# 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 Frequency from' .. To' .. ' values allow one to change the frequency range from its default value f=0 to f=f_Nyquist=0.5/dt, where dt is the sampling interval entered by the user. Selected frequencies must fall within the Nyquist range or a warning is given. The selected frequency range will determine the frequency interval over which other options (e.g., median smoothing, robust red noise fit--see below) will operate. The number of frequencies over the interval f=0 to 0.5/dt is the first power of 2 greater than the number of datapoints, or 1024--whichever value is larger. The latter choice provides a visually smooth interpolated spectrum when the number of datapoints is small. A selected subrange will contain a proportionately smaller number of frequency points.

Please note that the null hypothesis is effectively re-determined whenever sub-interval of the spectrum is analyzed. In this case an AR(1) spectrum is fitted only to that range, which will in general change the confidence levels, and thus, potentially, the threshold for "reshaping" of the spectrum, and detecting a harmonic peak!

• Entering a new number in the Resolution' box resets the resolution half-bandwidth delta f = p f_Rayleigh (where f_Rayleigh=1/(N*dt) is the minimum possible spectral resolution) from its default value p=2.
• Entering a new value in the Number of Tapers' box resets the number of windowing functions used in spectral estimation. This value cannot exceed the 2p-1 where p is the integer entered in the Resolution' box. A lower value can however be set for Number of Tapers' if the user wants to be especially conservative regarding potential spectral leakage bias.

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''

• red noise (default)
• locally white
• white

The choice of red noise''assumes a noise background that consists of a temporally integrated Gaussian white noise or AR(1)'' noise process. This null hypothesis is strongly motivated for dynamical reasons in the study of geophysical phenomena, and represents the default option of the Toolkit.

The choice of locally white noise'' assumes a colored noise process that varies slowly but arbitrarily with frequency. This choice is recommended if there is a priori reason to believe that the noise background has a complex structure.

The choice of white noise'' represents a good null hypothesis if absolutely nothing is known a priori about the physics or dynamics of the process producing the noise background.

There are three possible Signal Assumptions''

The choice either'' indicates that the spectrum should be tested both for the presence of narrowband signals whose significance is measured by their amplitude relative to the estimated noise background, and for the presence of harmonic'' signals which are significant as measured by the Thomson variance ratio test for periodic signals (F-test). This is the default choice.

The choice narrowband'' will test the spectrum only for the presence of narrowband signals, the former of the two possible signal detection procedures described above.

The choice harmonic'' will test the spectrum only for the presence of periodic signals, the latter of the two possible signal detection procedures described above.

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

• high-resolution

The choice adaptive'' indicates that the adaptive MTM spectrum estimation procedure, most resistant to broadband spectral leakage, is to be employed. This is the default option.

The choice high-resolution'' indicates that the high-resolution MTM spectrum, involving a simple weighted average of the contributions of independent eigenspectra, should be calculated.

There are two possible Normalization conventions of the spectrum:

• N (default)
• none

The choice N'' represents a convention in which the spectrum is calculated per unit time by dividing by the length of the data series in time units N'' (ie, a power spectral density). This is the default convention, as it is throughout the Toolkit.

The choice none'' indicates a standard Fourier convention of a finite length time series in which the spectrum scales with the number of data points.

The Reshape Threshold'' menu:

Here the threshold for significance of harmonic peak detection (90%,95%,99%,99.5% and 99.9%) in the F-test'' can be changed from its default setting of 95%.

A Reshaping'' procedure is used to separate the continuous and harmonic portions of the spectrum. When the Signal Assumption option is set to Either, the detected harmonic peak will be reshaped only if it is ALSO signficant to the estimated noise background at a Reshape Threshold level.

For the Harmonic'' option all F-test significant peaks will be reshaped. The Reshaping'' procedure is not performed if Narrowband option is selected.

The user should be aware that chance statistics should lead to a 5% rate of spurious harmonic signal detection at the 95% level, corresponding to roughly 2-3 false positives over the Nyquist interval for a timeseries of length N=100 time units, underscoring Thomson's (1982) general warning that one should interpret with great caution significances of less than about 1-1/N.

The Noise Estimation'' menu allows two choices of the way in which the noise background is estimated:

The robust'' noise background choice employs a robust estimator (median smooth) to find an optimal background fit to the empirical spectrum for measuring the signficance of narrowband signals. This choice guards against the contamination of noise paramater estimation by narrowband signal and significant trend contributions, and represents the default option.

The raw'' option estimates noise parameters directly from the unfiltered or raw'' time series. This option, albeit more traditional, is generally discouraged. It should be selected, however, if the user is interested in signals whose significance is measured only by the Thomson variance ratio test for periodicity, without regard to their significance as measured by their amplitude relative to the estimated noise background.

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.

The log fit'' employs a criterion which minimizes the misfit of the robustly estimated background with the log of the spectral density. This provides a better conceptual fit to the spectrum when dynamic ranges are large, and is the suggested choice. It is the default option.

The linear fit'' employs a criterion which will weight the robust fit of the noise background by the amplitude of the spectrum. This choice is recommended if it is more important to fit the low-frequency, high-power part of noise background.

Any results which show great sensitivity to the choice of log fit'' and linear fit'', or to the value of the median smoothing window f_smooth over that range should be interpreted with some caution.

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.

The first 4 choices share common spectral density units and can be selected simultaneously, while the harmonic peak test'' has variance-ratio units and cannot be displayed simultaneously with the other choices.

The choice raw spectrum''indicates that the raw MTM spectrum should be displayed. Four additional smooth curves are shown, in increasing vertical progression, for the median, 50%, 90%, 95%, and 99% significance levels relative to the estimated noise background.

The choice reshaped'' indicates the estimated continuous MTM spectrum or reshaped spectrum'' (the spectrum with estimated contributions of harmonic signals removed). As above, signficance curves are also shown.

The choice harmonic spectrum'' indicates that the estimated harmonic or periodic component of the spectrum should be displayed. If the reshaped spectrum'' is also checked, the plot shows the estimated continous spectrum, along with a curve that rises above it to indicate the portion of the spectrum above the continuous background associated with a period signal. The latter component is shown as a narrow spike of breadth equal to the spectral estimation bandwidth.

The choice smoothed'' will plot the median-smoothed spectrum, if it was computed.

The choice harmonic peak test'' indicates that the variance-ratio test for significance of harmonic signal detection should be shown as a function of frequency. Also shown are the median, 90,95,and 99% confidence levels for significant detection of a periodic signal relative to the assumption of locally-white noise.

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 significant 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 theharmonic 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 clickMake 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.