📊Probabilistic Decision-Making Unit 10 – Time Series Analysis & Forecasting

Key Concepts and Terminology

• Time series defined as a sequence of data points collected at regular time intervals
• Stationarity refers to the statistical properties of a time series remaining constant over time
  • Stationary time series have constant mean, variance, and autocorrelation
• Trend represents the long-term increase or decrease in the data over time
• Seasonality refers to recurring patterns or cycles within a fixed period (hourly, daily, monthly, etc.)
• Autocorrelation measures the correlation between a time series and a lagged version of itself
• White noise is a series of uncorrelated random variables with zero mean and constant variance
• Differencing involves computing the differences between consecutive observations to remove trend and seasonality

Time Series Components and Patterns

• Time series can be decomposed into trend, seasonal, and residual components
• Trend component captures the overall long-term direction of the series (upward, downward, or stable)
• Seasonal component represents regular, predictable patterns that repeat over fixed periods
  • Seasonal patterns can be additive (constant amplitude) or multiplicative (amplitude varies with level)
• Cyclical component refers to irregular fluctuations or cycles with varying length and magnitude
• Residual or irregular component is the remaining part after removing trend, seasonal, and cyclical components
• Identifying and modeling these components is crucial for understanding the underlying patterns and making accurate forecasts
• Additive decomposition assumes components are added together, while multiplicative decomposition assumes they are multiplied

Statistical Methods for Time Series Analysis

• Autocorrelation function (ACF) measures the correlation between a time series and its lagged values
  • ACF plot helps identify the presence and strength of autocorrelation at different lags
• Partial autocorrelation function (PACF) measures the correlation between a time series and its lagged values, while controlling for the effect of intermediate lags
  • PACF plot helps determine the order of an autoregressive model
• Ljung-Box test assesses the overall randomness of a time series by testing for the presence of autocorrelation
• Augmented Dickey-Fuller (ADF) test checks for the presence of a unit root and determines if a time series is stationary
• Seasonal decomposition of time series (STL) separates a time series into trend, seasonal, and residual components
• Spectral analysis examines the frequency domain of a time series to identify periodic patterns and cycles
• Cross-correlation measures the similarity between two time series as a function of the lag between them

Forecasting Techniques and Models

• Moving average (MA) models predict future values based on the average of past observations
  • Simple moving average (SMA) assigns equal weights to past observations
  • Weighted moving average (WMA) assigns different weights to past observations based on their recency
• Exponential smoothing models assign exponentially decreasing weights to past observations
  • Simple exponential smoothing (SES) is suitable for data with no clear trend or seasonality
  • Holt's linear trend method extends SES to capture trends in the data
  • Holt-Winters' method incorporates both trend and seasonality components
• Autoregressive (AR) models predict future values based on a linear combination of past values
• Autoregressive moving average (ARMA) models combine AR and MA components to capture both short-term and long-term dependencies
• Autoregressive integrated moving average (ARIMA) models extend ARMA to handle non-stationary data by applying differencing
• Seasonal ARIMA (SARIMA) models account for seasonal patterns in addition to trend and autocorrelation
• Vector autoregression (VAR) models capture the dynamic relationships among multiple time series variables

Evaluating Forecast Accuracy

• Forecast accuracy measures how well a model's predictions match the actual values
• Mean absolute error (MAE) calculates the average absolute difference between the forecasted and actual values
• Mean squared error (MSE) calculates the average squared difference between the forecasted and actual values
  • Root mean squared error (RMSE) is the square root of MSE and is in the same unit as the data
• Mean absolute percentage error (MAPE) expresses the average absolute error as a percentage of the actual values
• Theil's U statistic compares the accuracy of a forecast model to that of a naive forecast (using the previous value as the prediction)
• Rolling origin evaluation assesses the model's performance by iteratively updating the training data and making predictions on a rolling basis
• Cross-validation techniques, such as k-fold cross-validation, help assess the model's generalization ability and prevent overfitting

Applications in Decision-Making

• Demand forecasting helps businesses anticipate future customer demand for products or services
  • Accurate demand forecasts enable efficient resource allocation, inventory management, and production planning
• Sales forecasting predicts future sales volumes, revenue, and growth rates
  • Sales forecasts inform strategic decisions, budgeting, and target setting
• Economic forecasting estimates future economic indicators (GDP, inflation, unemployment rates, etc.)
  • Economic forecasts guide policy decisions, investment strategies, and business planning
• Energy load forecasting predicts future energy consumption patterns to optimize power generation and distribution
• Traffic volume forecasting helps transportation authorities plan infrastructure, manage congestion, and allocate resources
• Financial market forecasting aims to predict future prices, returns, and volatility of financial assets (stocks, bonds, currencies, etc.)
• Supply chain forecasting optimizes inventory levels, production schedules, and logistics based on anticipated demand and lead times

Advanced Topics and Current Trends

• Deep learning models, such as recurrent neural networks (RNNs) and long short-term memory (LSTM) networks, capture complex non-linear patterns in time series data
• Ensemble methods combine multiple forecasting models to improve accuracy and robustness
  • Techniques include bagging, boosting, and stacking
• Hierarchical forecasting reconciles forecasts at different levels of aggregation (product, region, etc.) to ensure consistency
• Bayesian forecasting incorporates prior knowledge and updates predictions based on new data using Bayesian inference
• Functional time series analysis treats each observation as a function rather than a scalar value
• Transfer learning leverages knowledge from related time series to improve forecasting performance on a target series
• Anomaly detection identifies unusual or unexpected patterns in time series data, which can indicate errors, fraud, or system failures
• Real-time forecasting updates predictions dynamically as new data becomes available, enabling quick response to changing conditions

Practical Tips and Common Pitfalls

• Preprocess the data by handling missing values, outliers, and inconsistencies
  • Techniques include interpolation, smoothing, and robust estimation methods
• Stationarize the data if necessary by removing trend and seasonality through differencing or transformation
• Select an appropriate forecasting model based on the characteristics of the data and the problem at hand
  • Consider factors such as trend, seasonality, autocorrelation, and available historical data
• Validate the model's assumptions and assess its goodness-of-fit using diagnostic tests and residual analysis
• Choose an appropriate forecast horizon based on the decision-making context and the reliability of long-term predictions
• Monitor forecast accuracy over time and update the model periodically to adapt to changes in the underlying patterns
• Be cautious of overfitting, which occurs when a model is too complex and fits the noise rather than the true signal
  • Regularization techniques and cross-validation can help mitigate overfitting
• Interpret the results in the context of the problem domain and communicate the limitations and uncertainties of the forecasts to stakeholders