Canonical Correlation Analysis for finding patterns in coupled fields

First CCA pattern of Sea Level Pressure (SLP) and Sea Surface Temperature (SST) monthly anomalies for the region between -180 °W to -70 °W and +30 °N to -30 °S.

The following post demonstrates the use of Canonical Correlation Analysis (CCA) for diagnosing coupled patterns in climate fields. The method produces similar results to that of  Maximum Covariance Analysis (MCA), but patterns reflect maximum correlation rather than maximum covariance. Furthermore, the output of the model is a combination of linear models that can be used for field prediction.

This particular method was illustrated by Barnett and Preisendorfer (1997) - it constructs a CCA model based on a truncated subset of EOF coefficients (i.e. "principle components") instead of using the original field (as with MCA). This truncation has several benefits for the fitting of the model - First, one reduces the amount of noise in the problem by eliminating the higher EOF modes, which represent poorly organized, small-scale features of the fields. Second, by using orthogonal functions, the algebra of the problem is simplified (see von Storch and Zweiers 1999 for details). Bretherton etal. (1992) reviewed several techniques for diagnosing coupled patterns and found the Barnett and Preisendorfer method (hereafter "BPCCA") and MCA to be the most robust.

Exponentiation of a matrix (including pseudoinverse)

The following function "exp.mat" allows for the exponentiation of a matrix (i.e. calculation of a matrix to a given power). The function follows three steps:
1) Singular Value Decomposition (SVD) of the matrix
2) Exponentiation of the singular values
3) Re-calculation of the matrix with the new singular values

The most common case where the method is applied is in the calculation of a "Moore–Penrose pseudoinverse", or a matrix raised to the power of -1. In this case, the function returns the same result as the "pseudoinverse" function from the corpcor package (although maybe not as nicely implemented). This should also be analogous to the pinv function in Matlab.

Less common is the need to calculate other powers of matrices. For example, I use this function to calculate the square root of a matrix (and square root of the inverse of a matrix) within Canonical Correlation Analysis (CCA). I hope to write a post shortly on the use of CCA in identifying coupled patterns in climate data.

Below is the code for the exp.mat function as well as some demonstrations of its use in R.

A ridiculous proof of concept: xyz interpolation

Ridiculous Orb

This is really the last one on this theme for a while... I had alluded to a combination of methods regarding xyz interpolation at the end of my last post and wanted to demonstrate this in a final example.

The ridiculousness that you see above involved two interpolation steps. First, a thin plate spline interpolation ("Tps" function of the fields package) is applied to the original random xyz field of distance to Mecca. This fitted model is then used to predict values at a new grid of 2° resolution. Finally, in order to avoid plotting polygons for each grid (which can be slow for fine grids), I obtain their projected coordinates with the mapproject function. Using these projected coordinates and their respective z values, a second interpolation is done with the "interp" function of the akima package onto a relatively fine grid of 1000x1000 positions. The result is a smooth field that can then be overlayed on the map using the "image" function (very fast).

So you may ask - When is this even necessary? I would say that it really only makes sense for projecting a filled.contour-type plot for relatively sparse geographic data. Be warned - for large amounts of xyz data, the interpolation algorithms can take a long time.

A couple of functions, found within this blog, are needed to reproduce the plot (earth.dist, color.palette).

the code to reproduce the figure...

XYZ geographic data interpolation, part 3

This will be probably be a final posting on interpolation of xyz data as I believe I have come to some conclusions to my original issues. I show three methods of xyz interpolation:
1. The quick and dirty method of interpolating projected xyz points (bi-linear)
2. Interpolation using Cartesian coordinates (bi-linear)
3. Interpolation using spherical coordinates and geographic distances (thin plate spline)

XYZ geographic data interpolation, part 2

Having recently received a comment on a post regarding geographic xyz data interpolation, I decided to return to my original "xyz.map" function and open it up for easier interpretation. This should make the method easier to adapt and follow.

The above graph shows the distance to Mecca as interpolated from 1000 randomly generated lat/lon data using the "interp" function of the akima package. Several functions, found within this blog, are needed to reproduce the plot (pos2coord, earth.dist, new.lon.lat, color.palette, val2col, image.scale). One thing you will notice is the strip of uninterpolated area within the stereographic projection. This is a problem that I have yet to resolve and has to do with the fact that the interpolation is not considering the connection along the 180° longitude line. This will probably require some other type of interpolation based on geographic distances rather than Cartesian coordinates.


