A Deeper Dive into Principal Component Analysis

This post is meant to be a continuation of Dave Gold’s introductory post on Principal Component Analysis, which is an excellent explanation on how to conduct a PCA and visualize the principal components. The goal of this post is to elaborate on how to proceed after you have conducted a PCA and to address some common questions and concerns associated with the method.

Performing a PCA in R

Often times, you will perform a PCA on large datasets that contain many variables and/or many observations. One such dataset that will be used as an example is the Living Blended Drought Atlas (LBDA) which is a reconstruction of the Palmer Drought Severity Index (PDSI) over the contiguous United States from 1473-2005. This dataset contains 4968 columns, or variables, each of which is a grid cell over the U.S., and 533 rows, each of which is a yearly observation. We will call this dataset, X. In matrix notation, we will denote the PCA analysis formula as:

U=XW    (1)

where X is the dataset, W is the weighting matrix, whose columns are the key patterns in the data, and U is the matrix whose columns are the resulting principal components (PCs). You can perform a PCA on this dataset with a single function in R, prcomp.

 PCA=prcomp(X, scale=TRUE/FALSE) 

The first input into the function is your data matrix and the second input is used to declare if your dataset should be scaled to have a unit variance before the PCA is conducted. There are various other inputs into the function, listed here, that can be included if necessary.

prcomp returns three sets of results in a list that we have called “PCA”:

  1. sdev: the standard deviations of the principal components (if you square them, you get the eigenvalues of the covariance/correlation matrix)
  2. rotation: the loading matrix whose columns are the eigenvectors (W in the equation above)
  3. x: the rotated data or your PCs (The columns of U in the equation above)

And that’s it! You have the results of the PCA. Now comes the more difficult part: interpreting them.

How do I choose how many PCs to keep?

The dimensions associated with equation (1) are as follows:

    U=XW

(nxk)=(nxk)(kxk) 

If the number of observations is much larger than the number of variables in the dataset, i.e. n>>k, then the PCA will return k distinct eigenvectors. In our case, since n is smaller than our number of variables, the most non-zero eigenvectors that the PCA will return is n. Either way, we have many supposedly distinct patterns. How do we decide how many of those patterns to keep?

The answer is not always clear and most often subjective and case-dependent. One common tool used is a scree plot or an eigenvalue spectrum.

Scree Plot

Each column of our W matrix is a distinct, independent pattern, also called an empirical orthogonal function (EOF). Each EOF is responsible for explaining some amount of variance in the dataset. A scree plot, shown in Figure 1, allows you visualize this variance breakdown.

Picture1

Figure 1: Scree Plot

 

On the x-axis of the scree plot is the EOF (we’ve chosen to keep 10) and on the y-axis is the total variance explained by that EOF. Each variance is equivalent to the eigenvalue associated with its respective eigenvector. You can find the eigenvalues/variances by squaring of the results of sdev that is returned by prcomp.

Generally speaking, one can look for the “elbow” of the scree plot to determine at what point to truncate the EOFs and retain up to the EOF before the elbow. At about the 5th EOF, the graph starts to level off and all subsequent EOFs start to contribute about the same amount of variance. Therefore, the elbow of the graph is located at about the 5th EOF and you will retain the first 4 EOFs.

North’s Rule of Thumb

North’s Rule of Thumb is more of a precise way of truncating and involves creating confidence intervals around your estimates of the variance. The rule states that you should truncated EOFs only when the confidence intervals of the variances start to intersect. At this point, the eigenvectors are considered too close to be interpretable and spacing might be due to sampling error rather than a clear distinction between the variances [1].

Rotated EOFs

At some point, your EOFs might start to exhibit patterns that can be hard to interpret or attribute to a physical phenomenon. It is not uncommon for these types of patterns to result from pure noise in your data, especially if you are analyzing latter EOFs that explain a very small amount of the variance [2].

Rotating EOFs is a practice that is done to simplify the patterns obtained in EOFs and make them more interpretable. A varimax orthogonal rotation can be used to determine an optimum rotation matrix that maximizes the variance in the columns of W. The variance of the columns is maximized by driving some of the loadings to zero and trying to maximize the values of other loadings. In R, this is done using the varimax function.

 my.varimax=varimax(PCA$rotation[,1:10]) 

In the above command, my input is the first 10 EOFs from the original weighting matrix. The result, my.varimax, is a list with the following components:

  1. loadings: the resulting rotated loading matrix
  2. rotmat: the rotation matrix

The new loading matrix, Wrot , is still orthogonal after the rotation, and the eigenvectors are, therefore, still orthonormal. However, multiplication of the rotated loading matrix by the original dataset, X, to obtain a new U, results in principal components are not guaranteed to be independent. This can be seen through further inspection of the correlation matrix associated with U. Unfortunately, this is a tradeoff associated with obtaining EOFs that are simpler and easier to interpret.

Sources:

[1] http://yyy.rsmas.miami.edu/users/bmapes/teaching/MPO581_2011/EOF_chapter_DelSole.pdf

[2] Hannachi, A., Jolliffe, I.T., and Stephenson, D.B. (2007), Empirical orthogonal functions and related techniques in atmospheric science: A review, International Journal of Climatology, 27, 1119-1152.

*All information or figures not specifically cited came from class notes and homework from Dr. Scott Steinschneider’s class, BEE 6300: Environmental Statistics 

Advertisements

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Google+ photo

You are commenting using your Google+ account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s