**Notation**

Contingency table N (I´ J)

Row mass = row sums/grand total = *n _{i}*

Column mass = column sums/grand total = *n*_{+j}/*n*

Correspondence matrix is defined as the original table (or matrix) N divided
by the grand total *n*.

The matrix of row profiles can also be defined as the rows of the
correspondence matrix ** P** divided by their respective row sums (

Matrix of row profile *= D*

where *D*_{r} is the diagonal
matrix of row masses.

The matrix of column profiles consists of the columns of the correspondence
matrix *P *divided by their respective column sums.

Matrix of column profiles = *D*_{c}^{–
1} *P*

where *D*_{c} is the diagonal
matrix of the column masses.

The correspondence analysis problem is to find a low-dimensional approximation to the original data matrix that represents both the row and column profiles

*R*=* D*_{r}* *^{–1}
*P*

** C**=

In a low *k*-dimensional subspace, where *k* is
less than *I* or *J*. These two *k*-dimensional subspaces
(one for the row profiles and one for the column profiles) have a geometric
correspondence that enables us to represent both the rows and columns in the
same display.

Since we wish to graphically represent the distances between row (or column)
profiles, we orient the configuration of points at the center of gravity of
both sets. The centroid of the set of row points in
its space is the vector of column masses. The centroid
of the set of column point in its space is ** r**, the vector of row
masses. This is the average column profile.

To perform the analysis with respect to the center of gravity, ** P**
is centered "symmetrically" by rows and columns,

The singular value decomposition (SVD) is defined as the decomposition of an
I´ J matrix* *A* *as the
product of three matrices

** A**=

where the matrix G is a diagonal matrix of positive numbers in decreasing order:

g _{1 }_{³ }g _{2 }_{³ }……g _{n
}_{³ }0 (2)

where *k* is the rank of ** A**, and
the columns of the matrices

*U*^{T}** U**=

where *U** ^{T} *is the
transpose of

g _{1}, g _{2}, ……,g _{k }are
called singular values.

Columns of ** U** (

Columns of ** V** (

Consider a set of *I* points in *J*-dimensional space, where coordinates
are in the rows of the matrix ** Y** with masses

(** x** –

Let *D*_{m} and *D*_{q}_{
}be the diagonal matrices of point masses and dimension weights
respectively

Let **m** be the vector of point messes (we
have already assumed that ):

*I *^{T}** m= I**

where ** I** is the vector of ones.

Any low-dimensional configuration of the points can be derived directly from the singular value decomposition of the matrix:

(5)

where is the centroid
of the rows of **Y**.

Applying singular value decomposition to the above equation, we find that
principal coordinates of row points (*i.e. *projections of row profiles
onto principal axes) are contained in the following matrix:

** F**=

The coordinates of the points in an optimal a -dimensional subspace are contained in the first a columns. The principal axes of this space are contained in the matrix

** A **=

Here, we have two special cases of the above general result, *viz*. *Row
problem* and *Column* *problem. *These problems involve the
reduction of dimensionality of the row profiles and the column profiles, where
each set of points has its associated masses and Chi-square distances. Both these problems reduce to singular value
decomposition of the same matrix of standardized residuals.

*Row problem*

The row problem consists of a set of *I* profiles in the rows of **=
D**

*r*^{T}
*D** _{r}* -

where *c *^{T} is the row
vector of the column masses

The matrix ** A** in (Equation 5) can be
written as

** A = D_{r}^{1/2}(D_{r}^{-1}P-IC^{T})D_{c}^{-1/2}** (7)

which can be rewritten as

** A = D_{r}^{-1/2}
(P-yc^{T})D_{r}^{-1/2}** (8)

*Column problem*

The column problem consists of a set of *J* profiles in the columns of *P**D*_{c- }^{1} with masses ** c** in
the diagonal of

By transposing the matrix ** P D_{c- }^{1}**
of column profiles, we obtain

The matrix in Equation (5)

** **(9)

can be written as

This is the transpose of the matrix derived for ** A**., the row
problem. It follows that both the row and column problems can be solved by
singular value decomposition of the same matrix of standardized residuals:

(10)

The elements of this *I**´ J*
matrix are:

(11)

It can be easily seen that the centroid of these profiles is:

(the row vector of *r*
masses)

The matrix in Equation 5 is thus reduced to

(12)

It can be easily seen that the matrix A is the transpose of the matrix derived for the row problem. These results imply that both the row problem and column problems are solved by computing the singular value decomposition of the same matrix (i.e. the matrix of the standard residuals).

(13)

whose elements are:

** **(14)

It follows from Equation ( 10 ) that the Chi-square
statistic can be decomposed into *I **´
J *components of the form:

The sum of squares of the elements of ** A** is the total inertia
of the contingency table.

Total inertia =

which is the chi-square statistic divided by *n*.

Thus, there are *k *= min [*I*-1, *J*-1] dimensions in the
solution. The squares of the singular values of *A**i.e.* the
eigenvalues of *A*^{T}** A**
or

The principal coordinates of the row problem are:

G (15)

or in the scalar notation:

(16)

The principal coordinates of the columns are obtained from:

G

or in the scalar notation:

The standard coordinates of the rows are the principal coordinates divided
by their respective singular values, *i.e.*

*X=F*** G ^{-1}= ** (17)

or in the scalar notation

The standard coordinates of the columns are the principal coordinates divided by their respective singular values:

*Y=G*** G ^{-1}= D_{c}^{-1/2}V ** (18)

*i.e.*

Each principal inertia l * _{k}*
is decomposed into components for each row i:

or in the matrix notation

(19)

The contribution of the rows to the principal inertia l * _{k}* is equal to:

For the i^{th} row, the inertia components for all *k *axes sum
up to the row inertia of the *i*^{th} row:

The left hand side of the above equation is identical to the sum of squared
elements in the *i*^{th} row of *A*

or

(20)

There are *k* = min [*I*-1, *J*-1] dimensions in the
solution. The square of the singular values of A, are denoted by are called
singular values.

The principal coordinates of the rows are obtained using [Equation (6)], for the row problem.

(21)

or in scalar notation:

Similarly the principal coordinates of the columns are obtained using Equation (6), for the column problem.

(22)

*i.e.*

The standard coordinates of the rows are the principal coordinates divided by their respective singular values:

(23)

*i.e.*

The standard coordinates of the columns are the principal coordinates divided by their respective singular values:

*i.e.*