In this instructional video, the presenter explains how to construct and interpret the point cloud of categories in Multiple Correspondence Analysis (MCA). Starting from the complete disjunctive table, he defines category coordinates (xik) and shows that each category is a point in an I‑dimensional space (I = number of individuals) with a weight proportional to its frequency. He derives the variance of a category as its squared distance from the origin and demonstrates that rarer categories lie farther from the origin, yet their inertia (distance squared times weight) remains moderate, preventing extreme rare categories from dominating the analysis. The inertia of a variable is shown to be proportional to (Kj − 1)/J, where Kj is the number of categories of variable j and J the total number of variables, explaining why variables with many categories have higher inertia that is spread over a higher‑dimensional subspace. The total inertia of all variables equals the inertia of the individuals’ cloud (K/J − 1). Principal component analysis is then applied to the category cloud, yielding an optimal visual representation (the simultaneous representation) that mirrors the earlier individuals’ plot. Transition formulas are presented that relate the optimal representations of individuals and categories via a dilation factor 1/√λs (λ_s being the eigenvalue of dimension s). These formulas reveal the double barycentric property: a category is the pseudo‑barycenter of its individuals, and an individual is the pseudo‑barycenter of its categories. The video concludes by linking these concepts to the interpretation of MCA plots and previewing the next step—using supplementary variables for further analysis.
Chapters
We have seen how to build a point cloud of individuals, and how to interpret it using the variables categories. In this video, we are going to see how to build a point cloud of categories, and then how to get an optimal representation of them. We will discover the link between the optimal representation of individuals and the optimal representation of categories by way of some transition formulas. Let's go back to the data table showing the individuals and the categories. In row i and column k, we have xik, which is equal to yik over pk minus 1. Here, yik equals 1 if the individual is in category k, and 0 otherwise. And pk is the proportion of individuals in category k, and we subtract 1 to center each column. Let's look at the point cloud of categories, that is, for the columns. Each column takes 1 value per individual, of which there are I. It's therefore a set of I numbers. From this point of view, it's a point in I-dimensional space, where each dimension corresponds to an individual. Now, here is the ith individual, which is associated with a weight of 1/i. The category is represented by the point Mk with coordinates xik. Remember that the weight of a category is proportional to how common it is in the data. When we move on to the whole set of categories, we have a cloud Nk of points. This point cloud has the following property: for a given individual i, the sum of the coordinates xik of the categories of a given variable is equal to 0. Therefore, the origin is the same thing as the center of gravity of the categories of a given variable. As the property holds for all the variables, the center of gravity of the point cloud N_k is itself the origin. Let's now calculate the variance of category k. This is equal to the square of the distance between this category and the origin in the space R^i. If we look at the square of this, it's indeed the sum of the squares of the coordinates weighted by one over i. Now what happens if we try to state this variance in terms of the complete disjunctive table? This is what we get. When all has been calculated, we get one over p k minus one. This means that the distance between a point m k representing category k. And the origin becomes larger as the category becomes, becomes less common. In multiple correspondence analysis, uncommon categories are located far from the origin. Let's look at how this distance varies as a function of PK for a few values. We can see that when PK moves from 1/2 to 1/5, the distance doubles. When PK moves from 1/5 to 1/10, it increases another 50%. Clearly, this is something important to remember, that the distance increases greatly with rarity. But don't forget, in principal component method, it's inertia and not distance that counts when constructing the dimensions. So let's look at the inertia of the case category. The inertia is the square of the distance between k and the origin multiplied by the category's weight. It is clear to see that when a category is rare, the distance increases but the weight decreases. There is therefore a pull in both directions. When all calculations are done, we get 1 minus P_k over J, which shows that rare categories have high inertia. They are therefore going to have an influence on the results coming out of the multiple correspondence analysis. Let's look at how the inertia varies in terms of frequency with J 10 variables. When PK goes from 1/2 to 1/10, the inertia practically doubles, passing from 0.05 to 0.09. Clearly, rare categories have an important influence. On the other hand, if we look at what happens when we move from 1/10 to 1/100, that is, from a rare category to an extremely rare one, the inertia barely increases. So basically, this means that MCA really does recognize rare categories but doesn't over-exaggerate the influence of extremely rare ones. Now let's calculate the distance between pairs of categories and express this distance as a function of the elements y_ik and y_ik' of the complete disjunctive table. This distance can be expressed uniquely as a function of p_k times p_k' and of the proportion of individuals that are in both categories k and k'. We call this proportion p_kk'. We see that the greater the number of individuals that are in only one of the categories, The greater the distance between the 2 categories. Now, let's calculate the inertia of the jth variable. This just means summing the inertias of that variable's categories. So, we get the inertia of the category k, and sum over the k_j categories of the jth variable. The resulting sum is k_j minus 1, over J. Therefore, we see that a variable's inertia is proportional to the number of categories it