E. R. Tracy

A REVIEW OF SYMBOLIC ANALYSIS OF EXPERIMENTAL DATA

This review covers the group of data-analysis techniques collectively referred to as symbolization or symbolic time-series analysis. Symbolization involves transformation of raw time-seriesmeasurements (i.e., experimental signals) into a series of discretized symbols that are processed to extract information about the generating process. In many cases, the degree of discretization can be quite severe, even to the point of converting the original data to single-bit values. Current approaches for constructing symbols and detecting the information they contain are summarized. Novel approaches for characterizing and recognizing temporal patterns can be important for many types of experimental systems, but this is especially true for processes that are nonlinear and possibly chaotic. Recent experience indicates that symbolization can increase the efficiency of finding and quantifying information from such systems, reduce sensitivity to measurement noise, and discriminate both specific and general classes of proposed models. Examples of the successful application of symbolization to experimental data are included. Key theoretical issues and limitations of the method are also discussed.

Symbol statistics and spatio-temporal systems

Abstract We consider the problem of estimating parameters from time-series observations of spatio-temporal systems. Two types of models are considered: (a) a one-dimensional coupled map lattice with nearest neighbor diffusive coupling; and (b) the complex Ginzburg-Landau equation in one and two spatial dimensions. Model parameters are to be estimated using time-series observations from only a few sites. A symbolic partition of the time series is introduced and the probabilities of observing various symbol sequences in the data are measured. The parameter fitting is accomplished by adjusting parameters of the model until it produces time series whose symbol sequences have the same probabilities as the data. We show that it is possible to reliably estimate the parameters from a single time series when the spatio-temporal dynamics is “turbulent”, i.e. it displays a wide range of space and time scales with no discernible patterns.

Ray-based methods in multidimensional linear wave conversion

A tutorial introduction to the topic of linear wave conversion in multiple spatial dimensions is provided. The emphasis is on physical concepts, particularly those features of multidimensional conversion that are new and different from the more familiar “mode conversion” problem in one spatial dimension. After introductory comments, a brief review of WKB theory for vector wave equations in the absence of conversion is provided in order to introduce notation, terminology, and geometrical ideas. A primary theme of the discussion is that, although WKB (ray-based) methods break down in conversion regions, the ray geometry in the conversion region can be used to develop local wave equations that govern the two coupled wave channels undergoing conversion. These methods can be incorporated into ray-tracing algorithms providing, for the first time, the ability to follow the “ray splitting” associated with linear conversion in multidimensions, including the amplitude and phase changes associated with the conversion.

Data compression and information retrieval via symbolization

Converting a continuous signal into a multisymbol stream is a simple method of data compression which preserves much of the dynamical information present in the original signal. The retrieval of selected types of information from symbolic data involves binary operations and is therefore optimal for digital computers. For example, correlation time scales can be easily recovered, even at high noise levels, by varying the time delay for symbolization. Also, the presence of periodicity in the signal can be reliably detected even if it is weak and masked by a dominant chaotic/stochastic background. (c) 1998 American Institute of Physics.

Modeling and synchronizing chaotic systems from experimental data

Abstract The inverse problem of extracting evolution equations from chaotic time series measured from continuous systems is considered. The resulting equations of motion form an autonomous system of nonlinear ordinary differential equations (ODEs). The vector fields are modeled in the manner of implicit Adams integration using a basis set of polynomials that are constructed to be orthonormal on the data. The fitting method uses the Rissanen minimum description length (MDL) criterion to determine the optimal polynomial vector field. It is then demonstrated that one can synchronize the model to an experimentally measured time series. In this case synchronization is used as a nontrivial test for the validity of the models.

