# Quantification of intrinsic quality of a principal dimension in
correspondence analysis and taxicab correspondence analysis

Vartan Choulakian, Université de Moncton, Canada  email:
<EMAIL_ADDRESS>
(August 2021)

###### Abstract

Collins(2002, 2011) raised a number of issues with regards to correspondence
analysis (CA), such as: qualitative information in a CA map versus
quantitative information in the relevant contingency table; the interpretation
of a CA map is difficult and its relation with the % of inertia (variance)
explained. We tackle these issues by considering CA and taxicab CA (TCA) as a
stepwise Hotelling/Tucker decomposition of the cross-covariance matrix of the
row and column categories into four quadrants. The contents of this essay are:
First, we review the notion of quality/quantity in multidimensional data
analysis as discussed by Benzécri, who based his reflections on Aristotle.
Second, we show the importance of unravelling the interrelated concepts of
dependence/heterogeneity structure in a contingency table; and to picture them
two maps are needed. Third, we distinguish between intrinsic and extrinsic
quality of a principal dimension; the intrinsic quality is based on the signs
of the residuals in the four quadrants, hence to the interpretability.
Furthermore, we provide quantifications of the intrinsic quality and use them
to uncover structure in particular in sparse contingency tables. Finally, we
emphasize the importance of looking at the residual cross-covariance values at
each iteration.

Key words: dependence/heterogeneity; correspondence analysis; contribution
map; residual cross-covariance; intrinsic quality.

AMS 2010 subject classifications: 62H25, 62H30

In memoriam: Jean-Paul Benzécri (28 February 1932, 24 November 2019)

##  1 Introduction

In this essay we comment on Collins’ (2002, 2011) statement ”correspondence
analysis makes you blind”. Collins stated it twice as a reply to Whitlark and
Smith (2001) and Bock (2011a), who analyzed two different brand image count
data sets by correspondence analysis (CA). Both Whitlark and Smith (2002) and
Bock (2011b) in their reply to Collins (2002, 2011) accepted without
hesitation the insightful and original observations described by Collins. Beh
and Lombardo (2014, p.131-132) provide a cursory report on these dialogues.

Table 1: WS brand-attribute count table.
---
Attribute
Company | innovative | leader | solution | rapport | efficient | relevant | essential | trusted
Oracle | 155 | 157 | 109 | 133 | 151 | 96 | 35 | 170
Nokia | 375 | 350 | 274 | 318 | 351 | 284 | 91 | 408
Fedex | 476 | 675 | 550 | 669 | 748 | 627 | 307 | 754
A | 86 | 66 | 105 | 110 | 117 | 76 | 30 | 122
B | 30 | 21 | 25 | 37 | 40 | 20 | 9 | 43
C | 18 | 12 | 11 | 16 | 17 | 12 | 2 | 18
D | 25 | 23 | 33 | 36 | 34 | 28 | 12 | 35
E | 21 | 20 | 21 | 26 | 27 | 18 | 9 | 36
F | 190 | 307 | 305 | 332 | 355 | 309 | 131 | 392
G | 18 | 16 | 16 | 25 | 21 | 18 | 10 | 29
H | 408 | 549 | 467 | 551 | 613 | 523 | 239 | 624
I | 143 | 225 | 194 | 191 | 206 | 184 | 121 | 248

Table 2: Seriated WS brand-attribute data structure uncovered by Collins.
---
Attribute
Company | trusted | efficient | rapport | leader | relevant | solution | innovative | essential | average
Fedex | 754 | 748 | 669 | 675 | 627 | 550 | 476 | 307 | 601
H | 624 | 613 | 551 | 549 | 523 | 467 | 408 | 239 | 497
Nokia | 408 | 351 | 318 | 350 | 284 | 274 | 375 | 91 | 306
F | 392 | 355 | 332 | 307 | 309 | 305 | 190 | 131 | 290
I | 248 | 206 | 191 | 225 | 184 | 194 | 143 | 121 | 189
Oracle | 170 | 151 | 133 | 157 | 96 | 109 | 155 | 35 | 126
A | 122 | 117 | 110 | 66 | 76 | 105 | 86 | 30 | 89
D | 35 | 34 | 36 | 23 | 28 | 33 | 25 | 12 | 28
B | 43 | 40 | 37 | 21 | 20 | 25 | 30 | 9 | 28
E | 36 | 27 | 26 | 20 | 18 | 21 | 21 | 9 | 22
G | 29 | 21 | 25 | 16 | 18 | 16 | 18 | 10 | 19
C | 18 | 17 | 16 | 12 | 12 | 11 | 18 | 2 | 13
average | 240 | 223 | 204 | 202 | 183 | 176 | 162 | 83 | 184

###  1.1 WS brand-attribute count data

Table 1, from Whitlark and Smith (2001), shows a brand-attribute count data.
Whitlark and Smith (2001) analyzed it by CA resulting in a map, Figure 1, very
similar to the Taxicab CA (TCA) map in Figure 2; they interpreted Figure 1
using the adjusted chi-square residuals by arguing that it corrects
”inaccuracies introduced by the dimensionality problem”. TCA is a l1 variant
of CA, see Choulakian (2006). The interpretation of Figures 1 and 2 does not
correspond to the simple seriated structure uncovered by Collins (2002) given
in Table 2, where the rows and columns of Table 1 are permuted according to
the row and column averages in descending order. Collins interpreted the
seriated structure in Table 2 as:

”a relief map of a plain sloping steeply down from north (FedEx) to south
(Brand C) and less steeply from west (Trusted) to east (Essential). On the
plain would be a molehill (Nokia is more innovative). But isn’t the table easy
enough?”

Note that this simple structure uncovered by Collins is not found in Figures 1
and 2. Why? Lemma 1 provides a partial answer: The brands and the attributes
are not independent, because of the existence of multiple ’molehill’s, the
most important molehill being (Nokia is more innovative) as mentioned by
Collins. The molehills (in bold in Table 2) represent positive associations,
and can be analyzed and visualized in a complementary form in Figures (1 and
4) or in Figures (2 and 3). Figures 3 and 4 are TCA and CA contribution plots,
which help us to interpret Figures 1 and 2. This is a summary of the paper.

Figure 1: CA map of WS data.

Figure 2: TCA map of WS data.

Figure 3: TCOV (TCA contribution map of WS data.

Figure 4: CA contribution map of WS data.

###  1.2 Issues raised by Collins

Collins (2002) summarized his arguments against CA in Whitlark and Smith
(2001) paper as:

”The popularity of CA and other techniques seems to arise from two feelings:
The data are too complex to be handled by the human brain, and pictures
communicate more than tables. In fact, the analysis of brand ”image” data like
that given by the authors is easy, and quantitative patterns are best
communicated through simple tables supported by words. A picture like the
correspondence ”map” shown by W&S may say something qualitative about patterns
in the data, but it says nothing quantitative or testable. Not even, for
example, that the two dimensions of the map account for nearly 90% of the
variance of the data, as reported in the text.”

The highlighted parts are ours. There are two main points raised by Collins.

First, qualitative patterns in CA map Figure 1 do not reflect quantitative
patterns in Table 2. Friendly and Kwan (2011) divided statisticians (data
analysts) in two categories: graph-people and table-people. Clearly Collins
belongs to the group of table-people. In this paper, we will show that both,
tables and maps, are needed for global and local analysis. Additionally, in
section 2 we will discuss the important notions of quality and quantity in
data analysis, based on Benzécri’s reflections.

Second, the interpretation of a CA map is difficult and its relation with the
% of inertia (variance) explained. Here, we have to discuss two interrelated
issues in CA. a) It concerns the interpretation of CA maps, which as we
mentioned, was raised by Whitlark and Smith (2001); then clearly stated by
Bock (2011b, pp. 587–588), and further discussed by Bock (2017) in a R-Blog
titled ” How to Interpret Correspondence Analysis Plots (It Probably Isn’t the
Way You Think)”; in the R-blog are discussed nine complex issues for the
interpretation of CA maps. Beh and Lombardo (2014, p.132) summarize it as: in
CA ”the inability of the principal coordinate to provide a meaningful
interpretation of the distance between a row and column point in these plots”.
Greenacre (2013) also proposed contribution biplots to tackle this issue. b)
In this paper, we will distinguish between the intrinsic quality and the
extrinsic quality of a principal dimension, and introduce indices that
quantify the intrinsic quality of a principal axis. In CA maps ”variance
accounted for” reflects the extrinsic quality. The intrinsic quality of a
principal dimension examines the four quadrants of the residual cross-
covariance matrix. We tackle these issues by considering CA and TCA as a
stepwise Hotelling/Tucker decomposition of the residual cross-covariance
matrix of the row and column categories into four quadrants.

###  1.3 Hotelling and Tucker decompositions

In mathematics, a data set $\mathbf{X}=(x_{ij})$ for $i=1,...,I$ and
$j=1,...,J$ can be interpreted as three kinds of mapping, see Benzécri (1973,
p.56) and Choulakian (2016a). First, as a linear mapping: X:
R${}^{J}\rightarrow$ R${}^{I};$ second, as a linear mapping: XT:
R${}^{I}\rightarrow$ R${}^{J};$ third, as a bilinear mapping X: (R${}^{I},$
R${}^{I})\rightarrow$ R. Hotelling’s (1933) principal components analysis
(PCA) is developed within the first two settings, while Hotelling’s (1936)
canonical correlation analysis is developed within the third setting, where X
represents a cross-covariance matrix. Benzécri (1973) emphasized the
development of CA within the first two settings as a weighted PCA method. In
this paper, we shall emphasize the third setting.

It is well known that CA is a particular kind of Hotelling’s canonical
correlation analysis, see for instance Goodman (1991), where the two sets of
variables are the indicator sets of the categories of the two nominal
variables. Another method, similar in perspective to canonical correlation
analysis is Tucker’s (1958) interbattery analysis, which maximizes the
covariance measure between the linear combination of the two indicator sets of
quantitative variables. When CA did not produce interpretable maps of
contingency tables, Tenenhaus and Augendre (1996) proposed the Tucker
interbattery analysis as an alternative to CA.

The parameters in Hotelling’s canonical correlation and Tucker’s covariance
analyses are generally estimated by singular value decomposition (SVD). When
we estimate the parameters by TaxicabSVD (TSVD) introduced by Choulakian
(2006) in place of SVD, surprisingly we notice that these two analyses
complement each other because they are linearly related, see equation (17).
For further details, see Choulakian, Simonetti and Gia (2014).

Figure 3, named TCov map, displays taxicab interbattery analysis map of WS
data. Figures 2 and 3 (TCA and TCov maps) are different (similarly Figures 1
and 4 (CA and CA contribution maps) are different): thus they provide
different information to us, sometimes confusing (for instance observe the
positions of the brands B, E and F in Figures 2 and 3). One of the major
novelties of this paper is that, we interpret Figure 3, the taxicab
interbattery analysis TCov map, as TCA contribution map, and consequently we
provide a new perspective on the interpretation of the associated TCA map
Figure 2 via Lemma 6. For the interpretation of the row and column labels on
the TCov map, we shall use a quantification of the intrinsic quality of a
principal dimension, named quality of signs of residuals (QSR) index; which
will be complimented by a look at the seriated residual covariance matrix.
Then we extend the development of these ideas to CA; where we also discuss
sparse contingency tables having the quasi-two blocks diagonal structure,
which, according to Benzécri (1973, p.188-190), is quite common. Greenacre
(2013) introduced and discussed CA contribution biplots, but did not relate
them to CA maps; Lemma 6 accomplishes this task.

