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Correspondence Analysis (part 3/5): Inertia and percentage of inertia – We've now seen how to plot point clouds for
the rows, and point clouds for the columns, and how to project them onto graphical plots.
In correspondence analysis, like all principal component methods, the first indices we want
to look at are the percentages of inertia.
We'll see in this section that in correspondence
analysis, the inertia is a rather particular thing. Let's start by talking about the percentage of inertia. Like in all principal component methods,
percentages of inertia are the first indices we look at.
The question we want to ask is the following: we have a point cloud representation.
How good is it? Generally speaking, the quality of the representation
is measured with respect to the ratio of the projected inertia over the total inertia.
Usually, we multiply this by 100 to get a percentage. In the precise case of a cloud N_I and the s-th dimension, the quality of representation
of the cloud N_I onto the s-th dimension is the projected inertia of N_I onto this dimension,
divided by the total inertia of N_I. We can write this in a way you've already seen,
with lambda_s over the sum of the lambda_k. As we just said, this is then given as a percentage.
Let's look at the percentages of inertia in our example. We see that in the first dimension, the percentage of inertia is 54.75, which is quite high.
We can therefore say that the 1st dimension represents 54.75% of the deviation from independence. In the 2nd dimension, the percentage of inertia is 24.60. Thus, the first two dimensions together represent almost 79% of the deviation from independence. Essentially, this means that we can just stop,
and interpret these two dimensions only. Here's a mathematical property of what we have done: the projected inertia, i.e, the eigenvalues,
can be added up from each dimension to the next. This comes from the fact that the dimensions
are orthogonal. Therefore, the sum of all the eigenvalues,
or the sum of the projected inertia if you like, equals the total inertia of the point cloud N_I. This is true for all principal
component methods. In the particular case of correspondence analysis,
this total inertia equals Phi². Let's do a few calculations using correspondence
analysis on our example: we can multiply the total inertia, 0.1522, by n,
the sum of the table, which is 570, and we get a chi² value of 86.75. Given the number of degrees of freedom here, the p-value is 2.77 * 10e-6, which is tiny. There is clearly a highly significant connection
between the countries and the prize categories in our example. Here is something else we can do with
the percentages of inertia. As these decrease as the rank of the dimension increases, we can use this decreasing sequence
as a guide to find a cut-off, i.e., choose the number of dimensions to keep.
As an example, here is the decreasing sequence of eigenvalues of a correspondence analysis
on a contingency table facing off 10 white wines from the Loire region with 30 descriptors.
The wines are in the rows, the descriptors in the columns, and xij is the number of times
the descriptor j was associated with wine i. When we look at the decreasing sequence of eigenvalues using a bar plot, it's clear that
the first two eigenvalues are much larger than the others.
The first two dimensions therefore dominate in terms of inertia, suggesting that the best
way to interpret the data is to just look at the plane defined by these two dimensions.
In correspondence analysis, it is useful to distinguish between the inertia, and the percentage
of inertia, because the inertia are themselves interesting. They make up part of phi², which gives a global measure of the link between two variables. In correspondence analysis, the following theoretical result is very important: the eigenvalues are always between 0 and 1. Recall that in principal components analysis,
where the variables are standardized, it's different, because the first eigenvalue is
automatically greater than or equal to 1. What does it mean to have an eigenvalue of
1 in correspondence analysis? This limit case is quite interesting, as we
will see. And what does the data look like in this case?
Well, it's like this: We can separate the rows into two blocks,
I1 and I2. The columns can also be separated into two
blocks, J1 and J2. And suppose that this double division represents
exclusivity between blocks, i.e., rows of block I1 are only linked with J1, and not
at all with J2. And rows in block I2 are exclusively linked
with J2, and not at all with J1. This represents a very strong association,
because we have an exclusive link between categories of one variable and those of the
other. What does this look like graphically?
We end up with this graph. The first dimension, corresponding to the
eigenvalue of 1, perfectly separates the block I1 from I2.
I.e., inside I1, no distinctions are made, and we perfectly separate J1 and J2.
Inside J1, no distinctions are made. These curious inertia, let's look at them
again in another data set. This data is about recognizing three tastes:
sweet, sour and bitter. The experimental design is: for each taste,
we have asked ten people to try and recognize the taste of a sample they are given.
Here is the little data table of results. Let's read it row by row.
The sweet sample was identified as sweet 10 times out of 10, and never mistaken for sour
or bitter. The sour sample was never taken for sweet,
but was once mistaken for bitter. Similarly, the bitter sample was never taken
for sweet, but 3 times mistaken for sour. When we now do correspondence analysis on
this table, we obtain the first eigenvalue of 1, which is the signal telling us we have
a diagonal block structure in the data. What this means is: if we look back at the
table, we see that all the non-zero data is found in blocks along the diagonal.
