📊Predictive Analytics in Business Unit 5 – Time Series Analysis & Forecasting

Key Concepts

• Time series data consists of observations collected at regular intervals over time (hourly, daily, monthly, yearly)
• Stationarity assumes the statistical properties of the data remain constant over time
  • Constant mean, variance, and autocorrelation structure
• Autocorrelation measures the correlation between a variable and its lagged values
• Partial autocorrelation measures the correlation between a variable and its lagged values after removing the effect of intermediate lags
• White noise is a series of uncorrelated random variables with zero mean and constant variance
• Differencing transforms non-stationary data into stationary data by computing the differences between consecutive observations
• Decomposition separates a time series into its components trend, seasonality, and residuals

Data Preparation

• Handling missing values through interpolation, forward-filling, or backward-filling
• Dealing with outliers using statistical methods (Z-score, IQR) or domain knowledge
• Transforming data to stabilize variance using logarithmic, square root, or Box-Cox transformations
• Scaling data to a common range (min-max scaling) or standardizing to zero mean and unit variance
• Creating lagged variables to capture the effect of past observations on the current observation
• Splitting data into training, validation, and testing sets for model development and evaluation
• Resampling data to change the frequency of observations (upsampling or downsampling)

Trend and Seasonality

• Trend represents the long-term increase or decrease in the data over time
  • Can be linear, exponential, or polynomial
• Seasonality refers to regular, periodic fluctuations in the data
  • Can be additive (constant amplitude) or multiplicative (amplitude varies with the level of the series)
• Detecting trend using visual inspection, moving averages, or regression analysis
• Identifying seasonality using visual inspection, autocorrelation plots, or Fourier analysis
• Removing trend and seasonality to obtain stationary residuals
  • Differencing, detrending, or seasonal adjustment techniques (STL decomposition, SEATS)
• Modeling trend and seasonality explicitly in time series models (SARIMA, Holt-Winters)

Time Series Models

• Autoregressive (AR) models predict future values based on a linear combination of past values
  • Order $p$ determines the number of lagged values used
• Moving Average (MA) models predict future values based on a linear combination of past forecast errors
  • Order $q$ determines the number of lagged errors used
• Autoregressive Moving Average (ARMA) models combine AR and MA components
  • Suitable for stationary data
• Autoregressive Integrated Moving Average (ARIMA) models extend ARMA to handle non-stationary data
  • Differencing order $d$ determines the number of times the data is differenced to achieve stationarity
• Seasonal ARIMA (SARIMA) models incorporate seasonal AR, MA, and differencing terms
• Exponential Smoothing (ES) models use weighted averages of past observations to forecast future values
  • Simple ES, Holt's linear trend ES, Holt-Winters' seasonal ES

Forecasting Techniques

• Recursive forecasting uses the model to predict one step ahead, then updates the model with the actual value before predicting the next step
• Direct forecasting trains separate models for each forecast horizon
• Rolling forecasting uses a fixed window of historical data to train the model and updates the window as new observations become available
• Ensemble forecasting combines predictions from multiple models to improve accuracy and robustness
  • Simple averaging, weighted averaging, or stacking
• Hierarchical forecasting reconciles forecasts at different levels of aggregation (bottom-up, top-down, or middle-out approaches)
• Forecast combination methods (Bates-Granger, Newbold-Granger, Holt-Winters) assign weights to individual forecasts based on their historical performance

Model Evaluation

• Splitting data into training and testing sets to assess model performance on unseen data
• Cross-validation techniques (rolling origin, time series cross-validation) for time-dependent data
• Evaluation metrics for point forecasts
  • Mean Absolute Error (MAE), Mean Squared Error (MSE), Root Mean Squared Error (RMSE), Mean Absolute Percentage Error (MAPE)
• Evaluation metrics for probabilistic forecasts
  • Coverage probability, prediction intervals, Continuous Ranked Probability Score (CRPS)
• Residual diagnostics to check model assumptions
  • Ljung-Box test for autocorrelation, Jarque-Bera test for normality, Engle's ARCH test for heteroscedasticity
• Comparing models using information criteria (AIC, BIC) or forecast accuracy measures
• Backtesting to evaluate model performance on historical data

Real-World Applications

• Demand forecasting in supply chain management (inventory optimization, production planning)
• Sales forecasting in retail and e-commerce (promotional planning, pricing strategies)
• Energy demand forecasting for utilities (electricity, gas, water)
• Traffic volume forecasting for transportation planning and management
• Financial market forecasting (stock prices, exchange rates, volatility)
• Economic forecasting (GDP growth, inflation, unemployment rates)
• Disease incidence and prevalence forecasting in healthcare (epidemic modeling)

Advanced Topics

• Multivariate time series models (Vector Autoregression, Vector Error Correction)
• State space models and Kalman filtering for dynamic systems
• Bayesian time series models (Bayesian structural time series, dynamic linear models)
• Neural network-based models (Recurrent Neural Networks, Long Short-Term Memory, Gated Recurrent Units)
• Deep learning architectures for time series (Temporal Convolutional Networks, Transformers)
• Functional time series models for high-dimensional data (functional autoregressive models)
• Nonlinear time series models (Threshold AR, Smooth Transition AR, Markov Switching models)
• Forecasting with exogenous variables (ARIMAX, SARIMAX, regression with ARIMA errors)