The lecture explains how correspondence analysis (CA) represents categorical data geometrically by constructing point clouds of row and column profiles in high‑dimensional space. Each row profile is a point in \(\mathbb{R}^j\) with coordinates \(f{ij}/f{i\cdot}\); the collection of these points forms the row cloud \(Ni\). A mean profile (the marginal column distribution) serves as the cloud’s centre of gravity (G) when rows are weighted by their marginal frequencies. An analogous column cloud \(Nj\) is built in \(\mathbb{R}^i\) with the mean column profile as its centre. Distances between profiles and the centre are measured with the chi‑squared distance, a weighted Euclidean metric that incorporates the inverse of marginal column (or row) probabilities. The inertia of a cloud – the weighted sum of squared chi‑squared distances – equals \(\chi^2/N\) and quantifies deviation from independence. A key duality result shows that row‑cloud inertia equals column‑cloud inertia, meaning rows and columns have symmetric roles in CA. CA proceeds by projecting the clouds onto successive orthogonal axes that maximise projected inertia; each axis’s inertia is an eigenvalue (\(\lambda_s\)). The first two axes define a plane on which both rows and columns are plotted (a biplot). The speaker interprets the biplot using examples (UK close to the origin, Italy far from it) and discusses how to label dimensions by examining the distribution of row and column points, illustrating the scientific‑vs‑humanities split and the physics‑chemistry versus economics separation. The video concludes by summarising the visualization of row and column clouds and previewing further topics on inertia and simultaneous representations.
Chapters
We have seen that correspondence analysis works with respect to an independence model. Now, to further describe correspondence analysis, we are going to work with essentially geometric ideas and construct point clouds in space. First, let's build the cloud of row profiles. A row profile is a set of j numbers which can be seen as a point in j-dimensional space R^j. Each dimension corresponds to a category j of the variable V2. Therefore, in the jth dimension, the row profile i has coordinate value fij over fi point. If we then consider all of the row profiles i together, we obtain a cloud of row profiles, points in R^j, which we can call Ni. To this point cloud, we can add what we called the mean profile as another point. Its jth coordinate is simply f_j-point. We call this G for center of gravity. In fact, G can be seen as the center of gravity of the cloud N_i as long as we associate each point i with a weight proportional to its marginal value f_i-point. In this space, what interests us the most is to compare the positions of the row profiles i with the mean profile. For this, we are first going to take the mean profile and translate it to the origin, and the other points along with it. In this space, we need to be able to, to calculate distances. The distance measure used in correspondence analysis is called the chi-squared distance. It greatly resembles the usual Euclidean distance. Indeed, it involves a sum of differences. We calculate the difference between profiles i and i prime, that is between f_ij over f_i point and f_i prime j over f_i prime point. We take the square and then sum over all j. The difference between this and the Euclidean distance is that here each dimension j is associated with a weight 1 over f point j. The justification for this distance measure will become clear later in the course. The center of gravity of the cloud corresponding to the mean profile has coordinates f.j. Therefore, the distance between it and profile i is simply this distance. Just like for the cloud of row profile points, we can build the cloud of column profile points. A column profile is a set of i values So it becomes a point in I-dimensional space. In this space, each dimension corresponds to a category i of variable V1. For this category, the coordinates of the column profile are f_ij over f_point_j. There are J column profiles in all, which together make up the cloud of points N_J. We then add the mean profile g_j to the cloud, which has coordinates f_i-point in the ith dimension. This mean profile can be considered the cloud's center of gravity as long as we assign to each column profile j a weight corresponding to its marginal sum, or more precisely, its marginal probability, which here is f-point_j. In this space, we will be greatly interested in the distance between the column profiles and the mean profile. To simplify, we translate the center of gravity to the origin, and the other points along with it. To calculate distances in this space, we use, as for the cloud of raw profiles, the chi-squared distance. Here, this represents a sum of squares of differences in coordinates, where each difference is weighted by 1 over fi points. The cloud's center of gravity, corresponding to the mean profile, as coordinate f_i point. Therefore, the distance between profile j and the mean profile is simply the square root of this quantity. So what happens if there is independence? Well, if so, the conditional probability equals the marginal probability, which means that all the profiles are the same as the mean profile. In other words, the cloud ends up just being a single point at the origin. And this is true for both point clouds. Therefore, the further the data is from independence, the further the profiles are from the origin. Let's now calculate the inertia of the point cloud, Ni, with respect to its center of gravity. The inertia of a point cloud is the sum of the inertia of each