###  1.4 Organisation

This paper is organized as follows: Section 2 sketches Benzécri’s reflections
on quality and quantity in data analysis. Section 3 presents an overview of
taxicab singular value decomposition (TSVD); section 4 presents preliminaries.
In section 5 we develop the main subject matter, the quantification of the
intrinsic quality of a principal dimension in CA and TCA. Section 6 presents
applications. Finally we conclude in section 7.

##  2 Quantity and Quality

Benzécri (1982, 1988) has two papers on quality and quantity; in the second he
discussed the relationship between quantity and quality historically, starting
with Aristotle and finishing it with his description within the philosophy of
data analysis, aka CA framework. Here we quote from Benzécri (1988, section
1.7):

”Pour l’analyse des données, nous retenons d’abord, suivant Aristote, que ”le
caractère propre de la quantité qu’on peut lui attribuer l’égal et l’inégal”,
tandis que, ”semblabe ou dissemblable se dit uniquement des qualités”. De ce
point de vue, une description multidimensionnelle est toujours qualitative
même si elle comporte que des variables numériques précises, parceque la
multiplicité des descriptions possibles est telle qu’on rencontrera jamais
d’égales, mais seulement de semblables”.

The following two definitions and the corollary provide a succinct summary of
the quote.

Definition 1 (Aristotle on quantity): X is a quantitative variable if, given
two realizations $x$ and $y$ of X, then either $x=y$ or $x\neq y.$

Definition 2 (Aristotle on quality): X is a qualitative variable if, given two
realizations $x$ and $y$ of X, then either $x$ is similar to $y$ or $x$ is
dissimilar to $y.$

Corollary 3 (Benzécri): Any multidimensional description is always qualitative
even though its components are precisely numerical.

Similar ideas also are expressed in a forward essay by Benzécri in Murtagh
(2005).

Benzécri’s schematic conceptual formulation of data analysis is the following
directed diagram

$\text{Quality(data)}\longrightarrow\text{Quantity(factors)}\longrightarrow\text{Quality(clusters).}$

The first step: Quality$($data$)\longrightarrow\ \ $Quantity$($factors$)$ is
done by dimension reduction. The nature of each factor (latent variable) is
quantitative and there are almost always more than one factor. Even though
each latent variable is quantitative, but its interpretation is qualitative:
According to Benzécri (1988, section 1.3, in comments on Descartes): ”Toute
qualité n’est que l’expression d’un rapport de quantités”; that is, quality is
the expression of a ratio of quantities. For interpretation of a principal
dimension, we apply Aristotle/Benzécri principle. Aristotle in his book
PHYSICS defined ”principles are contraries” and cited as examples taken from
his predecessors ”hot and cold”, ”the rare and the dense” and ”plenum and
void” see Aristotle (1960, p.14). In CA, Benzécri (1973, p.227) following
Aristotle based the interpretation of a principal dimension on contraries
(dichotomies, oppositions) and gradations, where an opposition or a gradation
represent a latent variable. In another context, Choulakian (2014, 2016b) used
Euclid’s principle of contradiction for interpretation of the first principal
dimension for the analysis of rank data.

The second step: Quantity$($factors$)\longrightarrow\ \ $Quality$($clusters)
is done by usual methods such as k-means.

Murtagh (2005, section 1.1), described Benzécri’s paradigm “a tale of three
metrics”; which clearly characterizes the diagram where : the chi-squared and
the Euclidean metrics are for the first step, and the ultrametrics for the
second step. A similar description to the above diagram is also stated by De
Leeuw (2005). This fact also is reflected in the first printed work, Benzécri
(1973), titled DATA ANALYSIS; which is composed of two volumes: The first
volume’s subtitle is La Taxonomie; the second volume’s subtitle is Analyse des
Correspondances.

##  3 An overview of taxicab singular value decomposition

Consider a matrix $\mathbf{X}$ of size $I\times J$ and $rank(\mathbf{X})=k$.
Taxicab singular value decomposition (TSVD) of X is a decomposition similar to
SVD of X; see Choulakian (2006, 2016a).

In TSVD the calculation of the dispersion measures $(\delta_{\alpha})$,
principal axes ($\mathbf{u}_{\alpha},\mathbf{v}_{\alpha})$ and principal
scores $(\mathbf{a}_{\alpha},\mathbf{b}_{\alpha})$ for $\alpha=1,...,k$ is
done in an stepwise manner. We put $\mathbf{X}_{1}=\mathbf{X}=(x_{ij})$ and
$\mathbf{X_{\alpha}}$ be the residual matrix at the $\alpha$-th iteration.

The variational definitions of the TSVD at the $\alpha$-th iteration are

$\displaystyle\delta_{\alpha}$ $\displaystyle=$
$\displaystyle\max_{\mathbf{u\in\mathbb{R}}^{J}}\frac{\left|\left|\mathbf{X_{\alpha}u}\right|\right|_{1}}{\left|\left|\mathbf{u}\right|\right|_{\infty}}=\max_{\mathbf{v\in\mathbb{R}}^{I}}\
\frac{\left|\left|\mathbf{X_{\alpha}^{\prime}v}\right|\right|_{1}}{\left|\left|\mathbf{v}\right|\right|_{\infty}}=\max_{\mathbf{u\in\mathbb{R}}^{J},\mathbf{v\in\mathbb{R}}^{I}}\frac{\mathbf{v}^{\prime}\textbf{X}\mathbf{{}_{\alpha}u}}{\left|\left|\mathbf{u}\right|\right|_{\infty}\left|\left|\mathbf{v}\right|\right|_{\infty}},$
$\displaystyle=$ $\displaystyle\max||\mathbf{X_{\alpha}u||}_{1}\ \
\text{subject to }\mathbf{u}\in\left\\{-1,+1\right\\}^{J},$ $\displaystyle=$
$\displaystyle\max||\mathbf{X_{\alpha}^{\prime}v||}_{1}\ \ \text{subject to
}\mathbf{v}\in\left\\{-1,+1\right\\}^{I},$ $\displaystyle=$
$\displaystyle\max\mathbf{v}^{\prime}\mathbf{X_{\alpha}u}\text{ \ subject to \
}\mathbf{u}\in\left\\{-1,+1\right\\}^{J},\mathbf{v}\in\left\\{-1,+1\right\\}^{I}.$

The $\alpha$-th principal axes are

$\mathbf{u}_{\alpha}\
=\arg\max_{\mathbf{u}\in\left\\{-1,+1\right\\}^{J}}\left|\left|\mathbf{X_{\alpha}u}\right|\right|_{1}\text{
\ \ and \ \ }\mathbf{v}_{\alpha}\
=\arg\max_{\mathbf{v}\in\left\\{-1,+1\right\\}^{I}}\left|\left|\mathbf{X_{\alpha}^{\prime}v}\right|\right|_{1}\text{,}$
(1)

and the $\alpha$-th principal vectors are

$\mathbf{a}_{\alpha}=\mathbf{X_{\alpha}u}_{\alpha}\text{ \ and \
}\mathbf{b}_{\alpha}=\mathbf{X_{\alpha}^{\prime}v}_{\alpha}.$ (2)

Furthermore the following relations are also useful

$\mathbf{u}_{\alpha}=sgn(\mathbf{b}_{\alpha})\text{ \ and \
}\mathbf{v}_{\alpha}=sgn(\mathbf{a}_{\alpha}),$ (3)

where $sgn(.)$ is the coordinatewise sign function, $sgn(x)=1$ if $x>0,$ and
$sgn(x)=-1$ if $x\leq 0.$

The $\alpha$-th taxicab dispersion measure $\delta_{\alpha}$ can be
represented in many different ways

$\displaystyle\delta_{\alpha}$ $\displaystyle=$
$\displaystyle\mathbf{v}_{\alpha}^{\prime}\mathbf{X_{\alpha}u}_{\alpha}$
$\displaystyle=$
$\displaystyle\left|\left|\mathbf{X_{\alpha}u}_{\alpha}\right|\right|_{1}=\left|\left|\mathbf{a}_{\alpha}\right|\right|_{1}=\mathbf{a}_{\alpha}^{\prime}\mathbf{v}_{\alpha},$
$\displaystyle=$
$\displaystyle\left|\left|\mathbf{X_{\alpha}^{\prime}v}_{\alpha}\right|\right|_{1}=\left|\left|\mathbf{b}_{\alpha}\right|\right|_{1}=\mathbf{b}_{\alpha}^{\prime}\mathbf{u}_{\alpha}.$

The $(\alpha+1)$-th residual correspondence matrix is

$\mathbf{X_{\alpha+1}}=\mathbf{X_{\alpha}-a}_{\alpha}\mathbf{b}_{\alpha}^{\prime}/\delta_{\alpha}.$
(5)

An interpretation of the term
$\mathbf{a}_{\alpha}\mathbf{b}_{\alpha}^{\prime}/\delta_{\alpha}$ in (5) is
that, it represents the best rank-1 approximation of the residual
correspondence matrix $\mathbf{X_{\alpha}}$, in the sense of taxicab norm.

Thus TSVD of X corresponds to

$x_{ij}=\sum_{\alpha=1}^{k}a_{\alpha}(i)b_{\alpha}(j)/\delta_{\alpha},$ (6)

a decomposition similar to SVD, but where the vectors
$(\mathbf{a}_{\alpha},\mathbf{b}_{\alpha})$ for $\alpha=1,...,k$ are
conjugate, a weaker property than orthogonality. That is

$\mathbf{a}_{\alpha}^{\prime}sgn(\mathbf{a}_{\beta})=\mathbf{b}_{\alpha}^{\prime}sgn(\mathbf{b}_{\beta})=0\text{
for }\alpha>\beta.$

In TSVD, the calculation of the principal component weights,
$\mathbf{u}_{\alpha}$ and $\mathbf{v}_{\alpha},$ and the principal scores,
$\mathbf{a}_{\alpha}$ and $\mathbf{b}_{\alpha},$ can be accomplished by two
algorithms. The first one is based on complete enumeration based on equation
(1). The second one is based on iterating the transition formulae (2,3). This
is an ascent algorithm; that is, it increases the value of the objective
function at each iteration, see Choulakian (2006, 2016a). The iterative
algorithm could converge to a local maximum; so it should be restarted from
several initial configurations. The rows or the columns of the data can be
used as starting values.

