  The state of a stochastic process evolving over a time $t$ is typically
assumed to lie on a normal distribution whose width scales like $t^{1/2}$.
However, processes where the probability distribution is not normal and the
scaling exponent differs from $\frac{1}{2}$ are known. The search for possible
origins of such "anomalous" scaling and approaches to quantify them are the
motivations for the work reported here. In processes with stationary
increments, where the stochastic process is time-independent, auto-correlations
between increments and infinite variance of increments can cause anomalous
scaling. These sources have been referred to as the $\it{Joseph}$ $\it{effect}$
the $\it{Noah}$ $\it{effect}$, respectively. If the increments are
non-stationary, then scaling of increments with $t$ can also lead to anomalous
scaling, a mechanism we refer to as the $\it{Moses}$ $\it{effect}$. Scaling
exponents quantifying the three effects are defined and related to the Hurst
exponent that characterizes the overall scaling of the stochastic process.
Methods of time series analysis that enable accurate independent measurement of
each exponent are presented. Simple stochastic processes are used to illustrate
each effect. Intraday Financial time series data is analyzed, revealing that
its anomalous scaling is due only to the Moses effect. In the context of
financial market data, we reiterate that the Joseph exponent, not the Hurst
exponent, is the appropriate measure to test the efficient market hypothesis.
