  Oil price data have a complicated multi-scale structure that may vary with
time. We use time-frequency analysis to identify the main features of these
variations and, in particular, the regime shifts. The analysis is based on a
wavelet-based decomposition and analysis of the associated scale spectrum. The
joint estimation of the local Hurst exponent and volatility is the key to
detect and identify regime shifting and switching of the oil price. The
framework involves in particular modeling in terms of a process of
`multi-fractional' type so that both the roughness and the volatility of the
price process may vary with time. Special epochs then emerge as a result of
these degrees of freedom, moreover, as a result of the special type of spectral
estimator used. These special epochs are discussed and related to historical
events. Some of them are not detected by standard analysis based on maximum
likelihood estimation. The paper presents a novel algorithm for robust
detection of such special epochs and multi-fractional behavior in financial or
other types of data. In the financial context insight about such behavior of
the asset price is important to evaluate financial contracts involving the
asset.
