g. Multi-Taper Method

If you pressed the `Multi-Taper Method' button, new menu window should appear resembling Figure 1.

Figure 1. Control panel for Multi-Taper Method.

There are three (3) options that are changed by the user explicitly entering chosen values of parameters:

Selecting the `Frequency Range' button will allow you to change the frequency range from its default value f=0 to f=f_Nyquist=0.5/dt (the Nyquist freqency, where dt is the sampling interval of the data read in and equal to dt=1 by default if no sampling interval was provided). Either the lower `from' frequency f1 or upper `to' frequency f2 can be independently changed by entering the appropriate values in the boxes below. Selected frequencies must fall within the original full default 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.
Entering a new number in the `Resolution' box resets the resolution half-bandwidth delta f = p f_Rayleigh (where f_Rayleigh is the minimum possible spectral resolution) from its default value p=2. Changing this choice will automatically change the default value of `# of tapers' to its maximum value 2p-1 (see below), admitting a lower-variance spectrum estimate at the cost of decreased spectral resolution. The largest value that can be entered is p=8.
Entering a new value in the `# of tapers' box resets the number of windowing functions used in spectral estimation. As indicated above, 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 `# of tapers' if the user wants to be especially conservative regarding potential spectral leakage bias.

There are six (6) clickable buttons, some of which may not be active depending on what actions have been taken:

Clicking the `OK'button indicates that you are happy with the present settings, and are ready to proceed with calculation (or recalculation if parameters have been changed since a previous execution of `OK') of the spectrum. Upon the selection of `OK' the spectrum will be calculated and displayed. It may take several seconds for the calculation of the spectrum. The spectrum is plotted in a new graphic window and should look something like Figure 2.
.

Figure 2. MTM spectrum for `soi' (. X-axis units are cycles per month, and y axis is spectral density .
Selecting the `Reconstruction...' button allows the user to perform time reconstructions of any of the signals identified as signficant in the spectrum analysis. A menu contains an itemized list of the central frequencies of narrowband signals with associated statistical significanes for signals significant at greater than the 90% level relative to the specified null hypothesis (a maximum of the 40 most significant signals are stored). The user can select one or more of these signals for reconstruction, or a group of them to be silmultaneously reconstructed and summed, by clicking on the appropriate frequencies. The selected signals will be highlighted in the list. To select a group of signals, hold down the control key while selecting each one. The group of selected signals should appear highlighted
The `display'button in the menu will produce a plot of the reconstructed signal along with the raw data, while the `save' button will allow the user to save the reconstructed signal as a file.
Selecting the `Save' or `Save As...' buttons allows the user to save the spectrum in a file.

There are four (4) menu items below the `MTM preferences' heading which allow the user to change various settings from their defaults:

Selecting the ``Null hypothesis'' menu item will produce a list of possible choices which can be selected. This includes several subgroups each from which one choice is selected:

The first subgroup of options is:

red noise

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.

The second subgroup of options is:

robust

raw

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.

The third subgroup of options represents a choice between the two options:

log fit

linear fit

AND a choice of the item

median smoothing

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.

The ``median smoothing'' option is enforced for the ``robust'' noise background estimation and is thereby a default. The default is selection of 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. This value can be varied by the user within the range 2 p f_Rayleigh to 0.25 f_N. As with the choice of ``log fit'' and ``linear fit'', any results which show great sensitivity to the value of the median smoothing window f_smooth over that range should be interpreted with some caution.

Selecting the ``Signal Assumptions'' menu item will produce a list of (3) mutally exclusive choices which can be selected.

The choice ``either'' indicates that the spectrum should

for the vertical axis of the displayed spectra:

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. 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.

Selecting the ``Display'' menu item provides a list of different conventions for displaying the results of the spectral analysis procedure:

The first group represents two mutually-exclusive choices for the vertical axis of the displayed spectra:

log scale

linear scale

The choice ``log scale''indicates that spectra should be plotted by the conventional log(spectral density) vs frequency dependence, and is the default setting.

The choice ``linear scale'' conversely plots results using a linear vertical axis vs. frequency. This convention is more sensible for displaying the results of the ``harmonic peak test'' (see below).

The second group represents the choice of spectral analyses or diagnostics to be displayed:

reshaped vs. harmonic spectrum

raw spectrum

harmonic spectrum

reshaped spectrum

harmonic peak test

Note that the first 4 choices share common spectral density units and can be selected silmultaneously, while the 5th display item exhibits variance-ratio units and cannot be displayed silmutaneously with the other choices.

The choice ``reshaped vs. harmonic spectrum'' indicates the the estimated continuous MTM spectrum or ``reshaped spectrum'' (the spectrum with estimated contributions of harmonic signals removed) should be shown along with the harmonic component of the spectrum. This is accomplished by showing the estimated continous spectrum by a solid curve, along with a dashed 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. 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. This is the default display item.

The choice ``raw spectrum''indicates that the raw MTM spectrum should be displayed. 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, as a function of frequency.

The choice ``reshaped spectrum'' indicates that the reshaped spectrum (see above) should be shown on its own, along with confidence intervals.

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.

Selecting the ``Spectrum'' menu item provides a list of choices regarding the details of how the MTM spectrum is estimated:

The first group represents a choice between two methods of estimating the MTM spectrum:

adaptive

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.

The 2nd choice represents a choice between three mutually exclusive possible normalization conventions of the spectrum:

none

1/dt

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. This is the default convention.

The choice ``N'' represents a convention in which the spectrum is calculated per unit time by dividing by the length of the dataseries in in time units ``N'' (ie, a power spectral density).

The choice ``1/dt'' represents a convention in which the spectrum is scaled by the sampling interval (dt), approriate for the spectral estimate of a presumed time-limited signal such as a pulse.

The 3rd item regards the ``reshaping procedure'':

The user indicates whether or not the ``reshaping'' procedure (used to separate the continuous and harmonic portions of the spectrum) should be performed. The procedure is performed by default. The user also selects a treshold for significance of harmonic peak detection (90%,95%,99%) in the reshaping procedure. The default setting is 95% significance. 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.

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URL: users.guide.mtm.html
This file last modified: May 5, 1997