  An algorithm is described that can generate random variants of a time series
or image while preserving the probability distribution of original values and
the pointwise Holder regularity. Thus, it preserves the multifractal properties
of the data. Our algorithm is similar in principle to well-known algorithms
based on the preservation of the Fourier amplitude spectrum and original values
of a time series. However, it is underpinned by a dual-tree complex wavelet
transform rather than a Fourier transform. Our method, which we term the
Iterated Amplitude Adjusted Wavelet Transform (IAAWT) method can be used to
generate bootstrapped versions of multifractal data and, because it preserves
the pointwise Holder regularity but not the local Holder regularity, it can be
used to test hypotheses concerning the presence of oscillating singularities in
a time series, an important feature of turbulence and econophysics data.
Because the locations of the data values are randomized with respect to the
multifractal structure, hypotheses about their mutual coupling can be tested,
which is important for the velocity-intermittency structure of turbulence and
self-regulating processes.
