d. Multi-Taper Method

Selecting the `Multi-Taper Method' button from the Analysis Tools menu on the main panel launches the following window (shows its state after pressing Get Default Values button, see below):

Figure 4: MTM window.

Having specified the data vector to be analyzed (here our 'data' vector with the SOI time series) and the sampling interval, there are three MTM parameters that need to be specified. Again, a Get Default Values button is provided as a guide, as well as a means to initialize properly the `Frequency range' and a 'smoothing window width' in MTM Options(see below) for the sampling interval not equal to one.
When changing the values of the sampling interval, the user has to ensure correct scaling of the `Frequency range' and a 'smoothing window width' by changing them manually or by using Get Default Values button.

The tapers themselves can be analyzed using the Tapers pull-down menu, which gives the following window (shows the state after selecting the vector and pressing Get Default Values button):

Figure 5: MTM Tapers window.

This allows one to compute and plot the tapers that are generated for given choices of parameters. 

There are several options in the MTM Options pull-down menu which allow the user to change various settings from their defaults:

Figure 6: MTM options.

There are three possible choices of ``Null hypothesis'' There are three possible ``Signal Assumptions'' 

The ``Spectrum'' menu item provides two alternatives regarding the details of how the MTM spectrum is estimated:

There are two possible Normalization conventions of the spectrum:

The ``Reshape Threshold'' menu: The ``Noise Estimation'' menu allows two choices of the way in which the noise background is estimated: Under ``Robust Settings'', it is possible to modify the way that the ``robust'' noise background is estimated. The default is to select a noise background based on the fit to a median smooth of the raw spectrum with a smoothing window of width f_smooth frequency units that is the larger of either f_Nyquist/6 or the full spectral resolution bandwith 2 p f_Rayleigh. This generally insures that the overall structure of the spectrum over the full ``Nyquist'' interval is recognized in the optimal background fit, while assuring that the fit is resistant to the influence of narrowband features in the spectrum.

The median smoothing window width can be varied by the user within the range 2 p f_Rayleigh to 0.25 f_N. In addition, the ``Misfit Criterion'' can be set to either a log fit or a linear fit

To calculate the spectrum of our SOI series, we set the parameter for the ``Reshape Threshold'' to 90%, and click the Compute button. Then, we click the Plot button to view the spectrum which should look like the following:

Figure 7: MTM Spectrum.

The MTM `Plot Options' pull-down menu on the main MTM panel controls launches the following window with default settings:

Figure 8: MTM Plot Options.

As we can see from the Fig.7 and Components Frequency Selection list of MTM Reconstruction window in Fig.9 (see below), the MTM analysis identifies two highly significant peaks, one centered at f=0.0146, and another centered at f=0.0342. The signals are significant at well above the 99% level. We associate thease peaks with the low-frequency LF(band) and high frequency (HF) band ENSO signals. A weak trend is also identified at f=0.001. The black peaks indicate harmonic signals selected in the Reshaping procedure. There are 8 such peaks including both an LF and HF ENSO band peak. Selecting the `Reconstruction' pull-down menu on the main MTM panel allows the user to perform time reconstructions of any of the signals identified as signficant in the spectrum analysis, via the following window: 

Figure 9: MTM Reconstruction window

If Signal Assumptions option is set to either or narrowband ,the Component(s) Frequency panel in Fig.9 will contain a list of the central frequencies of narrowband signals identified as significant at greater than the 90% level relative to the specified null hypothesis while for the harmonic option it will contain the ``harmonic'' signals which are significant as measured by the F-test for periodic signals. A maximum of the 40 most significant signals are stored. The user can select one or more of these signals using the mouse for reconstruction, and then click Make Selection button to fill in the Selected Frequencies field. Names are required for the ``RCs matrix'' and the ``RC-sum vector''. Upon output, the former contains all the RCs requested, with their sum in the latter vector. The ``Plot'' button plots the RC-sum against the original timeseries, with a mean average removed. Reconstructing the selected SOI frequencies as shown in Fig.9, we obtain the following filtered series:

Figure 10: MTM Reconstructed signal.

We leave to a user to include the trend component in reconstruction, and to observe the difference.

This file last modified: March 31, 2000