Good morning. Today, we will discuss correspondence
analysis. Correspondence analysis is a multivariate analysis, multivariate analysis. It has two
components; one - it deals with categorical data, it deals with categorical data and then it converts this data to certain distance, distance. Using those distance values, it creates perceptual map, create perceptual map. So, the one of the advantage of correspondence analysis is that that it can it can work with categorical data. By categorical data we mean the nominal or ordinal data, nominal and or
ordinal data. Actually, in correspondence analysis, the
data are observed in terms of counts or frequency, in terms of frequency. For example, let there
are three age group of people, age 1, age 2 and age 3. Now, we can see their food habits,
food habits, habit 1, habit 2 and habit 3. Now, if you see a particular population and
we may find that in general, this age group with this food habit there may be n 11, n
12, n 13, n 21, n 22, n 23, 31, 32, 33. Now, this n ij, this is number of observations
or I can say counts or say that frequency that is frequency data.
The question comes, that is there any relationship between these frequencies observed with the
variable called age, which is categorized into three groups. Is there any relationship
between the observations here, for example this with the food habits? Is there relationship
between age and food habit in realizing this frequency counts. So, all those things can
be captured through correspondence analysis. Now, if we go back to the history of correspondence
analysis, we will find that correspondence analysis was developed by Jean Paul Benzecri
and his colleagues in 1960s. This is a geometric approach on multivariate descriptive data
analysis. So, the way Jean Paul Benzecri and his team developed the correspondence analysis
that same approach we will follow here. So, let us see that what a correspondence
analysis can do. Correspondence analysis is a dimension reduction technique similar to
factor analysis, but extends factor analysis in two counts; handling of categorical variable,
particularly measured in nominal scale and developing perceptual maps of extracted components.
So, what I mean to say here? So, what I mean to say here is that you require
developing a contingency table. Contingency table or cross table, both are same. When
you say at contingency table or cross table, this is basically bivariate table and here
you see although age is a continuous variable, but age is made as it is a nominal variable
by categorizing into three age groups, but food habit is definitely a nominal variable.
So, if you want to use correspondence analysis, you have to have the variables of interest
in nominal scale. Now, if you, if you use ordinal data also,
it will be treated as nominal data. If you use continuous data, then you convert them
to nominal data. The sole purpose is to understand whether there is correspondence between between
the observed frequencies and the variables considered either independently or jointly.
It is similar to in case of ANOVA. We have seen that there are two things in ANOVA. One
is main effects and interaction effects. Now, main effects are related to the factors like
your age and food habits, interaction effect between the factors in similar way, but in
MANOVA. In ANOVA, what is required? It is required,
the data, the response variables will be continuous in nature here it is different and correspondence
analysis has also relations similarity with log linear model. In log linear model, we
also try to find out main effects and interaction effects using certain parameter, but there
definitely are a lot of differences between all the correspondence analyses between these
models. First of all, correspondence analysis uses
frequency data. Log linear model also uses frequency data, but log linear model will
not go for dimension reduction. Correspondence analysis will go for dimension reduction.
Also, it is one way a principal component analysis or similar to factor factor analysis.
Like in another way, it is having also a capability to identify the relationship between the categories
across variables, categories between variables. It also has the potential to create map. By
seeing the map, we are in a position to talk about the association along the categories
of the variables and between the categories of two variables; the associations can be
carefully interpreted. Now, see, what are the different names we
find in the literature related to correspondence analysis? Dual scaling, method of reciprocal
averages, optimal scaling, canonical analysis of contingency tables, categorical discriminant
analysis, homogeneity analysis, quantification of qualitative data, what is in, what is mean
and seen also that in these different names and that similar to correspondence analysis
are, in fact the correspondence analysis is used in the literature and also published.
Now, correspondence analysis another important issue here is that it captures the non linearity
between the variables represented in the contingency table. So, let us see that how correspondence analysis
works. Now, this one, we will be representing in terms of correspondence analysis process. That process model some is inputs, then activities,
then outputs. Inputs means what correspondence analysis requires. So, if you see that it
requires some certain number of variables, which are categorical variables, then you
have to create a contingency table, contingency table of frequency data frequency data. Then
what it will do? It will do three things; captures measure of the association, captures
measure of association, then compute feet measures and then create perceptual maps.
So, what are the outputs? Then there are some one is the perceptual maps, perceptual maps.
