A Practical Guide to Wavelet 2

Wavelets

      In the previous section we saw how one measure of the stationarity of a time series was to calculate the running variance using a fixed-width window. Although we pointed out the disadvantage of using a fixed-width window, one could repeat the analysis with a variety of window widths. By smoothly varying the window width one could then build a picture of the changes in variance versus both time and window width. The obvious problem with this technique is the simple "boxcar" shape of the window function, which introduces edge effects such as ringing. As mentioned earlier, using such a black-box-car, we still have no information on what is going on within the box, but only recover the average energy.

      In Figure 2a we see an example of a wave "packet", of finite duration and with a specific frequency. One could imagine using such a shape as our window function for our analysis of variance. This "wavelet" has the advantage of incorporating a wave of a certain period, as well as being finite in extent. In fact, the wavelet shown in Figure 2a (called the Morlet wavelet) is nothing more than a Sine wave (green curve in Figure 2b) multiplied by a Gaussian envelope (red curve).

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Fig 2.

(a) Morlet wavelet of arbitrary width and amplitude, with time along the x-axis. (b) Construction of the Morlet wavelet (blue dashed) as a Sine curve (green) modulated by a Gaussian (red).

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      Assuming that the total width of this wavelet is about 15 years, we can find the correlation between this curve and the first 15 years of our time series shown in Figure 1. This single number gives us a measure of the projection of this wave packet on our data during the 1876-1890 period, i.e. how much [amplitude] does our 15-year period resemble a Sine wave of this width [frequency]. By sliding this wavelet along our time series one can then construct a new time series of the projection amplitude versus time.

      Finally, one can then vary the "scale" of the wavelet by changing its width. This is the real advantage of wavelet analysis over a moving Fourier spectrum. For a window of a certain width, the sliding FFT is fitting different numbers of waves, i.e. there can be many high-frequency waves within a window, while the same window can only contain a few (or less than one) low-frequency waves. The wavelet analysis always uses a wavelet of the exact same shape, only the size scales up or down with the size of the window.

      In addition to the amplitude of any periodic signals, we would also like information on the phase. In practice, the Morlet wavelet shown in Figure 2a is defined as the product of a complex exponential wave and a Gaussian envelope:

(2.1) . . . . . . . . . .

where Psi is the wavelet value at non-dimensional time eta, and w₀ is the wavenumber. This is the basic wavelet function, but we now need some way to change the overall size as well as slide the entire wavelet along in time. We thus define the "scaled wavelets" as:

(2.2) . . . . . . . . . .

where s is the "dilation" parameter used to change the scale, and n is the translation parameter used to slide in time. The factor of s^-1/2 is a normalization to keep the total energy of the scaled wavelet constant.

      We are given a time series X, with values of x_n, at time index n. Each value is separated in time by a constant time interval dt. The wavelet transform W_n(s) is just the inner product (or convolution) of the wavelet function with our original timeseries:

(2.3) . . . . . . . . . .

where the asterisk (*) denotes complex conjugate.

      The above integral can be evaluated for various values of the scale s (usually taken to be multiples of the lowest possible frequency), as well as all values of n between the start and end dates. A two-dimensional picture of the variability can then be constructed by plotting the wavelet amplitude and phase.

      Figure 3 shows the power (absolute value squared) of the wavelet transform for the NINO3 SST data. The (absolute value)² gives information on the relative power at a certain scale and a certain time. A plot of the amplitude and phase would show the actual oscillations of the individual wavelets, rather than just their magnitude.

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Fig 3.

(a) Time series of El Niño sea surface temperature. (b) The wavelet power spectrum, using the Morlet wavelet. The x-axis is the wavelet location in time. The y-axis is the wavelet period in years. The black contours are the 10% significance regions, using a red-noise background spectrum. The red areas indicate that high El Niño activity occurred during 1880-1920 and 1965-present, while 1920-1960 was relatively calm.

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      Comparing Figures 1 and 3 it is now much clearer that there was large power in the 2-7 year ENSO period during both the earlier and latter parts of this century. In addition we can now see hints of a 16-year oscillation as well as power at even lower frequencies.

      The wavelet transform also gives information on changes in frequency that may have occured. Thus, from 1960-1990 the ENSO time band (2-7 years) seems to have undergone a slow oscillation in period from a 3-year period between events back in 1965 up to about a 5-year period in the early 1980s.