##  4 Preliminaries

Let $\mathbf{P=N/}n=(p_{ij})$ of size $I\times J$ be the associated
correspondence matrix of a contingency table N, where
$n=\sum_{i=1}^{I}\sum_{j=1}^{J}N(i,j)$. We define as usual
$p_{i\ast}=\sum_{j=1}^{J}p_{ij}$ , $p_{\ast j}=\sum_{i=1}^{I}p_{ij},$ the
vector $\mathbf{r=(}p_{i\ast})\in\mathbb{R}^{I},$ the vector
$\mathbf{c=(}p_{\ast j})\in\mathbb{R}^{J}$, and
$\mathbf{D}_{I}=Diag(\mathbf{r})$ the diagonal matrix having diagonal elements
$p_{i\ast},$ and similarly $\mathbf{D}_{J}=Diag(\mathbf{c}).$ We suppose that
$\mathbf{D}_{I}$ and $\mathbf{D}_{J}$ are positive definite metric matrices of
size $I\times I$ and $J\times J$, respectively; this means that the diagonal
elements of $\mathbf{D}_{I}$ and $\mathbf{D}_{J}$ are strictly positive. Let

$\displaystyle\mathbf{P}^{(1)}$ $\displaystyle=$
$\displaystyle(\mathbf{P}-\mathbf{rc}^{\top})$ $\displaystyle=$
$\displaystyle\mathbf{Cov}(\mathbf{P})$

or elementwise

$p_{ij}^{(1)}=p_{ij}-p_{i\ast}p_{\ast j}$ (8)

be the residual matrix with respect to the independence model. $p_{ij}^{(1)}$
is the cross-covariance between the categories of the $i$th nominal row
variable and the $j$th nominal column variable.

The independence assumption $p_{ij}^{(1)}=0$ can also be interpreted in
another way as

$(\frac{p_{ij}}{p_{i\ast}p_{\ast j}}-1)=0,$ (9)

which can be reexpressed as

$\displaystyle\frac{1}{p_{i\ast}}(\frac{p_{ij}}{p_{\ast j}}-p_{i\ast})$
$\displaystyle=$ $\displaystyle 0$ $\displaystyle=$
$\displaystyle\frac{1}{p_{\ast j}}(\frac{p_{ij}}{p_{i\ast}}-p_{\ast j});$

this is the row and column homogeneity models. Benzécri (1973, p.31) named the
vector ($\frac{p_{ij}}{p_{\ast j}}$ for $i=1,...,I$ and $j$ fixed) the profile
of the $j$th column; and the element $\frac{p_{ij}}{p_{i\ast}p_{\ast j}}$ the
density function of the probability measure $(p_{ij})$ with respect to the
product measure $p_{i\ast}p_{\ast j}$. The element
$\frac{p_{ij}}{p_{i\ast}p_{\ast j}}$is named Pearson ratio in Goodman (1996)
and Beh and Lombardo (2014, p.123).

###  4.1 Estimation of the parameters by SVD

Suppose the independence assumption
$\mathbf{Cov(P)}=\mathbf{P}^{(1)}=\mathbf{0}$ is not true, then each of the
two equivalent model formulations (8,10) can be generalized to explain the
nonindependence by adding bilinear terms, where
$k=rank(\mathbf{P}^{(1)}\mathbf{)}$.

a) Cov (cross-covariance) decomposition:

$p_{ij}-p_{i\ast}p_{\ast
j}=\sum_{\alpha=1}^{k}a_{\alpha}(i)b_{\alpha}(j)/\sigma_{\alpha}.$ (11)

This is an interbattery analysis proposed by Tucker (1958). Tenenhaus and
Augendre (1996) estimated the parameters in (11) by singular value
decomposition (SVD) of the matrix $\mathbf{Cov(P).}$ The parameters in (11)
satisfy the following equations

$\sigma_{\alpha}^{2}=\sum_{i=1}^{I}|a_{\alpha}(i)|^{2}=\sum_{j=1}^{J}|b_{\alpha}(j)|^{2}\text{\
\ \ for }\alpha=1,...,k;$ (12)

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{I}a_{\alpha}(i)a_{\beta}(i)=\sum_{j=1}^{J}b_{\alpha}(j)b_{\beta}(j)\text{\
\ for }\alpha\neq\beta$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{I}a_{\alpha}(i)=\sum_{j=1}^{J}b_{\alpha}(j)\text{\ \
for }\alpha=1,...,k.$

b) CA (correspondence analysis) decomposition

$\frac{p_{ij}}{p_{i\ast}p_{\ast
j}}-1=\sum_{\alpha=1}^{k}f_{\alpha}(i)g_{\alpha}(j)/\sigma_{\alpha}.$ (14)

This decomposition has many interpretations. Essentially, for data analysis
purposes Benzécri (1973) interpreted it as weighted principal components
analysis of row and column profiles. Another useful interpretation, comparable
to Tucker interbattery analysis, is Hotelling(1936)’s correlation analysis,
see Lancaster (1958) and Goodman (1991). The parameters in (14) satisfy the
following equations

$\sigma_{\alpha}^{2}=\sum_{i=1}^{I}|f_{\alpha}(i)|^{2}p_{i\ast}=\sum_{j=1}^{J}|g_{\alpha}(j)|^{2}p_{\ast
j}\text{\ \ \ for }\alpha=1,...,k;$ (15)

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{I}f_{\alpha}(i)f_{\beta}(i)p_{i\ast}=\sum_{j=1}^{J}g_{\alpha}(j)g_{\beta}(j)p_{\ast
j}\text{\ \ for }\alpha\neq\beta$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{I}f_{\alpha}(i)p_{i\ast}=\sum_{j=1}^{J}g_{\alpha}(j)p_{\ast
j}\text{\ \ for }\alpha=1,...,k.$

The above two decompositions given in (11) and (14) are cross-covariance
based. There are also association (log ratio) based decompositions see Goodman
(1991, 1996) or Greenacre and Lewi (2009).

###  4.2 Estimation of the parameters by TSVD

First, we estimate the parameters ($a_{\alpha}(i),\ b_{\alpha}(j),$
$\delta_{\alpha})$ in (11) by TSVD; then the parametrs in (14) will be
linearly related by

$a_{\alpha}(i)=p_{i\ast}f_{\alpha}(i)\ ,\ \ b_{\alpha}(j)=p_{\ast
j}g_{\alpha}(j)\text{ and }\delta_{\alpha}=\sigma_{\alpha}.$ (17)

The parameters $a_{\alpha}(i)$ and $b_{\alpha}(j)$ in (11) are the principal
coordinates of the TCov decomposition and they satisfy

$\delta_{\alpha}=\sum_{i=1}^{I}|a_{\alpha}(i)|=\sum_{j=1}^{J}|b_{\alpha}(j)|\text{\
\ \ for }\alpha=1,...,k;$ (18) $\displaystyle 0$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{I}a_{\alpha}(i)sgn(a_{\beta}(i))=\sum_{j=1}^{J}b_{\alpha}(j)sgn(b_{\beta}(j))\text{\
\ for }\alpha>\beta$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{I}a_{\alpha}(i)=\sum_{j=1}^{J}b_{\alpha}(j)\text{\ \
for }\alpha=1,...,k.$

Similarly, the parameters $f_{\alpha}(i)$ and $g_{\alpha}(j)$ in (14) are the
principal coordinates of the TCA decomposition and they satisfy

$\delta_{\alpha}=\sum_{i=1}^{I}|f_{\alpha}(i)|p_{i\ast}=\sum_{j=1}^{J}|g_{\alpha}(j)|p_{\ast
j}\text{\ \ \ for }\alpha=1,...,k;$ (19) $\displaystyle 0$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{I}f_{\alpha}(i)f_{\beta}(i)p_{i\ast}=\sum_{j=1}^{J}g_{\alpha}(j)g_{\beta}(j)p_{\ast
j}\text{\ \ for }\alpha>\beta$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{I}f_{\alpha}(i)p_{i\ast}=\sum_{j=1}^{J}g_{\alpha}(j)p_{\ast
j}\text{\ \ for }\alpha=1,...,k.$

Let $\mathbf{P}^{(m)}=(p_{ij}^{(m)})$ be the $m$th residual correspondence
matrix, where

$p_{ij}^{(m+1)}=p_{ij}-p_{i\ast}p_{\ast
j}-\sum_{\alpha=1}^{m}a_{\alpha}(i)b_{\alpha}(j)/\delta_{\alpha}\text{\ \ \
for\ \ }m=1,...,k-1.$ (20)

Similarly, let $\mathbf{D}^{(m)}=(d_{ij}^{(m)})$ be the $m$th residual density
matrix, where

$d_{ij}^{(m+1)}=\frac{p_{ij}}{p_{i\ast}p_{\ast
j}}-1-\sum_{\alpha=1}^{m}f_{\alpha}(i)g_{\alpha}(j)/\delta_{\alpha}\text{\ \ \
for\ \ }m=1,...,k-1.$ (21)

Let $S\cup\overline{S}=I$ be an optimal binary partition of $I$, and
$T\cup\overline{T}=J$ be an optimal binary partition of $J,$ such that
$S=\left\\{i:a_{\alpha}(i)\geq 0\right\\}$ and $T=\left\\{j:b_{\alpha}(j)\geq
0\right\\}.$ Besides (18), the taxicab dispersion $\delta_{\alpha}$ will
additionally be related to the TCov principal coordinates $a_{\alpha}(i)$ and
$b_{\alpha}(j)$ in (11) by the following useful equations:

$\displaystyle\delta_{\alpha}/2$ $\displaystyle=$ $\displaystyle\sum_{i\in
S}a_{\alpha}(i)=-\sum_{i\in\overline{S}}a_{\alpha}(i)$ $\displaystyle=$
$\displaystyle\sum_{j\in
T}^{I}b_{\alpha}(j)=-\sum_{j\in\overline{T}}b_{\alpha}(j).$

$\displaystyle\delta_{\alpha}/4$ $\displaystyle=$ $\displaystyle\sum_{(i,j)\in
S\times
T}p_{ij}^{(\alpha)}=\sum_{(i,j)\in\overline{S}\times\overline{T}}p_{ij}^{(\alpha)}$
$\displaystyle=$ $\displaystyle-\sum_{(i,j)\in\overline{S}\times
T}p_{ij}^{(\alpha)}=-\sum_{(i,j)\in S\times\overline{T}}p_{ij}^{(\alpha)}.$

Equations (22, 23) follow from the fact that $\mathbf{P}^{(\alpha)}$ for
$\alpha=1,...,k$ is a double-centered matrix, see Choulakian and Abou-Samra
(2020). The quantification of the intrinsic quality of a principal dimension
is based on (23).

###  4.3 An observation

The TCov principal coordinates, $a_{\alpha}(i)$ and $b_{\alpha}(j),$ are
uniformly weighted, see equation (18); meanwhile TCA principal coordinates,
$f_{\alpha}(i)$ and $g_{\alpha}(j),$ are marginally weighted, see equation
(19). What is the consequence to this? The answer to this question is:
Benzécri’s principle of distributional equivalence, which states that CA (and
TCA) results are not changed if two proportional columns or rows are merged
into one. This has the practical consequence that the effective size of sparse
and large data sets can be smaller than the observed size; for further details
concerning sparse contingency tables see Choulakian (2017).

