6.4 Integrated ARIMA models

ARIMA models are powerful tools for analyzing time series data, combining autoregressive, differencing, and moving average components. They can handle both stationary and non-stationary data, making them versatile for various applications like stock prices and weather forecasting.

Understanding the orders (p,d,q) in ARIMA models is crucial. The AR order (p) relates current observations to past ones, differencing order (d) achieves stationarity, and MA order (q) incorporates past error terms. These components work together to capture complex patterns in time series data.

Integrated ARIMA Models

Definition of ARIMA models

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• ARIMA models combine autoregressive (AR), differencing (I), and moving average (MA) components to analyze time series data
  • AR component captures the relationship between an observation and a certain number of lagged observations (previous values in the series)
  • I component involves differencing the time series to achieve stationarity (constant mean, variance, and autocovariance over time)
  • MA component models the error term as a linear combination of error terms occurring at various times in the past (residuals)
• ARIMA models are denoted as ARIMA(p,d,q)
  • p is the order of the AR term (number of lagged observations)
  • d is the degree of differencing (number of times the series is differenced)
  • q is the order of the MA term (number of lagged error terms)
• ARIMA models handle both stationary and non-stationary time series data
• Useful for modeling time series with trends, seasonal patterns, and short-term dependencies (stock prices, weather data)

Orders in ARIMA models

• The order of an ARIMA model is determined by three parameters: p, d, and q
• p represents the number of autoregressive terms in the model
  • An AR(p) model relates the current observation to the past p observations
  • AR(1) model relates the current observation to the immediately preceding observation (lag of 1)
• d represents the number of differencing operations applied to the time series to achieve stationarity
  • Differencing involves computing the differences between consecutive observations
  • A time series that requires d differencing operations to become stationary is integrated of order d, denoted as I(d)
• q represents the number of moving average terms in the model
  • An MA(q) model relates the current observation to the past q error terms
  • MA(1) model relates the current observation to the immediately preceding error term (lag of 1)

Stationarity and Parameter Estimation

Differencing for stationarity

• Stationarity is a crucial assumption for many time series models, including ARIMA
  • A stationary time series has constant mean, variance, and autocovariance over time
• Differencing transforms a non-stationary time series into a stationary one
  1. Compute the first difference by subtracting each observation from its preceding observation
  2. If the first difference is not stationary, repeat the process to obtain higher-order differences
• The order of differencing (d) in an ARIMA model is determined by the number of differencing operations required to achieve stationarity
• Once stationarity is achieved, the differenced time series can be used for ARIMA modeling
  • The original time series can be reconstructed by applying the inverse of the differencing operation to the model's predictions (cumulative sum)

Parameter estimation for ARIMA

• After determining the orders (p, d, q) of an ARIMA model, estimate the model parameters using appropriate statistical methods
• Least squares estimation minimizes the sum of squared residuals between the observed values and the model's predicted values
  • Residuals are the differences between the observed values and the model's predictions
  • Objective: find parameter values that minimize the sum of squared residuals
• Maximum likelihood estimation finds parameter values that maximize the likelihood function
  • Likelihood function measures the probability of observing the given data under the assumed model
  • Objective: find parameter values that make the observed data most likely
• Both methods aim to find the best-fitting parameter values for the ARIMA model
• Statistical software packages provide tools for estimating ARIMA model parameters (R, Python)

Interpretation of ARIMA results

• Interpreting ARIMA results involves understanding the impact of each component (AR, I, and MA) on the model's predictions and performance
• Autoregressive (AR) terms capture the relationship between an observation and its lagged values
  • Positive AR coefficients: an increase in past values leads to an increase in the current value
  • Negative AR coefficients: an increase in past values leads to a decrease in the current value
  • Magnitude of AR coefficients determines the strength of the relationship between current and lagged values
• Differencing (I) component removes trends and seasonality from the time series
  • Interpreting results of a differenced series requires considering the order of differencing and its impact on the original time series
  • Higher orders of differencing may remove more complex patterns but can make the series more difficult to interpret
• Moving average (MA) terms capture the relationship between an observation and past forecast errors
  • Positive MA coefficients: a positive error in the past leads to an increase in the current value
  • Negative MA coefficients: a positive error in the past leads to a decrease in the current value
  • Magnitude of MA coefficients determines the strength of the relationship between current value and past errors
• Assess overall performance of an ARIMA model using evaluation metrics
  • Mean squared error (MSE), root mean squared error (RMSE), or mean absolute error (MAE)
  • Lower values of these metrics indicate better model performance
• Interpret ARIMA results in the context of the problem domain, considering practical implications of the model's predictions and limitations of the modeling approach