Generating FARIMA Time Series Using the Definition Method

Generating FARIMA (Fractional Autoregressive Integrated Moving Average) time series using the definition method involves creating a generalized stochastic process that incorporates three key components: Autoregressive (AR), Fractional Integration (I), and Moving Average (MA) components. In implementation, this typically requires coding fractional differencing using binomial expansion or Fast Fourier Transform (FFT) methods to handle non-integer differencing parameters. During HURST parameter estimation, we perform statistical analysis on the time series to quantify long-range dependence characteristics. Common implementation approaches include Rescaled Range (R/S) analysis or detrended fluctuation analysis (DFA) algorithms, which measure how autocorrelations decay polynomially rather than exponentially. This long-range dependence indicates that past values in the time series significantly influence present and future values, making it crucial for accurate future value prediction. Therefore, by estimating HURST parameters for FARIMA time series through appropriate statistical methods and corresponding MATLAB/Python functions (e.g., arfima simulation routines and hurst exponent calculators), we gain deeper insights into the dynamic characteristics of time series. This provides a more precise foundation for future forecasting and analytical applications, particularly in fields like econometrics and network traffic analysis where long-memory processes are prevalent.