##  5 Main developments

Let L be a permutation matrix such that the coordinates of
$\mathbf{r}_{L}=\mathbf{Lr}$ are in decreasing order,
$\mathbf{r}_{L}(i)\geq\mathbf{r}_{L}(i+1)$ for $i=1,...,I-1.$ Similarly, M be
a permutation matrix such that the coordinates of $\mathbf{c}_{M}=\mathbf{Mc}$
are in decreasing order, $\mathbf{c}_{M}(j)\geq\mathbf{c}_{M}(j+1)$ for
$j=1,...,J-1.$

We consider the matrix

$\mathbf{LP}^{(1)}\mathbf{M}^{\top}=\mathbf{L}(\mathbf{P}-\mathbf{rc}^{\top})\mathbf{M}^{\top}.$
(24)

We have the following easily proved result

Lemma 4: Let $\mathbf{S=LPM}^{\top}.$ A necessary condition for the
independence model, $\mathbf{LP}^{(1)}\mathbf{M}^{\top}=\mathbf{0}$ or
$p_{ij}-p_{i\ast}p_{\ast j}=0,$ is that

$\mathbf{S}(i,j)\geq\mathbf{S}(i,j+1)\text{ \ \ for\ \ \ }j=1,...,J-1\text{
and }i\text{ fixed}$ (25)

and

$\mathbf{S}(i,j)\geq\mathbf{S}(i+1,j)\text{ \ \ for\ \ \ }i=1,...,I-1\text{
and }j\text{ fixed.}$ (26)

Remark: Relations (25 and 26) characterize Robinson matrices used for
seriation of artifacts or sites in archeology. That is why we named
$\mathbf{S,}$ see Table 2, seriated contingency table following its seriated
row $\mathbf{r}_{L}$ and column $\mathbf{c}_{M}$ marginals.

Lemma 5: TSVD of $\mathbf{P}$ is equivalent to TCov($\mathbf{P})=$ TSVD of
$\mathbf{P}^{(1)}.$

Lemma 6 states that the $\alpha$th row TCA (or CA) principal factor score
$\mathbf{f}_{\alpha}(i)$ is the weighted covariance of the $\alpha$th residual
density function $\mathbf{D}^{(\alpha)}(i,:)$ with the $\alpha$-th principal
axis $\mathbf{u}_{\alpha}$; where $\mathbf{D}^{(\alpha)}(i,:)$ is the ith row
of $\mathbf{D}^{(\alpha)}$ and $\mathbf{D}^{(\alpha)}(:,j)$ is the jth column
of $\mathbf{D}^{(\alpha)}.$

Lemma 6: In CA and TCA

$\displaystyle\mathbf{f}_{\alpha}(i)$ $\displaystyle=$
$\displaystyle\mathbf{D}^{(\alpha)}(i,:)\mathbf{D}_{J}\mathbf{u}_{\alpha}$
$\displaystyle=$ $\displaystyle
cov(\mathbf{D}^{(\alpha)}(i,:),\mathbf{u}_{\alpha})$

and

$\displaystyle\mathbf{g}_{\alpha}(j)$ $\displaystyle=$
$\displaystyle\mathbf{v}_{\alpha}^{\top}\mathbf{D}_{I}\mathbf{D}^{(\alpha)}(:,j)$
$\displaystyle=$ $\displaystyle
cov(\mathbf{D}^{(\alpha)}(:,j),\mathbf{v}_{\alpha}),$

where $\mathbf{u}_{\alpha}$ represents the $\alpha$-th standardized principal
axis in each method. In CA, $\mathbf{u}_{\alpha}=$
$\frac{\mathbf{g}_{\alpha}}{\sigma_{\alpha}}$and
$\mathbf{v}_{\alpha}=\frac{\mathbf{f}_{\alpha}}{\sigma_{\alpha}}.$ In TCA,
$\mathbf{u}_{\alpha}=sign(\mathbf{b}_{\alpha})=sign(\mathbf{g}_{\alpha})$ and
$\mathbf{v}_{\alpha}=sign(\mathbf{a}_{\alpha})=sign(\mathbf{f}_{\alpha})$ for
$\alpha=1,...,k$, see equations (3 and 17).

Proof: Here, we provide a proof for TCA. We use the transition formula (2) for
$m=1,...,k-1,$

$\displaystyle\mathbf{a}_{m+1}(i)$ $\displaystyle=$
$\displaystyle\sum_{j}p_{ij}^{(m+1)}\mathbf{u}_{m+1}(j)$ $\displaystyle=$
$\displaystyle\sum_{j}\left[p_{ij}-p_{i\ast}p_{\ast
j}-\sum_{\alpha=1}^{m}a_{\alpha}(i)b_{\alpha}(j)/\delta_{\alpha}\right]\mathbf{u}_{m+1}(j)$
$\displaystyle=$ $\displaystyle\sum_{j}\left[p_{ij}-p_{i\ast}p_{\ast
j}-\sum_{\alpha=1}^{m}a_{\alpha}(i)b_{\alpha}(j)/\delta_{\alpha}\right]\frac{p_{i\ast}p_{\ast
j}}{p_{i\ast}p_{\ast j}}\mathbf{u}_{m+1}(j)$ $\displaystyle=$
$\displaystyle\sum_{j}\left[\frac{p_{ij}}{p_{i\ast}p_{\ast
j}}-1-\sum_{\alpha=1}^{m}f_{\alpha}(i)g_{\alpha}(j)/\delta_{\alpha}\right]p_{i\ast}p_{\ast
j}\mathbf{u}_{m+1}(j),\text{ by (17),}$ $\displaystyle\mathbf{f}_{m+1}(i)$
$\displaystyle=$ $\displaystyle\sum_{j}\left[\frac{p_{ij}}{p_{i\ast}p_{\ast
j}}-1-\sum_{\alpha=1}^{m}f_{\alpha}(i)g_{\alpha}(j)/\delta_{\alpha}\right]p_{\ast
j}\mathbf{u}_{m+1}(j)$ $\displaystyle=$
$\displaystyle\sum_{j}d_{ij}^{(m+1)}p_{\ast j}\mathbf{u}_{m+1}(j),$

which is the required result (27).

Remark: In CA, due to (16), equation (27), similarly (28), can further be
simplified to

$\displaystyle\mathbf{f}_{m+1}(i)$ $\displaystyle=$
$\displaystyle\sum_{j}\frac{p_{ij}}{p_{i\ast}p_{\ast j}}p_{\ast
j}\mathbf{u}_{m+1}(j)$ $\displaystyle=$ $\displaystyle
cov(\mathbf{D}(i,:),\mathbf{u}_{m+1}),$

a well known result in Bastin et. al. (1980, p.157) or Goodman(1991, p. 1105,
eq. A.1.3).

###  5.1 Quantifying the intrinsic quality of a taxicab principal axis

Within the Euclidean framework a measure of the quality of a principal
dimension is the proportion (or percentage) of the residual variance explained
(or inertia in the case of CA)

$\%(\text{explained residual variance by dimension }\alpha)=\frac{100\
\sigma_{\alpha}^{2}}{\sum_{\beta=\alpha}^{k}\sigma_{\beta}^{2}}.$

This is an extrinsic measure of quality, because it compares the dispersion of
a principal axis $\sigma_{\alpha}^{2}$ with the residual dispersion
$\sum_{\beta=\alpha}^{k}\sigma_{\beta}^{2}$. In the above equation replacing
the l2 terms by the corresponding l1 terms, we obtain the measure of intrinsic
quality $QSR_{\alpha}$ expressed in Definition 7.

Let $S\cup\overline{S}=I$ be an optimal binary partition of $I$, and similarly
$T\cup\overline{T}=J$ be an optimal binary partition of $J$ for the $\alpha$th
principal dimension. Thus the data set is divided into four quadrants. We
define a new index showing the quality of signs of the residuals (QSR) in each
quadrant of the $\alpha$th residual cross-covariance matrix
$\mathbf{P}^{(\alpha)}$ for $\alpha=1,...,k$ in (20).

Definition 7: For $\alpha=1,...,k,$ the measure of the quality of signs of the
residuals in the quadrant $E\times F\subseteq I\times J$ is

$\displaystyle QSR_{\alpha}(E,F)$ $\displaystyle=$
$\displaystyle\frac{\sum_{(i,j)\in E\times F}p_{ij}^{(\alpha)}}{\sum_{(i,j)\in
E\times F}|p_{ij}^{(\alpha)}|},\text{ \ and by (23)}$ $\displaystyle=$
$\displaystyle\frac{\delta_{\alpha}/4}{\sum_{(i,j)\in E\times
F}|p_{ij}^{(\alpha)}|}\text{ \ \ for }(E,F)=(S,T)\text{ and
(}\overline{S},\overline{T})$ $\displaystyle=$
$\displaystyle\frac{-\delta_{\alpha}/4}{\sum_{(i,j)\in E\times
F}|p_{ij}^{(\alpha)}|}\text{ \ \ for }(E,F)=(\overline{S},T)\text{ and
(}S,\overline{T}).$

Similarly, a quantification of the quality of signs of the optimal cut of
dimension $\alpha$ is

$QSR_{\alpha}=\frac{\delta_{\alpha}}{\sum_{(i,j)}|p_{ij}^{(\alpha)}|}\text{.}$

Remark: The computation of the elements of $QSR_{\alpha}(E,F)$ are done easily
in the following way. We note that the $\alpha$th principal axis can be
written as

$\mathbf{u}_{\alpha}=\mathbf{u}_{\alpha+}+\mathbf{u}_{\alpha-}\text{\ \ \ \
by\ \ (1)},$

where
$\mathbf{u}_{\alpha+}=(\mathbf{u}_{\alpha}+\mathbf{1}_{J})/2\in\left\\{0,1\right\\}^{J}$
and
$\mathbf{u}_{\alpha-}=(\mathbf{u}_{\alpha}-\mathbf{1}_{J})/2\in\left\\{-1,0\right\\}^{J};$
similarly

$\mathbf{v}_{\alpha}=\mathbf{v}_{\alpha+}+\mathbf{v}_{\alpha-}\text{\ \ \ \ \
by \ (1)},$

where
$\mathbf{v}_{\alpha+}=(\mathbf{v}_{\alpha}+\mathbf{1}_{I})/2\in\left\\{0,1\right\\}^{I}$
and
$\mathbf{v}_{\alpha-}=(\mathbf{v}_{\alpha}-\mathbf{1}_{I})/2\in\left\\{-1,0\right\\}^{I}$,
and $\mathbf{1}_{I}$ designates a column vector of 1’s of size $I.$ So