Precision Enhancement of MALDI-TOF-MS Using High Resolution Peak Detection and Label-Free Alignment*

We have developed an automated procedure for aligning peaks in multiple TOF spectra that eliminates common timing errors and small variations in spectrometer output. Our method incorporates high‐resolution peak detection, re‐binning, and robust linear data fitting in the time domain. This procedure aligns label‐free (uncalibrated) peaks to minimize the variation in each peaks location from one spectrum to the next, while maintaining a high number of degrees of freedom. We apply our method to replicate pooled‐serum spectra from multiple laboratories and increase peak precision (t/σt) to values limited only by small random errors (with σt less than one time count in 89 out of 91 instances, 13 peaks in seven datasets). The resulting high precision allowed for an order of magnitude improvement in peak m/z reproducibility. We show that the CV for m/z is 0.01% (100 ppm) for 12 out of the 13 peaks that were observed in all datasets between 2995 and 9297 Da.

A Bayesian network approach to feature selection in mass spectrometry data

BackgroundTime-of-flight mass spectrometry (TOF-MS) has the potential to provide non-invasive, high-throughput screening for cancers and other serious diseases via detection of protein biomarkers in blood or other accessible biologic samples. Unfortunately, this potential has largely been unrealized to date due to the high variability of measurements, uncertainties in the distribution of proteins in a given population, and the difficulty of extracting repeatable diagnostic markers using current statistical tools. With studies consisting of perhaps only dozens of samples, and possibly hundreds of variables, overfitting is a serious complication. To overcome these difficulties, we have developed a Bayesian inductive method which uses model-independent methods of discovering relationships between spectral features. This method appears to efficiently discover network models which not only identify connections between the disease and key features, but also organizes relationships between features--and furthermore creates a stable classifier that categorizes new data at predicted error rates.ResultsThe method was applied to artificial data with known feature relationships and typical TOF-MS variability introduced, and was able to recover those relationships nearly perfectly. It was also applied to blood sera data from a 2004 leukemia study, and showed high stability of selected features under cross-validation. Verification of results using withheld data showed excellent predictive power. The method showed improvement over traditional techniques, and naturally incorporated measurement uncertainties. The relationships discovered between features allowed preliminary identification of a protein biomarker which was consistent with other cancer studies and later verified experimentally.ConclusionsThis method appears to avoid overfitting in biologic data and produce stable feature sets in a network model. The network structure provides additional information about the relationships among features that is useful to guide further biochemical analysis. In addition, when used to classify new data, these feature sets are far more consistent than those produced by many traditional techniques.

On the nonlinear Schro¨dinger limit of the Korteweg–de Vries equation

Abstract Using the multiscale approach of Zakharov and Kuznetsov it is shown that the nonlinear Schrodinger periodic scattering data is related to the Korteweg-de Vries periodic scattering data via an average over the Korteweg-de Vries carrier oscillation. This allows a complete elucidation of the physical meaning of the nonlinear Schrodinger scattering data, conservation laws, theta function solutions and reality constraint.

Mode conversion in the Gulf of Guinea

Linear mode conversion is the partial transfer of wave energy from one wave type (a) to another (b) in a weakly non-uniform background state. For propagation in one dimension (x), the local wavenumber k x j of each wave (j = a,b) varies with x; if these are equal at some x R , the waves are locally in phase, and resonant energy transfer can occur. We model wave propagation in the Gulf of Guinea, where wave a is an equatorially trapped Rossby-gravity (Yanai) wave, and wave b is a coastal Kelvin wave along the (zonal) north coast of the Gulf, both propagating in zonal coordinate x. The coupling of the waves is due to the overlap of their eigenfunctions (normal modes in y, the meridional coordinate). We derive coupled mode equations from a variational principle, and obtain an analytic expression for the wave-energy conversion coefficient, in terms of the wave frequency and the scale length of the thermocline depth.

Reconstruction of chaotic signals using symbolic data

Abstract We discuss the reconstruction of dynamical systems from noisy time-series. In particular, we consider the use of the symbol statistics (coarse-grained signal data) as the target for reconstruction. The statistics of symbol sequences is relatively insensitive to moderate amounts of measurement noise (σ(noise)/σ(signal) ≈ 10–20%), while larger amounts produce a substantial bias. We show that it is possible to produce robust reconstructions even when σ(noise)/σ(signal) ≈ O(1). Our study shows that even at such high noise levels the procedure is convergent : i.e. the accuracy of parameter estimates is limited only by the amount of data and computer time available.