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In the Jeopardy! game show contestants are presented with questions formulated as answers that require answers in the form questions. For example, if a contestant selects “Normality for $200” she might be shown the following clue:

“The average \frac{x_1+x_2+\cdots + x_n}{n},”

to which she would reply “What is the maximum likelihood estimate for the mean of independent identically distributed Gaussian random variables from which samples x_1,x_2,\ldots,x_n have been obtained?” Host Alex Trebek would immediately exclaim “That is the correct answer for $200!”

The process of doing mathematics involves repeatedly playing Jeopardy! with oneself in an unending quest to understand everything just a little bit better. The purpose of this blog post is to provide an exposition of how this works for understanding principal component analysis (PCA): I present four Jeopardy clues in the “Normality” category that all share the same answer: “What is principal component analysis?” The post was motivated by a conversation I recently had with a well-known population geneticist at a conference I was attending. I mentioned to him that I would be saying something about PCA in my talk, and that he might find what I have to say interesting because I knew he had used the method in many of his papers. Without hesitation he replied that he was well aware that PCA was not a statistical method and merely a heuristic visualization tool.

The problem, of course, is that PCA does have a statistical interpretation and is not at all an ad-hoc heuristic. Unfortunately, the previously mentioned population geneticist is not alone; there is a lot of confusion about what PCA is really about. For example, in one textbook it is stated that “PCA is not a statistical method to infer parameters or test hypotheses. Instead, it provides a method to reduce a complex dataset to lower dimension to reveal sometimes hidden, simplified structure that often underlie it.” In another one finds out that “PCA is a statistical method routinely used to analyze interrelationships among large numbers of objects.” In a highly cited review on gene expression analysis PCA is described as “more useful as a visualization technique than as an analytical method” but then in a paper by Markus Ringnér  titled the same as this post, i.e. What is principal component analysis? in Nature Biotechnology, 2008, the author writes that “Principal component analysis (PCA) is a mathematical algorithm that reduces the dimensionality of the data while retaining most of the variation in the data set” (the author then avoids going into the details because “understanding the details underlying PCA requires knowledge of linear algebra”). All of these statements are both correct and incorrect and confusing. A major issue is that the description by Ringnér of PCA in terms of the procedure for computing it (singular value decomposition) is common and unfortunately does not shed light on when it should be used. But knowing when to use a method is far more important than knowing how to do it.

I therefore offer four Jeopardy! clues for principal component analysis that I think help to understand both when and how to use the method:

1. An affine subspace closest to a set of points.

Suppose we are given numbers x_1,\ldots,x_n as in the initial example above. We are interested in finding the “closest” number to these numbers. By “closest” we mean in the sense of total squared difference. That is, we are looking for a number m such that \sum_{i=1}^n (m-x_i)^2 is minimized.

This is a (straightforward) calculus problem, solved by taking the derivative of the function above and setting it equal to zero. If we let f(m) = \sum_{i=1}^n (m-x_i)^2 then f'(m) = 2 \cdot \sum_{i=1}^n (m-x_i) and setting f'(m)=0 we can solve for m to obtain m = \frac{1}{n} \sum_{i=1}^n x_i.

The right hand side of the equation is just the average of the n numbers and the optimization problem provides an interpretation of it as the number minimizing the total squared difference with the given numbers (note that one can replace squared difference by absolute value, i.e. minimization of \sum_{i=1}^n |m-x_i|, in which case the solution for is the median; we return to this point and its implications for PCA later).

Suppose that instead of n numbers, one is given n points in \mathbb{R}^p. That is, point is {\bf x}^i = (x^i_1,\ldots,x^i_p). We can now ask for a point {\bf m} with the property that the squared distance of {\bf m} to the n points is minimized. This is asking for min_{\bf m} \sum_{i=1}^n ||{\bf m}-{\bf x}^i||_2.

The solution for m can be obtained by minimizing each coordinate independently, thereby reducing the problem to the simpler version of numbers above, and it follows that {\bf m} = \frac{1}{n} \sum_{i=1}^n {\bf x}^i.

This is 0-dimensional PCA, i.e., PCA of a set of points onto a single point, and it is the centroid of the points. The generalization of this concept provides a definition for PCA:

Definition: Given n points in \mathbb{R}^p, principal components analysis consists of choosing a dimension k < p and then finding the affine space of dimension with the property that the squared distance of the points to their orthogonal projection onto the space is minimized.

This definition can be thought of as a generalization of the centroid (or average) of the points. To understand this generalization, it is useful to think of the simplest case that is not 0-dimensional PCA, namely 1-dimensional PCA of a set of points in two dimensions:

PCA_Figure1In this case the 1-dimensional PCA subspace can be thought of as the line that best represents the average of the points. The blue points are the orthogonal projections of the points onto the “average line” (see, e.g., the red point projected orthogonally), which minimizes the squared lengths of the dashed lines. In higher dimensions line is replaced by affine subspace, and the orthogonal projections are to points on that subspace. There are a few properties of the PCA affine subspaces that are worth noting:

  1. The set of PCA subspaces (translated to the origin) form a flagThis means that the PCA subspace of dimension k is contained in the PCA subspace of dimension k+1. For example, all PCA subspaces contain the centroid of the points (in the figure above the centroid is the green point). This follows from the fact that the PCA subspaces can be incrementally constructed by building a basis from eigenvectors of a single matrix, a point we will return to later.
  2. The PCA subspaces are not scale invariant. For example, if the points are scaled by multiplying one of the coordinates by a constant, then the PCA subspaces change. This is obvious because the centroid of the points will change. For this reason, when PCA is applied to data obtained from heterogeneous measurements, the units matter. One can form a “common” set of units by scaling the values in each coordinate to have the same variance.
  3. If the data points are represented in matrix form as an n \times p matrix X, and the points orthogonally projected onto the PCA subspace of dimension are represented as in the ambient p dimensional space by a matrix \tilde{X}, then \tilde{X} = argmin_{M:rk(M)=k} ||X-M||_2. That is, \tilde{X} is the matrix of rank k with the property that the Frobenius norm ||X-\tilde{X}||_2 is minimized. This is just a rephrasing in linear algebra of the definition of PCA given above.