Then second output is different feet measures and actually we will get many numerical numerical
measures, numerical measures for interpretation for interpretation of ca results. So, that
means if you want to use correspondence analysis for some purposes, what you have to do? You
have to first find out what are the variables of interest, list those variables, collect
data, then what will be the data structure? Data structure will be similar to like this
i 1 to n number of observations. Now, your categorical variable 1, categorical variable
2, so like categorical variable p, then what is required? What will be required?
Suppose observation 1, it may fall with one categorical variable with some category some
category. Suppose it is basically xc 1 category, xc 2 category xcp category, similarly different
categories. So, when there are more than two variables with multiple categories, then you
require multiple correspondence analyses to include all, all the categorical variables.
But, if we consider only two variables like this x 1, x 2 with this category, then you
will be able to develop suppose you write x 1 here and this side x 2. You know it has
category, category 1 category 2 to suppose category k and it has category, category 1,
category 2, suppose category kk. Then you will finally, get a frequency table
like this n 11, n 12, n 1n, I think it is k. So, it will be k, a similar table will
be, so that means your data will be coming like this. Then I will show you one example
and and from this data, you take two variables at a time with them. These are the categories
of the variables, related variables. Using this and n, we will be able to find out this
correspondence table or contingency table. We will be able to find out this contingency
table. Now, a sum total of all these n values that
will be capital N because we have considered n number of observations. This is small n,
sorry not capital N, small n, we are measuring this small n, so n 11, n 12, what I do? I
mean I mean to say that sum summation i equal to suppose if I say this is k, j equal to
11 to capital K nij, this will be n sum total of all will be n. So, this is our data. Now,
from this data this way, you are getting the contingency table or correspondence table.
If you have more than two variables, two categorical variables, you have to go for multiple correspondence
or contingency table and or we can say the nested contingency tables.
Then, what we are trying to say that what are the activities that will be performed?
You have to first find out the measure of association. How do you find out this measure
of association, this is an issue and we will discuss next. Then once you find out the association
measure, and then you will you will, using those values, you do some action and then
some results you get. Based on these results, you find out the feet measures and finally,
maps you create. So, let us start with a case study because
it will help us to understand ca, an organization which manufactures diagnostic medical equipment.
This is our case. The organization does assembly of final device and has outsourced all the
components required for the final assembly. This one is one of our case study one paper;
that is reference is Aravindan S and Maiti 2012. A framework for integrated analysis
of quality defects in supply chain, ASQ quality management journal, volume 19, issue 1, page
number 34 to 52. Now, let us see that what is this case study in totality? That we found out that that compare, if you
see the what are the different components that sheet metal component, plastic and non
metal component, fabrication and machining, circuit board and electronics, cables and
hamess, these are the different components where ultimately defects per million, defective
parts per million that is captured from six sigma point of view. What we found out that
the sheet metal component has been the highest, highest defects. So, it is a case. So, we
can say this is the sheet metal case we have to consider for improvement. Then, when we capture the defect data, unrelated
data with respect to the sheet metal component, what you find that there are four things,
four categorical variables, which can be used to capture the that data set totally in totality.
By this, I am not saying that there are no other variables which are contributing to
the sheet metal defects. It is there, but in for the present study what has happened
with discussion with the personnel, who are basically working there in this particular
company. And who are dealing day to day activities related to this assembly operation and with
discussion with the, we find that these are the things that can be considered.
One is for sheet metal component; there is part number, box structure component, double
bend component, single bend component, flat component. That means four categories. There
are different vendors. Actually, that company having more than fifty vendors, around seventy
something vendors are there, but four numbers of vendors, they are supplying this sheet
metal component, so vendor 1, vendor 2, vendor 3, vendor 4. This is another categorical variable.
Then, we our definitely, we want to minimize the defects. So, in order to minimize the
defects, we what we have done? We have first seen that what types of defects are generated
during assembly operations or after assembly final product when it is sent to the customers,
what type of defects are reported plus in house that quality checking what type of defects
are identified? Based on this, we found that the category of defects. Anyway, this is also
categorical variable having one two three four five six seven eight nine different categories.
Then, it is obvious that there may be seasonal effect, which was which was found after discussion
with the personnel; personnel involved in this assembly operation January February to
September because we have collected data from January to September of a particular year.