has, minus 1. For example, in a questionnaire, the variable sex has 2 categories, and therefore an inertia of 1 over J, that is, 1 over the number of variables. In contrast, the region variable has 21 categories, So the total inertia of this variable is 20 over J. At a first glance, this possible disparity between inertias, varying here by a factor of 20, seems worrying. It might make us consider only working with variables with more or less the same number of categories. However, no need to worry. Multiple correspondence analysis easily deals with these kinds of differences. Indeed, a variable j with Kj categories corresponds to Kj indicator functions in the complete disjunctive table, and therefore corresponds to a subspace with Kj-1 dimensions. The -1 comes from the fact that all of these Kj indicator functions are linked because their sum is 1. And we have seen that the inertia of the set of categories for a given variable j is proportional to K_{j-1}, which is precisely the dimension of the subspace in which the inertia is found. In other words, variables with many categories have high inertia, but this inertia is, in a way, diluted into a correspondingly large dimensional subspace. This explains why the variable sex can really only be strongly linked with one dimension. Only one dimension can strongly separate men and women. In contrast, the region variable could be linked with many dimensions, with, for example, one dimension separating the north from the south, another the west from the east, etc., etc. And to bring this to an end, we can calculate the total inertia by taking the sum of the inertias of all the variables. And surprise! We get the same value as the inertia of the point cloud of individuals, that is K over J minus 1. As has now become natural to us, we are going to get to work on this point cloud of categories using principal component method. That means sequentially constructing orthogonal dimensions that maximize the inertia. This is how the visual representation of the categories comes out. It's the optimal representation. Notice how this graph strongly resembles the one we constructed earlier when we were using the categories to interpret the individuals plot. So it's not completely the same. Nevertheless, we can interpret this new graph in mostly the same way: a first dimension, or rather a first diagonal, that separates individuals doing lots of activities from those doing few, and a second diagonal dividing generally young person activities from older person ones. In a similar kind of way, to what we did when studying the individuals, we can project individuals onto this plot of the categories. An individual can be considered like the set of categories they are in, so we can place an individual at the barycenter of its categories. We see that all the individuals end up in the center of the cloud, which makes sense because they are essentially representing means here. Now, let's take a pause and a stock of all these graphs we have plotted and try to see if we can find any connections between them. First, let's go back to the optimal representation of the point cloud of individuals that we constructed in the previous video. To interpret it, we have placed each category at the barycenter of the individuals in it. If we note gsk the coordinate of the kth category in the s dimension, we have this formula: yik equals 1 if the ith individual is in category k and 0 otherwise. So here, we have the sum of the coordinates of the individuals in category k divided by the number of individuals in category k. That is, we are calculating the mean coordinates of the individuals in category k. Now we look again the optimal representation of the categories that we built earlier in the current video. We see that the categories' coordinates are dilated, or more spread out, than in the graph on the left. Now let's project each individual onto the barycenter of the categories it's in. The coordinate value of the ith individual on the eth dimension is therefore the mean of the categories of that individual. In this formula, y_ik which we multiply with g_sk corresponds to the sum of the coordinates of the categories of the ith individual, and since an individual is in only one category per variable, and there are J variables, we divide by J to have the mean of the coordinates of the categories of that individual. The 2 graphs, on the left and on the right, are different, but one is just a dilated version of the other. In the graph on the left, if we fix the individual's position but dilate by a factor of 1 over the square root of lambda_s, this is what we get. Lambda_s is the eigenvalue associated with the eth dimension, and it's clearly a dilation because lambda_s is always less than 1. When writing this relationship, we just have to introduce the factor of 1 over the square root of lambda_s. In the graph on the right, we fix the positions of the categories and dilate now the point cloud of individuals again by this coefficient of 1 over the square root of lambda_s. In the transition formula, we only introduce this term of 1 over the square root of lambda_s. And what do we see? Well, with these 2 dilations, the graph on the left is identical to the graph on the right. This is the graph which we call the simultaneous representation. It's the graph which pops out of a software package when we run MCA. The representation of the individuals is optimal, and so is that of the categories. Both transition formulas show how to move from one representation to the other and help our interpretation of the graph. The formula on the left says that a category is at the pseudo-barycenter of the individuals that are in it. The formula on the right says that an individual is at the pseudo-barycenter of the categories it is in. Thus, an individual will be found on the same side as the categories it's in and away from the categories it's not in. Similarly, a category will be near the individuals that are in it, and away from individuals that are not. This symmetry between the 2 properties means that we call it a double barycentric property. So we have now seen how to build the point clouds of individuals and categories. In the next video, we'll have a look at some interpretation aids and consider ways to use supplementary information like the supplementary variables we saw earlier when doing MCA.
