# SSA example: Low Signal-to-Noise ratio

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'. 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 ValuesCompute and Plotbuttons, 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 theCompute 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-squaredquantitative 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.