Surrogate data testing

Surrogate data testing^[1] (or the method of surrogate data) is a statistical proof by contradiction technique similar to permutation tests^[2] and parametric bootstrapping. It is used to detect non-linearity in a time series.^[3] The technique involves specifying a null hypothesis ${\displaystyle H_{0}}$ describing a linear process and then generating several surrogate data sets according to ${\displaystyle H_{0}}$ using Monte Carlo methods. A discriminating statistic is then calculated for the original time series and all the surrogate set. If the value of the statistic is significantly different for the original series than for the surrogate set, the null hypothesis is rejected and non-linearity assumed.^[3]

The particular surrogate data testing method to be used is directly related to the null hypothesis. Usually this is similar to the following: The data is a realization of a stationary linear system, whose output has been possibly measured by a monotonically increasing possibly nonlinear (but static) function.^[1] Here linear means that each value is linearly dependent on past values or on present and past values of some independent identically distributed (i.i.d.) process, usually also Gaussian. This is equivalent to saying that the process is ARMA type. In case of fluxes (continuous mappings), linearity of system means that it can be expressed by a linear differential equation. In this hypothesis, the static measurement function is one which depends only on the present value of its argument, not on past ones.