In this synthetic example we follow **msignal.tkt** project in **Examples/Multivariate Small Signal **folder of **kSpectra** **Examples**.

Recently, Groth and Ghil (2011) have demonstrated that a classical **M-SSA** analysis suffers from a degeneracy problem, namely the eigenvectors (also called empirical orthogonal functions or EOFs) do not separate well between distinct oscillations when the corresponding eigenvalues are similar in size. This problem is a shortcoming of principal component analysis in general, not just of **M-SSA** in particular. In order to reduce mixture effects and to improve the physical interpretation, Groth and Ghil (2011) have proposed a subsequent **Varimax Rotation** of the **MSSA ST-EOFs**. of the spatio-temporal EOFs (**ST-EOFs**) of the **M-SSA**. To avoid a loss of spectral properties (Plaut and Vautard 1994), they have introduced a slight modification of the common **varimax rotation** that does take the spatio-temporal structure of **ST-EOFs** into account.

Here we demonstrate** MSSA **analysis with **Varimax Rotation** on a low signal-to-noise multivariate dataset:

The synthetic test series consists of 6 spatial channels, eachwith a length of 130 data points; this series represents a sum of **4 distinct spatio-temporal oscillatory modes**, contaminated by large-amplitude temporal red noise. These modes account for similar variance, and **MSSA **with **Varimax Rotation** helps their detection and reconstruction without mixing**. **Each of the oscillatory modes has a distinct frequency in time (from a low-frequency of **f=0.133** to a high-frequency of **f=0.435**); they also have varying amplitude and phase shift across the spatial channels:

Our task will be detect and reconstruct these ** signal **modes by using

We start **kSpectra**, and go to **Data I/O** in **Tools**. Using **Finder**, we simply double-click **msignal.tkt** file in** MSSA Prediction** folder. It contains **x **(full data). Then we go to **PCA/MSSA** in **Tools**, select** x**** **from **Data **Pop-up menu, change the **Window **value to **M=25**, select **'BK' **method** **for** Covariance, MSSA **tool option**, Chi-squared** as signfi ance test, check **VarImax Rotation** box and set name in **Spectrum** box to **mssa**:

Then click **Compute**, followed by **Plot,** to obtain **MSSA** spectral estimate:

Here the eigenspectrum and red-noise error bars are plotted against the dominant frequencies associated with each MSSA mode. The **four pairs **above error bars **correspond to 4 oscillatory modes detected as signficant** by **MSSA**. Note that these modes have similar eigenvalues, and therefore **varimax rotation** is expected to be helpful.

The frequencies and variances captured by each mode are displayed in a components table of **Advanced** options panel.

Select a pair of rows corresponding to particular oscillatory pair in **MSSA components** table** (8 and 10 for Mode 1; 3 and 6 for Mode 2; 7 and 9 for Mode 3; 4 and 5 for Mode 4), set a name in Result section, **and then click **Compute. **To** **see **2-D** plot of **Reconstructed mode ***in all channels***, **choose** 2-D Fill Plot **option as above**, **and** **click **Plot **in** Result **section to obtain Figures below. The oscillatory modes and their modulatoin, both in amplitude and phase shift between the spatial channels, is reproduced reasonably well (compare with reference modes above):

As a comparison,** MSSA without Varimax rotation **(repeat steps above but leave **Varimax Rotation** box unchecked) results in a similar looking spectrum except that **ST-EOFs** of low-frequency modes are now mixed (notice frequency shift of **ST-MSSA** mode from **f=0.165** to **f=0.133** resulting in a triple):

Reconstruction of the two low-frequency modes (**Mode 1 and 2**) invetably results in a very mixed and modulated patterns that don't resemble much reference patterns (see above), and the high-frequency modes (**Modes 3 and 4**) are negatively impacted as well.