Multifractal analysis: Pitfalls of standard procedures and alternatives

When applied to measures that are not multifractal at all, standard methods of multifractal analysis (MFA) may produce results that make the measures appear multifractal. This is because such methods, which are based on the scaling properties of moments of the measure of interest, tend to produce envelopes of the actual spectrum that look like the {{α},f({α})} spectra of legitimate multifractal measures. We show examples and consequences of this property through the application of MFA to analytic functions, for which the actual spectrum can be easily evaluated, and to measures obtained from empirical data. It is shown specifically that application of MFA to the ``width function'' of river basins has led in the past to erroneous conclusions. We develop tools to distinguish between legitimate and spurious multifractal spectra, in part by using existing MFA procedures in unusual ways and in part by modifying such procedures. Finally, we discuss the extension of MFA to properties other than the scaling of local integrals. We illustrate this possibility by considering increments and integrals of increments of a function. We find that width functions are to a first approximation monofractal relative to differences, with an {α} of about 0.3. This means that at small scales, such functions behave like fractional Brownian motion with a fractal dimension of about 1.7. This is in essential agreement with previous findings. Numerical issues in the application of MFA algorithms are briefly discussed.