Wavelet Methods For Time Series Analysis Percival Pdf 25
Vaniria Setser
This chapter presents a set of tools, which allow gathering information about the frequency components of a time series. In a first step, we discuss spectral analysis and filtering methods. Spectral analysis can be used to identify and to quantify the different frequency components of a data series. Filters permit to capture specific components (e.g., trends, cycles, seasonalities) of the original time series. Both spectral analysis and standard filtering methods have two main drawbacks: (i) they impose strong restrictions regarding the possible processes underlying the dynamics of the series (e.g., stationarity) and (ii) they lead to a pure frequency-domain representation of the data, i.e., all information from the time-domain representation is lost in the operation.
In a second step, we introduce wavelets, which are relatively new tools in economics and finance. They take their roots from filtering methods and Fourier analysis, but overcome most of the limitations of these two methods. Their principal advantages derive from (i) combined information from both time domain and frequency domain and (ii) their flexibility as they do not make strong assumptions concerning the data-generating process for the series under investigation.
As an example, let us consider an economic variable, whose evolution is fully determined by the state of the economy. A complete business cycle lasts on average 36 months and therefore f = 1/36 months.
This paper addresses a bias problem in the estimate of wavelet power spectra for atmospheric and oceanic datasets. For a time series comprised of sine waves with the same amplitude at different frequencies the conventionally adopted wavelet method does not produce a spectrum with identical peaks, in contrast to a Fourier analysis. The wavelet power spectrum in this definition, that is, the transform coefficient squared (to within a constant factor), is equivalent to the integration of energy (in physical space) over the influence period (time scale) the series spans. Thus, a physically consistent definition of energy for the wavelet power spectrum should be the transform coefficient squared divided by the scale it associates. Such adjusted wavelet power spectrum results in a substantial improvement in the spectral estimate, allowing for a comparison of the spectral peaks across scales. The improvement is validated with an artificial time series and a real coastal sea level record. Also examined is the previous example of the wavelet analysis of the Nio-3 SST data.
The rest of this paper is arranged as follows. The bias problem is further illustrated with a real ocean time series in section 2. In section 3, theoretical derivations are followed to shed light on what is underlying a power spectral analysis. A physically consistent definition of energy, and hence an alternate formalism of power spectrum, are proposed, which allows for a solution of the problem. The improvement of the biased spectrum by the new formalism is validated in sections 4, 5, and 6, respectively, with idealized and real world examples. A summary and discussion are then provided in section 7.
Time series of hourly coastal sea level at St. Petersburg, Florida, are from the National Oceanic and Atmospheric Administration/National Ocean Service (NOAA/NOS) ( -ops.nos.noaa.gov) from January 1993 through December 2005. After quality control, the sea level record is de-tided by removing the four major tidal constituents: M2, S2, K1, and O1, using the T_Tide Harmonic Analysis Toolbox of Pawlowicz et al. (2002). It is then 48-h low-pass filtered, 12-h subsampled, and adjusted for the inverse barometer effect. Now the time series contains mainly subtidal sea level variations (Fig. 1, top). The air pressure data are from two NOAA/National Data Buoy Center (NDBC) stations, 42036 and VENF1 (Venice, Florida) ( ), and from University of South Florida surface buoys on the West Florida Shelf. The locations of the sea level and meteorological stations can be found in Fig. 1 of Liu and Weisberg (2005a).
The MATLAB wavelet package of TC98 is used to analyze the subtidal sea level time series. Following the guide of TC98, the wavelet parameters are chosen as follows. The wavelet base function is chosen to be Morlet, which is often used in analyzing geophysical data. A start scale of 2 days is specified since this is the smallest actual time scale for the 48-h low-pass-filtered time series. The spacing between the discrete scales, dj, is chosen as 1/8; that is, there are 8 suboctaves per octave. The total number of scales is determined by both dj and j1, where j1 is the number of the octaves; here it is set to be 9. Thus, there are dj j1 + 1 = 73 scales ranging from 2 to 1024 days. The wavelet transform is converted to wavelet power spectrum as instructed in the sample MATLAB program.
