time series

Time series forecasting predicts future values based on historical temporal patterns, enabling data-driven decision making in domains from financial markets to supply chain planning. Problem formulation: given historical observations x₁, x₂, ..., xₜ, predict future values xₜ₊₁, xₜ₊₂, ..., xₜ₊ₕ (h = forecast horizon). Classical methods: (1) ARIMA (AutoRegressive Integrated Moving Average—linear, assumes stationarity), (2) exponential smoothing (Holt-Winters—trend and seasonality), (3) Prophet (Facebook—additive model with trend, seasonality, holidays). Deep learning approaches: (1) RNNs/LSTMs (sequential processing, capture long-term dependencies), (2) Transformers (attention-based—Temporal Fusion Transformer, Informer for long sequences), (3) N-BEATS (pure deep learning, interpretable blocks), (4) DeepAR (probabilistic, handles multiple related series). Key considerations: (1) univariate vs. multivariate (single vs. multiple related time series), (2) point forecasts vs. probabilistic (confidence intervals, quantiles), (3) short-term vs. long-term horizon, (4) handling missing data and irregularly sampled series. Feature engineering: (1) lag features (past values), (2) rolling statistics (moving averages, std), (3) time-based features (day of week, month, holidays), (4) external regressors (weather, promotions). Applications: (1) finance (stock prices, risk management), (2) demand forecasting (inventory optimization, capacity planning), (3) energy (load forecasting, renewable generation), (4) IoT (sensor predictions, anomaly detection), (5) healthcare (patient monitoring, epidemic forecasting). Evaluation metrics: MAE, RMSE, MAPE, quantile loss (for probabilistic forecasts). Modern approaches combine classical statistical methods with deep learning for robust, interpretable forecasting systems.