We investigate the box dimension of a time series having an inverse power-law spectra in a high dimensional Euclidean space. The time series can be random (colored noise) or deterministic. Both isotropic and anisotropic cases are included in our investigation. We study both the graph dimension and trail dimension of the time series. We show that with the same inverse power-law spectra, the deterministic series has a lower graph dimension than that of the colored noise, though they both can have fractal dimensions. We also derive a sharp upper bound on the trial dimension of the time series.