So, we could not collect beyond that because ultimate in this assembly operation where
you have consulted our study that started in this month and that was that up to September
that data was available. So, by saying this, what I mean to say that all variables are
categorical in nature. Now, let us see that how do we develop contingency
table. So, as there are four categorical variables and we want to take two at a time, so that
means we will be able to find out 4 C 2 contingency tables. So, 6 contingency tables that mean
six two way two way relationships will be formed out of these four variables. Let us see one after another here categorical
data in terms of frequency. So, this is the categories of defects, these are the vendors
and what does it mean that vendor 1 supply components having this BD. BD means BD is
hole missing bending dimension. I think it is basically bend and dent, B is bend and
dent D stands for deburring, DM dimension related, FI fastener installation, so like
this, so 150 bend dent defects can be attributed to vendor 1. Similarly, 142 responsible vendor
2, for 146 vendor 3 and vendor 4, 57. So, if you see this particular column, what
you find that ultimately vendor 4 is having less number of defects related to bend and
dent in comparison to other 3 vendors that is vendor 1, 2 and 3. They are almost equally,
equally contributing towards defects related to bend and dent. Similar, in similar manner,
you will be able to find out this. This that means deburring defects, dimension related
defects like this. So, if I go for the total, what you found out that that vendor 1 1076,
vendor 2 1141, vendor 3 1043, vendor 4 1181, but here in first case, we are finding out
vendor 4 is least contributing. Whereas, if I see the total, we are able to see that this
vendor 4 is contributing the maximum although with respect to vendor 2.
The difference is not that significant. So, that means there are some relationship may
be. What I mean to say as here it is less that means somewhere it is contributing more.
For example, if you see the dimension related things, vendor 4 is contributing 269 defects
compared to others where vendor 1 is 207. So, in this manner, what I mean to, what I
mean to say that, so there may be relationship between the vendors and categories of defects.
So, this table is known as contingency table. Now, you can generate similar contingency
table for what are those things? First one, we have done category of defects
versus vendor, this is one. Then category of defect versus you can go for part number,
then your category of defect versus your time, then your vendor versus part number, then
you can say vendor versus time, then number is, sixth one is your vendor, part number
versus time, six tables. Now, six simple correspondence analyses can be done. So, we can say this
is M1, M2, so like this. So, there will be M6 model 6, M stands for model. Now, with respect to the first model, what
we have is the category versus vendor. We have put few questions. What are those questions?
What are the similarities and differences among the 4-vendors with respect to the 8-category
of defects. So, we are saying that there are 8 categories 1, 2, 3, 4, 5, 6, 7, 8, 8 categories
of defects. Now, that means if I want to see the vendor position, then you are basically
getting an eight dimensional issue coordinate. That means what I mean to say? I mean to say that category of defects that
is BD, that D, similarly W, so if it dimension 1, dimension 2, this is dimension 8, not visible
here, not visible here, but it is very difficult. What will happen? Now, the vendor 1 has a
position suppose somewhere here, so this one is having eight coordinate values that means
vendor 1 is measured using this eight categories in terms of one that values are 157, 137,
207, 91 76, 210, 185, 20. Vendor 2 coordinate values are like this, vendor 3 coordinate
values are like this, vendor 4 coordinate values are like this.
What we are saying that then what are the similarities and differences among four vendors.
So, we using this, this row, this row, this row values, this row values, we want to see
whether there is differences between vendor 1, vendor 2, vendor 3, and vendor 4. Are they
similar? Are they dissimilar? Note only vendor 1 to vendor 4 four with BD or with D or with
DM or FI, we are talking about vendor similarity or differences between vendor 1, vendor 2,
vendor 3 and vendor 4 in terms of all the defects. Second one is what are the similarities
and differences among the eight categories of defects with respect to four vendors? What
does it mean? Let us simply say that we have four vendors. We are now considering it four dimensional
system that vendor 1, vendor 2, vendor 3, vendor 4, 1, 2, 3, and 4. Now, for a particular
defect, you may be say seeing somewhere here in a four dimensional phase, BD is there.
Now, what is this BDs point, then that coordinates, then 150, 142, 146 and 57.
Now, if I, if we want to know that our this question, what are the similarities, differences
among eight categories of defect with respect to four vendors, so you want to see BD, D,
DM, FI, P, S, W, they are similar or dissimilar or if similar, how much similar or how much
dissimilar that this vendors with respect to, sorry with respect this, these as well
as this eight categories with respect to this four vendors, how they were similar and dissimilar.