$\displaystyle QSR_{\alpha}(S,T)$ $\displaystyle=$ $\displaystyle
QSR_{\alpha}(\mathbf{v}_{\alpha+},\mathbf{u}_{\alpha+})$ $\displaystyle=$
$\displaystyle\frac{\delta_{\alpha}/4}{\mathbf{v}_{\alpha+}^{\prime}abs(\mathbf{X}_{\alpha})\mathbf{u}_{\alpha+}}>0,$
$\displaystyle QSR_{\alpha}(\overline{S},\overline{T})$ $\displaystyle=$
$\displaystyle QSR_{\alpha}(\mathbf{v}_{\alpha-},\mathbf{u}_{\alpha-})$
$\displaystyle=$
$\displaystyle\frac{\delta_{\alpha}/4}{\mathbf{v}_{\alpha-}^{\prime}abs(\mathbf{X}_{\alpha})\mathbf{u}_{\alpha-}}>0,$
$\displaystyle QSR_{\alpha}(S,\overline{T})$ $\displaystyle=$ $\displaystyle
QSR_{\alpha}(\mathbf{v}_{\alpha+},\mathbf{u}_{\alpha-})$ $\displaystyle=$
$\displaystyle\frac{\delta_{\alpha}/4}{\mathbf{v}_{\alpha-}^{\prime}abs(\mathbf{X}_{\alpha})\mathbf{u}_{\alpha+}}<0,$
$\displaystyle QSR_{\alpha}(\overline{S},T)$ $\displaystyle=$ $\displaystyle
QSR_{\alpha}(\mathbf{v}_{\alpha-},\mathbf{u}_{\alpha+})$ $\displaystyle=$
$\displaystyle\frac{\delta_{\alpha}/4}{\mathbf{v}_{\alpha+}^{\prime}abs(\mathbf{X}_{\alpha})\mathbf{u}_{\alpha-}}<0,$

where $abs(\mathbf{X}_{\alpha})=(|X_{\alpha}(i,j)|).$

To interpret the above indices, we recall from elementary probability theory
the definition of association between two events by defining an index of
association $ass(i,j)=p_{ij}-p_{i\ast}p_{\ast j}$ for $i=1,...,I$ and
$j=1,...,J$.

a) When $ass(i,j)=0$, then the $i$th category of the row variable and the
$j$th category of the column variable are not associated (independent).

b) When $ass(i,j)>0$, then the $i$th category of the row variable and the
$j$th category of the column variable are attractively or positively
associated; that is, the event (i,j) occurs more than by chance.

c) When $ass(i,j)<0$, then the $i$th category of the row variable and the
$j$th category of the column variable are repulsively or negatively
associated; that is, the event (i,j) occurs less than by chance.

Based on these, the interpretation of the indices becomes evident: for
instance, $QSR_{\alpha}(S,T)>0$ measures the intensity of the attractive
association between the subsets $S$ and $T;$ while
$QSR_{\alpha}(\overline{S},T)<0$ measures the intensity of the repulsive
association between the subsets $\overline{S}$ and $\ T.$

Allard et al. (2020) used the QSR index to choose between two competing
methods of data analysis, TCA and taxicab log-ratio analysis of contingency
tables and compositional data.

Notation:

$QSR_{\alpha}(+)=\left\\{QSR_{\alpha}(\mathbf{u}_{\alpha+},\mathbf{v}_{\alpha+}),QSR_{\alpha}(\mathbf{u}_{\alpha-},\mathbf{v}_{\alpha-})\right\\}$

$QSR_{\alpha}(-)=\left\\{QSR_{\alpha}(\mathbf{u}_{\alpha+},\mathbf{v}_{\alpha-}),QSR_{\alpha}(\mathbf{u}_{\alpha-},\mathbf{v}_{\alpha+})\right\\}\vskip
12.0pt plus 4.0pt minus 4.0pt$

We have the following easily proved result

Lemma 8: a) For $\alpha=1,...,k,$ $QSR_{\alpha}=1$ if and only if
$QSR_{\alpha}(S,T)=-QSR_{\alpha}(S,\overline{T})=-QSR_{\alpha}(\overline{S},T)=QSR_{\alpha}(\overline{S},\overline{T})=1.$

b) For $\alpha=k,$ $QSR_{\alpha}=1.$

c)
($QSR_{\alpha}(S,T)+|QSR_{\alpha}(S,\overline{T})|+|QSR_{\alpha}(\overline{S},T)|+QSR_{\alpha}(\overline{S},\overline{T}))/4\geq
QSR_{\alpha}.$

The proof of part c, is based on the arithmetic-harmonic means inequality
which states that for four strictly positive real numbers $a,b,c$ and $d$

$\frac{a+b+c+d}{4}\geq\frac{4}{\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}};$

equality is attained when $a=b=c=d$.

###  5.2 Quantifying the intrinsic quality of a principal axis in CA

Let $\mathbf{D}^{(m)}=(d_{ij}^{(m)})$ be the $m$th residual density matrix in
CA,

$\displaystyle d_{ij}^{(m+1)}$ $\displaystyle=$
$\displaystyle\frac{p_{ij}}{p_{i\ast}p_{\ast
j}}-1-\sum_{\alpha=1}^{m}f_{\alpha}(i)g_{\alpha}(j)/\sigma_{\alpha}$
$\displaystyle=$ $\displaystyle\frac{p_{ij}}{p_{i\ast}p_{\ast
j}}-1-\sum_{\alpha=1}^{m}\sigma_{\alpha}v_{\alpha}(i)u_{\alpha}(j)\text{\ \ \
for\ \ }m=1,...,k-1,$

where $\mathbf{u}_{\alpha}=\frac{\mathbf{g}_{\alpha}}{\sigma_{\alpha}}$ and
$\mathbf{v}_{\alpha}=\frac{\mathbf{f}_{\alpha}}{\sigma_{\alpha}}$ represent
the $\alpha$th standardized principal axis coordinates in CA.

Let $\mathbf{Q}^{(m)}=(q_{ij}^{(m)})$ be the $m$th residual cross-covariance
matrix in CA obtained from (29),

$\displaystyle q_{ij}^{(m+1)}$ $\displaystyle=$ $\displaystyle
p_{i\ast}p_{\ast j}d_{ij}^{(m+1)}$ $\displaystyle=$ $\displaystyle
p_{ij}-p_{i\ast}p_{\ast j}-\sum_{\alpha=1}^{m}\sigma_{\alpha}p_{i\ast}p_{\ast
j}v_{\alpha}(i)u_{\alpha}(j)\text{\ \ \ for\ \ }m=1,...,k-1.$

####  5.2.1 $CA\\_QSR$ $\ \ \ indices$

Let $S\cup\overline{S}=I$ be an optimal binary principal axis partition of
$I$, and similarly $T\cup\overline{T}=J$ be an optimal principal axis
partition of $J$ by CA. Thus the residual covariance matrix is divided into
four quadrants: $S=\left\\{i:v_{\alpha}(i)\geq 0\right\\}$ and
$T=\left\\{j:u_{\alpha}(j)\geq 0\right\\}.$ Based on the observation that both
$q_{ij}^{(m+1)}$ and $p_{ij}^{(m+1)}$ are double centered, we can quantify the
intrinsic quality of CA principal dimension by replacing $p_{ij}^{(m+1)}$ by
$q_{ij}^{(m+1)}$ in subsection 5.1, and obtain $CA\\_QSR$ measures.

Definition 9: For $\alpha=1,...,k,$ the CA measure of the quality of signs of
the residuals in the quadrant $E\times F\subseteq I\times J$ is

$\displaystyle CA\\_QSR_{\alpha}(E,F)$ $\displaystyle=$
$\displaystyle\frac{\sum_{(i,j)\in E\times F}q_{ij}^{(\alpha)}}{\sum_{(i,j)\in
E\times F}|q_{ij}^{(\alpha)}|}$ $\displaystyle=$
$\displaystyle\frac{\varpi_{\alpha}/4}{\sum_{(i,j)\in E\times
F}|q_{ij}^{(\alpha)}|}\text{ \ \ for }(E,F)=(S,T)\text{ and
(}\overline{S},\overline{T})$ $\displaystyle=$
$\displaystyle\frac{-\varpi_{\alpha}/4}{\sum_{(i,j)\in E\times
F}|p_{ij}^{(\alpha)}|}\text{ \ \ for }(E,F)=(\overline{S},T)\text{ and
(}S,\overline{T}).$

for $E=S$ and $\overline{S},$ and, $F=T$ and $\overline{T}$. Similarly the CA
measure of the quality of signs of principal dimension $\alpha$ is

$CA\\_QSR_{\alpha}=\frac{\varpi_{\alpha}}{\sum_{(i,j)}|q_{ij}^{(\alpha)}|},$

where

$\displaystyle\varpi_{\alpha}$ $\displaystyle=$ $\displaystyle
sign(\mathbf{f}_{\alpha}^{\prime})\ \mathbf{Q}^{(\alpha)}\
sign(\mathbf{g}_{\alpha})$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{I}\sum_{j=1}^{J}sign(f_{\alpha}(i))sign(g_{\alpha}(j))\
q_{ij}^{(\alpha)}$ $\displaystyle=$ $\displaystyle 4\sum_{(i,j)\in E\times
F}sign(f_{\alpha}(i))sign(g_{\alpha}(j))\ q_{ij}^{(\alpha)}\text{,}$

for $E=S$ and $\overline{S},$ and, $F=T$ and $\overline{T}.$

Note that equations (31) and (23) are similar, and they follow from the
important observation that both residual cross-covariance matrices
$q_{ij}^{(m)}$ and $p_{ij}^{(m)}$ for $m=1,...,k$ are double centered.
Furthermore, the $CA\\_QSR$ indices satisfy the three properties in Lemma 8.

We have the following

Lemma 10: a) When both CA and TCA produce the same binary partition (cut) of
the set of column and the set of row variables of the cross-covariance matrix
$\mathbf{P}^{(1)}$, then $CA\\_QSR_{1}=QSR_{1}$ and$\ \ \
CA\\_QSR_{1}(E,F)=QSR_{1}(E,F)$ for $E=S$ and $\overline{S},$ and, $F=T$ and
$\overline{T}$. Thus $\delta_{1}=\varpi_{1}.$

The proof is evident based on the fact that $p_{ij}^{(1)}=$
$q_{ij}^{(1)}=p_{ij}-p_{i\ast}p_{\ast j}.$ Note that for $\alpha\neq 1,$
$p_{ij}^{(\alpha)}\neq$ $q_{ij}^{(\alpha)}.$

b) When CA and TCA produce different binary partitions of the set of column or
the set of row variables of the cross-covariance matrix, then
$\varpi_{1}<\delta_{1}.$

The proof is evident, for TCA maximizes $\delta_{1}.$

####  5.2.2 Absolute and relative contributions of a quadrant in CA

A better known decomposition, similar to (31), where CA is interpreted as
Hotelling’s canonical correlation analysis, is:

$\displaystyle\sigma_{\alpha}$ $\displaystyle=$
$\displaystyle\mathbf{v}_{\alpha}^{\prime}\ \mathbf{P}\ \mathbf{u}_{\alpha}$
(32) $\displaystyle=$ $\displaystyle\mathbf{v}_{\alpha}^{\prime}\
\mathbf{Q}^{(\alpha)}\ \mathbf{u}_{\alpha}\ \ \ \ \text{by (16),}$
$\displaystyle=$ $\displaystyle\sum_{i=1}^{I}\sum_{j=1}^{J}\
v_{\alpha}(i)u_{\alpha}(j)q_{ij}^{(\alpha)}$ $\displaystyle=$
$\displaystyle\sum_{F=T,\overline{T}}\sum_{E=S,\overline{S}}\sum_{(i,j)\in
S\times T}v_{\alpha}(i)u_{\alpha}(j)q_{ij}^{(\alpha)}.$

Equation (32) shows that, the dispersion measure
$\sigma_{\alpha}=corr(\mathbf{v}_{\alpha},\mathbf{u}_{\alpha})$ is a weighted
correlation measure, and is decomposed into four positive parts, each part
representing the absolute contribution of a quadrant $E\times F$ to
$\sigma_{\alpha}$. From the four positive parts in (32), we define the signed
absolute (respectively relative) contribution of the quadrant $sACQ$
(respectively $sRCQ$) to $\sigma_{\alpha}$

$sACQ_{\alpha}(E,F)=\sum_{(i,j)\in E\times
F}sign(v_{\alpha}(i))sign(u_{\alpha}(j))v_{\alpha}(i)u_{\alpha}(j)q_{ij}^{(\alpha)},$

and

$sRCQ_{\alpha}(E,F)=sACQ_{\alpha}(E,F)/\sigma_{\alpha},$

for $E=S$ and $\overline{S},$ and, $F=T$ and $\overline{T}.$

Similarly we define the signed residual contribution $sRES_{\alpha}$ to be

$sRES_{\alpha}=\sum_{F=T,\overline{T}}\sum_{E=S,\overline{S}}sACQ_{\alpha}(E,F),$

which can be interpreted as a global index of attractive or repulsive
association of a CA principal dimension.

