Robert Cawley

Multifractal decompositions of Moran fractals

We present a rigorous construction and generalization of the multifractal decomposition for Moran fractals with infinite product measure. The generalization is specified by a system of nonnegative weights in the partition sum. All the usual (smooth) properties of the f(α) theory are recovered for the case that the weights are equal to unity. The generalized spectrum, f(α, w), is invariant to a group of gauge transformations of the weights, and, in addition, need no longer be concave. In case the fractal is a Cantor set generated by an iterated function system of similarities, α is the pointwise dimension of the measure. We discuss properties of some examples.

SNR perfomance of a noise reduction algorithm applied to coarsely sampled chaotic data

Abstract We describe results of application to coarsely sampled Lorenz time series of an algorithm for noise reduction. The method does not depend on having detailed prior information about system dynamics. Systematic numerical studies reveal linear growth in peak values of maximum SNR improvement with the logarithm of trajectory length.

Predictability in time series

Abstract We introduce a technique to characterize and measure predictability in time series. The technique allows one to formulate precisely a notion of the predictable component of given time series. We illustrate our method for both numerical and experimental time series data.

Detection and diagnosis of dynamics in time series data: Theory of noise reduction

We describe a general four‐step approach to chaotic noise reduction: embedding, data state vector alteration, disembedding and iteration. In this way, a noise reduction algorithm may be regarded as a repeated application of an operator A:v(t)→v(t) on a space of scalar time series. We suggest that systematics of the response of a time series to iteration of A can be studied to estimate quantitatively optimal algorithm parameters, such as best embedding trial dimension, d=dpk, and number of iterations, nM=nM(d), and other quantities, parameters depending on A, to achieve maximum improvement.

Chaotic noise reduction by local‐geometric‐projection with a reference time series

We describe a family of algorithms for chaotic noise reduction when a ‘‘reference’’ time series is assumed to be given. These algorithms are variations of the local‐geometric‐projection (LGP) algorithm first introduced in. Signal‐to‐noise ratio (SNR) improvements achieved depend on the quality of the reference time series, the degree to which information available from knowledge of the reference data is used, and on the SNR initially present in the data. We report results for cases where the reference time series is the original noise‐free time series, and where it is taken to be a noise reduced version of the given noisy time series.

Noise reduction for chaotic data by geometric projection

We present a method for noise reduction that does not depend on detailed prior knowledge of system dynamics. The method has performed reasonably well for known maps and flows. Also, we present an empirically based technique to estimate the initial signal-to-noise ratio for time series whose dynamical origin may be unknown.

Dimension Measurements from Cloud Radiance

Infrared emissions from clouds exhibit chaotic behavior as a function of angular distance at a fixed time. Preliminary results for dimensions of the graphs of intensity vs angle for emissions at 3–5 µm and 8–12 µm are reported for a sample cloud.

Applications of a statistical test for ‘‘smooth’’ dynamics

We give results of a few simple applications of a statistical test for ‘‘smoothness’’ of embedded time series recently introduced by the authors. The method, which is applicable to both map and flow data, exploits an arbitrariness in the choice of vector field for computation of a statistic forming the basis of the test. The statistic we choose is a natural extension to the general vector field setting of Kaplan and Glass’s Λ‐statistic, although that specific choice is not essential to the method. Unavoidable uncertainties in Λ due to finite numerics are mitigated by the device of employing maximum and minimum values of Λ over a set of many randomly chosen vector fields. We examine properties under the test of examples chosen to illustrate the variety of effects that can occur in implementation of the test. Although we have focussed our investigations on low values of embedding trial dimension, the method seems likely to be generally reliable if appropriate data requirements are met.

Detecting smoothness in noisy time series

We describe the role of chaotic noise reduction in detecting an underlying smoothness in a dataset. We have described elsewhere a general method for assessing the presence of determinism in a time series, which is to test against the class of datasets producing smoothness (i.e., the null hypothesis is determinism). In order to reduce the likelihood of a false call, we recommend this kind of analysis be applied first to a time series whose deterministic origin is at question. We believe this step should be taken before implementing other methods of dynamical analysis and measurement, such as correlation dimension or Lyapounov spectrum.

Method to Discriminate Against Determinism in Time Series Data

We describe a general, systematic method for assessing the presence or absence of determinism in time series. Our method is rooted in the standard engineering paradigm of hypothesis testing. Our application of this procedure is novel, however, for we test given data sets against the class of data sets that produce smoothness. That is, our null hypothesis is that of determinism. We highlight two inherently interactive key features of our approach which conspire to make this treatment promising, the use of a smoothness detector and of chaotic noise reduction.