🔮Forecasting Unit 2 – Time Series Analysis

Key Concepts and Definitions

• Time series data consists of observations collected sequentially over time at regular intervals (hourly, daily, monthly, yearly)
• Forecasting predicts future values based on historical patterns and trends in the data
  • Utilizes mathematical models to capture underlying patterns and extrapolate future values
• Trend represents the long-term increase or decrease in the data over time
• Seasonality refers to recurring patterns or cycles within a fixed period (weekly, monthly, quarterly)
• Cyclical patterns are irregular fluctuations that occur over longer periods, often influenced by economic or business cycles
• Irregular or residual component captures random fluctuations not explained by trend, seasonality, or cyclical patterns
• Autocorrelation measures the correlation between a time series and its lagged values
  • Positive autocorrelation indicates persistence, while negative autocorrelation suggests mean reversion

Components of Time Series

• Level indicates the average value of the series over time
• Trend component captures the long-term upward or downward movement in the data
  • Can be linear, exponential, or polynomial in nature
• Seasonal component represents regular, repetitive patterns within a fixed time period
  • Additive seasonality assumes constant seasonal effects across the series
  • Multiplicative seasonality assumes the magnitude of seasonal effects varies with the level of the series
• Cyclical component captures irregular fluctuations or cycles that extend beyond a single seasonal period
• Irregular or residual component represents random, unpredictable fluctuations not captured by other components
• Decomposition techniques (additive or multiplicative) separate a time series into its individual components for analysis and modeling

Data Preparation and Visualization

• Data cleaning involves handling missing values, outliers, and inconsistencies in the time series data
  • Interpolation methods (linear, spline) estimate missing values based on surrounding observations
  • Outlier detection techniques (Z-score, Tukey's method) identify and treat extreme values
• Resampling changes the frequency of the time series data (upsampling or downsampling)
• Smoothing techniques (moving average, exponential smoothing) reduce noise and highlight underlying patterns
• Plotting the time series helps visualize trends, seasonality, and other patterns
  • Line plots display observations over time
  • Seasonal subseries plots group data by seasonal periods to identify recurring patterns
• Autocorrelation plots (ACF) and partial autocorrelation plots (PACF) assess the correlation structure and identify significant lags
• Lag plots compare the time series against lagged values to detect autocorrelation patterns

Stationarity and Transformations

• Stationarity assumes the statistical properties of a time series remain constant over time
  • Mean, variance, and autocorrelation structure should be time-invariant
• Non-stationary series exhibit changing mean, variance, or autocorrelation over time
• Differencing removes trend and seasonality by computing the differences between consecutive observations
  • First-order differencing calculates the change between adjacent observations
  • Seasonal differencing removes seasonal patterns by subtracting values from the same season in the previous cycle
• Logarithmic transformation stabilizes the variance of a time series with increasing or decreasing variability
• Power transformations (Box-Cox) help achieve normality and stabilize variance
• Unit root tests (Augmented Dickey-Fuller, KPSS) assess the presence of stationarity or non-stationarity in a time series

Time Series Models

• Autoregressive (AR) models predict future values based on a linear combination of past values
  • AR(p) model includes p lagged values as predictors
• Moving Average (MA) models predict future values based on a linear combination of past forecast errors
  • MA(q) model includes q lagged forecast errors as predictors
• Autoregressive Moving Average (ARMA) models combine AR and MA components
  • ARMA(p, q) model includes p AR terms and q MA terms
• Autoregressive Integrated Moving Average (ARIMA) models extend ARMA to handle non-stationary series through differencing
  • ARIMA(p, d, q) model applies d differences to achieve stationarity before fitting an ARMA(p, q) model
• Seasonal ARIMA (SARIMA) models capture both non-seasonal and seasonal patterns
  • SARIMA(p, d, q)(P, D, Q)m model includes seasonal AR, differencing, and MA terms at seasonal lag m
• Exponential Smoothing (ES) models use weighted averages of past observations to forecast future values
  • Simple ES assigns equal weights to all past observations
  • Holt's linear trend method captures level and trend components
  • Holt-Winters' method incorporates level, trend, and seasonality

Model Selection and Evaluation

• Information criteria (AIC, BIC) assess the trade-off between model fit and complexity
  • Lower values indicate better model performance
• Cross-validation techniques (rolling origin, k-fold) evaluate model performance on unseen data
• Residual analysis examines the differences between observed and predicted values
  • Residuals should be uncorrelated, normally distributed, and have constant variance
• Ljung-Box test assesses the presence of autocorrelation in the residuals
• Forecast accuracy measures compare predicted values with actual observations
  • Mean Absolute Error (MAE) calculates the average absolute difference between predicted and actual values
  • Mean Squared Error (MSE) measures the average squared difference between predicted and actual values
  • Mean Absolute Percentage Error (MAPE) expresses the average absolute error as a percentage of the actual values
• Forecast horizon determines the number of future periods to predict
  • Short-term forecasts predict a few periods ahead
  • Long-term forecasts extend further into the future

Forecasting Techniques

• Naive methods use simple assumptions for forecasting
  • Random walk assumes the next value is equal to the current value plus random noise
  • Seasonal naive method uses the value from the same season in the previous cycle
• Exponential smoothing methods assign higher weights to more recent observations
  • Single exponential smoothing (SES) is suitable for series without trend or seasonality
  • Double exponential smoothing (Holt's method) captures level and trend components
  • Triple exponential smoothing (Holt-Winters' method) incorporates level, trend, and seasonality
• ARIMA forecasting uses the fitted ARIMA model to generate future predictions
  • Iterative approach uses predicted values as inputs for subsequent forecasts
• Ensemble methods combine predictions from multiple models to improve forecast accuracy
  • Simple averaging takes the mean of predictions from different models
  • Weighted averaging assigns different weights to models based on their performance
• Forecast combination techniques (Bates-Granger, Newbold-Granger) optimize the weights assigned to individual models in an ensemble

Practical Applications and Case Studies

• Demand forecasting predicts future product demand to optimize inventory and supply chain management
  • Retail sales forecasting helps plan staffing, inventory levels, and promotions
• Economic forecasting predicts macroeconomic variables (GDP, inflation, unemployment) to guide policy decisions
  • Central banks use economic forecasts to set monetary policy and interest rates
• Energy load forecasting predicts electricity demand to optimize power generation and distribution
  • Short-term load forecasting helps balance supply and demand in real-time
  • Long-term load forecasting aids in capacity planning and infrastructure investments
• Financial market forecasting predicts asset prices, volatility, and risk for investment and risk management
  • Stock price forecasting helps investors make informed trading decisions
  • Exchange rate forecasting is crucial for international trade and currency risk management
• Weather forecasting predicts future weather conditions to support decision-making in various sectors
  • Agriculture relies on weather forecasts for crop planning and irrigation scheduling
  • Transportation and logistics use weather forecasts to optimize routes and ensure safety
• Disease outbreak forecasting predicts the spread and impact of infectious diseases to guide public health interventions
  • Epidemic models (SIR, SEIR) simulate disease transmission dynamics
  • Forecasts inform resource allocation, vaccination strategies, and containment measures