####  5.2.3 Two blocks diagonal contingency tables

Here we discuss the particular case of contingency tables which have two
blocks diagonal structure; that is, for binary partitions $I=I_{1}\cup I_{2}$
and $J=J_{1}\cup J_{2},$ $p_{ij}=0$ for $(i,j)\in I_{1}$x$J_{2\text{ }}$ and
$(i,j)\in I_{2}$x$J_{1\text{ }}.$ Then a well-known result due to Benzécri
(1973, p. 188-190) or Bastin et. al.(1980, pp. 174-179) is

Theorem 10: (Benzécri): A contingency table has two blocks diagonal structure
if and only if $\sigma_{1}=1;$ moreover, $f_{1}(i)=g_{1}(j)=c_{1}$ for
$(i,j)\in I_{1}$x$J_{1\text{ }}$ and $f_{1}(i)=g_{1}(j)=c_{2}$ for $(i,j)\in
I_{2}$x$J_{2\text{ }},$ where $c_{1}$ and $c_{2}$ are constants.

Corollary 11: If $\sigma_{1}=1,$ then

a)
$CA\\_QSR_{1}(\mathbf{v}_{1+},\mathbf{u}_{1-})=CA\\_QSR_{1}(\mathbf{v}_{1-},\mathbf{u}_{1+})=-1;$

b)
$sACQ_{1}(\mathbf{v}_{1+},\mathbf{u}_{1-})=sACQ_{1}(\mathbf{v}_{1-},\mathbf{u}_{1+}).\vskip
12.0pt plus 4.0pt minus 4.0pt$

Remark:

a) Benzécri (1973, p. 188-190) generalized the result of Theorem 10 to
k-blocks diagonal contingency tables.

b) Benzécri (1973, p.189-190) observed that it is rare to have $\sigma_{1}=1,$
but not uncommon to have $\sigma_{1}^{2}\geq 0.7;$ then in these cases the
structure of the contingency table may be either quasi-two blocks diagonal,
where few cells will be nonzero in the quadrants $I_{1}$x$J_{2\text{ }}$ and
$I_{2}$x$J_{1\text{ }};$ or not, see Benzécri (1973, p.246) and Choulakian and
de Tibeiro (2013).

Rodent species abundance data set, discussed in the next section provides an
example of a sparse contingency table having quasi-two blocks diagonal
structure; furthermore, it shows that $\sigma_{1}=1$ is a sufficient condition
but not necessary to have $minCA\\_QSR_{1}(-)=-1.$

###  5.3 Two new unified formulas

Choulakian (2006) showed that both CA and TCA satisfy few fundamental
identical formulas -such as data reconstruction formula (14) and the
transition formulas (27, 28) of Lemma 6, even though mathematically they are
different. This paper extends the similarity of both methods by showing that
the dispersion measures also can be represented in a common form. For
$\alpha=1,...,k$

$\varpi_{\alpha}\text{ and }\delta_{\alpha}=\sum_{(i,j)\in I\times
J}sign(f_{\alpha}(i))sign(g_{\alpha}(j))p_{i\ast}p_{\ast j}d_{ij}^{(\alpha)},$

and

$\sigma_{\alpha}\text{ and }\delta_{\alpha}=\sum_{(i,j)\in I\times
J}v_{\alpha}(i)u_{\alpha}(j)p_{i\ast}p_{\ast j}d_{ij}^{(\alpha)}.$

###  5.4 Uncomparability of CA and TCA contribution maps

CA and TCA maps, Figures 1 and 2, are comparable because they are based on the
same data reconstruction formula (14)

$\frac{p_{ij}}{p_{i\ast}p_{\ast
j}}-1=\sum_{\alpha=1}^{k}f_{\alpha}(i)g_{\alpha}(j)/\gamma_{\alpha},$

where
$\gamma_{\alpha}^{2}=\sigma_{\alpha}^{2}=\sum_{i=1}^{I}p_{i\ast}f_{\alpha}(i)^{2}=\sum_{j=1}^{J}p_{\ast
j}g_{\alpha}(j)^{2}$ in CA and
$\gamma_{\alpha}=\delta_{\alpha}=\sum_{i=1}^{I}p_{i\ast}|f_{\alpha}(i)|=\sum_{j=1}^{J}p_{\ast
j}|g_{\alpha}(j)|$ in TCA. Figures 1 and 2 are obtained by plotting the
coordinates ($f_{1}(i),f_{2}(i))$ and ($g_{1}(j),g_{2}(j)).$

Figures 3 and 4 representing TCA and CA contribution maps are not comparable,
because they do not represent the same object.

TCA contribution (TCov) is the factoring of the cross-covariance matrix by
TSVD:

$p_{ij}-p_{i\ast}p_{\ast
j}=\sum_{\alpha=1}^{k}a_{\alpha}(i)b_{\alpha}(j)/\delta_{\alpha},$

where
$\delta_{\alpha}=\sum_{i=1}^{I}|a_{\alpha}(i)|=\sum_{j=1}^{J}|b_{\alpha}(j)|$
for $\alpha=1,...,k.$

CA contribution is the factoring of the chi-square residuals by SVD:

$\frac{p_{ij}-p_{i\ast}p_{\ast j}}{\sqrt{p_{i\ast}p_{\ast
j}}}=\sum_{\alpha=1}^{k}a_{\alpha}(i)b_{\alpha}(j)/\sigma_{\alpha},$

where
$\sigma_{\alpha}^{2}=\sum_{i=1}^{I}|a_{\alpha}(i)|^{2}=\sum_{j=1}^{J}|b_{\alpha}(j)|^{2}$
for $\alpha=1,...,k;$ this is symmetric scaling, different from the one used
by Greenacre (2013).

Figures 3 and 4 are obtained by plotting the coordinates ($a_{1}(i),a_{2}(i))$
and ($b_{1}(j),b_{2}(j)).$ We observe that TCov map really represents TCA
contribution plot, while CA contribution map represents only the orderings of
the points according to their contributions. A really representative CA
contribution plot should be based on the coordinates $a_{\alpha}(i)^{2}\
$sign($a_{\alpha}(i))\ $or $b_{\alpha}(j)^{2}\ $sign($b_{\alpha}(j)).$

##  6 Applications

In this section First we revisit WS data set in detail; then consider briefly
two other data sets.

###  6.1 WS brand-attribute count data

Table 3 displays the quality of signs of residuals ($QSR$) measures in % of WS
data: We use it to choose the number of principal dimensions. Our interest
focuses on $maxQSR_{\alpha}(+),$ because maps essentially reflect attractive
association between two optimal subsets of the row and column categories. The
first two with values of 100% and above 89.29% are significant compared to the
3rd with value of 63.55%. QSR values, $CA\\_QSR_{2}=75.6\%$
$>QSR_{2}=72.99\%,$ show that the CA map is slightly preferable to the TCA
map. Given that CA and TCA maps Figures 1 and 2 are very similar, we use TCov-
TCA framework, introduced in this essay, to interpret this data set; for there
is a lot of literature on the interpretation of CA maps, starting with the
pioneering work of Benzécri (1973, Vol.2, chapter 2) and the few references
that we cited in the introduction.

Table 3: QSR values (in %) of WS data for the first 4 dimensions.
---
$\alpha$ | $QSR_{\alpha}(+)$ | $QSR_{\alpha}(-)$ | $QSR_{\alpha}$ | $\delta_{\alpha}$
1 | (100, 52.05) | (-52.65, -87.07) | 67.01 | 0.0476
2 | (65.62, 89.29) | (-84.01, -60.77) | 72.99 | 0.0318
3 | (62.50, 63.55) | (-78.88, -62.65) | 66.25 | 0.0203
4 | (58.13, 64.54) | (-46.90, -80.23) | 60.17 | 0.0130
$\alpha$ | $CA\\_QSR_{\alpha}(+)$ | $CA\\_QSR_{\alpha}(-)CA\\_$ | $CA\\_QSR_{\alpha}$ | $\varpi_{\alpha}$
1 | (100, 52.05) |  ( -52.65,-87.07) | 67.01 | 0.0476
2 | (67.95, 89.80) | (-90.24, -62.60) | 75.60 | 0.0312
3 | (58.20, 44.56) | (-49.68, -59.46) | 52.24 | 0.0155
4 | (88.57, 48.59) | (-57.37, -69.26) | 62.76 | 0.0130

Figure 3 represents taxicab interbattery analysis TCov map, which can also be
interpreted as TCA contribution map. In Figure 3, brand $i$ is represented by
the principal coordinates $(a_{1}(i),a_{2}(i)),$ and attribute $j$ is
represented by the principal coordinates $(b_{1}(j),b_{2}(j)).$

Table 4: WS Covariance matrix ($\times$10${}^{4})$ seriated along its first
TCOV principal coordinates.
---
Attribute
Company | innovative | trusted | rapport | efficient | solution | leader | relevant | essential | $\mathbf{a}_{1}(i)$
Nokia | 60 | 5 | $-12$ | $-12$ | $-11$ | $8$ | $-12$ | $-27$ | $129$
Oracle | 25 | 3 | $-3$ | $-1$ | $-6$ | $11$ | $-16$ | $-12$ | $57$
A | 4 | 3 | $7$ | $5$ | $11$ | $-18$ | $-7$ | $-6$ | $15$
B | 3 | 4 | $3$ | $3$ | $-1$ | $-6$ | $-4$ | $-2$ | $13$
E | 1 | 4 | $1$ | $0$ | $0$ | $-2$ | $-2$ | $-1$ | $10$
C | 4 | 0 | $1$ | $1$ | $-1$ | $-1$ | $-1$ | $-2$ | $8$
G | 1 | 2 | $2$ | $-1$ | $-1$ | $-3$ | $-1$ | $1$ | $6$
D | $0$ | $-1$ | $3$ | $0$ | $3$ | $-5$ | $0$ | $0$ | $-2$
I | $-13$ | $1$ | $-10$ | $-13$ | $8$ | $10$ | $-2$ | $\mathbf{20}$ | $-25$
F | $-37$ | $8$ | $6$ | $2$ | $\mathbf{16}$ | $-6$ | $\mathbf{12}$ | $0$ | $-58$
H | $-17$ | $-13$ | $1$ | $6$ | $-4$ | $3$ | $\mathbf{17}$ | $8$ | $-60$
Fedex | $-30$ | $-16$ | $2$ | $11$ | $-14$ | $9$ | $\mathbf{17}$ | $\mathbf{20}$ | $-93$
$\mathbf{b}_{1}(j)$ | $194$ | $44$ | $-4$ | $-10$ | $-18$ | $-23$ | $-86$ | $-98$ |