That is what is written here, the similarities and differences amongst the eight categories
of defects with respect to the four vendors. Now, obviously the third question is what
is the relationship between the vendor and the category of defects? These are the vendors
and this category of defect. So, when you talk about the first one that what are the
similarities and differences among the four vendors with respect to the eight categories
of defects, it is similar to main effects of vendor. Second one is similar to main effects
of categories of defect bit. Third question is talking about the interactions between
vendor and defects. Those interactions we are considering here.
Now, the fourth one that can these relationships be represented graphically in a joint low
dimensional space because you have seen that eight vendors, eight categories of defects.
If we consider and if you measure vendor 1 to vendor 4, then a particular vendor has
eight coordinates, eight dimensional issues. So, but again, when we are measuring the defects
category, each of the category with respect to vendors is four dimensional issues. Now,
question is the, more the dimension, the complexity is more difficult, very difficult to make
decisions. So, what we require? We require reducing the dimension as well.
So, that is why that is the fourth question can these relationships be represented graphically
in a joint to low dimensional space. It is possible if there is a relationship amongst
the categories of the defects amongst the vendors as well as between categories of defects
and vendors. If there is relationship dependency, then the reduction is possible, low dimensional
reduction is possible. If it is not there, they are independent, then you cannot reduce
the dimension; either you have to go by the vendor wise, that four dimensional issue or
category wise eight dimensional issue. You cannot reduce reduction. Reduction is possible
only when there is some dependency, some similarity between the anti similarities.
So, now we want to see that again repeat the questions. Question one is our that we are
basically four vendors, your target is vendor similarity and dissimilarity, in question
two category of defects similarity and dissimilarity, question three what you are trying to say?
You are trying to find out relationship between vendor and defect category of defects and
question four; you want to reduce the dimensional reduction. So, what are the answers to these
questions? How do we know? Question number one is answered by row profiles,
row profiles. Please keep in mind that here along the row;
vendors are there along the rows, first row vendor 1 like this. That is why when we are
talking about vendor similarity and dissimilarity; I am talking about row profile, but if you
change interchange, suppose the categories of defects will come along the rows and vendor
along the columns. Then when we talk about row profile, it will be related to the category
of defects. So, it is the way you design the contingency table and accordingly the row
profile and column profile will give you the concerned reply or responses or answers.
Now, for categories of defect is column, row profile is R and it will be column profile
that is C. Now, is there any relationship between these or not? That we will be capturing
through weighted chi square distance. The final one that is dimension reduction possible
or not, you will be capturing through singular value decomposition SVD. Actually, what happened?
SVD will you has different new dimensions, hidden dimensions, latent dimensions and then
we will consider two of the most prominent dimensions. Then we create perceptual map
using this dimensions, perceptual maps leaving this two where we are seeing answering the
fourth one. Now, let us see that what is row profile? So, there are different steps to perform correspondence
analysis. The contingency table what you see earlier, this one, we are defining this as
X, capital X. That 4 vendors, 1, 2, 3, 4, 8 defects, 1 to 8, everywhere some value is
there. This is X and this value is x11, x12, x18, like this, similarly x41, x42, and x48.
This is basically frequency data contingency table, whatever way you can define like this. Now, what is our step one? Now, what is our
step one? Step one is you develop correspondence matrix. You see what is it in there obtain
correspondence matrix Z where each element in Z is that xij divided by capital N, where
capital N is the total number of observations. So, what I mean to say here? I mean to say
this 4441, this is the total number of defects found for this particular case for I think
up to September that is nine months data. Then this is capital N, 150 here, this is
x 11, 137 is x 12, so we are now creating z11, which will be 150 by 4441. That means
each of the element in this x table here will be divided by the total that is what we are
talking about here. So, 0.34 is nothing but 150 divided by 4441 and 4441 when divided
by 4441, this will be 1. So, this is your first step. Then so that mean if I write here
that what is our step one. Step one is let us found out zij which is
xij by N. Now, go to step two, divide each element of
matrix Z by the respective row mass. If you see what is row mass here, now you go back
to the original table 1076 divided by 4441, this is what is your row mass? So, that means
row total divided by total, so first step one is this step two each of this elements
will be divided by row mass. So, that is why, what happened 0.034 divided by 0.242, it is
giving you 0.139. So, each of the elements of the Z matrix is divided by their corresponding
row mass. So, vendor 1 case, each of the element here, these elements are divided by this vendor
2 case. Each of these elements of z matrix is divided by 0.257, so like this.
