Determining Local Singularity Strengths and their Spectra with the Wavelet Transform

We present a robust method of estimating the effective strength of singularities (the effective Hoelder exponent) locally at an arbitrary resolution. The method is motivated by the multiplicative cascade paradigm, and implemented on the hierarchy of singularities revealed with the wavelet transform modulus maxima tree. In addition, we illustrate the direct estimation of the scaling spectrum of the effective singularity strength, and we link it to the established partition function based multifractal formalism. We motivate both the local and the global multifractal analysis by showing examples of computer generated and real life time series

Wavelet methods in (financial) time-series processing

We briefly describe the major advantages of using the wavelet transform for the processing of financial time series on the example of the S&P index. In particular, we show how to uncover local the scaling (correlation) characteristics of the S&P index with the wavelet based effective H'older exponent [1, 2]. We use it to display the local spectral (multifractal) contents of the S&P index. In addition to this, we analyse the collective properties of the local correlation exponent as perceived by the trader, exercising various time horizon analyses of the index. We observed an intriguing interplay between such (different) time horizons. Heavy oscillations at shorter time horizons which seem to be accompanied by a steady decrease of correlation level for longer time horizons, seem to be characteristic patterns before the biggest crashes of the index. We find that this way of local presentation of scaling properties may be of economic importance

Rule discovery: tough, not meaningless

`Model free' rule discovery from data has recently been subject to considerable criticism, which has cast a shadow over the emerging discipline of time series data mining. However, other than in data mining, rule discovery has long been the subject of research in statistical physics of complex phenomena. Drawing from the expertise acquired therein, we suggest explanations for the two mechanisms of the apparent `meaninglessness' of rule recovery in the reference data mining approach. One reflects the universal property of self-affinity of signals from real life complex phenomena. It further expands on the issue of scaling invariance and fractal geometry, explaining that for ideal scale invariant (fractal) signals, rule discovery requires more than just comparing two parts of the signal. Authentic rule discovery is likely to look for the possible `structure' pertinent to the failure mechanism of the (position and/or resolution-wise) invariance of the time series analysed. The other reflects the redundancy of the `trivial' matches, which effectively smoothes out the rule which potentially could be discovered. Orthogonal scale space representations and appropriate redundancy suppression measures over autocorrelation operations performed during the matches are suggested as the methods of choice for rule discovery

Revealing local variability properties of human heartbeat intervals with the local effective Hölder exponent

The local effective H'older exponent has been applied to evaluate the variability of heart rate locally at an arbitrary position (time) and resolution (scale). The local effective H'older exponent [8, 9] used is effectively insensitive to local polynomial trends in heartbeat rate due to the use of the Wavelet Transform Modulus Maxima technique. Also the variability so obtained is compatible in the sense of distribution to the Multifractal Spectra of the analysed heart rate time series. This provides the possibility of standardising the variability estimation for comparison between different patients and between different recordings for one patient. The previously reported global correlation behaviour [1] is captured in the effective H'older exponent based, local variability estimate. This includes discriminating healthy and sick (congestive heart failure patients) on the basis of both the central (Hurst) exponent and the width of the multifractal spectra. In addition to this, we observed intriguing patterns of individual response in variability records to daily activities. A moving average filtering of H'older exponent based variability estimates was used to enhance these fluctuations. We find that this way of local presentation of scaling properties may be of clinical importance

Wavelets and the Unborn Child.

During labour, the attending medical staff use fetal heart rate recordings for evaluation of fetal well being and may base immediate intervention, such as a Caesarean section or taking a fetal scalp blood sample, on this. Using characteristics derived in real-time from the heart rate, obstetricians can predict a good outcome very well. However, in cases of fetal heart rate patterns considered `bad' by the obstetrician, at least half of these turn out to have been false alarms and the (operative) intervention unnecessary. Decision making can be improved by providing relevant information contained in the heart rate on a more solid, objective basis, making it independent of the personal experience of the specialist. This is enabled by recent progress in the modelling and analysis of heartbeat inter-beat dynamics, using the most advanced methods of signal processing (wavelet transform). CWI is tackling the mathematical side of this problem in cooperation with the Academic Medical Centre in Amsterdam (W.J. van Wijngaarden) and the Institute of Information and Computing Sciences of Utrecht University (R. Castelo). After mimicing the obstetrician's expert knowledge, the ultimate goal is to provide better than human performance by automated learning of predictive models

Local Effective Hölder Exponent Estimation on the Wavelet Transform Maxima Tree

We present a robust method of estimating an effective H\"older exponent locally at an arbitrary resolution. The method is motivated by the multiplicative cascade paradigm, and implemented on the hierarchy of singularities revealed with the wavelet transform modulus maxima tree. In addition, we illustrate the possibility of the direct estimation of the scaling spectrum of the effective H\"older exponent, and we link it to the established partition functions based multifractal formalism. We motivate both the local and the global multifractal analysis by showing examples of computer generated and real life time series

Revealing Local Variability Properties of Human Heartbeat Intervals with the Local Effective Hölder Exponent

textabstractThe local effective H'older exponent has been applied to evaluate the variability of heart rate locally at an arbitrary position (time) and resolution (scale). The local effective H'older exponent [8, 9] used is effectively insensitive to local polynomial trends in heartbeat rate due to the use of the Wavelet Transform Modulus Maxima technique. Also the variability so obtained is compatible in the sense of distribution to the Multifractal Spectra of the analysed heart rate time series. This provides the possibility of standardising the variability estimation for comparison between different patients and between different recordings for one patient. The previously reported global correlation behaviour [1] is captured in the effective H'older exponent based, local variability estimate. This includes discriminating healthy and sick (congestive heart failure patients) on the basis of both the central (Hurst) exponent and the width of the multifractal spectra. In addition to this, we observed intriguing patterns of individual response in variability records to daily activities. A moving average filtering of H'older exponent based variability estimates was used to enhance these fluctuations. We find that this way of local presentation of scaling properties may be of clinical importance

Direct multifractal spectrum calculation from the wavelet transform

We present a direct method of calculation of the multifractal spectrum from the wavelet decomposition. Information pertinent to singular structures in time series is captured by the WTMM method and the local effective Hoelder exponent is evaluated locally for each singular point of the time series. The direct multifractal spectrum is obtained from the scaling of the histograms of the local effective Hoelder exponent. In addition, we illustrate the possibility of estimation of the spectrum from the entire continuous wavelet transform

Oversampling the Haar wavelet transform

The Haar wavelet representation and a number of related representations have been shown to be a simple and powerful technique for similarity matching of time series. In this report, we extend the standard formulation to the translation invariant oversampled system. This makes possible a particularly efficient incremental scheme for coefficient calculation. As an additional benefit, the oversampled scheme provides for easy incremental update of the decomposition on new input samples. The system is further extended over higher order scaling functions of smoother character and over wavelets with more vanishing moments