As an example, we apply **MSSA** to the near-global data set of monthly sea-surface temperatures (SSTs) anomalies data set for **1950-2004, from 30° S to 60° N, on a 10° latitude x 10° longitude** grid (International Research Institute for Climate and Society (IRI), see **sst_mssa.tkt** project in Examples/Oceanography folder)

This translates into a dataset with **360** channels (**36 longitude x10 latitude**), **648** months long. The channels with **NaN** values in all rows indicate land mask, and there are** 247** channels with actual data. The input data is a matrix with a name **"sst", **which has **648** rows (time-coordinate) and** 360** columns (space coordinate). To setup a **2-D** grid for display of the spatial coordinate, we factor number of columns (**cols**) and set **col1** and **col2 **values for **sst **matrix as** 36 **and **10**, respectively.

Then, all relevant **MSSA (PCA) **results for this matrix will be obtained and can plotted using **2-D** grid for spatial coordinate. For **Fill** or **Contour 2-D** plots, such dataset can be plotted in **col1-col2**, **row-col1** and **row-col2 **coordinates**. **For ** Fill **and **col1-col2 **options** **we obtain the following plot**: **

Then we can browse through the 3rd dimension of this dataset -- **row (time), **by using **Graph Controls/Axes/2D** option:

Selecting the **`MSSA/PCA'** option from the **Tools **menu on the main panel launches the following window:

Having selected the data to be analyzed (here `**sst**') and the sampling interval, user can choose* analysis* type, i.e.

Main options for **MSSA** are temporal **Window Length**, type of **Significance
Test**, **Covariance **method, and number of spatial** EOFs **channels (for **PCA->MSSA**). The number of ** Components** specifies how many **MSSA** components will be retained for **Reconstruction/Prediction**. Results of** PCA**/**MSSA **are stored in matrix with a name specified in **Spectrum** field. In addition,** T-EOFs**, ** ST-EOFs** and **T-PCs** are stored in matrices with names obtained by prefixing "**eof**_", "**steof**_" and "**pc**_" to a **Spectrum **name,** **and can be accessed in** Data I/O **tool.
If results from several** **analyses have been stored in different matrices, the parameters used in a particular computation will be restored in GUI by simply selecting correspondent matrix from a **Spectrum** pop-up list. **Get Default Values **button is provided as a guide.

In general, it is better to choose the **Reduced** **Covariance** estimator for data with many spatial channels *L* and large temporal window *M, *so when *ML* > *N*' it is more efficient to diagonalize the *reduced* *N*' x *N*' *covariance matrix*, rather than the *LM* x *LM*

**Varimax Rotation** has been introudced initiallyfor better physical interpretation of EOFs from PCA. Recently, Groth and Ghil (2011) have demonstrated that a classical M-SSA analysis suffers from a degeneracy problem, with eigenvectors not well separating 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 ST-EOFs. To avoid a loss of spectral properties (Plaut and Vautard 1994), they have introduced a slight modification of the common **varimax rotation** that takes the spatio-temporal structure of ST-EOFs into account.

V**arimax rotation** is applied to leading number of **MSSA or PCA components** specified in **Components **field if the V**arimax Rotation** box is checked. **MSSA** with **Varimax Rotation** is demonstrated on Multivariate Small Signal example.

For the example below we don't apply **varimax rotation** so we leave the box unchecked as above.

To demonstrate the usage of ** N'-windows** vs.

**None****Monte Carlo****Chi-squared**

Choosing ** Chi-squared** in the **Significance tests** menu, for `**Reduced**` covariance and ** M=N'=324** we obtain the following plot:

Here the data and red-noise error bars are plotted against the dominant frequencies associated with each MSSA mode. The dominant frequencies of MSSA modes are computed only for the ** Monte-Carlo** or approximate, but much faster,** Chi-squared** test. The frequencies and variances captured by each mode are displayed in a components table of **Advanced** options panel.
For ** M=N'=324** and '

The quasi-biennial pair (modes 13 and 14) 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 will use the eigenmodes provided by the covariance matrix of the **AR(1) **process. So, we select **AR(1)** instead of **Data** basis in **MSSA** **Test Options** of **Advanced Options** panel:

, and repeat computation 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 0-0.07 cy/month frequency interval of interest. 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(~0.033 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 maximum spectral resolution of 1/*M* = 1/*N*' =1/324 months.

It is useful to verify the above results with the **Monte-Carlo **test, using randomly generated red-noise surrogates**. ** For large datasets such computation may take a long time**. **

In contrast to `**Reduced` **covariance**, **when using `**Broomhead &King` **option** **and** M=60**, both the quasi-quadrennial (modes 2 and 3 in this case) and quasi-biennial (mode 7 and 8) pairs appear to be well resolved and significant:

The quasi-quadrennial pair accounts ~ 23% of variance of input data, while less energetic quasi-biennial accounts for ~ 7%.

We can check whether a pair of selected MSSA modes are indeed in phase quadrature and form oscillatory pair, by plotting their **PCs**, *and/or* inspecting their** EOFs**. We can do so by selecting rows from** MSSA components** table, setting a **plot **option, i.e.** 1-D or 2-D (Fill or Contour) **at the bottom of **Advanced MSSA** panel and hitting a **Plot PCs **or** Plot EOFs** button.

The following plot is for the **EOF **of** 2nd MSSA component (**`**Broomhead &King` **option** **and** M=60) **using

This plot shows how a **single****MSSA** **EOF** is * expressed* in

We can see that the 3rd mode is mostly energetic in the 1st channel as well, by selecting 2nd and 3rd row of** MSSA components** table, setting **Plot Options** as above and hit **Plot EOFs** button**: **

The associated PCs are also in phase quadrature, as expected:

For overview of prediction **Options** (**AR order** and **Lead**) see MSSA prediction.

We can reconstruct contributions from selected **MSSA** components in **Components** table of **Advanced** panel in original** Grid** or

The name of matrix with reconstruction is specified by the user in** Result **box at the bottom of

* *

If **2-D** **Plot** option is chosen (**Contour** or **Fill**), reconstructed data *only* will be plotted in** 2-D** time-space coordinates; the figure below shows contribution of the oscillatory pairs in all input PCs from** PCA**, using **1-D** **plot **and** PCA reconstruction **options:

The figure below shows reconstruction of original gridded data (**Grid** option), displayed with a ** 2-D** **Fill **plot** **option. Since the spatial dimension of **gridded** input data has been **2-D** factored (i.e **x*y** or *c ol1*col2*) using

Then we can browse through the **Z** dimension of this dataset --** time, **by using slider, stepper or text field in **Graph Controls/Axes/2D**:

The **Movie** button makes a QuickTime movie animation of the **2-D Active Plot** along the dataset's Z dimension (here time) and exports it to a file using **Save As** panel.

Linear transformation can be applied to the **Z** dimension of **2D** plots in **Axes/2D** settings. Parameters of the transformation has to be set first. To return to the default settings, use transformation factors **1.0 and 0.0 (**as** **in** Z=1*Z + 0), or Defaults **button**. **Above panels show how the transformation is applied to have time in calendar years, instead of months. User can also adjust the **Max** and **Min** values of the plotted field in **Graph Controls/Axes/2D.**