####  6.1.1 Interpretation of the first principal dimension of TCov map

Table 4 displays the cross-covariance matrix seriated along its first TCov
principal coordinates; where we clearly also observe the four principal
quadrants for $\alpha=1.$

Let $T=\left\\{innovative,trusted\right\\}$ and
$S=\left\\{Nokia,Oracle,A,B,C,G\right\\}$ for the first principal dimension.
In Table 4 we observe that the covariance values in $S\times T$ quadrant are
all positive; and that is the reason that max$QSR_{1}(+)=100\%,$ the highest
attainable value. Similarly one observes that for the first two principal
dimensions $minQSR_{\alpha}(-)$ are $-87.07\%$ and $-84.04\%$, relatively
significant values compared to the lower bound $-100\%$. The first taxicab
dispersion measure, displayed in Table 3, is
$\delta_{1}=0.0476=\displaystyle\sum\limits_{i=1}^{12}|a_{1}(i)|=\displaystyle\sum\limits_{i=1}^{8}|b_{1}(j)|$
by (18). The last column and the last row of Table 4 display the signed
absolute contributions $a_{1}(i)$ and $b_{1}(j)$ to $\delta_{1}$, that
according to Benzécri (1973, p.47) assist in the interpretation of the first
factor. So the relative contribution (RC) of innovative to the first factor is
$RC_{1}(innovative)=194/476=0.41\leq 0.5$, a very high value indeed. Similarly
the $RC_{1}(relevant$ or $essential)=(98+86)/476=0.39\leq 0.5$, a very high
value. In $TCov(\mathbf{P})$, the attainable upper bound of a RC of a
coordinate $a_{\alpha}(i)$ or $b_{\alpha}(j)$ is 0.5 by equation (22). So the
first principal dimension represents the factor opposing (innovative
associated with Nokia, and with lesser degree with Oracle) to (relevant-
essential associated with the brands Fedex, H and F). Furthermore, the cross-
covariance matrix in Table 4 informs us more about this opposition: cov(Nokia,
innovative)=60 and cov(Oracle, innovative)=25, while cov(F, innovative)=-37,
cov(Fedex, innovative)=-30, and cov(H, innovative)=-17. Note that in
particular the intensity of the negative covariances, representing the three
major repulsive associations, can not be assessed in Figure 2. So, to assess
quantitatively an association between a row and a column, one has to follow
Collins advice and look at the value in a table of numbers.

Furthermore, examining the seriated Table 2, we see that the last three brands
Fedex, H and F do not have any molehills: they satisfy equation (25) of Lemma
4, and they have quite large marginal weights.

####  6.1.2 Interpretation of the second principal dimension of TCov map

Table 5 displays the residual cross-covariance matrix P(2) seriated along its
2nd TCov principal coordinates: It shows that the second principal or latent
variable is based on the opposition between (leader-innovative associated
essentially with Nokia) and (solution-rapport-trusted associated with brands F
and A).

Figure 2 shows that each of the brands $\left\\{C,G,D,E,B\right\\}$ have very
small almost insignificant contributions either to the first or to the second
principal dimensions; and this is also evident in Tables 4 and 5.

Table 5: WS P(2) matrix ($\times$10${}^{4})$ seriated along its second TCOV
principal coordinates.
---
Attribute
Company | leader | innovative | essential | relevant | efficient | trusted | rapport | solution | $\mathbf{a}_{2}(i)$
Nokia | 14 | 7 | $-0$ | $12$ | $-9$ | -7 | -11 | -6 | 65
fedex | 5 | 8 | $-1$ | $+0$ | $-9$ | -8 | 2 | -17 | 29
I | 9 | -3 | 15 | -7 | -14 | 3 | -10 | 7 | 28
H | -0 | 8 | -4 | 6 | 5 | -8 | 0 | -7 | 19
Oracle | 14 | 2 | -0 | -6 | 0 | -2 | -3 | -4 | 17
C | -1 | 0 | -1 | 1 | 1 | -0 | 1 | -1 | -1
G | -3 | -2 | 2 | 1 | -1 | 2 | 2 | -1 | -4
E | -2 | -3 | 1 | -1 | 0 | 3 | 1 | 0 | -9
D | -5 | 1 | -1 | -0 | -0 | -1 | 3 | 3 | -10
B | -5 | -2 | 1 | -2 | 4 | 2 | 3 | -1 | -18
A | -17 | -2 | -3 | -4 | 5 | 2 | 7 | 12 | -52
F | -9 | -13 | -12 | 1 | 0 | 13 | 6 | 14 | -66
$\mathbf{b}_{2}(j)$ | 83 | 42 | 24 | 10 | -18 | -42 | -45 | -53 |

####  6.1.3 TCA map

Equation (17) shows that the TCA map, Figure 2, is a change of scale of the
TCov map, Figure 3. In Figure 2, brand $i$ is represented by the principal
coordinates $(f_{1}(i),f_{2}(i)),$ and attribute $j$ is represented by the
principal coordinates $(g_{1}(j),g_{2}(j)).$ The first row and column
principal coordinates are given in Table 6 in decreasing order, and
accordingly the $(density-1=\frac{p_{ij}}{p_{i\ast}p_{\ast j}}-1$) values of
the attribute and brand categories are displayed in Table 6. However, the
relative position of some brands and attributes in Figure 2 are completely
different in Figure 3. For instance, in TCA map Figure 2 on dimension 1 the
brand C seems to be much more important than the brands Oracle or Nokia: By
Lemma 6, the covariance of brand C with the first factor is $f_{1}(C)=13/100,$
which is much larger than $f_{1}(Nokia)=9.3/100$; while in TCov map Figure 3
it is the opposite, the contributions are $a_{1}(C)=8/10000$ and
$a_{1}(Nokia)=129/10000$. This aspect is the cause of confusion and difficulty
in the interpretation of CA or TCA maps. Lemma 6 helps us to explain this
fact: The brand $C$ is strongly associated with the latent variable
innovative, but it does not contribute to the construction of this latent
variable. While Nokia is moderately associated with the latent variable
innovative, even though it constructs this latent variable; because it also
constructs the 2nd principal dimension.  That is, the TCov map helps us to
identify rows or columns that essentially contribute to the formation of a
latent variable, while the TCA map shows us the rows and the columns which are
highly associated with the latent variables defined by the TCov decomposition.

Table 6: WS (density -1) matrix ($\times$ $100)$ seriated along the first TCA
principal dimension.
---
Attribute
Company | innovative | trusted | rapport | efficient | solution | leader | relevant | essential | $f_{1}(i)$
C | 54 | 4 | $9$ | $6$ | $-13$ | $-17$ | $-9$ | $-67$ | $13$
Oracle | 40 | 4 | $-4$ | $-1$ | $-9$ | $14$ | $-23$ | $-38$ | $10$
B | 21 | 17 | $19$ | $17$ | $-7$ | $-32$ | $-28$ | $-29$ | $10$
E | 7 | 24 | $6$ | $0$ | $-1$ | $-18$ | $-19$ | $-10$ | $9.4$
Nokia | 39 | 2 | $-6$ | $-6$ | $-6$ | $4$ | $-7$ | $-34$ | $9.3$
G | 7 | 16 | $18$ | $-10$ | $-12$ | $-24$ | $-5$ | $16$ | $6.8$
A | 10 | 5 | $12$ | $8$ | $24$ | $-32$ | $-14$ | $-25$ | $3.8$
D | $1$ | $-5$ | $15$ | $-1$ | $22$ | $-26$ | $0$ | $-6$ | $-1.5$
H | $-7$ | $-4$ | $0$ | $2$ | $-2$ | $1$ | $6$ | $7$ | $-2.7$
I | $-14$ | $1$ | $-9$ | $-10$ | $7$ | $9$ | $-2$ | $42$ | $-2.9$
Fedex | $-10$ | $-4$ | $1$ | $3$ | $-4$ | $3$ | $5$ | $13$ | $-3.4$
F | $-25$ | $4$ | $3$ | $1$ | $10$ | $-3$ | $7$ | $0$ | $-4.5$
$g_{1}(j)$ | $17.6$ | $2.7$ | $-0.3$ | $-0.7$ | $-1.5$ | $-1.6$ | $-6.9$ | $-17.3$ |

####  6.1.4 WS data set by CA

Table 3 displays the $CA\\_QSR$ values for the WS data set: $CA\\_QSR_{1}$ and
$QSR_{1}$ values are identical by Lemma 10; $CA\\_QSR_{2}$ values being a
little bit better than the corresponding $QSR_{2}$ values. This is the main
reason that both CA and TCA maps, Figures 1 and 2, are very similar.

Table 7 represents the $sACQ$ and $sRSQ$ values for the WS data set, which
reflect, somewhat in a different way, the $CA\\_QSR$ values. For the first
principal dimension, there is a positively associated quadrant which
contributes $44.15\%$ to $\sigma_{1}$, and a negatively associated quadrant
whose contribution to $\sigma_{1}$ is $27.89\%.$ Furthermore, we note that in
the first principal dimension globally the excess attractive association is
quite small $100sRES_{1}/\sigma_{1}=9.9\%.$ For the second principal
dimension, there is a negatively associated quadrant which contributes
$48.43\%$ to $\sigma_{2}$; a postively associated quadrant whose contribution
to $\sigma_{2}$ is $28.43\%;$ and a small excess repulsive association
$100sRES_{2}/\sigma_{2}=-12.14\%$.

Table 7: sACQ and sRCQ of WS data for the first 4 dimensions of CA.
---
$\alpha$ | $100sACQ_{\alpha}(+)$ | $100sACQ_{\alpha}(-)$ | $100sRES_{\alpha}$ | $100\sigma_{\alpha}$
1 | (4.02, 0.98) | (-1.56, -2.54) | 0.90 | 9.10
2 | (0.76, 1.39) | (-2.38, -0.37) | -0.59 | 4.90
3 | (1.13, 0.29) | (-0.29, -1.02) | 0.11 | 2.72
4 | (0.48, 0.42) | (-0.52, -0.73) | -0.35 | 2.15
$\alpha$ | $100sRCQ_{\alpha}(+)$ | $100sRCQ_{\alpha}(-)$ | $100sRES_{\alpha}/\sigma_{\alpha}$ | $100\sigma_{\alpha}/\sigma_{\alpha}$
1 | (44.15, 10.80) |  ( -17.16,-27.89) | 9.90 | 100
2 | (15.50, 28.43) | (-48.43, -7.46) | -12.14 | 100
3 | (41.48, 10.50) | (-10.69, -37.34) | 3.95 | 100
4 | (22.39, 19.38) | (-24.25, -33.98) | -16.47 | 100

Here, we also want to remind the reader that CA and TCA of two other data sets
of marketing research, discussed in Bock (2011a) and Bendixen (1996), produced
similar results to the analysis of WS data set by both methods.