For instance, here, sweetness is only ever perceived as sweetness, and no other taste
is ever perceived as sweet. We therefore have this eigenvalue of 1.
The corresponding plot's first dimension therefore perfectly separates sweet and perceived sweetness,
with one point on top of the other, with bitter, perceived bitterness, sour, and perceived
sourness all close together on the other side. Now, let's move on to look at the second dimension.
This separates bitter and perceived bitterness on one side, from sour and perceived sourness
on the other. It basically shows that bitter is usually
perceived as bitter, and sour as sour. And yes, this is exactly what happened.
BUT, when we look back at the data table, it's a little confusing.
Sour isn't always perceived as such, and nor is bitter. How can we tease this out of the math? Well, quite simply by looking at the eigenvalues.
If there had been no perception errors between sour and bitter, the eigenvalue would have
been 1. But here it's only 0.375, so,
much less than 1. This is the indicator that there has been
confusion between sour and bitter, i.e., from time to time, one is perceived as the other.
This is clearly visible in the graphical output. On the first dimension, we have a kind of
perfect situation, in that we clearly see the separation between sweet on one side,
and sour and bitter on the other, with a much larger inertia in this separation than between
sour and bitter. These two are much closer to each other than
with sweet. So, the plot clearly shows that there has
been much more confusion between sour and bitter than between either of these two and
sweet. Let's now look at another data table in which
the confusion between the categories is increased. i.e., this time, sour was only perceived as
such 7 times instead of 9. And bitterness was only correctly detected
5 times. Still, sweetness was always correctly detected,
and nothing else was perceived as sweetness. So, how does this affect the math?
Well, we still find the first eigenvalue of 1, corresponding to the perfect separation
between sweetness and the others. But now, for the second dimension, the eigenvalue
has dropped from 0.375 to 0.04. Now, look at the graphical output.
It still looks similar to the previous one. The first dimension still perfectly separates
sweet from the others. What's changed is that the separation between
sour and bitter is much less than before. This is the way the plot tells us that there
has been a lot of confusion between sour and bitter. This is what leads to the small eigenvalue of 0.04 associated with this 2nd dimension.
How about in the most extreme case, where there is no confusion at all, and the data
table is diagonal? Well, we'd end up with two eigenvalues equal
to 1. So what have we learned from the two examples
we see here? We see that in both, the second dimension
is always showing the same thing: it's separating bitter and perceived bitterness from sour
and perceived sourness. So, basically, the second dimension means
the same thing in both examples. It's showing that bitter is mostly perceived
as bitter, and sour as sour. But now, from one example to the other, overall
it's not the same thing, because, in the first example, we can say that recognition is good,
whereas in the second, it's pretty poor. This teaches us that the plot indeed shows
the contrast between sour and bitter, but says nothing about the strength of this contrast.
It's the eigenvalue that's going to tell us whether the separation is strong or not.
If it's equal to 1, the separation is very strong, and in fact represents a total separation,
an exclusive relationship. If on the other hand, the eigenvalue is tiny,
like we see in the second example with 0.04, we've struggled to get a majority of the predictions
right. Bitterness is only correctly recognized half
the time, and sourness 7 times in 10. Really, this means that in correspondence
analysis, we should always start by looking at the eigenvalues, because it shows whether
the links we find in the data are weak or strong. The graphical output doesn't show us this, because it doesn't give information on the
strength of links, only their type. Here, the link is simply that sour is mostly
associated with perceived sourness, and bitter with perceived bitterness. Let's now go back and have a look at
the Nobel prize data. We've already talked about the percentages of inertia of 55 and 25, and how these were
large compared to the next dimensions. We therefore decided to keep just these two
when moving to the interpretation phase. However, the actual values for the inertia
are 0.083 and 0.037, which are quite weak, especially compared with the value 1.
We would have had a value of 1 if there had been at least one exclusive association between
categories and countries. For example, all the prizes for one category
went to only one country, and that country had no prizes in any other category. Clearly, therefore, we are far away from this situation. Indeed, the Nobel prizes for the various categories
are well spread out across the countries. If we now look at the sum of the inertias, and knowing that in correspondence analysis,
this sum is Phi², it can have at most a value of 5, if each of the inertias equals 1. We are very far from this case.
Looking at the table again, we are clearly extremely far from having exclusive links
between the categories of the two variables. We’ve seen that inertias have a specific
meaning in correspondence analysis and should be analyzed before interpreting. In the next video we’ll see how correspondence analysis allows us to simultaneously look
at the rows and the columns on the same plot. .
How to split 24 marbles in the ratio 3:5 – .
More on Percents, Convert 1/5, 3/5, 1/3, & 2/3 to Decimals, Mark Fractions on Number Line – .