At this point it is useful to mention some terminology confusion associated with PCA. Unfortunately there is no standard for describing the various parts of an analysis. What I have called the “PCA subspaces” are also sometimes called “principal axes”. The orthogonal vectors forming the flag mentioned above are called “weight vectors”, or “loadings”. Sometimes they are called “principal components”, although that term is sometimes used to refer to points projected onto a principal axis. In this post I stick to “PCA subspaces” and “PCA points” to avoid confusion.

Returning to Jeopardy!, we have “Normality for $400” with the answer “An affine subspace closest to a set of points” and the question “What is PCA?”. One question at this point is why the Jeopardy! question just asked is in the category “Normality”. After all, the normal distribution does not seem to be related to the optimization problem just discussed. The connection is as follows:

2. A generalization of linear regression in which the Gaussian noise is isotropic.

PCA has an interpretation as the maximum likelihood parameter of a linear Gaussian model, a point that is crucial in understanding the scope of its application. To explain this point of view, we begin by elaborating on the opening Jeopardy! question about Normality for $200:

The point of the question was that the average of n numbers can be interpreted as a maximum likelihood estimation of the mean of a Gaussian. The Gaussian distribution is

f(x,\mu,\sigma) = \frac{1}{\sqrt{2 \pi \sigma^2}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}. Given the numbers x_1,\ldots,x_n, the likelihood function is therefore

L(\mu,\sigma) = \prod_{i=1}^n \frac{1}{\sqrt{2 \pi \sigma^2}} e^{-\frac{(x_i-\mu)^2}{2\sigma^2}}. The maximum of this function is the same as the maximum of its logarithm, which is

log L(\mu,\sigma) = \sum_{i=1}^n \left( log \frac{1}{\sqrt{2 \pi \sigma^2}} -\frac{(x_i-\mu)^2}{2\sigma^2} \right). Therefore the problem of finding the maximum likelihood estimate for the mean is equivalent to that of finding the minimum of the function

S(\mu) = \sum_{i=1}^n (x_i-\mu)^2. This is exactly the optimization problem solved by 0-dimensional PCA, as we saw above. With this calculation at hand, we turn to the statistical interpretation of least squares:

PCA_Figure2

Given n points \{(x_i,y_i)\}_{i=1}^n in the plane (see figure above),  the least squares line y=mx+b (purple in figure) is the one that minimizes the sum of the squares \sum_{i=1}^n \left( (mx_i+b) - y_i \right)^2. That is, the least squares line is the one minimizing the sum of the squared vertical distances to the points. As with the average of numbers, the least squares line has a statistical interpretation: Suppose that there is some line y=m^{*}x+b^{*} (black line in figure) that is unknown, but that “generated” the observed points, in the sense that each observed point was obtained by perturbing the point m^{*}x_i +b^{*} vertically by a random amount from a single Gaussian distribution with mean 0 and variance \sigma^2. In the figure, an example is shown where the blue point on the unknown line “generates” the observed red point; the Gaussian is indicated with the blue streak around the point. Note that the model specified so far is not fully generative, as it depends on the hidden points m^{*}x_i +b^{*} and there is no procedure given to generate the x_i. This can be done by positing that the x_i are generated from a Gaussian distribution along the line y=m^{*}x+b (followed by the points y_i generated by Gaussian perturbation of the y coordinate on the line). The coordinates x_i can then be deduced directly from the observed points as the Gaussian perturbations are all vertical. The relationship between the statistical model just described and least squares is made precise by a theorem (which we state informally, but is a special case of the Gauss-Markov theorem):

Theorem (Gauss-Markov): The maximum likelihood estimate for the line (the parameters and b) in the model described above correspond to the least squares line.

The proof is analogous to the argument given for the average of numbers above so we omit it. It can be generalized to higher dimensions where it forms the basis of what is known as linear regression. In regression, the x_i are known as independent variables and y the dependent variable. The generative model provides an interpretation of the independent variables as fixed measured quantities, whereas the dependent variable is a linear combination of the independent variables with added noise. It is important to note that the origins of linear regression are in physics, specifically in work of Legendre (1805) and Gauss (1809) who applied least squares to the astronomical problem of calculating the orbits of comets around the sun. In their application, the independent variables were time (for which accurate measurements were possible with clocks; by 1800 clocks were accurate to less than 0.15 seconds per day) and the (noisy) dependent variable the measurement of location. Linear regression has become one of the most (if not the most) widely used statistical tools but as we now explain, PCA (and its generalization factor analysis), with a statistical interpretation that includes noise in the x_i variables, seems better suited for biological data.

