# Distortion in Correspondence Analysis and in Taxicab Correspondence
Analysis: A Comparison

Choulakian V, Université de Moncton, Canada<EMAIL_ADDRESS>
(March 2024)

###### Abstract

Distortion is a fundamental well-studied topic in dimension reduction papers,
and intimately related with the underlying intrinsic dimension of a mapping of
a high dimensional data set onto a lower dimension. In this paper, we study
embedding distortions produced by Correspondence Analysis and its robust
$l_{1}$ variant Taxicab Correspondence analysis, which are visualization
methods for contingency tables. For high dimensional data, distortions in
Correspondence Analysis are contractions; while distortions in Taxicab
Correspondence Analysis could be contractions or stretchings. This shows that
Euclidean geometry is quite rigid, because of the orthogonality property;
while Taxicab geometry is quite flexible, because the orthogonality property
is replaced by the conjugacy property.

Key words: Distortion; contraction or stretching; intrinsic dimension; taxicab
correspondence analysis; data visualization.

AMS 2010 subject classifications: 62H25, 62H30

##  1 Introduction

There are two kinds of dimensionality reduction clearly mentioned in Freksen
(2021): First, Johnson-Lindenstrauss type where data are projected using
random projections; second, PCA type where data are projected using optimal
projections. In our case correspondence analysis (CA) and taxicab CA (TCA) are
PCA type.

Distortion is a fundamental concept well-studied in dimension reduction
papers, and intimately related with the underlying intrinsic dimension of a
mapping of a high dimensional data set onto a lower dimension; see in
particular, Johnson and Lindenstrauss (1982, 1984), Bourgain (1985), Vankadara
and von Luxburg (2018), Freksen (2021), Agrawal et al. (2021) and Meila and
Zhang (2023).

Vankadara and von Luxburg (2018) define the commonly used notion of global
distortion as ”Distortion measures aim to quantify the deviation of an
embedding from isometry”.

Meila and Zhang (2023, subsection 5.1) define local distortion, where they
state : ”Many embedding algorithms tend to contract regions of M (manifold)
where the data are densely sampled and to stretch the sparsely sampled
regions.” This observation seems to apply to TCA too but in visualization; see
the Taxicab parabola of type 1 in Figure 2 representing the structure of the
data set in Example 1. We note that there are 2 types of parabolas in Taxicab
geometry, see Figures 2 and 4.

The intrinsic dimension is an essential number for data visualization and
interpretation in PCA type analyses. We propose a new simple criterion within
TCA.

In this paper, we compare essential properties of distortion of the row or
column profiles of a contingency table from their barycenters. This will help
us also to deduce the lower and upper bounds of TCA intrinsic dimension of a
data set.

So the contents of this paper are: In section 2, we present an outline of CA
and TCA; section 3 presents main results; section 4 discusses applications;
and we conclude in section 5.

Benzécri (1973, volumes 1 and 2) is the reference book on CA. Among others,
Greenacre (1984) is one of the best representative of Benzécri’s approach; Beh
and Lombardo (2014) present a panoramic review of CA. Taxicab CA is a robust
$l_{1}$ variant of CA, see Choulakian (2006, 2016, 2021).

##  2 Preliminaries

Let $\mathbf{N}=(n_{ij})$ for $i=1,...,I$ and $j=1,...,J$ be a 2-way
contingency table and $\mathbf{P=N/}n=(p_{ij})$ of size $I\times J$ the
associated correspondence matrix (probability table) of N, where
$n=\sum_{i,j}n_{ij}$. We define as usual $p_{i+}=\sum_{j=1}^{J}p_{ij}$ the
marginal probability of the $i$th row, and similarly
$p_{+j}=\sum_{i=1}^{I}p_{ij}$ the marginal property of the $j$th column.

The CA association index of P is

$\displaystyle\Delta_{ij}$ $\displaystyle=(\frac{p_{ij}}{p_{i+}p_{+j}}-1)$
$\displaystyle=\frac{1}{p_{i+}p_{+j}}(p_{ij}-p_{i+}p_{+j})$
$\displaystyle=\frac{1}{p_{i+}}(\frac{p_{ij}}{p_{+j}}-p_{i+})$
$\displaystyle=\frac{1}{p_{+j}}(\frac{p_{ij}}{p_{i+}}-p_{+j}),$

where: $\frac{p_{ij}}{p_{i+}p_{+j}}$ is density function of $p_{ij}$ with
respect to $p_{i+}p_{+j},$ $\frac{p_{ij}}{p_{+j}}$ is the profile of the $j$th
column and similarly $\frac{p_{ij}}{p_{i+}}$ is the profile of the $i$th row.

