⏳Intro to Time Series Unit 9 – Seasonal ARIMA Models

What's the Deal with Seasonal ARIMA?

• Seasonal ARIMA (SARIMA) models extend regular ARIMA models to handle seasonal patterns in time series data
• Incorporates both non-seasonal and seasonal components to capture complex patterns and dependencies
• Useful for data exhibiting regular, repeating patterns over fixed periods (monthly sales, quarterly earnings)
• Combines autoregressive (AR), differencing (I), and moving average (MA) components with seasonal counterparts
• Requires careful identification of seasonal patterns and appropriate model specification for accurate forecasting
• Enables businesses to make informed decisions based on seasonal trends and fluctuations (inventory management, resource allocation)
• Provides a powerful framework for understanding and predicting seasonal behavior in various domains (economics, finance, weather)

The Building Blocks: AR, I, and MA

• AR (autoregressive) component captures the relationship between an observation and a certain number of lagged observations
  • Assumes the current value depends on its own past values
  • Specified by the order $p$, indicating the number of lag terms included
• I (differencing) component removes the trend and makes the time series stationary
  • Applies differencing operator $\nabla$ to eliminate non-stationarity
  • Specified by the order $d$, indicating the number of times the series is differenced
• MA (moving average) component captures the relationship between an observation and past forecast errors
  • Assumes the current value depends on the past errors or residuals
  • Specified by the order $q$, indicating the number of lagged forecast errors included
• The combination of AR, I, and MA components forms the basis of ARIMA models
• Orders $(p, d, q)$ are determined through careful analysis of autocorrelation and partial autocorrelation plots
• Selecting appropriate orders is crucial for capturing the underlying patterns and achieving accurate forecasts

Seasonality: Why It Matters

• Seasonality refers to regular, repeating patterns in time series data over fixed periods (yearly, quarterly, monthly)
• Ignoring seasonality can lead to inaccurate forecasts and poor decision-making
• Seasonal patterns can arise from various factors (weather, holidays, business cycles)
• Identifying and modeling seasonality is essential for understanding the underlying dynamics of the data
• Seasonal components in SARIMA models capture the recurring patterns and improve forecast accuracy
• Enables businesses to anticipate and plan for seasonal fluctuations (demand planning, resource allocation)
• Helps in detecting anomalies and unusual behavior deviating from the expected seasonal patterns
• Seasonal adjustments can reveal underlying trends and facilitate comparisons across different time periods

SARIMA Components Breakdown

• SARIMA models incorporate both non-seasonal and seasonal components
• Non-seasonal components:
  • AR($p$): Autoregressive component of order $p$
  • I($d$): Differencing component of order $d$
  • MA($q$): Moving average component of order $q$
• Seasonal components:
  • Seasonal AR($P$): Autoregressive component of order $P$ for the seasonal part
  • Seasonal I($D$): Differencing component of order $D$ for the seasonal part
  • Seasonal MA($Q$): Moving average component of order $Q$ for the seasonal part
  • Seasonal period $m$: The number of periods per season (12 for monthly data, 4 for quarterly data)
• The complete SARIMA model is denoted as SARIMA($p,d,q$)($P,D,Q$)$_m$
• Each component contributes to capturing different aspects of the time series behavior
• Seasonal components are applied at the seasonal lags determined by the seasonal period $m$
• Interaction between non-seasonal and seasonal components allows for modeling complex patterns and dependencies

Identifying Seasonal Patterns

• Visual inspection of time series plots can reveal apparent seasonal patterns
• Seasonal subseries plots display observations from the same season together, highlighting seasonal behavior
• Autocorrelation function (ACF) and partial autocorrelation function (PACF) plots can indicate seasonal lags
  • Significant spikes at seasonal lags in ACF suggest the presence of seasonality
  • PACF can help determine the order of seasonal AR terms
• Seasonal differencing can remove seasonal patterns and make the series stationary
  • Applying seasonal differencing of order $D$ at lag $m$ eliminates seasonal non-stationarity
• Statistical tests (Seasonal Kendall test, Friedman test) can formally assess the presence of seasonality
• Domain knowledge and understanding of the underlying process can guide the identification of seasonal patterns
• Iterative process of model fitting and diagnostic checks helps refine the identification of seasonal components

Model Selection and Fitting

• Selecting the appropriate SARIMA model involves determining the orders $(p,d,q)(P,D,Q)_m$
• Iterative process of model specification, estimation, and diagnostic checking
• Use ACF and PACF plots to identify potential orders for non-seasonal and seasonal components
• Consider multiple candidate models and compare their performance
• Estimation methods (maximum likelihood, least squares) are used to estimate the model parameters
• Information criteria (AIC, BIC) help in model selection by balancing goodness of fit and model complexity
  • Lower values of AIC and BIC indicate better model fit
• Residual analysis is performed to assess the adequacy of the fitted model
  • Residuals should exhibit no significant autocorrelation and follow a white noise process
• Cross-validation techniques can be used to evaluate the model's performance on unseen data
• Iterative refinement of the model based on diagnostic checks and domain knowledge
• Parsimony principle suggests selecting the simplest model that adequately captures the underlying patterns

Diagnostic Checks: Is Your Model Any Good?

• Diagnostic checks assess the adequacy and validity of the fitted SARIMA model
• Residual analysis is a key component of diagnostic checking
  • Residuals should be uncorrelated and normally distributed
  • ACF and PACF plots of residuals should show no significant autocorrelation
  • Ljung-Box test can formally test for residual autocorrelation
• Normality of residuals can be assessed using histograms, Q-Q plots, and statistical tests (Shapiro-Wilk test)
• Homoscedasticity (constant variance) of residuals can be checked using residual plots against fitted values or time
• Overfitting should be avoided by selecting parsimonious models and using cross-validation
• Stability of model parameters can be assessed by examining their significance and confidence intervals
• Forecast accuracy measures (MAPE, RMSE) evaluate the model's performance on test data
• Visual inspection of fitted values against actual values helps assess the model's goodness of fit
• Diagnostic checks provide insights into the model's limitations and areas for improvement

Real-World Applications

• SARIMA models find applications in various domains where seasonal patterns are prevalent
• Retail and e-commerce: Forecasting sales, demand planning, and inventory management based on seasonal trends
• Finance and economics: Modeling and forecasting economic indicators (GDP, inflation), stock prices, and trading volumes
• Energy and utilities: Predicting energy consumption, electricity demand, and load forecasting considering seasonal patterns
• Tourism and hospitality: Forecasting tourist arrivals, hotel occupancy rates, and revenue management strategies
• Weather and climate: Modeling and predicting temperature, precipitation, and other meteorological variables with seasonal variations
• Healthcare: Analyzing and forecasting disease incidence, hospital admissions, and resource allocation based on seasonal patterns
• Supply chain management: Optimizing inventory levels, production planning, and logistics considering seasonal demand fluctuations
• Marketing and advertising: Planning promotional campaigns and allocating budgets based on seasonal consumer behavior