The statistical interpretation of least squares can be extended to a similar framework for PCA. Recall that we first considered a statistical interpretation for least squares where an unknown line y=m^{*}x+b^{*} “generated” the observed points, in the sense that each observed point was obtained by perturbing the point m^{*}x_i +b^{*} vertically by a random amount from a single Gaussian distribution with mean 0 and variance \sigma^2. PCA can be understood analogously by replacing vertically by orthogonally (this is the probabilistic model of Collins et al., NIPS 2001 for PCA). However this approach is not completely satisfactory as the orthogonality of the perturbation is is not readily interpretable. Stated differently, it is not obvious what physical processes would generate points orthogonal to a linear affine subspace by perturbations that are always orthogonal to the subspace. In the case of least squares, the “vertical” perturbation corresponds to noise in one measurement (represented by one coordinate). The problem is in naturally interpreting orthogonal perturbations in terms of a noise model for measurements. This difficulty is resolved by a model called probabilistic PCA (pPCA), first proposed by Tipping and Bishop in a Tech Report in 1997, and published in the J. of the Royal Statistical Society B 2002,  and independently by Sam Roweis, NIPS 1998, that is illustrated visually in the figure below, and that we now explain:

PCA_Figure3

 

In the pPCA model there is an (unknown) line (affine space in higher dimension) on which (hidden) points (blue) are generated at random according to a Gaussian distribution (represented by gray streak in the figure above, where the mean of the Gaussian is the green point). Observed points (red) are then generated from the hidden points by addition of isotropic Gaussian noise (blue smear), meaning that the Gaussian has a diagonal covariance matrix with equal entries. Formally, in the notation of Tipping and Bishop, this is a linear Gaussian model described as follows:

Observed random variables are given by t = Wx + \mu + \epsilon where x are latent (hidden) random variables, W is a matrix describing a subspace and Wx+\mu are the latent points on an affine subspace (\mu corresponds to a translation). Finally, \epsilon is an error term, given by a Gaussian random variable with mean 0 and covariance matrix \sigma^2 I. The parameters of the model are W,\mu and \sigma^2. Equivalently, the observed random variables are themselves Gaussian, described by the distribution t \sim \mathcal{N}(\mu,WW^T + \psi) where \psi \sim \mathcal{N}(0,\sigma^2I). Tipping and Bishop prove an analogy of the Gauss-Markov theorem, namely that the affine subspace given by the maximum likelihood estimates of W and \mu is the PCA subspace (the proof is not difficult but I omit it and refer interested readers to their paper, or Bishop’s Pattern Recognition and Machine Learning book).

It is important to note that although the maximum likelihood estimates of W,\mu in the pPCA model correspond to the PCA subspace, only posterior distributions can be obtained for the latent data (points on the subspace). Neither the mode nor the mean of those distributions corresponds to the PCA points (orthogonal projections of the observations onto the subspace). However what is true, is that the posterior distributions converge to the PCA points as \sigma^2 \rightarrow 0. In other words, the relationship between pPCA and PCA is a bit more subtle than that between least squares and regression.

The relationship between regression and (p)PCA is shown in the figure below:

PCA_Figure4In the figure, points have been generated randomly according to the pPCA model. the black smear shows the affine space on which the points were generated, with the smear indicating the Gaussian distribution used. Subsequently the latent points (light blue on the gray line) were used to make observed points (red) by the addition of isotropic Gaussian noise. The green line is the maximum likelihood estimate for the space, or equivalently by the theorem of Tipping and Bishop the PCA subspace. The projection of the observed points onto the PCA subspace (blue) are the PCA points. The purple line is the least squares line, or equivalently the affine space obtained by regression (y observed as a noisy function of x). The pink line is also a regression line, except where is observed as a noisy function of y.

A natural question to ask is why the probabilistic interpretation of PCA (pPCA) is useful or necessary? One reason it is beneficial is that maximum likelihood inference for pPCA involves hidden random variables, and therefore the EM algorithm immediately comes to mind as a solution (the strategy was suggested by both Tipping & Bishop and Roweis). I have not yet discussed how to find the PCA subspace, and the EM algorithm provides an intuitive and direct way to see how it can be done, without the need for writing down any linear algebra:

PCA_Figure6

The exact version of the EM shown above is due to Roweis. In it, one begins with a random affine subspace passing through the centroid of the points. The “E” step (expectation) consists of projecting the points to the subspace. The projected points are considered fixed to the subspace. The “M” step (maximization) then consists of rotating the space so that the total squared distance of the fixed points on the subspace to the observed points is minimized. This is repeated until convergence. Roweis points out that this approach to finding the PCA subspace is equivalent to power iteration for (efficiently) finding eigenvalues of the the sample covariance matrix without computing it directly. This is our first use of the word eigenvalue in describing PCA, and we elaborate on it, and the linear algebra of computing PCA subspaces later in the post.

Another point of note is that pPCA can be viewed as a special case of factor analysis, and this connection provides an immediate starting point for thinking about generalizations of PCA. Specifically, factor analysis corresponds to the model t \sim \mathcal{N}(\mu,WW^T + \psi) where the covariance matrix \psi is less constrained, and only required to be diagonal. This is connected to a comment made above about when the PCA subspace might be more useful as a linear fit to data than regression. To reiterate, unlike physics, where some coordinate measurements have very little noise in comparison to others, biological measurements are frequently noisy in all coordinates. In such settings factor analysis is preferable, as the variance in each coordinate is estimated as part of the model. PCA is perhaps a good compromise, as PCA subspaces are easier to find than parameters for factor analysis, yet PCA, via its pPCA interpretation, accounts for noise in all coordinates.

A final comment about pPCA is that it provides a natural framework for thinking about hypothesis testing. The book Statistical Methods: A Geometric Approach by Saville and Wood is essentially about (the geometry of) pPCA and its connection to hypothesis testing. The authors do not use the term pPCA but their starting point is exactly the linear Gaussian model of Tipping and Bishop. The idea is to consider single samples from n independent identically distributed independent Gaussian random variables as one single sample from a high-dimensional multivariate linear Gaussian model with isotropic noise. From that point of view pPCA provides an interpretation for Bessel’s correction. The details are interesting but tangential to our focus on PCA.

