3. TOOLKIT DEMONSTRATION

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), see Low Signal project in Examples folder of the distribution. 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). The red line in the figure below is the first column of the dataset - the combined data+noise, while the blue is embedded oscillatory signal. Our task will be to identify this weak oscillation signal in this data.

Test Dataset

To read and plot this dataset, we use the Data I/O and Utilities tools. First we load the dataset as Matrix in Data I/O. Second, we use the 'Manage Data' function in the 'Utilities' tool to extract the first column of the matrix into the vector 'data'. Results from Blackmat-Tukey FFT indicate that we need to apply MTM and SSA methods to answer conclusively the question about oscillatory origin of the peak at frequency of 0.18:

Selecting the MTM from the Analysis tool, and after clicking Default, Compute and Plot buttons, the MTM Spectrum of the data is plotted:

MTM Analysis

We see that MTM correctly isolates a significant oscillatory signal (red peak) at the correct frequency ~ f=0.18. See the MTM Demo for the details of the MTM analysis. To confirm our findings we use the SSA analysis tool. After clicking the Default button, we change SSA window to 40, check the Strong FFT and Same Frequency pairing options in Advanced Options window, and then click the Compute and Plot buttons. 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, 'Data' EOFs basis and test against a pure red-noise null-hypothesis by living Include EOFs field blank. After computing we obtain the following plot:

SSA: Chi-Squared Test with 'Data' EOFs

Using 'Null-Hyp Eofs' basis (AR(1) basis option ) confirms that there is 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 exercise 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.