Celebratio Mathematica

I. Daubechies, J. Lu, and H.-T. Wu: “Synchrosqueezed wavelet transforms: An empirical mode decomposition-like tool,” Appl. Comput. Harmon. Anal. 30 : 2 (2011), pp. 243–261. MR 2754779 Zbl 1213.42133 ArXiv 0912.2437 article

The EMD algorithm is a technique that aims to decompose into their building blocks functions that are the superposition of a (reasonably) small number of components, well separated in the time-frequency plane, each of which can be viewed as approximately harmonic locally, with slowly varying amplitudes and frequencies. The EMD has already shown its usefulness in a wide range of applications including meteorology, structural stability analysis, medical studies. On the other hand, the EMD algorithm contains heuristic and ad hoc elements that make it hard to analyze mathematically.

In this paper we describe a method that captures the flavor and philosophy of the EMD approach, albeit using a different approach in constructing the components. The proposed method is a combination of wavelet analysis and reallocation method. We introduce a precise mathematical definition for a class of functions that can be viewed as a superposition of a reasonably small number of approximately harmonic components, and we prove that our method does indeed succeed in decomposing arbitrary functions in this class. We provide several examples, for simulated as well as real data.

Jianfeng Lu	Related
Hau-Tieng Wu	Related

H.-T. Wu, P. Flandrin, and I. Daubechies: “One or two frequencies? The synchrosqueezing answers,” Adv. Adapt. Data Anal. 3 : 1–2 (April 2011), pp. 29–39. MR 2835580 Zbl 1234.94018 article

Hau-Tieng Wu	Related
Patrick Flandrin	Related

H.-T. Wu, S.-S. Hseu, M.-Y. Bien, Y. R. Kou, and I. Daubechies: “Evaluating the physiological dynamics via synchrosqueezing: Prediction of the ventilator weaning,” IEEE Trans. Biomed. Eng. 61 : 3 (March 2014), pp. 736–744. article

Oscillatory phenomena abound in many types of signals. Identifying the individual oscillatory components that constitute an observed biological signal leads to profound understanding about the biological system. The instantaneous frequency (IF), the amplitude modulation (AM), and their temporal variability are widely used to describe these oscillatory phenomena. In addition, the shape of the oscillatory pattern, repeated in time for an oscillatory component, is also an important characteristic that can be parametrized appropriately. These parameters can be viewed as phenomenological surrogates for the hidden dynamics of the biological system. To estimate jointly the IF, AM, and shape, this paper applies a novel and robust time-frequency analysis tool, referred to as the synchrosqueezing transform (SST). The usefulness of the model and SST are shown directly in predicting the clinical outcome of ventilator weaning. Compared with traditional respiration parameters, the breath-to-breath variability has been reported to be a better predictor of the outcome of the weaning procedure. So far, however, all these indices normally require at least 20 min of data acquisition to ensure predictive power. Moreover, the robustness of these indices to the inevitable noise is rarely discussed. We find that based on the proposed model, SST and only 3 min of respiration data, the ROC area under curve of the prediction accuracy is \( 0.76 \). The high predictive power that is achieved in the weaning problem, despite a shorter evaluation period, and the stability to noise suggest that other similar kinds of signal may likewise benefit from the proposed model and SST.

Hau-Tieng Wu	Related
Mauo-Ying Bien	Related
Yu Ru Kou	Related
Shu-Shua Hseu	Related

Y. G. Wang, H.-T. Wu, I. Daubechies, Y. Li, E. H. Estes, and E. Z. Soliman: “Automated J wave detection from digital 12-lead electrocardiogram,” J. Electrocardiol. 48 : 1 (January–February 2015), pp. 21–28. article

In this report we provide a method for automated detection of J wave, defined as a notch or slur in the descending slope of the terminal positive wave of the QRS complex, using signal processing and functional data analysis techniques. Two different sets of ECG tracings were selected from the EPICARE ECG core laboratory, Wake Forest School of Medicine, Winston Salem, NC. The first set was a training set comprised of 100 ECGs of which 50 ECGs had J-wave and the other 50 did not. The second set was a test set (\( n = 116 \) ECGs) in which the J-wave status (present/absent) was only known by the ECG Center staff. All ECGs were recorded using GE MAC 1200 (GE Marquette, Milwaukee, Wisconsin) at 10 mm/mV calibration, speed of 25 mm/s and 500 HZ sampling rate. All ECGs were initially inspected visually for technical errors and inadequate quality, and then automatically processed with the GE Marquette 12-SL program 2001 version (GE Marquette, Milwaukee, WI). We excluded ECG tracings with major abnormalities or rhythm disorder. Confirmation of the presence or absence of a J wave was done visually by the ECG Center staff and verified once again by three of the coauthors. There was no disagreement in the identification of the J wave state. The signal processing and functional data analysis techniques applied to the ECGs were conducted at Duke University and the University of Toronto. In the training set, the automated detection had sensitivity of 100% and specificity of 94%. For the test set, sensitivity was 89% and specificity was 86%. In conclusion, test results of the automated method we developed show a good J wave detection accuracy, suggesting possible utility of this approach for defining and detection of other complex ECG waveforms.

Yi (Grace) Wang	Related
Hau-Tieng Wu	Related
E. Harvey Estes	Related
Elsayed Z. Soliman	Related
Yabing Li	Related

I. Daubechies, Y. G. Wang, and H.-t. Wu: “ConceFT: Concentration of frequency and time via a multitapered synchrosqueezed transform,” Philos. Trans. Roy. Soc. A 374 : 2065 (2016). Article ID 20150193, 19 pages. MR 3479895 Zbl 1353.42031 ArXiv 1507.05366 article

Yi (Grace) Wang	Related
Hau-Tieng Wu	Related

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