We are therefore ready to return to Jeopardy!, where we have “Normality for $600” with the answer “A generalization of linear regression in which the Gaussian noise is isotropic” and the question “What is PCA?”

3. An orthogonal projection of points onto an affine space that maximizes the retained sample variance.

In the previous two interpretations of PCA, the focus was on the PCA affine subspace. However in many uses of PCA the output of interest is the projection of the given points onto the PCA affine space. The projected points have three useful related interpretations:

  1. As seen in in section 1, the (orthogonally) projected points (red -> blue) are those whose total squared distance to the observed points is minimized.
  2. What we focus on in this section, is the interpretation that the PCA subspace is the one onto which the (orthogonally) projected points maximize the retained sample variance.
  3. The topic of the next section, namely that the squared distances between the (orthogonally) projected points are on average (in the l_2 metric) closest to the original distances between the points.

PCA_Figure5

 

The sample variance of a set of points is the average squared distance from each point to the centroid. Mathematically, if the observed points are translated so that their centroid is at zero (known as zero-centering), and then represented by an n \times p matrix X, then the sample covariance matrix is given by \frac{1}{n-1}X^tX and the sample variance is given by the trace of the matrix. The point is that the jth diagonal entry of \frac{1}{n-1}X^tX is just \frac{1}{n-1}\sum_{i=1}^n (x^i_j)^2, which is the sample variance of the jth variable. The PCA subspace can be viewed as that subspace with the property that the sample variance of the projections of the observed points onto the subspace is maximized. This is easy to see from the figure above. For each point (blue), Pythagoras’ theorem implies that d(red,blue)^2+d(blue,green)^2 = d(red,green)^2. Since the PCA subspace is the one minimizing the total squared red-blue distances, and since the solid black lines (red-green distances) are fixed, it follows that the PCA subspace also maximizes the total squared green-blue distances. In other words, PCA maximizes the retained sample variance.

The explanation above is informal, and uses a 1-dimensional PCA subspace in dimension 2 to make the argument. However the argument extends easily to higher dimension, which is typically the setting where PCA is used. In fact, PCA is typically used to “visualize” high dimensional points by projection into dimensions two or three, precisely because of the interpretation provided above, namely that it retains the sample variance. I put visualize in quotes because intuition in two or three dimensions does not always hold in high dimensions. However PCA can be useful for visualization, and one of my favorite examples is the evidence for genes mirroring geography in humans. This was first alluded to by Cavalli-Sforza, but definitively shown by Lao et al., 2008, who analyzed 2541 individuals and showed that PCA of the SNP matrix (approximately) recapitulates geography:

PCA_Lao

Genes mirror geography from Lao et al. 2008: (Left) PCA of the SNP matrix (2541 individuals x 309,790 SNPs) showing a density map of projected points. (Right) Map of Europe showing locations of the populations .

In the picture above, it is useful to keep in mind that the emergence of geography is occurring in that projection in which the sample variance is maximized. As far as interpretation goes, it is useful to look back at Cavalli-Sforza’s work. He and collaborators who worked on the problem in the 1970s, were unable to obtain a dense SNP matrix due to limited technology of the time. Instead, in Menozzi et al., 1978 they performed PCA of an allele-frequency matrix, i.e. a matrix indexed by populations and allele frequencies instead of individuals and genotypes. Unfortunately they fell into the trap of misinterpreting the biological meaning of the eigenvectors in PCA. Specifically, they inferred migration patterns from contour plots in geographic space obtained by plotting the relative contributions from the eigenvectors, but the effects they observed turned out to be an artifact of PCA. However as we discussed above, PCA can be used quantitatively via the stochastic process for which it solves maximum likelihood inference. It just has to be properly understood.

To conclude this section in Jeopardy! language, we have “Normality for $800” with the answer “A set of points in an affine space obtained via projection of a set of given points so that the sample variance of the projected points is maximized” and the question “What is PCA?”

4. Principal component analysis of Euclidean distance matrices.

In the preceding interpretations of PCA, I have focused on what happens to individual points when projected to a lower dimensional subspace, but it is also interesting to consider what happens to pairs of points. One thing that is clear is that if a pair of points are projected orthogonally onto a low-dimensional affine subspace then the distance between the points in the projection is smaller than the original distance between the points. This is clear because of Pythagoras’ theorem, which implies that the squared distance will shrink unless the points are parallel to the subspace in which case the distance remains the same. An interesting observation is that in fact the PCA subspace is the one with the property where the average (or total) squared distances between the points is maximized. To see this it again suffices to consider only projections onto one dimension (the general case follows by Pythagoras’ theorem). The following lemma, discussed in my previous blog post, makes the connection to the previous discussion:

Lemma: Let x_1,\ldots,x_n be numbers with mean \overline{x} = \frac{1}{n}\sum_i x_i. If the average squared distance between pairs of points is denoted D = \frac{1}{n^2}\sum_{i,j} (x_i-x_j)^2 and the variance is denoted V=\frac{1}{n}\sum_i (x_i-\overline{x})^2 then V=\frac{1}{2}D.

What the lemma says is that the sample variance is equal to the average squared difference between the numbers (i.e. it is a scalar multiple that does not depend on the numbers). I have already discussed that the PCA subspace maximizes the retained variance, and it therefore follows that it also maximizes the average (or total) projected squared distance between the points. Alternately, PCA can be interpreted as minimizing the total (squared) distance that is lost, i,e. if the original distances between the points are given by a distance matrix D and the projected distances are given by \tilde{D}, then the PCA subspace minimizes \sum_{ij} (D^2_{ij} - \tilde{D}^2_{ij}), where each term in the sum is positive as discussed above.

