## 3. TOOLKIT DEMONSTRATION

##
f. A Very Low Signal-to-Noise Dataset

We assume that the reader is already familiar with the main SOI demo example.
To demonstrate Toolkit capabilities to detect amplitude and phase-modulated
oscillations with a very low signal to noise ratio,
we use here the
test dataset of Allen and Smith (1996). This synthetic test series consists of randomly-generated damped oscillations bursts superimposed on large amplitude AR(1) noise.
The period of the oscillations is 5.5 units, which corresponds to the frequency f=0.181 .
The first column of the file contains the resulting time series, which is a sum of the
noise (the second column)
and oscillatory signal (the third column).
To read this dataset, we use the **'Read Matrix'** function from the **File/Data**
pull-down menu. We specify the name of the matrix to store the data in as - 'mat'.

*Figure 18. Read Matrix*

We can plot the matrix 'mat' using the **'Matrix'** function of the **Plot**
pull-down menu on the main window.
First, we select the name of the matrix 'mat' from the list (Fig.19), using the
arrow to the right of **'Select Matrix** box in Fig 20.

*Figure 19. Matrix List*

and then specify the columns of the matrix to plot.

*Figure 20. Plot Matrix*

After pressing the **Plot** button, the following graphics window is launched.

*Figure 21. Test Dataset*

Here the black line is the first column of the dataset - the combined data signal (see above). Our
task will be to identify weak green oscillation signal in this data.
First we use the **'Matrix/Vector'** function in the **'File/Data'** pull-
down menu to load the first column of
the matrix into the vector 'data'.

*Figure 22. Matrix/Vector*

Selecting the **Multi-taper Method** from the ** Analysis Tools** on the main
window, and after clicking **Get Default Values**, **Compute** and
**Plot** buttons, the MTM Spectrum of the data is plotted:

*Figure 23. MTM Analysis *

We see that MTM correctly isolates a significant
oscillatory signal (red line ) at the correct frequency ~ f=0.18. See
the MTM Demo for the details of
of the MTM analysis.
To confirm our findings we use the **SSA** analysis tool.
After clicking the **Get Default Values** button, we change
**SSA window** to 40, check the **Strong FFT** and **Same Frequency**
pairing options in **Test Options** window, and then click the
**Compute** and
**Plot** buttons. We obtain the following SSA eigenvalue spectrum using the **Error
Bars ** significance test:

*Figure 24. SSA spectrum *

The pairing tests in the **LogFile** show that pairs 1,2 and 8,9 are
candidates for oscillatory signals.
To test further, we choose the **Chi-squared** quantitative significance test, and
test against a pure red-noise null-hypothesis. We obtain the following plot:

*Figure 25. SSA: Chi-Squared Test*

The results of both the 'Data Eofs' and 'Null-hyp. Eofs' bases on the above plot
indicate significantly elevated variance in EOFS 8 and 9, which
form the pair at close to the same frequency as the weak oscillatory
signal in our dataset.

As an excercise to the reader, the components with EOFS 8-9 can be
reconstructed, and then passed to the MEM tool to check the frequency.
Combining the results of MTM and SSA analysis, we conclude that our
data contains an oscillatory signal at f=0.18.

##
A FINAL CAUTION FROM THE AUTHORS:

While the simple graphical interface provided by the Toolkit makes it easy
to apply sophisticated techniques in a black-box manner, users are urged
to fully understand the underlying theoretical assumptions to avoid misapplicati
on.
No test is proof that an oscillation exists, and the line between noise
and signal can be subjective and transitory.

*This file last modified: 9/25/03*