###  6.2 Faust data set

Faust (2005) analyzed by CA a two-mode affiliation network, (0-1) matrix
$\mathbf{Z=(}z_{ij})$ of size $22\times 15$; where the 22 rows represent 22
countries and the 15 columns the regional trade and treaty organizations in
the American continent. The country $i$ is a member of the organization $j$ if
$z_{ij}=1$; and $z_{ij}=0$ means the country $i$ is not a member of the
organization $j$. Choulakian and Abou-Samra (2020) compared the CA and TCA
maps, where they found that the TCA map is much more interpretable than the
corresponding CA map because of the existence of some influential columns and
rows (outliers?) that dominated the second CA principal dimension. Table 8
displays the QSR measures for both methods. $QSR_{1}=63.67\%$ is of comparable
value to $CA\\_QSR_{1}=61.25\%.$ However the corresponding values for the
second dimension are completely different: $QSR_{2}=57.96\%$ and
$CA\\_QSR_{2}=33.41\%,$ a significant difference of $24.55\%$. This is also
reflected in the $sRCQ_{2}$ values reported in Table 9, where a positively
associated quadrant contributes $78.56\%$ to $\sigma_{2}=0.5331$; while in TCA
this value is constant and equals 25%. Furthermore,
$100sRES_{2}/\sigma_{2}=63.41\%$ represents a significant excess of attractive
association.

Table 8: QSR values (in %) of Faust data for the first 4 dimensions.
---
$\alpha$ | $QSR_{\alpha}(+)$ | $QSR_{\alpha}(-)$ | $QSR_{\alpha}$ | $\delta_{\alpha}$
1 | (56.52, 53.79) | (-67.43, -85.43) | 63.67 | 0.4266
2 | (43.78, 70.80) | (-68.79, -57.12) | 57.96 | 0.2751
3 | (65.51, 44.38) | (-78.00, -82.57) | 63.77 | 0.2659
4 | (61.45, 55.04) | (-67.63, -44.33) | 55.72 | 0.1782
$\alpha$ | $CA\\_QSR_{\alpha}(+)$ | $CA\\_QSR_{\alpha}(-)CA\\_$ | $CA\\_QSR_{\alpha}$ | $\varpi_{\alpha}$
1 | (69.71, 44.86) |  ( -59.20,-84.94) | 61.25 | 0.4104
2 | (17.51, 63.72) | (-49.87, -37.22) | 33.41 | 0.1570
3 | (60.29, 43.54) | (-81.95, -45.06) | 54.09 | 0.2195
4 | (33.26, 55.77) | (-46.57, -54.61) | 45.57 | 0.1636

Table 9: sACQ and sRCQ of Faust data for the first 4 dimensions of CA.
---
$\alpha$ | $100sACQ_{\alpha}(+)$ | $100sACQ_{\alpha}(-)$ | $100sRES_{\alpha}$ | $100\sigma_{\alpha}$
1 | (22.56, 10.21) | (-11.13, -15.90) | 5.74 | 59.80
2 | (1.68, 41.89) | (-5.75, -4.00) | 33.81 | 53.31
3 | (4.55, 25.96) | (-8.26, -6.22) | 16.02 | 44.99
4 | (8.32, 11.90) | (-11.00, -8.17) | 1.06 | 39.40
$\alpha$ | $100sRCQ_{\alpha}(+)$ | $100sRCQ_{\alpha}(-)$ | $100sRES_{\alpha}/\sigma_{\alpha}$ | $100\sigma_{\alpha}/\sigma_{\alpha}$
1 | (37.72, 17.08) |  ( -18.62,-26.58) | 9.60 | 100
2 | (3.14, 78.56) | (-10.79, -7.50) | 63.41 | 100
3 | (10.11, 57.70) | (-18.36, -13.83) | 35.62 | 100
4 | (21.13, 30.21) | (-27.92, -20.74) | 2.68 | 100

###  6.3 Rodent species abundance data

Table 10 displays the seriated Rodent species abundance data of size
$28\times\ 9$, where 9 species of rodents have been counted at each of 28
sites in California. The data set is from Quinn and Keough (2002), but is
available in the R package TaxicabCA. Choulakian (2017) presented a detailed
analysis of this data set by CA and TCA. Given that, $\sigma_{1}^{2}\geq
0.7465,$ a value greater than 0.7 marked by Benzécri, we clearly observe the
quasi-two blocks diagonal structure in Table 10.

Table 10: CA seriated rodent species abundance data. |
---|---
Sites | rod1 | rod2 | rod6 | rod8 | rod3 | rod5 | rod9 | rod4 | rod7 | $f_{1}(site)$
24 | 1 |  |  |  |  |  |  |  |  | 3.49
17 | 3 |  |  |  |  |  |  |  |  | 3.49
21 | 2 | 1 |  |  |  |  |  |  |  | 3.35
10 | 1 | 2 |  |  |  |  |  |  |  | 3.21
9 | 3 | 8 |  |  |  |  |  |  |  | 3.18
14 | 1 | 3 |  |  |  |  |  |  |  | 3.17
7 |  | 11 |  |  |  |  |  |  |  | 3.07
11 |  | 9 |  |  |  |  |  |  |  | 3.07
15 |  | 11 |  |  |  |  |  |  |  | 3.07
25 |  | 5 |  |  |  |  |  |  |  | 3.07
22 |  | 3 |  |  |  |  |  |  |  | 3.07
8 |  | 16 |  |  |  |  |  |  |  | 3.07
16 |  | 4 |  |  |  |  |  |  |  | 3.07
1 |  | 13 | 2 |  | 3 | 1 |  | 1 |  | 1.85
20 | 3 |  |  |  | 27 |  |  | 1 |  | -0.4
12 |  | 3 | 16 | 7 | 1 | 5 |  |  |  | -0.5
3 |  | 4 | 9 |  | 36 | 2 |  |  |  | -0.13
13 |  | 4 | 12 |  | 39 | 4 |  |  |  | -0.17
4 |  | 4 | 30 | 18 | 53 | 5 | 3 | 1 |  | -0.28
18 |  | 2 | 14 | 4 | 78 | 10 |  |  |  | -0.34
5 |  | 2 | 16 |  | 63 | 11 |  | 21 |  | -0.37
23 |  |  | 8 | 2 |  | 2 |  |  |  | -0.38
26 |  |  | 11 | 2 | 22 |  |  |  |  | -0.39
27 |  |  | 9 | 1 | 29 | 10 |  |  |  | -0.40
19 |  |  |  |  | 1 |  |  |  |  | -0.41
28 |  |  | 1 |  | 10 |  |  | 1 |  | -0.42
6 |  | 1 | 8 | 2 | 48 | 12 | 2 | 35 | 12 | -0.44
2 |  | 1 | 16 | 2 | 57 | 9 | 3 | 65 | 8 | -0.45
$g_{1}(rod)$ | 3.02 | 2.65 | -0.32 | -0.32 | -0.36 | -0.37 | -0.44 | -0.48 | -0.52 |

Furthermore,

a) $CA\\_QSR_{1}(\mathbf{v}_{1+},\mathbf{u}_{1+})=0.965,$
$CA\\_QSR_{1}(\mathbf{v}_{1-},\mathbf{u}_{1-})=0.190,$

$CA\\_QSR_{1}(\mathbf{v}_{1-},\mathbf{u}_{1+})=-0.943,$
$CA\\_QSR_{1}(\mathbf{v}_{1+},\mathbf{u}_{1-})=-1.$

This shows that the quality of signs in the 4th quadrant of the
$Cov(\mathbf{P})$( not shown), is very poor, 0.190 ; and the result in
Corollary 11a is approximately satisfied, -1 and -0.943.

b) $sACQ_{1}(\mathbf{v}_{\alpha+},\mathbf{u}_{\alpha+})=0.674,\
sACQ_{1}(\mathbf{v}_{1-},\mathbf{u}_{1-})=0.014,$

$sACQ_{1}(\mathbf{v}_{1-},\mathbf{u}_{1+})=-0.084,\
sACQ_{1}(\mathbf{v}_{1+},\mathbf{u}_{1-})=-0.092.$

This shows that the result in Corollary 11b is approximately satisfied.

c) $sRCQ_{1}(\mathbf{v}_{\alpha+},\mathbf{u}_{\alpha+})=0.7798,\
sRCQ_{1}(\mathbf{v}_{1-},\mathbf{u}_{1-})=0.0158,$

$sRCQ_{1}(\mathbf{v}_{1-},\mathbf{u}_{1+})=-0.0974,\
sRCQ_{1}(\mathbf{v}_{1+},\mathbf{u}_{1-})=-0.1070.$

This shows that 78% of the contribution to $\sigma_{1}=0.864$ comes from the
2nd quadrant of Table 10; a very small proportion 1.58% comes from the 4th
quadrant of Table 10; and about 20% comes from the quasi-sparse blocks.

##  7 Conclusion

A crucial first step in data analysis of multivariate tables is the
preprocessing step: centering and/or scaling of the data. In the case of CA of
contingency tables, the row and column marginals are intricate part of the
centering and scaling of the method via the chi-square residuals defined for
the row and column profiles. In a pioneering work, Goodman (1991) and his
discussants compared the effects of marginal weighted scores and uniform
weighted scores in association and correlation models in the analysis of
contingency tables. A parallel to this problem is: Should we decompose the
covariance matrix (do Tucker interbattery analysis) or the correlation matrix
(do Hotelling canonical correlation analysis)?

This essay attempted to clarify mainly the following issues:

First, we showed that the aims of CA and TCA are different, but interrelated.
The aim of CA is to explain the heterogeneity of the row and column profiles
(row and column conditional distributions), from which as a byproduct we get a
view of the dependence structure of the cross-covariance matrix. While, the
aim of TCA is to explain the dependence structure of the cross-covariance
matrix, then as a by product obtain a view of the heterogeneity of the row and
column profiles. Empirical data have shown that the cross-covariance matrix
($p_{ij}-p_{i\ast}p_{\ast j})$ is much more robust than the chi-square
residual matrix ($\frac{p_{ij}-p_{i\ast}p_{\ast j}}{\sqrt{p_{i\ast}p_{\ast
j}}})$, a fact first observed by Tenenhaus and Augendre (1996).

Second, two maps are needed to fully picture the dependence/heterogeneity
structure in a contingency table; and Lemma 6 explains in a simple way the
relationship between the two maps.

Third, for a principal dimension we introduced the new concept of the
intrinsic quality and distinguished it from the often used extrinsic quality;
and related the intrinsic quality to the quality of signs of the residuals in
the four quadrants. Furthermore, we provided quantifications of the intrinsic
quality by introducing $QSR$ and $sACQ$ indices.

Fourth, we emphasized the importance of looking at the residual covariance
values at each iteration, a general procedure exemplified by Tukey (1977) in
exploratory data analysis.

    Acknowledgements

Choulakian’s research has been supported by NSERC of Canada. Choulakian thanks
Eric Beh for discussing this topic and furnishing few references; and Ahcène
Brahmi for help in editing.

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