This interpretation of PCA leads to an interesting application of the method to (Euclidean) distance matrices rather than points. The idea is based on a theorem of Isaac Schoenberg that characterizes Euclidean distance matrices and provides a method for realizing them. The theorem is well-known to structural biologists who work with NMR, because it is one of the foundations used to reconstruct coordinates of structures from distance measurements. It requires a bit of notation: is a distance matrix with entries d_{ij} and \Delta is the matrix with entries -\frac{1}{2}d^2_{ij}. {\bf 1} denotes the vector of all ones, and {\bf s} denotes a vector.

Theorem (Schoenberg, 1938): A matrix D is a Euclidean distance matrix if and only if the matrix B=(I-{\bf 1}{\bf s}')\Delta(I-{\bf s}{\bf 1}') is positive semi-definite where {\bf s}'{\bf 1} = 1.

For the case when {\bf s} is chosen to be a unit vector, i.e. all entries are zero except one of them equal to 1, the matrix B can be viewed as the Gromov transform (known as the Farris transform in phylogenetics) of the matrix with entries d^2_{ij}. Since the matrix is positive semidefinite it can be written as B=XX^t, where the matrix X provides coordinates for points that realize D. At this point PCA can be applied resulting in a principal subspace and points on it (the orthogonal projections of X). A point of note is that eigenvectors of XX^t can be computed directly, avoiding the need to compute X^tX which may be a larger matrix if n < p.

The procedure just described is called classic multidimensional scaling (MDS) and it returns a set of points on a Euclidean subspace with distance matrix \tilde{D} that best represent the original distance matrix D in the sense that \sum_{ij} (D^2_{ij} - \tilde{D}^2_{ij}) is minimized. The term multidimensional scaling without the “classic” has taken on an expanded meaning, namely it encapsulates all methods that seek to approximately realize a distance matrix by points in a low dimensional Euclidean space. Such methods are generally not related to PCA, but classic multidimensional scaling is PCA. This is a general source of confusion and error on the internet. In fact, most articles and course notes I found online describing the connection between MDS and PCA are incorrect. In any case classic multidimensional scaling is a very useful instance of PCA, because it extends the utility of the method to cases where points are not available but distances between them are.

Now we return to Jeopardy! one final time with the final question in the category: “Normality for $1000”. The answer is “Principal component analysis of Euclidean distance matrices” and the question is “What is classic multidimensional scaling?”

An example

To illustrate the interpretations of PCA I have highlighted, I’m including an example in R inspired by an example from another blog post (all commands can be directly pasted into an R console). I’m also providing the example because missing in the discussion above is a description of how to compute PCA subspaces and the projections of points onto them. I therefore explain some of this math in the course of working out the example:

First, I generate a set of points (in \mathbb{R}^2). I’ve chosen a low dimension so that pictures can be drawn that are compatible with some of the examples above. Comments following commands appear after the # character.

 set.seed(2)             #sets the seed for random number generation.
 x <- 1:100              #creates a vector x with numbers from 1 to 100
 ex <- rnorm(100, 0, 30) #100 normally distributed rand. nos. w/ mean=0, s.d.=30
 ey <- rnorm(100, 0, 30) # " " 
 y <- 30 + 2 * x         #sets y to be a vector that is a linear function of x
 x_obs <- x + ex         #adds "noise" to x
 y_obs <- y + ey         #adds "noise" to y
 P <- cbind(x_obs,y_obs) #places points in matrix
 plot(P,asp=1,col=1) #plot points
 points(mean(x_obs),mean(y_obs),col=3, pch=19) #show center

At this point a full PCA analysis can be undertaken in R using the command “prcomp”, but in order to illustrate the algorithm I show all the steps below:

 M <- cbind(x_obs-mean(x_obs),y_obs-mean(y_obs))#centered matrix
 MCov <- cov(M)          #creates covariance matrix

Note that the covariance matrix is proportional to the matrix $M^tM$. Next I turn to computation of the principal axes:

 eigenValues <- eigen(MCov)$values       #compute eigenvalues
 eigenVectors <- eigen(MCov)$vectors     #compute eigenvectors

The eigenvectors of the covariance matrix provide the principal axes, and the eigenvalues quantify the fraction of variance explained in each component. This math is explained in many papers and books so we omit it here, except to say that the fact that eigenvalues of the sample covariance matrix are the principal axes follows from recasting the PCA optimization problem as maximization of the Raleigh quotient. A key point is that although I’ve computed the sample covariance matrix explicitly in this example, it is not necessary to do so in practice in order to obtain its eigenvectors. In fact, it is inadvisable to do so. Instead, it is computationally more efficient, and also more stable to directly compute the singular value decomposition of M. The singular value decomposition of M decomposes it into M=UDV^t where is a diagonal matrix and both U and V^t are orthogonal matrices. I will also not explain in detail the linear algebra of singular value decomposition and its relationship to eigenvectors of the sample covariance matrix (there is plenty of material elsewhere), and only show how to compute it in R:

 d <- svd(M)$d          #the singular values
 v <- svd(M)$v          #the right singular vectors

The right singular vectors are the eigenvectors of M^tM.  Next I plot the principal axes:

 lines(x_obs,eigenVectors[2,1]/eigenVectors[1,1]*M[x]+mean(y_obs),col=8)

