Correspondence Analysis (part 4/5): Simultaneous representation

Now let's go back to interpreting the graphical output of correspondence analysis. This is so we can talk about another important feature of correspondence analysis. So far, we have commented on the plot in terms of the rows and in terms of the columns separately. However, correspondence analysis lets us simultaneously look at the rows and the columns on the same plot. Intuitively, this simultaneous representation is possible because the rows and columns are the same types of objects, categories of qualitative variables. This simultaneous representation of the rows and columns works in correspondence analysis due to what we call transition formulas or barycentric properties. Here, we are looking at the transition formula on the s dimension. Let's be clear what the notation means. f_s,i is the coordinate value for row i on the s dimension. Gsj is the coordinate value for column j on the s dimension. We can thus see how to link the row values with the column values. Let's look at the details. As there is a sum over the j, we know how to express the coordinate value for one of the rows with respect to the coordinate values of all the columns. fij over fi point, this is the profile's jth term. So, we are in the process of summing over the j, that is, over all the columns, the coordinate values of the columns, weighted by the elements of the i-th profile. So, in a way, we do a kind of mean of the coordinate values of the columns, and in this mean, column j is more dominant the more it has a large conditional probability for the i-th profile. Once we have calculated these centers of gravity, or barycenters if you like, we spread them out because the coefficient one over the square root of lambda s is by construction larger than one. We can write this transition formula in the following way: row i is at the barycenter of all the columns, as each column has been given the weight f i j over f i point. That is, the jth term of the ith row profile. This barycentric property can be seen intuitively in the following way. A row will be close to the columns with which it is most associated. That is, if fij over fi point is large, then gs of j plays an important role in calculating the coordinate value of i. In correspondence analysis, rows and columns play symmetric roles, so if we permute the roles played by each, we find the second barycentric property. Basically, a column is next to the rows with which it associates the most. It is this double barycentric property that lets us use the simultaneous representation we have seen, and therefore help us extract interesting information. Let's now apply these barycentric properties to a dataset we have seen before, recognizing the fundamental tastes sweet, sour, bitter. We're going to look at 2 datasets, one where there was little confusion between sour and bitter, with only 3 1 4 errors out of the 20, and another where there was more confusion between sour and bitter, with with 5 3 8 errors, that is, a mistake 40% of the time. So 20% confusion to the left, 40% to the right. We've already commented on the results of the correspondence analysis of these 2 data tables. The first eigenvalue is 1 in both cases due to the exclusive link between perceived and actual sweetness. So what consequences does this have on the barycenters? Recall for a moment the barycentric property. If lambda_s is 1, this means that one column will be at the exact barycenter of the rows. Take for example perceived sweetness. This is completely mixed up with sweet because it's only associated with sweet. On the other side of the plot, we have along the first dimension bitter, sour, perceived bitterness, and perceived sourness, which all have the same coordinate in the first dimension. Perceived bitterness is therefore exactly at the barycenter of sweetness, bitterness, and sourness because it is never linked with sweetness. As it is never associated with sweetness, the weight for sweetness is 0 in the calculation of the barycenter, for the position of perceived bitterness. Now, let's look at the second dimension. This shows clear associations between bitterness and perceived bitterness on one side, and sour and perceived sourness on the other. This is true for both plots we see here. The relative placements of all these points are completely the same to the left and to the right. This shows that it's this relative placement of the points that gives us information on their connections, that is, which categories are linked. But it says nothing about the strength of the links. For instance, is the association strong, very strong, almost exclusive? If we want to get information on the strength of the link, we have to look at the relevant eigenvalue, which is relatively high in the first case, 0.375, and quite weak in the, in the other, 0.042. The larger eigenvalue tells us that we have a much stronger association in that case. So, to be sure, we are far from having an exclusive association here, which would have given an eigenvalue of 1. Let's now take a precise look at how the barycentric property works by zooming in to the second dimension to take a closer look at the sourness and bitterness points. For the barycentric property, we must first calculate the barycenter and afterwards dilate or stretch the point by a factor of 1 over the square root of lambda_s. Take for example constructing the perceived bitterness point. To begin with, it's the barycenter of the sour and bitter points with coefficients 1 and 7. So if we calculate the barycenter of sour and bitter with coefficients 1/8 and 7/8, we land on the red circle, which is much closer to the bitter point than to the sour one. Let's do the same calculation for the right-hand side. This time, we calculate the barycenter of sour and bitter, whose coefficients are 3 and 5. We get the red circle, which is this time next to bitter, but only slightly. Thus, we see that looking at the barycenter gives us a good idea of the intensity of the link between the 2. What does correspondence analysis do next? It multiplies these values by 1 over the square root of lambda_s. This value, 1 over the square root of lambda_s, is quite different between the 2 examples, because it equals 1.6 in the first case, and 4.9 in the second. In other words, the point cloud of barycenters will be much more spread out in the right-hand side that is, when we have a weak link, than when we have a stronger one. In this way, correspondence analysis neutralizes the effect of the strength of the link and only keeps the information about the type of link. You might ask, why do we do this? The first reason is obvious. It's that we have exactly the same relationship between a row and a set of columns, and a column and a set of rows. We can clearly see the symmetric role played by the rows and columns. Another way to see this is to say, if we look at the barycenters when there is a weak link, we are going to have difficulty detecting the associations. Therefore, correspondence analysis spreads out the plot, a bit like a microscope, to make associations clearer. In conclusion, we can say that correspondence analysis says nothing about the significance of a link, Instead, the strength of a link is found in the eigenvalues, whereas the type of link is found in the relative positions of the points. Let's now use the double barycentric property of the Nobel Prize data. Recall that the center of gravity of the cloud is at the barycenter of the red points weighted by the mean profile. We should thus note down each category the weight of the category that maps the mean profile to the clouds' barycenter. Now look at Japan, for instance. Japan is the profile 26, 0, 9, 13, 4, and 48. It's therefore the barycenter of the red triangles that have been assigned these numbers. If we use a font size proportional to the weight of a category, we see that Japan has very strong weights for physics and chemistry compared with the size of these weights in the mean profile. Japan will therefore be positioned at the barycenter of the categories assigned these weights, which pulls Japan towards the bottom of the plot on the physics and chemistry side. This is indeed the barycentric property: a row point goes towards the columns it is the most associated with and away from columns it's the least associated with. Note that once the barycenter is calculated, the point is dilated with a coefficient 1 over the square root of lambda_s. This best explains why a barycenter can be outside of its points. Now, if we look at Italy, we know that it's much more associated with literature than economics, and so clearly its point is close to literature. Of course, we know that Italy has large percentages for physics and medicine too, But in reality, these are similar to that of the mean profile. Thus, these 2 percentages don't really push Italy away from the cloud's barycenter. Thanks to these barycentric properties, we can now simultaneously interpret the red and blue points, that is the rows and columns. We've already said that the first dimension separates the scientific prizes on the left from the others to the right. Now, we can also say that the US mostly gets scientific Nobel Prizes while the French and Italians are more likely to get the peace and literature prizes. The fact that Italy is more towards the edge than France shows that the Italians are even more specialized in the literature category than the French. Quantitatively, they get less literature prizes than the French But in Italy, the literature prize dominates with respect to the other categories. As for the 2nd dimension, it pushes physics and chemistry towards the bottom, and we see that it's the Germans and Japanese who obtain these prizes more than they obtain other prizes. We have seen how to interpret the simultaneous plot of correspondence analysis. In the next video, we'll see some useful interpretation aids.