R code to produce the above graph...

Maximal Information Coefficient (MIC)

Pearson r correlation coefficients for various distributions of paired data (Credit: Denis Boigelot, Wikimedia Commons)

A paper published this week in Science outlines a new statistic called the maximal information coefficient (MIC), which is able to equally describe the correlation between paired variables regardless of linear or nonlinear relationship. In other words, as Pearson's r gives a measure of the noise surrounding a linear regression, MIC should give similar scores to equally noisy relationships regardless of type.

Maximum Covariance Analysis (MCA)

Maximum Covariance Analysis (MCA) (Mode 1; scaled) of Sea Level Pressure (SLP) and Sea Surface Temperature (SST) monthly anomalies for the region between -180 °W to -70 °W and +30 °N to -30 °S.  MCA coefficients (scaled) are below. The mode represents 94% of the squared covariance fraction (SCF).

Maximum Correlation Analysis (MCA) is similar to Empirical Orthogonal Function Analysis (EOF) in that they both deal with the decomposition of a covariance matrix. In EOF, this is a covariance matrix based on a single spatio-temporal field, while MCA is based on the decomposition of a "cross-covariance" matrix derived from two fields.

Another aspect of speeding up loops in R

Any frequent reader of R-bloggers will have come across several posts concerning the optimization of code - in particular, the avoidance of loops.

Here's another aspect of the same issue. If you have experience programming in other languages besides R, this is probably a no-brainer, but for laymen, like myself, the following example was a total surprise. Basically, every time you redefine the size of an object in R, you are also redefining the allotted memory - and this takes some time. It's not necessarily a lot of time, but if you are having to do it during every iteration of a loop, it can really slow things down.

The following example shows three versions of a loop that creates random numbers and stores those numbers in a results object. The first example (Ex. 1) demonstrates the wrong approach, which is to concatenate the results onto the results object ("x") , thereby continually changing the size of x after each loop. The second approach (Ex. 2) is about 150x faster - x is defined as an empty matrix containing NAs, which is gradually filled (by row) during each loop. The third example (Ex. 3) shows another possibility if one does not know what the size of the results from each loop will be. An empty list is created of length equaling the number of loops. The elements of the list are then gradually filled with the loop results. Again, this is at least 150x faster than Ex. 1 (and I'm actually surprised to see that it may even be faster than Ex.2).

Define intermediate color steps for colorRampPalette

The following function, color.palette(), is a wrapper for colorRampPalette() and allows some increased flexibility in defining the spacing between main color levels. One defines both the main color levels (as with colorRampPalette) and an optional vector containing the number of color levels that should be put in between at equal distances.

     The above figure shows the effect on a color scale (see image.scale) containing 5 main colors (blue, cyan, white, yellow, and red).  The result of colorRampPalette (upper) produces an equal number of levels between the main colors. By increasing the number of intermediate colors between blue-cyan and yellow-red (lower), the number of color levels in the near white range is reduced. The resulting palette, for example, was better in highlighting the positive and negative values of an Emprical Orthogonal Function (EOF) mode.

Empirical Orthogonal Function (EOF) Analysis for gappy data

[Updates]: The following approach has serious shortcomings, which I have recently become aware of. In a comparison of gappy EOF approaches Taylor et al. (2013) [pdf] show that this traditional approach is not as accurate as others. Specifically, the approach of DINEOF (Data Interpolating Empirical Orthogonal Functions) proved to be the most accurate. I have outlined the DINEOF algorithm in another post [link]. and show a comparison of gappoy EOF methods here: http://menugget.blogspot.de/2014/09/pca-eof-for-data-with-missing-values.html. The R package "sinkr" now contains a version of the function ("eof") for easy installation: https://github.com/menugget/sinkr

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The following is a function for the calculation of Empirical Orthogonal Functions (EOF). For those coming from a more biologically-oriented background and are familiar with Principal Component Analysis (PCA), the methods are similar. In the climate sciences the method is usually used for the decomposition of a data field into dominant spatial-temporal modes.