This shows the first principal axis. Note that it passes through the mean as expected. The ratio of the eigenvectors gives the slope of the axis. Next

 lines(x_obs,eigenVectors[2,2]/eigenVectors[1,2]*M[x]+mean(y_obs),col=8)

shows the second principal axis, which is orthogonal to the first (recall that the matrix V^t in the singular value decomposition is orthogonal). This can be checked by noting that the second principal axis is also

 lines(x_obs,-1/(eigenVectors[2,1]/eigenVectors[1,1])*M[x]+mean(y_obs),col=8)

as the product of orthogonal slopes is -1. Next, I plot the projections of the points onto the first principal component:

 trans <- (M%*%v[,1])%*%v[,1] #compute projections of points
 P_proj <- scale(trans, center=-cbind(mean(x_obs),mean(y_obs)), scale=FALSE) 
 points(P_proj, col=4,pch=19,cex=0.5) #plot projections
 segments(x_obs,y_obs,P_proj[,1],P_proj[,2],col=4,lty=2) #connect to points

The linear algebra of the projection is simply a rotation followed by a projection (and an extra step to recenter to the coordinates of the original points). Formally, the matrix M of points is rotated by the matrix of eigenvectors W to produce T=MW. This is the rotation that has all the optimality properties described above. The matrix T is sometimes called the PCA score matrix. All of the above code produces the following figure, which should be compared to those shown above:

Rplot

There are many generalizations and modifications to PCA that go far beyond what has been presented here. The first step in generalizing probabilistic PCA is factor analysis, which includes estimation of variance parameters in each coordinate. Since it is rare that “noise” in data will be the same in each coordinate, factor analysis is almost always a better idea than PCA (although the numerical algorithms are more complicated). In other words, I just explained PCA in detail, now I’m saying don’t use it! There are other aspects that have been generalized and extended. For example the Gaussian assumption can be relaxed to other members of the exponential family, an important idea if the data is discrete (as in genetics). Yang et al. 2012 exploit this idea by replacing PCA with logistic PCA for analysis of genotypes. There are also many constrained and regularized versions of PCA, all improving on the basic algorithm to deal with numerous issues and difficulties. Perhaps more importantly, there are issues in using PCA that I have not discussed. A big one is how to choose the PCA dimension to project to in analysis of high-dimensional data. But I am stopping here as I am certain no one is reading at this far into the post anyway…

The take-home message about PCA? Always be thinking when using it!

Acknowledgment: The exposition of PCA in this post began with notes I compiled for my course MCB/Math 239: 14 Lessons in Computational Genomics taught in the Spring of 2013. I thank students in that class for their questions and feedback. None of the material presented in class was new, but the exposition was intended to clarify when PCA ought to be used, and how. I was inspired by the papers of Tipping, Bishop and Roweis on probabilistic PCA in the late 1990s that provided the needed statistical framework for its understanding. Following the class I taught, I benefited greatly from conversations with Nicolas Bray, Brielin Brown, Isaac Joseph and Shannon McCurdy who helped me to further frame PCA in the way presented in this post.

The Habsburg rulership of Spain ended with an inbreeding coefficient of F=0.254. The last king, Charles II (1661-1700), suffered an unenviable life. He was unable to chew. His tongue was so large he could not speak clearly, and he constantly drooled. Sadly, his mouth was the least of his problems. He suffered seizures, had intellectual disabilities, and was frequently vomiting. He was also impotent and infertile, which meant that even his death was a curse in that his lack of heirs led to a war.

None of these problems prevented him from being married (twice). His first wife, princess Henrietta of England, died at age 26 after becoming deeply depressed having being married to the man for a decade. Only a year later, he married another princess, 23 year old Maria Anna of Neuberg. To put it mildly, his wives did not end up living the charmed life of Disney princesses, nor were they presumably smitten by young Charles II who apparently aged prematurely and looked the part of his horrific homozygosity. The princesses married Charles II because they were forced to. Royals organized marriages to protect and expand their power, money and influence. Coupled to this were primogeniture rules which ensured that the sons of kings, their own flesh and blood and therefore presumably the best-suited to be in power, would indeed have the opportunity to succeed their fathers. The family tree of Charles II shows how this worked in Spain:

Charles

It is believed that the inbreeding in Charles II’s family led to two genetic disorders, combined pituitary hormone deficiency and distal renal tubular acidosis, that explained many of his physical and mental problems. In other words, genetic diversity is important, and the point of this blog post is to highlight the fact that diversity is important in education as well.

The problem of inbreeding in academia has been studied previously, albeit to a limited extent. One interesting article is Navel Grazing: Academic Inbreeding and Scientific Productivity by Horta et al published in 2010 (my own experience with an inbred academic from a department where 39% of the faculty are self-hires anecdotally confirms the claims made in the paper). But here I focus on the downsides of inbreeding of ideas rather than of faculty. For example home-schooling, the educational equivalent of primogeniture, can be fantastic if the parents happen to be good teachers, but can fail miserably if they are not. One thing that is guaranteed in a school or university setting is that learning happens by exposure to many teachers (different faculty, students, tutors, the internet, etc.) Students frequently complain when there is high variance in teaching quality, but one thing such variance ensures is that is is very unlikely that any student is exposed only to bad teachers. Diversity in teaching also helps to foster the development of new ideas. Different teachers, by virtue of insight or error, will occasionally “mutate” ideas or concepts for better or for worse. In other words, one does not have to fully embrace the theory of memes to acknowledge that there are benefits to variance in teaching styles, methods and pedagogy. Conversely, there is danger in homogeneity.

