Time series forecasting LSTM ARIMA models

Build time series forecasting models using statistical methods and deep learning for accurate predictions. Time series analysis: 1. Stationarity testing: Augmented Dickey-Fuller test, p-value <0.05 for stationarity. 2. Differencing: first-order differencing, seasonal differencing, achieve stationarity. 3. Decomposition: trend, seasonality, residuals, STL decomposition, seasonal pattern identification. Classical methods: 1. ARIMA modeling: AutoRegressive Integrated Moving Average, parameter selection (p,d,q). 2. Seasonal ARIMA: SARIMA(p,d,q)(P,D,Q,s), seasonal parameters, model selection using AIC/BIC. 3. Exponential smoothing: Holt-Winters method, alpha/beta/gamma parameters, trend and seasonality. Deep learning approaches: 1. LSTM networks: sequence modeling, forget gate, input gate, output gate mechanisms. 2. GRU (Gated Recurrent Unit): simplified LSTM, fewer parameters, faster training. 3. Transformer models: attention mechanism for sequences, positional encoding, parallel processing. Feature engineering: 1. Lag features: previous values, window sizes 3-12 periods, correlation analysis. 2. Moving averages: simple MA, exponential MA, different window sizes (7, 30, 90 days). 3. Seasonal features: month, quarter, day of week, holiday indicators, cyclical encoding. Model evaluation: 1. Mean Absolute Error (MAE): average prediction error, interpretable units. 2. Root Mean Square Error (RMSE): penalize large errors, same units as target. 3. Mean Absolute Percentage Error (MAPE): percentage error, scale-independent, <10% excellent. Cross-validation: time series split, walk-forward validation, expanding window, out-of-sample testing for reliable performance assessment.