As an example, we apply M-SSA to the near-global data set of monthly sea-surface temperatures from the Global Sea-Ice and Sea-Surface Temperature (GISST) data set *Rayner et al.-1995* for 1950-94, from 30S to 60N, on a 4-latitude by 5-longitude grid.

This translates into a dataset with 1295 channels, 540 months long. To read the GISST data we use the **Read Matrix** function from the**File/Data** menu on the main panel, which places the data into a matrix with a default name **"mat"** of 540x1295 size.

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):

Having specified the data to be analyzed (here `mat', the GISST time-series) and the sampling interval, the principal MSSA options to be specified are the **Window Length**, the type of **Significance Test**, **Covariance**, preprocessing **PCA** switch, and number of **PCA channels**(if necessary). The number of **MSSA Components** specifies how many components will be retained for future analyses. The **Get Default Values **button is provided as a guide.

### Window length

For comparison with previous ENSO studies and to demonstrate the usage of *N*'-windows vs. *M*-windows [see Eq. (11) and Fig.1, for the definition of ``complementary windows''], we set *M*=480, which yields an *N*'-window of 5 years, and *M*=270, which yields the *M*- and the *N*'-windows equal both to *N*/2.

**Note!**: The value in the **Window Length** is taken to equal to *N*' if **`Reduced'** option is chosen for **Covariance**, and *M* for other **Covariance**choices.

### Preprocessing with PCA

We set the **pre-PCA** switch to **Yes**, and the**No. of PCA channels** to **10** to prefilter data with standard PCA *Preisendorfer-1988* to retain the 10 leading spatial PCs that describe 55.2% of the variance, as we can see from the Log file:

This favors the association of larger decorrelation times with larger spatial scales, as expected for climatic *Fraedrich and Boettger-1978* and other geophysical fields, and the channels are uncorrelated at zero lag. After PCA the input data for MSSA has *L*=10 channels and length of*N*=540 months, and it is stored in the list of matrices as **SPC**; correspondent spatial EOFs are stored in a matrix **SEOF**

### Covariance Estimation

The method for estimating the lag-covariance matrix that is decomposed (diagonalized) by M-SSA is chosen by selecting either** `Reduced'** ,**`Vautard-Ghil'** , or `** Broomhead &King** ' from the **`Covariance' **menu on the main M-SSA control panel (Fig. 11).

We choose **Reduced** for the **Covariance** estimator, since in all cases *ML* > *N*' so that it is more efficient to diagonalize the *reduced* (N'xN') *covariance matrix* with elements given by Eq. (11), rather than the (MLxML) matrix whose elements are given by Eq. (6) (`** Broomhead &King**), or Eq. (1) **`Vautard-Ghil'**.

We set the **Window length** to 60 and 270 (**N'** for such option, see above).

### Significance test

There are three choices for **Significance test**

- None
- Monte Carlo
- Chi-squared

described below. If **None** is selected, the eigenspectrum is displayed in order of eigenvalue rank. The leading oscillatory pair over the entire domain has a quasi-quadrennial period for all values of *M*. The quasi-quadrennial pair (modes 2 and 3 in this case ), accounts for less then 30% vof ariance for *N*'=60, as we can see by opening the **Log File** shown in the next figure:

The smaller one of *M* and *N*' determines the approximate spectral resolution 1/*N*' or 1/*M*. Choosing *M*=*N*'=270 yields the maximum spectral resolution.

The dominant frequencies of MSSA modes are computed only in **Monte-Carlo** or apporximate, but much faster** Chi-squared** MSSA test. Choosing **Chi-squared** in the **Significance tests** menu, and *M*=*N*'=270 we obtain the following plot:

Here the red-noise surrogate projections are plotted against the dominant frequencies associated with each MSSA mode, and we have zoomed in on the 0-0.05 cy/month frequency interval of interest. Using *M*=*N*'=270 captures less variance - 12.1% - for the significant oscillatory quasi-quadriennial pair formed in this case by modes 3 and 4, as seen in the **Log File**:

### Monte-Carlo Test Options

The quasi-biennial pair (modes 14 and 15) does not pass the significance test when data eigenmodes are used as a basis for projections. However since the **Monte-Carlo** test is biased if we project onto the data eigenmodes, we project onto the eigenmodes provided by the covariance matrix of the AR(1) noise. So, we select **AR(1)** instead of **Data** option in **Test Options** pull-down menu:

to obtain the following result:

Here, the projections are plotted against the dominant frequencies associated with each noise eigenvector, and we have zoomed in on the 0-0.05 cy/month frequency interval of interest using the xmgr plotting tool. Since the latter are near-sinusoidal in this case, the resulting spectrum is closely related to a traditional Fourier power spectrum. Both the quasi-quadrennial(~0.023 cycle/month) and the quasi-biennial(~0.038 cycle/month) modes pass the test at the 95% level (as specified in **Test Options**). They are well separated in frequency by about 1/(20 months), which far exceeds the spectral resolution of 1/*M* = 1/*N*' =1/(270 months) ≈ 1/(22 years). The two modes are thus significantly distinct from each other spectrally, in agreement with the univariate SOI results using MEM and MTM, respectively.

We leave to a user to verify the above results with the **Monte-Carlo test** with 100 surrogates for *M*=*N*'=270, the maximum effective resolution. **Computation make take a while, so be patient!**

### Plot Options

We can check that ST-EOFs of oscillatory MSSA pairs are indeed in phase quadrature using **Plot Options** pull-down menu:

The following plot is for a quasi-quadriennial pair 3 and 4 and channel 1:

We leave as an exersise to check that a phase quadrature for a quasi-biennial pair (modes 14 and 15) is a mostly prominent for channel 2.

### Reconstruction

We can reconstruct the original multi-channel data or PCA components from selected MSSA modes using the **Reconstruction** pull-down menu in the MSSA window.

We can reconstruct either spatial PCs if PCA-prepocessing has been used or original data: choose **PCA** or **Grid** options, respectively. Here we show reconstruction in PCA space quasi-quadriennial pair for channel 1: