Constrained-realization Monte-Carlo method for hypothesis testing

Abstract

We compare two theoretically distinct approaches to generating articial (or \surrogate") data for testing hypotheses about a given data set. The rst and more straightforward approach is to t a single \best" model to the original data, and then to generate surrogate data sets that are \typical realizations" of that model. The second approach concentrates not on the model but directly on the original data; it attempts to constrain the surrogate data sets so that they exactly agree with the original data for a speci ed set of sample statistics. Examples of these two approaches are provided for two simple cases: a test for deviations from a gaussian distribution, and a test for serial dependence in a time series. Additionally, we consider tests for nonlinearity in time series based on a Fourier transform (FT) method and on more conventional autoregressive moving-average (ARMA) ts to the data. The comparative performance of hypothesis testing schemes based on these two approaches is found to depend on whether or not the discriminating statistic is pivotal. A statistic is \pivotal" if its distribution is the same for all processes consistent with the null hypothesis. The typical-realization method requires that the discriminating statistic satisfy this property. The constrained-realization approach, on the other hand, does not share this requirement, and can provide an accurate and powerful test without having to sacri ce exibility in the choice of discriminating statistic.

Surrogate Test for Pseudoperiodic Time Series Data

“…The method we describe here is simpler and tests a more specific null hypothesis. This method may be applied to test against the null hypothesis of a periodic orbit with uncorrelated noise in the very large number of experimental systems that exhibit pseudoperiodic behavior.By contrast, the three most successful, and widely applied, algorithms test for membership of the class of (i) independent and identical distributed (IID) noise processes, (ii) linearly filtered noise processes, and (iii) static monotonic nonlinear transformation of linearly filtered noise processes [1,8]. For time series data exhibiting strong pseudoperiodic behavior, the null hypotheses of IID or colored noise are obviously false.…”

“…By contrast, the three most successful, and widely applied, algorithms test for membership of the class of (i) independent and identical distributed (IID) noise processes, (ii) linearly filtered noise processes, and (iii) static monotonic nonlinear transformation of linearly filtered noise processes [1,8]. For time series data exhibiting strong pseudoperiodic behavior, the null hypotheses of IID or colored noise are obviously false.…”

Determinism in Financial Time Series

“…It is common practice to compare (nonlinear) statistic values of data to the values obtained from a scrambled time series. However, it is not sufficient to compare the data to a single scrambled time series: such a test has no power 14 (Theiler and Prichard 1996). A single surrogate time series provides only a (poor) estimate of the mean of the distribution for H and no estimate of the variance of that distribution.…”

Chaos emerging in soil failure patterns observed during tillage: Normalized deterministic nonlinear prediction (NDNP) and its application

“…In arriving at these conclusions, we conducted the conventional hypothesis testing for large scale and the Monte Carlo hypothesis testing for determining the significance levels. 31 …”