## 2 Materials and Methods
Continuous predictors underwent z-score normalization, enhancing model stability and convergence [40]. Prior to model building, initial exploratory data analysis was conducted to identify clear temporal trends, seasonal variations, and outliers. The time series was decomposed of crime rates, and autocorrelation analyses were performed to determine temporal dependencies and guide the selection of appropriate lagged predictors [5]. This preliminary analysis provided valuable findings, ensuring that subsequent models accurately reflected the data's temporal and structural characteristics. Our main modeling approach utilized a hybrid ARIMA-ANN model in two steps. First, an ARIMA model captured linear trends and seasonal patterns in the crime rate time series. Then, an artificial neural network (ANN) modeled the nonlinear residual patterns that remained unexplained by the ARIMA model. Determining the ARIMA model parameters involved careful examination of autocorrelation (ACF) and partial autocorrelation (PACF) analyses, alongside minimizing the Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC). Our diagnostic analyses consistently revealed significant short-term autocorrelation at lags of 1 to 3 months and annual seasonality at a lag of 12 months. Consequently, ARIMA models typically utilized parameter values of p and q between 1 and 2, with d set at 1, aligned with results from the Augmented Dickey-Fuller test for non-stationarity. This configuration effectively captured observed trends and seasonal structures, consistent with earlier research [5; 10; 16]. The ARIMA factor is represented as \(ARIMA(p,d,q)(P,D,Q)[s]\), using the ACF, PACF, and AIC/BIC results, and is applied in the first modeling step to the monthly crime rate time series (standardized number of offenses per 100,000 population):  
\[X_{t}=\mu+\sum_{i=1}^{p}\phi_{i}X_{t-i}+\sum_{j=1}^{q}\theta_{j}e_{t-j}+e_{t}, \tag{1}\]  
where \(X_{t}\) represents the crime rate at time \(t\), \(\phi_{i}\) and \(\theta_{j}\) are the autoregressive and moving average parameters, and \(e_{t}\) is the residual error term. Once the ARIMA model has been fitted, the residuals--representing the components of the crime rate not accounted for by linear or seasonal behavior--are input to a feed-forward artificial neural network (ANN). The ANN models the nonlinear relationships between the socioeconomic predictors and the crime rate and supports better classification of the monthly crime risk category [10]. As illustrated in Figure 1, the final prediction \(X_{t}^{HYBRIDY}\) is the sum of the ARIMA forecast \(X_{t}^{ARIMA}\) (linear and seasonal component of the crime rate) and a nonlinear correction generated by the ANN \(\dot{r}_{t}^{ANN}\):  
\[X_{t}^{HYBRIDY}=X_{t}^{ARIMA}+\dot{r}_{t}^{ANN} \tag{2}\]  
Following ARIMA model fitting, the residuals--indicating nonlinear patterns not explained by ARIMA--were analyzed using an ANN. The ANN was structured as a feed-forward multilayer perceptron optimized through practical validation procedures. Inputs included recent crime rates from the previous six months, the immediate ARIMA forecast, and up to four lags of the ARIMA residuals, justified by residual autocorrelation patterns. Extensive grid searches and five-fold time series cross-validation determined the optimal ANN configuration: a single hidden layer with 10 neurons, ReLU activation function, a learning rate of 0.005, L2 regularization of 0.01, and a dropout rate of 0.2 to minimize overfitting [40]. Training ceased after 50 epochs if performance did not improve. This configuration was validated using the final 20% of training data, preventing data leakage and ensuring reliable predictions. To provide comparative findings, several traditional multiclass classification methods--multinomial logistic regression, decision trees, random forests, and support vector machines (SVM)--were evaluated using identical predictors. Each classifier underwent thorough cross-validation and hyperparameter optimization [19].