Computing Multifractal Spectrum of 1D Time Series Using Box-Counting Method

We can compute the multifractal spectrum of one-dimensional time series using the box-counting method. This approach helps us better understand the complexity of time series and extract additional information from the data. For instance, we can calculate the fractal dimension of time series through multifractal spectrum analysis, which assists in determining sequence regularity and predicting future states. The implementation typically involves partitioning the time series into boxes of varying sizes (ε) and counting the probability measure (μ) within each box, followed by calculating the partition function χ(q,ε) = Σμ(q) for different moment orders q. Algorithmically, we compute the mass exponent τ(q) through linear regression of logχ(q,ε) versus log(ε), then derive the multifractal spectrum f(α) via Legendre transformation where α represents the singularity strength. Key functions in implementation would include box partitioning, probability distribution calculation, and multifractal parameter estimation. Simultaneously, we can utilize the multifractal spectrum to investigate both local characteristics and global structures of the sequence, thereby gaining deeper insights into its evolutionary patterns. Therefore, computing the multifractal spectrum for one-dimensional time series using the box-counting method enables better understanding and analysis of time series features and patterns, facilitating more effective research and practical applications in fields like signal processing and financial analysis.