From zero crossings to quantile‐frequency analysis of time series with an application to nondestructive evaluation

Abstract

Represented by the pioneering works of Professor Benjamin Kedem, zero crossings of time-series data have been proven useful for characterizing oscillatory patterns in many applications such as speech recognition and brainwave analysis. Robustness against outliers and nonlinear distortions is one of the advantages of zero crossings in comparison with traditional spectral analysis techniques. This paper introduces a new tool of spectral analysis for time-series data that goes beyond zero crossings. It is called quantile-frequency analysis (QFA). Constructed from trigonometric quantile regression, QFA transforms a time series into a bivariate function of quantile level and frequency variable. For each fixed quantile level, it corresponds to a periodogram-like function, called the quantile periodogram, which characterizes the oscillatory behavior of the time series round the quantile. By coupling QFA with functional principal component analysis, new dimension-reduced features are proposed for discriminant analysis of time series. The usefulness of these features is demonstrated by a case study of classifying real-world ultrasound signals for nondestructive evaluation of aircraft panels. Various machine learning classifiers are trained and tested by cross-validation. The results show a clear advantage of the QFA method over its ordinary-periodogram-based counterpart in delivering higher out-of-sample classification accuracy.

Quantile-Frequency Analysis and Deep Learning for Signal Classification

“…The results also show that proper tuning of the MLP classifier is essential to realize the benefit of quantile periodograms because poor choices of architectural parameters or learning rate can lead to inferior outcomes. Moreover, for most of the experimental settings, the MLP classifier with quantile periodograms as input outperforms the QFA-FPCA method in [15], the best accuracy scores of which lie within the range of 95.6%-97.2%, depending on the classifier and the number of principal components.…”

“…Table 4 illustrates this difficulty in our experiment. By treating each choice of architectural parameters as a trial, Table 4 shows the percentage of such trials for the MLP and CNN classifiers to achieve the test accuracy in different brackets above the 97.20 benchmark given by the non-DL classifiers in [15].…”

“…The superiority of QFA over the power spectrum in classifying these signals has been demonstrated in a recent publication [15] where simple linear and quadratic classifiers were applied to the coefficients of projection on a small number of FPCA principal components derived from the QFA arrays. The experiment in this paper further expands the previous study by considering more sophisticated MLP and CNN classifiers that take the QFA arrays directly as input without a separate preprocessing step for dimension reduction.…”

“…The performances of these classifiers are evaluated using a cross-validation scheme and compared with the performances of similar classifiers using the power spectrum as input instead of QFA. A comparison is also made between these classifiers and an QFA-based alternative considered in [15] that employs a combination of simple linear or quadratic classifiers with functional principal component analysis (FPCA). Our results show that a higher classification accuracy can be achieved by the proposed QFA-DL method compared to these contenders.…”

Quantile-Frequency Analysis and Spectral Measures for Diagnostic Checks of Time Series With Nonlinear Dynamics

“…This is sufficient for most practical purposes. However, there are situations such as one discussed in Li (2019) where it is necessary to include these values in the definition as follows: for ω = 0, it is natural to set q n (0, α ) := 0; for ω = π , it suffices to consider the 2‐parameter quantile regression problem while setting .…”