George Treviño

Using Wavelets to Detect Trends

Abstract Wavelets are a new class of basis functions that are finding wide use for analyzing and interpreting time series data. This paper describes a new use for wavelets—identifying trends in time series. The general signal considered has a quadratic trend. The inverted Haar wavelet and the elephant wavelet, respectively, provide estimates of the first-order and second-order coefficients in the trend polynomial. Unlike usual wavelet applications, however, this analysis requires only one wavelet dilation scale L, where L is the total length of the time series. Error analysis shows that wavelet trend detection is roughly half as accurate as least squares trend detection when accuracy is evaluated in terms of the mean-square error in estimates of the first-order and second-order trend coefficients. But wavelet detection is more than twice as efficient as least squares detection in the sense that it requires fewer than half the number of floating-point operations of least squares regression to yield the thr...

On wavelet analysis of nonstationary turbulence

Wavelets are new tools for turbulence analysis that are yielding important insights into boundary-layer processes. Wavelet analysis, however, has some as yet undiscussed limitations: failure to recognize these can lead to misinterpretation of wavelet analysis results. Here we discuss some limitations of wavelet analysis when applied to nonstationary turbulence. Our main point is that the analysis wavelet must be carefully matched to the phenomenon of interest, because wavelet coefficients obscure significant information in the signal being analyzed. For example, a wavelet that is a second-difference operator can provide no information on the linear trend in a turbulence signal. Wavelet analysis also yields no meaningful information about nonlinear behavior in a signal — contrary to claims in the literature — because, at any instant, a wavelet is a single-scale operator, while nonlinearity involves instantaneous interactions among many scales.

Averaging Intervals For Spectral Analysis Of Nonstationary Turbulence

We formulate a method for determining the smallest time interval Δ Tover which a turbulence time series can be averaged to decompose it intoinstantaneous mean and random components. From the random part the method defines the optimal interval (or averaging window) AW over which this part should be averaged to obtain the instantaneous spectrum. Both Δ T and AW vary randomly with time and depend on physical properties of the turbulence. Δ T also depends on the accuracy of the measurements and is thus independent of AW. Interesting features of the method are its real-time capability and the non-equality between AW and Δ T.

On the invariant functions of the turbulence bispectrum

Abstract It is established that the invariant functions of the turbulence bispectrum can themselves be represented in terms of basis functions which are independent of time. These basis functions are also scale-independent, and therefore identical for large eddies and small eddies alike. Dynamical implications of this result are discussed.

Isotropic analysis of grid-turbulence

Abstract The principles of self-similar isotropic turbulence are applied to the analysis of grid-decay. In particular it is theoretically established that energy-transfer occurs in only two distinct invariant modes. The decay constant of third-order correlations is posed in terms of the “amounts” of each mode present in the decay. This constant is determined empirically, and is found to be relatively insensitive to changes in Re. The related decay of turbulence intensity is computed.

Comment on “Time-frequency analysis with the continuous wavelet transform,” by W. Christopher Lang and Kyle Forinash [Am. J. Phys. 66 (9), 794–797 (1998)]

We read with much interest the paper by Lang and Forinash ~Ref. 1! but felt we needed to add words of caution regarding wavelet analysis. We state at the outset, though, it is not our intent to either blemish or deny the results of Lang and Forinash. We do, however, want to make clear that, when applied to well-defined problems, wavelets work wonders. When applied, on the other hand, to less well-defined problems, the wavelet transformdoes not always‘‘produce spectrograms which show the frequency content of sounds ~or other signals ! as a function of time in a manner analogous to sheet music.’’ The analysis in Ref. 1 is indeed accurate and to the point and produces the desired results in the cases described there because the signals analyzed were artificially created; consequently, their pitch and frequency content were known a priori . In phenomena where intensity and frequency content are not knowna priori, such as turbulent flows and other random signals, the wavelet transform often obscures significant information in the signal. As an example, a wavelet that is a second-difference operator, such as the French-hat wavelet, can provide no information on the linear trend in a signal. For the unfamiliar reader, the French-hat wavelet is defined as

An invariant theory approach to the problem of closure

2B(3a)12B(3a21)2B(3a22)%/2AL, where a5t/L and B(a) is the standard box function. That is, B(a)51 for 0<a<1, andB(a)50 otherwise. The wavelets problem for practitioners studying random turbulence is much like the problem posed by the professor in a chemistry class when she gives a beaker of liquid to a group of students and asks them to identify its contents. The students conduct a variety of tests which are known to reveal specific elements or compounds and then report their results. If the liquid contains elements for which there are no known tests, or if the students neglect to conduct a certain test, the students cannot identify all the contents. The same problem exists in identifying computer viruses—you can identify only the knownones. In Ref. 2 we demonstrate that wavelet analysis has limitations which are not widely appreciated; failure to recognize these can lead to misinterpretations. Reference 1 correctly points out the limitations of both the short-time Fourier transform and the Gabor transform, as well as the limitations the uncertainty principleimposes on both the Fourier transform and the wavelet transform. But the limitations on wavelet analysis we emphasize in Ref. 2 are just as fundamental. There we applied wavelet analysis to nonstationary turbulence data, but our results apply to any random signal in general. Our main point is that a given signal may contain components that are orthogonal to the analysis ~mother! wavelet; consequently, for a wavelet analysis to be viable, the analysis wavelet must be carefully matched to the phenomenon of interest. That is, you must have some a priori idea as to what scale elements are present in the signal and which wavelets are best suited for isolating them. Moreover, most wavelets are symmetric about the localization time~usually denoted as t0! and therefore assign the same weight to those elements of the signal forward in time from t0 by an amount as they do to those elements backward in time from t0 by an equal amount. In phenomena such as turbulence, where energy dissipation and its companion irreversibility are commonplace, such an assignation is plausible whent is small but cannot be a reliable characterization of the behavior when t is large; see Refs. 3, 4. One of the limitations Lang and Forinash ~Ref. 1! point out with respect to the Wigner distribution V

Current topics in nonstationary analysis : proceedings of the second workshop on nonstationary random processes and their applications

f (t,v)%—that a spectrogram based on V

Three-point velocity correlation for grid-turbulence

f (t,v)% will show interference artifacts or noise in regions where none should be—is not really a limitation at all. In their analysis it does produce the artifacts indicated, but in turbulent signals such artifacts are nonlinearities that appear routinely in signals encountered in nature. The source of the confusion is that V

Identifying Nonstationarity in Turbulence Series

f (t,v)% is denoted as theWigner distribution of f , the implication being that frequencies revealed by V