\log\widehat{y}_{i} \tag{3}\]  
In this context, the gradient computation for each parameter involves using the loss, with \(y\) representing the real value and \(\hat{y}\) representing the projected value. The gradients provide insights into the potential adjustments needed for the parameters. This process, known as backpropagation, occurs during the tuning phase or learning process. By carefully adjusting the network's weights, the loss over the entire dataset is minimized, thereby enhancing the model's generalization ability.