Andrews Curves

introduction

A long time ago, in a post about PCA (principal component analysis), I said that I did not know what Andrews curves were. (The suggestion was made that Andrews curves might help us decide how many principal components to keep. I did not understand how they were to be computed.)

Now I know. Let me show you. I will compute Andrews curves for what is called “the iris data”… for both the full data set (150 observations) and a reduced data set (30 observations). I will also show you a possible variant.

In addition, we will know that there are in fact three kinds of irises in the data – so we can assess how well the Andrews curves did. In practive, of course, we will be trying to figure out how many kinds of observations we have.

The data is here. The paper which explained Andrews curves to me is here. Andrews original paper is: Andrews D. Plots of high-dimensional data Biometrics 1972 28:125-136… but I haven’t found it freely available anywhere online.

In addition, there is a short and sweet webpage by one of the authors of the paper I read.
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Example: Is it a transition matrix? Part 2

We had three matrices from Jolliffe, P, V, and Q. They were allegedly a set of principal components P, a varimax rotation V of P, and a quartimin “oblique rotation” Q.

I’ll remind you that when they say “oblique rotation” they mean a general change-of-basis. A rotation preserves an orthonormal basis; a rotation cannot transform an orthonormal basis to a non-orthonormal basis, and that’s what they mean — a transformation from an orthonormal basis to a non-orthonormal basis, or possibly a transformation from a merely orthogonal basis to a non-orthogonal one. In either case, the transformation cannot be a rotation.

(It isn’t that complicated! If you change the lengths of basis vectors, it isn’t a rotation; if you change the angles between the basis vectors, it isn’t a rotation.)

Anyway, we showed in Part 1 that V and Q spanned the same 4D subspace of R^{10}\ .

Now, what about V and P? Let me recall them:
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Example: Is it a transition matrix? Part 1

This example comes from PCA / FA (principal component analysis, factor analysis), namely from Jolliffe (see the bibliography). But it illustrates some very nice linear algebra.

More precisely, the source of this example is:
Yule, W., Berger, M., Butler, S., Newham, V. and Tizard, J. (1969). The WPPSL: An empirical evaluation with a British sample. Brit. J. Educ. Psychol., 39, 1-13.

I have not been able to find the original paper. There is a problem here, and I do not know whether the problem lies in the original paper or in Jolliffe’s version of it. If anyone out there can let me know, I’d be grateful. (I will present 3 matrices, taken from Jolliffe; my question is, does the original paper contain the same 3 matrices?)

Like the previous post on this topic, this one is self-contained. In fact, it has almost nothing to do with PCA, and everything to do with finding — or failing to find! — a transition matrix relating two matrices.
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PCA / FA. A Map to my posts

The Posts

The purpose of this post is to provide guidance to a reader who has just discovered that I have a large pile of posts about principal components / factor analysis. This pile of posts might seem a very jungle, without any map.

Here, have a map.

As I finalize this post, it will be number 52 in PCA / FA. Here’s a list of the 52 posts, including the dates spanned by any group, and the number of posts in that group. (When the picture was taken, I didn’t know when this would be published. In fact, post 51 was scheduled but not yet published. Even more, post 51 did not even exist when the first picture was created.)

posts-table
transition/attitude matrices is a post that is sometimes relevant when we discuss “new data” in PCA, but it is not in the PCA / FA category.

“tricky prepro” is short for “tricky preprocessing”, and discusses the combination of constant row sums and covariance or correlation matrix.
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PCA / FA. Example 4 ! Davis, and almost everyone else

I would like to revisit the work we did in Davis (example 4). For one thing, I did a lot of calculations with that example, and despite the compare-and-contrast posts towards the end, I fear it may be difficult to sort out what I finally came to.

In addition, my notation has settled down a bit since then, and I would like to recast the work using my current notation.

The original (“raw”) data for example 4 was (p. 502, and columns are variables):

X_r = \left(\begin{array}{lll} 4 & 27 & 18 \\ 12 & 25 & 12 \\ 10 & 23 & 16 \\ 14 & 21 & 14\end{array}\right)
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PCA/FA Answers to some Basilevsky questions

Let us look at three of the questions I asked early in February, and answer two of them.

First, what do we know? What have we done?

We assume that we have data X, with the variables in columns, as usual. In fact, we assume that the data is at least centered, and possibly standardized.

We compute the covariance matrix

c = \frac{X^T X}{N-1}\ ,

then its eigendecomposition

c = v\ \Lambda^2\ v^T\ ,

where \Lambda^2 is the diaginal matrix of eigenvalues. We define the \sqrt{\text{eigenvalue}}-weighted matrix

A = v\ \Lambda\ .

Finally, we use A as a transition matrix to define new data Z:

X^T = A\ Z^T\ .

We discovered two things. One, the matrix A is the cross covariance between Z and X:

A = \frac{X^T Z}{N-1}\ .

I find this interesting, and I suspect that it would jump off the page at me out of either Harman or Jolliffe; that is, I suspect it is written there but it didn’t register.

Two, we discovered that we could find a matrix Ar which is the cross covariance between Zc and Xs. Read the rest of this entry »

PCA / FA Basilevsky: with data

Introduction

I am looking into Basilevsky because he did something I didn’t understand: he normalized the rows of the Ac matrix (which I denoted Ar). We discussed that, and we illustrated the computations, in the previous two posts. But we did those computations without having any data. I want to take a closer look, with data.

In contrast to As and Ac, which are eigenvector matrices, his Ar matrix is not. Nevertheless, as I said, his Ar is not without some redeeming value. In fact, all three of the A matrices have the same redeeming value.

I will show, first by direct computation and then by proof, that each of these A matrices is the cross covariance between X data and Z data.
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PCA / FA Basilevsky: what he did.

Introduction

The previous post discussed something interesting that Basilevsky did. (Search the bibliography, he’s there.) I can’t say I like it — because it leads to a basis which is non-orthogonal, and whose vectors are not eigenvectors of either the correlation or covariance matrices.

But I had to understand it.

I don’t know how widespread it is nowadays, but even if Basilevsky is the only author who does it, it’s another example of the lack of standardization (no pun intend, I swear) in PCA / FA. This branch of applied statistics is like the mythical Wild West: everybody’s got a gun and there are bullets flying all over the place. Law and order have not arrived yet.

OTOH, it’s nice to find something different in just about every book I open.
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PCA / FA Basilevsky on standardizing. Discussion.

Introduction and review

Basilevsky presents an extremely interesting idea. For all I know, it’s become common in the last 10-20 years, but I hadn’t seen it in any of the other books we’ve looked at.

I’ll tell you up front that he’s going to normalize the rows of an A matrix, specifically the A matrix computed from the eigendecomposition of the covariance matrix.

I’ll also tell you up front that I don’t see any good reason for doing it, but I’m not averse to finding such a reason someday.
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PCA / FA Example 9: scores & loadings

I want to look at reconstituting the data. Equivalently, I want to look at setting successive singular values to zero.

This example was actually built on the previous one. Before I set the row sums to 1, I had started with

t1 = \left(\begin{array}{lll} 1 & 1 & -3 \\ -1 & 2 & -2 \\ 1 & 3 & -1 \\ -1 & 4 & 1 \\ 1 & 5 & 4\end{array}\right)

I’m going to continue with Harmon’s & Bartholomew’s model: Z = A F, Z = X^T, X is standardized, A is an eigenvector matrix weighted by the square roots of the eigenvalues of the correlation matrix of X.

I want data with one eigenvalue so large that we could sensibly retain only that one. Let me show you how I got that.
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