This brings me to MOOCs. One of the great things about MOOCs is that they reach millions of people. Udacity claims it has 1.6 million “users” (students?). Coursera claims 7.1 million. These companies are greatly expanding the accessibility of education. Starving children in India can now take courses in mathematical methods for quantitative finance, and for the first time in history, a president of the United States can discreetly take a freshman course on economics together with its high school algebra prerequisites (highly recommended). But when I am asked whether I would be interested in offering a MOOC I hesitate, paralyzed at the thought that any error I make would immediately be embedded in the brains of millions of innocent victims. My concern is this: MOOCs can greatly reduce the variance in education. For example, Coursera currently offers 641 courses, which means that each courses is or has been taught to over 11,000 students. Many college courses may have less than a few dozen students, and even large college courses rarely have more than a few hundred students. This means that on average, through MOOCs, individual professors reach many more (2 orders of magnitude!) students. A great lecture can end up positively impacting a large number of individuals, but at the same time, a MOOC can be a vehicle for infecting the brains of millions of people with nonsense. If that nonsense is then propagated and reaffirmed via the interactions of the people who have learned it from the same source, then the inbreeding of ideas has occurred.

I mention MOOCs because I was recently thinking about intuition behind Bessel’s correction replacing n with n-1 in the formula for sample variance. Formally, Bessel’s correction replaces the biased formula

s^2_n = \frac{1}{n} \sum_{i=1}^n (x_i - \overline{x})^2

for estimating the variance of a random variable from samples x_1,\ldots,x_n with

s^2_{n-1} = \frac{1}{n-1} \sum_{i=1}^n (x_i-\overline{x})^2.

The switch from to n-1 is a bit mysterious and surprising, and in introductory statistics classes it is frequently just presented as a “fact”. When an explanation is provided, it is usually in the form of algebraic manipulation that establishes the result. The issue came up as a result of a blog post I’m writing about principal components analysis (PCA), and I thought I would check for an intuitive explanation online. I googled “intuition sample variance” and the top link was a MOOC from the Khan Academy:

The video has over 51,000 views with over 100 “likes” and only 6 “dislikes”. Unfortunately, in this case, popularity is not a good proxy for quality. Despite the title promising “review” and “intuition” for “why we divide by n-1 for the unbiased sample variance” there is no specific reason given why is replaced by n-1 (as opposed to another correction). Furthermore, the intuition provided has to do with the fact that x_i-\overline{x} underestimates x_i-\mu (where \mu is the mean of the random variable and \overline{x} is the sample mean) but the explanation is confusing and not quantitative (which it can easily be). In fact, the wikipedia page for Bessel’s correction provides three different mathematical explanations for the correction together with the intuition that motivates them, but it is difficult to find with Google unless one knows that the correction is called “Bessel’s correction”.

Wikipedia is also not perfect, and this example is a good one for why teaching by humans is important. Among the three alternative derivations, I think that one stands out as “better” but one would not know by just looking at the wikipedia page. Specifically, I refer to “Alternate 1” on the wikipedia page, that is essentially explaining that variance can be rewritten as a double sum corresponding to the average squared distance between points and the diagonal terms of the sum are zero in expectation. An explanation of why this fact leads to the n-1 in the unbiased estimator is as follows:

The first step is to notice that the variance of a random variable is equal to half of the expected squared difference of two independent identically distributed random variables of that type. Specifically, the definition of variance is:

var(X) = \mathbb{E}(X - \mu)^2 where \mu = \mathbb{E}(X). Equivalently, var(X) = \mathbb{E}(X^2) -\mu^2. Now suppose that Y is another random variable identically distributed to X and with X,Y independent. Then \mathbb{E}(X-Y)^2 = 2 var(X). This is easy to see by using the fact that

\mathbb{E}(X-Y)^2 = \mathbb{E}(X^2) + \mathbb{E}(Y^2) - 2\mathbb{E}(X)\mathbb{E}(Y) = 2\mathbb{E}(X^2)-2\mu^2.

This identity motivates a rewriting of the (uncorrected) sample variance s_n in a way that is computationally less efficient, but mathematically more insightful:

s_n = \frac{1}{2n^2} \sum_{i,j=1}^n (x_i-x_j)^2.

Of note is that in this summation exactly n of the terms are zero, namely the terms when i=j. These terms are zero independently of the original distribution, and remain so in expectation thereby biasing the estimate of the variance, specifically leading to an underestimate. Removing them fixes the estimate and produces

s_{n-1}^2 = \frac{1}{2n(n-1)} \sum_{i,j=1, i \neq j}^n (x_i-x_j)^2.

It is easy to see that this is indeed Bessel’s correction. In other words, the correction boils down to the fact that n^2-n = n(n-1), hence the appearance of n-1.

Why do I like this particular derivation of Bessel’s correction? There are two reasons: first, n-1 emerges naturally and obviously from the derivation. The denominator in s_{n-1}^2 matches exactly the number of terms being summed, so that it can be understood as a true average (this is not apparent in its standard form as s_{n-1}^2 = \frac{1}{n-1} \sum_{i=1}^n (x_i-\overline{x})^2. There is really nothing mysterious anymore, its just that some terms having been omitted from the sum because they were non-inofrmative. Second, as I will show in my forthcoming blog post on PCA, the fact that the variance of a random variable is half of the expectation of the squared difference of two instances, is key to understanding the connection between multi-dimensional scaling (MDS) and PCA. In other words, as my student Nicolas Bray is fond of saying, although most people think a proof is either right or wrong, in fact some proofs are more right than others. The connection between Bessel’s correction and PCA goes even deeper: as explained by Saville and Wood in their book Statistical Methods: A Geometric Approach n-1 can be understood to be a reduction in one dimension from the point of view of probabilistic PCA (Saville and Wood do not explicitly use the term probabilistic PCA but as I will explain in my PCA post it is implicit in their book). Finally, there are many subtleties to Bessel’s correction, for example it is an unbiased estimator for variance and not standard deviation. These issues ought to be mentioned in a good lecture about the topic. In other words, the Khan lecture is neither necessary nor sufficient, but unlike a standard lecture where the damage is limited to a small audience of students, it has been viewed more than 50,000 times and those views cannot be unviewed.

