⛽️Business Analytics Unit 9 – Time Series Analysis and Forecasting

Key Concepts in Time Series

• Time series data consists of observations collected sequentially over time at regular intervals (hourly, daily, monthly)
• Stationarity assumes the statistical properties of a time series remain constant over time
  • Constant mean, variance, and autocorrelation structure
• Autocorrelation measures the correlation between a time series and its lagged values
• Seasonality refers to regular, predictable patterns that repeat over fixed periods (weekly, monthly, yearly)
• Trend represents the long-term increase or decrease in the data over time
• White noise is a series of uncorrelated random variables with zero mean and constant variance
• Differencing transforms a non-stationary time series into a stationary one by computing the differences between consecutive observations

Components of Time Series Data

• Level indicates the average value of the time series over a specific period
• Trend captures the long-term increase or decrease in the data
  • Can be linear, exponential, or polynomial
• Seasonality represents regular, recurring patterns within a fixed time interval
  • Additive seasonality assumes the seasonal effect is constant over time
  • Multiplicative seasonality assumes the seasonal effect varies proportionally with the level of the series
• Cyclical component refers to irregular fluctuations lasting more than a year (business cycles, economic cycles)
• Irregular or residual component represents random, unpredictable fluctuations not captured by other components
• Decomposition techniques (additive, multiplicative) separate a time series into its individual components for analysis and modeling

Exploratory Data Analysis for Time Series

• Plotting the time series helps identify patterns, trends, seasonality, and outliers
• Summary statistics (mean, median, standard deviation) provide insights into the central tendency and dispersion of the data
• Rolling or moving averages smooth out short-term fluctuations and highlight long-term trends
• Autocorrelation function (ACF) measures the correlation between a time series and its lagged values
  • Helps determine the order of autoregressive terms in models
• Partial autocorrelation function (PACF) measures the correlation between a time series and its lagged values, controlling for shorter lags
  • Helps determine the order of moving average terms in models
• Augmented Dickey-Fuller (ADF) test assesses the stationarity of a time series
• Seasonal subseries plots help identify and visualize seasonal patterns in the data

Time Series Models and Techniques

• 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 time series
• Autoregressive Integrated Moving Average (ARIMA) models extend ARMA to handle non-stationary data through differencing
  • Order $(p, d, q)$ represents the AR order, differencing order, and MA order
• Seasonal ARIMA (SARIMA) models capture both non-seasonal and seasonal components
  • Order $(p, d, q)(P, D, Q)_m$ includes seasonal AR, differencing, and MA orders, and the seasonal period $m$
• Exponential smoothing methods (simple, Holt's, Holt-Winters) assign exponentially decreasing weights to past observations for forecasting

Forecasting Methods

• Naive methods use the most recent observation as the forecast for all future periods
  • Suitable for data with no clear trend or seasonality
• Drift method accounts for the average change between consecutive observations
• Simple exponential smoothing (SES) assigns exponentially decreasing weights to past observations
  • Suitable for data with no clear trend or seasonality
• Holt's linear trend method extends SES to capture trends in the data
  • Includes level and trend components with separate smoothing parameters
• Holt-Winters' seasonal method extends Holt's method to capture both trend and seasonality
  • Additive or multiplicative seasonality options
• ARIMA and SARIMA models are versatile and can handle a wide range of time series patterns
• Rolling origin evaluation helps assess the stability and accuracy of forecasting methods over time

Model Evaluation and Selection

• Split the data into training, validation, and testing sets for model development and evaluation
• Residual analysis examines the differences between observed and predicted values
  • Residuals should be uncorrelated, normally distributed, and have constant variance
• Mean Absolute Error (MAE) measures the average absolute difference between observed and predicted values
• Mean Squared Error (MSE) measures the average squared difference between observed and predicted values
  • Penalizes large errors more than MAE
• Root Mean Squared Error (RMSE) is the square root of MSE, providing an error metric in the same units as the data
• Mean Absolute Percentage Error (MAPE) expresses the average absolute error as a percentage of the observed values
• Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) balance model fit and complexity
  • Lower values indicate better models
• Cross-validation techniques (rolling origin, k-fold) assess model performance on unseen data and help prevent overfitting

Practical Applications in Business

• Demand forecasting predicts future product demand for inventory management and production planning
• Sales forecasting estimates future sales revenue for budgeting, resource allocation, and strategic decision-making
• Capacity planning forecasts the required resources (staff, equipment) to meet anticipated demand
• Financial forecasting projects future financial performance (revenue, expenses, cash flow) for budgeting and investment decisions
• Economic forecasting predicts macroeconomic indicators (GDP, inflation, unemployment) for policy-making and business strategy
• Energy demand forecasting helps utility companies plan electricity generation and distribution
• Predictive maintenance forecasts equipment failures for proactive maintenance scheduling and cost reduction

Advanced Topics and Trends

• Multivariate time series models (Vector Autoregression, Vector Error Correction) analyze relationships between multiple time series
• Neural networks and deep learning models (Recurrent Neural Networks, Long Short-Term Memory) capture complex, non-linear patterns
• Ensemble methods combine multiple models to improve forecasting accuracy and robustness
• Hierarchical forecasting reconciles forecasts at different levels of aggregation (product, region, company-wide)
• Bayesian methods incorporate prior knowledge and uncertainty into time series modeling and forecasting
• Functional time series models treat each observation as a function rather than a scalar value
• Transfer function models incorporate external variables (price, marketing spend) to improve forecasting accuracy
• Online learning and adaptive models continuously update parameters as new data becomes available