So, let us write down the step two, step two we are saying that zij when we divided by
row mass, keep in mind the corresponding row mass. You see that in the first step one,
the column values mass if you see that column mass is your column mass is this, 495 divided
by 4441, 474 divided by this. That means the column total divided by the total grand total.
So, column total divided by grand total is like this. If you sum up, row mass is 0.242,
0.257, 0.235, 0.266, this will lead to 1. If sum up the column masses, this will lead
to 1. So, while calculating row profile, first we
are finding out this that the correspondence matrix. Then what you are doing? You are basically
dividing each of the elements of the correspondence matrix by their corresponding row masses,
but see you are not considering column mass here because we are talking about row profiles. So, now I think you understand now that what
is your row profile. So, that vendor 1 position, if we consider in terms of row profiles, so
that mean with respect to this eight categories of defects. So, vendor 1 position is 0.139,
0.127, 0.192, 0.085 like this, vendor 2 like this, vendor 3 like this, vendor 4 like this.
So, under eight dimensions, you are basically putting that where is vendor 1. Getting me?
Where is vendor 2? So, you are getting these values. So, in the similar manner, you can
find out the column profiles. Step three is we want to find out, so this
is your row profile, you got in second stage. Third stage, you want to know the column profile.
What you will do? Again, you will basically zij by column mass that this work into basically,
so when I talk about divide each element of the correspondence matrix Z by the respective
column mass, this is the case. So, each one this is my that original correspondence matrix
Z, 0.034 is divided by 0.111. This is what is given here, 0.303.
Similarly, 0.287 like this. So, what I does it mean? That means that when you are talking
about column profile, we are talking about here different categories of defects. So,
in a four dimensional space, where the dimensions are created by the four vendors, where does
each category of defect lies? So, I think that the first that row profile will give
you similarity, dissimilarity and column of the row variables categories, column profile
will give you the similarity, dissimilarity between the column variable categories. Now,
we want to combine the two. So, we require having some other measure what is known as
step four. Here that weighted chi square distance, so
weighted chi square distance is calculated like this. Your step four is that weighted chi square
distance, which we are talking about as D and we are writing this D equal to Dr minus
half Z minus rc transpose into Dc minus half. So, this is basically r stands for row, c
stands for column. What way it is done? Now, Z, what is the Z matrix? This we have seen
that 4 categories of 4 vendors cross 8 categories of defects. Now, r which is row profile, which
is 4 cross 1, so that means this one is nothing but 0.242, 0.257, 0.235, 0.266.
Now, what is your c? c is 8 cross 1, 8 cross 1 that is 0.111, in the same manner that 0.025.
So, if I multiplied this two rc transpose, this will ultimately lead to 4 cross 1, 1
cross 8. So, 4 cross 8, which is matching with this one, this totality, it is a matrix
of dimension 4 cross 8. Then what is our Dr? You also require to know what is Dr? Dr here, it will be a diagonal matrix, where
the diagonal elements for these will be the row masses and for Dc, it will be the again
a diagonal matrix, diagonal elements will be the column masses, what is off diagonal
elements zero, both the cases. So, what will happen ultimately? Then by D minus half, you
are taking the inverse and then square root of the inverse basically what you are doing
here. So, if I say like this 4 cross 4, then 8 cross 8. Then this will be 4 cross 8. Ultimately,
your resultant matrix will be 4 cross 8, total matrix is 4 cross 8.
This is what is chi square distance and this chi square distance, if we use this formulation
where your rc, Dr, Dc everything is available because you have the contingency table, the
data is given to you. You see I have this is r, this one my c. Then I put create a 4
cross 4 matrix putting diagonal, all this values, we will be getting Dr; putting diagonal,
all these values, we will be getting Dc. Then you take that inverse square root of each
of the values there. Then you start multiply multiplying this with this. Then the resultant
will be multiplied with this. So, ultimately you will be getting this. This is what we
are talking about chi squared distance. What is the issue here is that you see r and
c? You are seeing r and c, r is this row mass and this column mass. These two you are multiplying,
these are again mass which is divided by the total. This two you are multiplying and you
are subtracting from from Z, original Z, this Z value, so this value minus this cross this.
So, if there is no independency between this two, vendor versus this from the marginal
probability concept point of view, what we know this joint probability here, this will
be the sum total of that is the multiple case of the marginal probability independent case.