point. The inertia of a point is the mass times the square of the distance to the center of gravity. So the inertia of a point i is f_i point times the square of the chi-squared distance between i and G_i. When we calculate this, we get an intermediate result, which is none other than phi squared, or if you prefer, chi-squared over N. The point cloud's inertia is therefore clearly an indicator of deviation from independence. As what interests us is this deviation from independence, we end up studying the inertia of N_i. This is the correspondence analysis point of view. The same reasoning holds for the cloud N_j, and we even have a surprising result: the inertia of N_i is equal to the inertia of N_j. This is an important thing to realize, wrapped up in what we call duality, a kind of paired equivalence. In fact, what this means is that it's the same thing to analyze the table in terms of rows or in terms of columns. This is a fundamental result in correspondence analysis. We see that in correspondence analysis, rows and columns have perfectly symmetric roles. This property distinguishes correspondence analysis from other methods like principal component analysis, for example. We have turned around our initial question on the link between qualitative variables into a question of deviation from independence. This deviation can be interpreted geometrically as the inertia of a point cloud N_i with respect to the origin. And how does correspondence analysis proceed? As all principal component methods, it simply projects the point cloud onto a series of axes, of dimensions, of maximum inertia. When we have a sequence of axes, we can take pairs of them and draw planes. Let's look at how we find the first plane. So the best planar representation of the point cloud Ni is found as we said, by projection onto some plane P. Here we have the point M_i which corresponds to the ith profile. This point M_i is projected onto the plane as the point H_i. In fact, what you see are the points H_i and you hope that they are not too different from the points M_i so that you can make some useful conclusions. So what criteria should we use to find the plane P? Remember that we want to best represent the inertia of the point cloud N_i, so we want to maximize the projected inertia. Let's have a closer look at this. Recall that the inertia of the point m_i is its mass f_i-point times the square of its distance to the origin. What we therefore want is to find the plane P so that the sum of the inertia of the projected points h_i over all i is maximal. To begin, we will find just one axis, u_1, with maximal inertia. Then a second one, u_2, that maximizes the inertia, with the constraint that it must be orthogonal to u_1. The pairing u1, u2 gives the plane P. As for notation, the inertia of the s-axis will be called lambda_s. We use the word lambda because this value is in fact an eigenvalue, and we say it's the eigenvalue of rank s because it turns out to be the s-largest. Here is the graphical output of correspondence analysis applied to our dataset. So it's a plane showing both the rows and columns projections. The rows countries are blue. The columns price categories are red. So how can we interpret this plot? Something we've gone on and on about is the distance to the center of gravity. So let's look at this for the UK. It's very close to the center of gravity. Look back at the raw profile for the UK, which is approximately 25%, 6%, 8%, 28%, 12%, and 22%, and compare this with the mean profile. We can see that it's pretty close: 25% versus 21%, 6% versus 11%, 8% versus 9%, 28% versus 25, 12 versus 9, and 22 versus 26%. Clearly, the UK's profile is close to the mean profile. On our graphical output, this means that the UK ends up close to the origin. How about Italy? Well, it's quite far from the origin, which means that its profile is quite far from the mean profile. Let's take a closer look at Italy. For example, 5% of the Italian prizes are in chemistry, while the mean value is 21%. And it's reversed for literature: 32% of Italy's prizes are in this, compared with an average of 9% over the G8 countries. Now, usually, when we have a plot, we want to label the dimensions. How can we do that here? Well, let's look at each group at a time. For example, let's start with the blue points representing the countries. We see a split between the North American countries to the left, the Latin countries, France, and Italy to the right, and Japan and Germany to the bottom. It's quite difficult to label the dimension using the countries unless we know that, for example, the North Americans are known for economics and and the French and Italians for literature. Now let's look at the red points, that is the prize categories. These are spread out so that the medicine and economics prizes are to the left and literature and peace to the right. The second vertical axis separates the physics and chemistry prizes for the economics one. We could hypothesize that the first dimension separates the scientific prizes from the others, while the second separates physics and chemistry from economics. In fact, the interpretation of the dimension is the same for both the rows and columns due to the duality principle we talked about earlier. In reality, we are studying the same table in terms of its rows or columns, but it's the same table which we compare with the one we would have if there was independence. So in summary, we have seen how to visualize the point cloud of the rows and the point cloud of the columns. In the next videos, we'll look more at 2 features of correspondence analysis: inertia and simultaneous representations.