In writing this blog post I pondered the irony of my call for added diversity in teaching while I preach my own idea (this post) to a large number of readers via a medium designed for maximal outreach. I can only ask that others blog as well to offer alternative points of view 🙂 and that readers inform themselves on the issues I raise by fact-checking elsewhere. As far as the statistics goes, if someone finds the post confusing, they should go and register for one of the many fantastic MOOCs on statistics! But I reiterate that in the rush to MOOCdom, providers must offer diversity in their offerings (even multiple lectures on the same topic) to ensure a healthy population of memes. This is especially true in Spain, where already inbred faculty are now inbreeding what they teach by MOOCing via Miriada X. Half of the MOOCs being offered in Spain originate from just 3 universities, while the number of potential viewers is enormous as Spanish is now the second most spoken language in the world (thanks to Charles II’s great-great-grandfather, Charles I).

May Charles II rest in peace.

One of my distinct memories from elementary school is going to “library class” to learn about the Dewey decimal classification and how to use a card catalog to find books. Searching for books efficiently was possible because cards in the catalog were sorted lexicographically.

It didn’t occur to me at the time, but the system required authors of books to be totally ordered. Without an ordering of authors in a book with multiple authors, there would be no way to decide where to place the card for the book in a catalog searchable by author. The practice of ordering authors on publications is evident in the oldest printed texts and has persisted to this day. I have never thought that it could be any other way.

However this past Wednesday I was visiting the University of Washington to deliver a seminar, and among the highlights of the visit was my meeting with the graduate students. I met 12 for lunch and two more came for dinner. Meeting with students is always my favorite part of a visit to a university. They have original and creative ideas, and most importantly, are not bound in their thought by archaic tradition. They frequently don’t know what one is supposed to think and how one is supposed to say it. They just think and speak!

One of the students I met on Wednesday was Vanessa Gray, a student of Doug Fowler, who in a conversation on authorship practices suggested to me the radical and brilliant idea that papers should be published without an ordering of authors.

Many journals now have a section called “Author contributions” where roles of individuals in collaborative projects can be described (many journals now require such descriptions). So why bother ordering the authors for a list underneath the title? As far as indexing and searching goes, Google and other search engines require only a set of authors, and not a specific ordering.

I agree with Vanessa that ending author ordering on publications would greatly improve fairness in the biological sciences, where many current projects involve complex assemblies of teams with complementary skills. “First authorship” is not well-defined when one author performed a large number of difficult experiments, and another developed novel algorithms and wrote complex software for analyzing the experiments. Similarly, “last authorship” fails as a concept when students are co-advised, or one principal investigator provides substantial funding on a project, while another is participating in doing the work. And recently, large consortium projects have completely destroyed any meaning of “author” by having hundreds, or even thousands of authors on projects. Even when there are relatively few authors people rarely credit anyone except the first and last authors, even if others did substantial work. In the recent ENCODE paper published in PNAS with 30 authors, it appears to me from the responses to my previous blog post about the paper that the 5th and 6th authors did a lot (majority?) of the work in putting together figures and results, yet I suspect the “credit” for the paper will go to the first author (the flip side in that case is that the first author is where blame is assigned as well).

There is also a theoretical justification for not ordering authors. Ordering of authors on a publication can be thought of as a ranking produced by “votes” of the participants in the project. Of course in practice not all votes are equal. In what is called dictatorship in social choice theory, PIs frequently make the decisions independently of how specific authors feel they may have contributed. This may work on a paper where there is a single PI (although it may be considered unfair by the graduate students and postdocs), however dictatorship as a system for determining authorship certainly breaks down when multiple PIs collaborate on a project. Arrow’s impossibility theorem explains that in the absence of dictatorship, there is a problem in producing a single ordering satisfying two other seemingly basic and essential fairness criteria. Informally, the theorem states that there is no authorship ordering system based on voting of contributing authors that can satisfy the following three criteria:

  • If every author thinks that X should be ordered before Y, then the author list should have X placed before Y.
  • For a fixed list of voting preferences regarding the ordering of X vs. Y, the ordering between X and Y in the author list will remain unchanged even if does not depend on the ordering of other pairs such as X and Z, Y and Z, or Z and W.
  • There is no “dictator”, i.e. no single author possesses the power to determine the author ordering.

Authors frequently have differing opinions about the impact of their own contribution to a publication, and therefore their preferences (votes) for author ordering are discordant. This means that any system for ordering authors will not satisfy everyone’s preferences, and in the sense of Arrow’s impossibility theorem will be unfair. One way around Arrow’s impossibility theorem is to specify authorship order without regard to authors’ preferences, for example by always ordering authors alphabetically (the Hardy-Littlewood rule). This method, usually the one used in the mathematical sciences, is also fraught with problems. Of course, listing author contributions for what they are is not entirely trivial. For example, different authors may have conflicting views about what it means to have “written the text of the paper”. But using words to describe contributions allow for much more detail about what each author did, and allows for nuanced contributions to be described (e.g., John and Jane were in the room when the initial idea for the project was discussed, but did not contribute anything afterwards).

To summarize, in the modern era of electronic publishing ordering of authors is unnecessary, and if it is unnecessary, then why confront Arrow’s theorem and inevitably produce orderings unfairly? Publications should just explain the author contributions. Time to end ordered authorship.

Yale_card_catalog The card catalog at Yale University’s Sterling Memorial Library (from Wikipedia).

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