The wavelet power spectrum of the subtidal St. Petersburg sea level is shown in the middle panel of Fig. 1. To our surprise, the spectrum is seriously biased in the frequency domain, so that the annual time scales outperform the synoptic weather time scales so much that the latter seems negligible in the wavelet power spectrum. This is unacceptable for the subtidal sea level fluctuations on the West Florida Shelf (WFS), where the synoptic winds play a dominant role on the inner WFS circulation and hence affect the coastal sea level fluctuation on synoptic time scales (e.g., Marmorino 1982; Mitchum and Sturges 1982; Cragg et al. 1983; Li and Weisberg 1999; He et al. 2004; Liu and Weisberg 2005a, b, 2007; Liu et al. 2007; Weisberg et al. 2005). Thus, the usefulness of the wavelet analysis seems to be doubtful for the subtidal sea level data.
Statistical significance testing is also provided in TC98. In the middle panel of Fig. 1, the black contour encloses regions of greater than 90% confidence for a red-noise process with a lag-1 coefficient of 0.9. It can be seen that these regions include both the synoptic and annual time scales; that is, the wavelet spectra are statistically significant on these two time scales. The biased wavelet spectrum needs improvement, at least visually, to be consistent with the statistical significance test.
To resolve the problem, we need to go back to the definition of energy in functional analysis. A rigorous treatment and detailed interpretation in the context of atmospheres and oceans is referred to by Liang and Robinson (2005). Some of the facts pertaining to wavelet transform are briefly presented in this section.
The wavelet analysis of the St. Petersburg sea level time series is revisited. The power at each point in the spectrum is divided by the corresponding scale, based on the energy definition (8). The result is very encouraging (Fig. 1, bottom). The spectrum values over the synoptic time scales are enlarged and are now comparable to those on the annual time scales; it is also more consistent with the 90% significance level (the black contours in Fig. 1, middle and bottom). This makes more sense in light of coastal oceanography. Over the synoptic weather band the inner WFS ocean circulation is mainly driven by the local winds (e.g., Liu and Weisberg 2005a, b). The synoptic weather winds are seasonally modulated, stronger in winter but weaker in summer half years. These are displayed in the wavelet spectrum as a seasonal modulation of synoptic weather band sea level variation: The synoptic weather band energy is higher and more significant in winter than in summer half years. We also noticed that the wavelet energy on the annual time scales is lower in 1998. This corresponds to an anomalous event of WFS circulation conditions, indicative of a large interannual variation due to the impact of the Loop Current near Dry Tortugas (Weisberg and He 2003; Weisberg et al. 2005). In a word, the rectified wavelet spectrum estimates are now consistent with the physics of the coastal sea level variability on the WFS.
The above formalism of wavelet power spectrum is further tested with an artificial time series composed of sine waves of known frequencies and amplitudes. Five sine waves, with a unit amplitude and periods of 1, 8, 32, 128, and 365 days, respectively, are summed to form an artificial time series as shown in Fig. 2 (top). The sample interval is 1 h, and the length of the time series is 13 yr; both are comparable with those of the St. Petersburg sea level time series. The TC98 MATLAB program is used again, and the wavelet parameters are chosen as those for the sea level analysis with the following exceptions: the start scale is 6 h, and j1 is set to be 12 so that the scale ranges from 6 h to 1024 days, covering the intrinsic periods of the sine waves.
It would be illustrative to check some previous wavelet results to see how Eq. (8) may make a difference. A good case study is the wavelet analysis of the Nio-3 SST data, which was given in TC98 as an example of wavelet analysis. This dataset is also well known in both meteorological and oceanographic communities. More detailed information about the dataset can be found in TC98.
7fc3f7cf58