  An increasing number of experimental studies employ single particle tracking
to probe the physical environment in complex systems. We here propose and
discuss new methods to analyze the time series of the particle traces, in
particular, for subdiffusion phenomena. We discuss the statistical properties
of mean maximal excursions, i.e., the maximal distance covered by a test
particle up to time t. Compared to traditional methods focusing on the mean
squared displacement we show that the mean maximal excursion analysis performs
better in the determination of the anomalous diffusion exponent. We also
demonstrate that combination of regular moments with moments of the mean
maximal excursion method provides additional criteria to determine the exact
physical nature of the underlying stochastic subdiffusion processes. We put the
methods to test using experimental data as well as simulated time series from
different models for normal and anomalous dynamics, such as diffusion on
fractals, continuous time random walks, and fractional Brownian motion.