So, in that case, what will happen? These cross these or this minus this cross, this
will become 0, distance will be 0. There is some distance that means there is some relationship
between these two sides. Any problem? I do not think there will be any problem, but is
a very straight forward thing. So, what is written here. You see D can be used to explain
the relationships between vendors and categories of defects this distance, but it is see it
is a really difficult proposition. It is very easy to say, but when you talk about vendor,
it has distance with respect to this all categories of defects.
Similarly, when you talk about the defects, it has also with respect to the vendors. So,
it is not straight forward as the way we thing, but even then but this is beautiful of of
see from the categorical data, nominal data where no nothing is available. Only some frequency
is available. Now, how nicely you are able to convert to it in a distance measures and
the chi square distance? So, if I know distance, distance is a position. I know one vendor
position, another vendor position in terms of distance. So, then it is possible for me
to see what is the difference in distance and some conclusions can be made. Then, what is required? Then we will go for
singular value decomposition after step four. Then, step five is your singular value decomposition.
Now, singular value decomposition, I think what is required here that it is a matrix
algebra issue. I think that any matrix can be decomposed depending on the nature of matrix.
If it is a square matrix, you can go for Eigen value, Eigen vector decomposition. When it
is not square, then you cannot go for that. So, singular value decomposition can be composed
is non square matrices. For example, in this case, our D is 4 into 8. Now, you want this
is not a square matrix, 4 cross cross 8 is not a square matrix.
What you want to do? We want to decompose this matrix and we we want this decomposition
is very, very interesting. I think in my PCA lecture physical component analysis, we have
seen that we have decomposed a matrix into Eigen value and Eigen vectors. The way that
lambda j sum total of j equal to 1 to p lambda j is a, is a transpose that is since we decompose
means one matrix p cross p matrix converted to like this one scalar and with vectors it
is also similar way it is done. Now, you see that SVD, what it does? It basically partitions
the D matrix into three matrices U, V and S. In case of Eigen value, Eigen vector you find out the Eigen vectors and Eigen values. Here,
three different matrices we are subtracting that decomposition is taking place where U
is a p cross k matrix, V is a q cross k matrix and S is a k cross k matrix with diagonal
elements. These S diagonal elements are s1 greater than equal to s2 greater than equal
to sk. k is the reduced dimension. What we have I am trying to say, we have 4 cross 8
matrix. Now, we are saying there is independence and
because of this feature, we are saying that this matrix easily can be, can have reduced
dimensions. So, that reduced dimension, what is reduced dimension? That k is minimum p
minus 1 q minus 1. What is p? p is suppose p is the rows, number of rows. We have 4 rows,
we have 8 columns. So, p minus 1 is 4 minus 1, 3, q minus 1 is 8 minus 1, 7. So, the k
can be minimum. k is minimum of 3 and 7, which is three.
Now, so that mean, what I mean to say this 4 cross 8 matrix, it can be decomposed into
lower dimensions that is the maximum, that dimensions will be 3 like principal component
we will be extract. We can be extracting 3 PCS from this D matrix chi squared distance
matrix. So, two components are always preferable because then the visual interpretation, the
graphing of the two components, all those things will be possible, but three component
also possible to map graphically. So, that means essentially what you are doing
then? Your D matrix, which is we are talking about D is p cross q matrix, this p cross
q matrix, this you are creating like this U that is p cross k, S k cross k and V k cross
q. So, if you say no it is then V transpose you write because we want U will be p cross
k, V will be q cross k and S definitely k cross k. The element of this S, the diagonal
element of this S, which are s1, s2 like sk, these are important things. These are the
things what we we want to extract. Getting me?
Now, see what we are, we are doing here, sk is the singular value. This sk value is known
as the singular value for the kth PC. So, that means what we are saying here? We are
saying that we are we are able to extract k PCs, k principal components. s1 is related
to first principal component, s2 is related to second principal component like this sk
is related to kth principal component. So, this is the singular, all singular values,
this singular value if you square it, you will be getting Eigen value. All of you know
by this time that Eigen values are the variance explains component.
What is the amount of variance of the original data set captured by the kth principal component
in this case is given by sk square. It is similar to lambda k where lambda k is the
Eigen value. So, that means Eigen value is the square of singular value. So, represent
the weighted variance explained by the kth principal component.
We will stop here. Then I will in the next class, I will continue with this singular
value decomposition approach and we will see with this particular case, how ultimately
will lead to developing perceptual maps means creating different principal components, their
coordinate values, then mapping them. Then finally, what will happen from the numerical
measures of CA as well as the maps, we want to infer about the defects in a design process
for this particular case what we have considered. Thank you very much.
