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 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 benefits of introducing **Varimax Rotation** into the ** M-SSA **analysis on a low signal-to-noise multivariate dataset. The synthetic test series consists of 6 spatial channels, each with 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 (see Matlab plotting scripts included with the data set).

To read
the data we use the **Read Matrix** function from the **File/Data** menu on the main panel, which places the** input file data** into a matrix with a default name **"mat"** of **130x6** size.

First, we perform analysis using **standard MSSA** without **Varimax Rotation**. Selecting the **`MSSA'** option from the **Analysis
Tools **menu on the main panel launches the following window, showing its state after pressing ** Get Default Values ** button, and specifying the parameters as described
below (leave for now **Varimax box unchecked**):

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

Here the eigenspectrum and red-noise error bars are plotted against the dominant frequencies associated with each **ST-EOF.** The **8 eigenvalues **above error bars **correspond to 4 oscillatory modes detected as signficant** by **MSSA using approximate Chi^2 test (as an exercise for the user apply more precise Monte-Carlo test to confirm these results)**. 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 **Log File**:

Notice a triplet at **f=0.357
** (**ST-EOFs** with rank # **5, 7** and **9**) which is a clear manifestation of mode mixing with standard **MSSA**, as discussed above. We will focus here on time-domain reconstruction of low-frequency modes, **f=0.165** (**ST-EOFs** with rank **# 3 and 4**) and **f=0.133** (**ST-EOFs** with rank # **8 and 10**) using the **Reconstruction** pull-down menu in the **MSSA** window as below, and store it in matrices** rcmat1** and **rcmat2**, respectively.

Now we repeat the **MSSA** analysis wirth **Varimax Rotation** turned on (**box checked **on main panel) which will be applied to the **leading MSSA components **(**10** in this case as specified on main **MSSA** panel), and obtain following **MSSA **spectral estimate:

Note that **Varimax rotation** removed mixing between high-frequency oscillatory modes, as also evident from the **Log file**:

Next, we perform **Reconstruction** of low--frequency modes at **f=0.133** (**ST-EOFs** with rank # **8 and 10**) and **f=0.165** (**ST-EOFs** with rank **# 3 and 4**) as above, but store them into** rcmat1var** and **rcmat2var** matrices, respectively. Using **File/Data ->Write Matrix** menu option, we write **rcmat1, rcmat2, rcmat1var and rcmat2var **into files with the same names.

By using extrenal **2-D **plotting package (**Matlab** in this case), we can compare these reconstructions with reference data (below). The **Varimax Rotation** restored both amplitude modulation and phase shift between the channels much better then standard **MSSA**.