The CA or Taxicab CA (TCA) decomposition is, for $k=$

$rank(p_{ij}-p_{i+}p_{+j}),$

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

calculated by generalized singular value decomposition (SVD) or taxicab SVD
(TSVD) of $\Delta_{ij}.$

A reference for the CA and TCA basic equations is Choulakian et al. (2014).

###  2.1 CA basic equations

The parameters $(f_{\alpha}(i),g_{\alpha}(j),\delta_{\alpha})$ satisfy: for
$\alpha,\beta=1,...,k=$

$rank(p_{ij}-p_{i+}p_{+j}):$

$\delta_{\alpha}^{2}=\sum_{i=1}^{I}f_{\alpha}^{\ \
2}(i)p_{i+}=\sum_{j=1}^{J}g_{\alpha}^{2}(j)p_{+j}$

$0=\sum_{i=1}^{I}f_{\alpha}(i)p_{i+}=\sum_{j=1}^{J}g_{\alpha}(j)p_{+j}$

$0=\sum_{i=1}^{I}f_{\alpha}(i)f_{\beta}(i)p_{i+}=\sum_{j=1}^{J}g_{\alpha}(j)g_{\beta}(j)p_{+j}$
for $\alpha\neq\beta.$

The last equatins show that the row factor scores $(f_{\alpha}(i))$ and the
column factor scores $(g_{\alpha}(j))$ for $\alpha=1,...,k$ are ORTHOGONAL;
which is an essential property of the EUCLIDEAN geometry.

The distance (also named Benzécri distance) between the $i$th row profile and
its barycenter is

$dist_{Benz}^{2}(\frac{P_{ij}}{P_{i+}},P_{+j})=\sum_{j=1}^{J}\frac{(\frac{P_{ij}}{P_{i+}}-P_{+j})^{2}}{P_{+j}}=\sum_{j=1}^{J}P_{+j}(\frac{P_{ij}}{P_{+j}P_{i+}}-1)^{2}=\sum_{\alpha=1}^{k}(f_{\alpha}(i))^{2}.$

Similarly, the distance between the $j$th column profile and its barycenter is

$dist_{Benz}^{2}(\frac{P_{ij}}{P_{+j}},P_{i+})=\sum_{i=1}^{I}\frac{(\frac{P_{ij}}{P_{+j}}-P_{i+})^{2}}{P_{i+}}=\sum_{i=1}^{I}P_{i+}(\frac{P_{ij}}{P_{+j}P_{i+}}-1)^{2}=\sum_{\alpha=1}^{k}(g_{\alpha}(j))^{2}.$

The total variance (also named inertia) is:

$weightedAve=\sum_{i,j}\frac{(p_{ij}-p_{i+}p_{+j})^{2}}{p_{i+}p_{+j}}=\sum_{i,j}P_{i+}P_{+j}(\frac{P_{ij}}{P_{+j}P_{i+}}-1)^{2}=$

$\ \ =\sum_{i=1}^{I}P_{i+}\
dist_{Benz}^{2}(\frac{P_{ij}}{P_{i+}},P_{+j})=\sum_{=1}^{J}P_{+j}\
dist_{Benz}^{2}(\frac{P_{ij}}{P_{+j}},P_{i+})=\sum_{\alpha=1}^{k}\delta_{\alpha}^{2}.$

###  2.2 TCA basic equations

The parameters $(f_{\alpha}(i),g_{\alpha}(j),\delta_{\alpha})$ satisfy: for
$\alpha,\beta=1,...,k=$

$rank(p_{ij}-p_{i+}p_{+j}):$

$\delta_{\alpha}=\sum_{i=1}^{I}|f_{\alpha}^{\
}(i)|p_{i+}=\sum_{j=1}^{J}|g_{\alpha}(j)|p_{+j}$

$0=\sum_{i=1}^{I}f_{\alpha}(i)p_{i+}=\sum_{j=1}^{J}g_{\alpha}(j)p_{+j}$

$0=\sum_{i=1}^{I}f_{\alpha}(i)\
sign(f_{\beta}(i))p_{i+}=\sum_{j=1}^{J}g_{\alpha}(j)\
sign(g_{\beta}(j))p_{+j}$ for $\alpha>\beta.$

The last equatins show that the row factor scores $(f_{\alpha}(i))$ and the
column factor scores $(g_{\alpha}(j))$ for $\alpha=1,...,k$ are CONJUGATE;
which is an essential property of the TAXICAB geometry.

The taxicab distance between the $i$th row profile and its barycenter is

$dist_{Taxi}(\frac{P_{ij}}{P_{i+}},P_{+j})=\sum_{j=1}^{J}P_{+j}|\frac{P_{ij}}{P_{+j}P_{i+}}-1|=\sum_{j=1}^{J}|(\frac{P_{ij}}{P_{i+}}-P_{+j})|\
\leq\sum_{\alpha=1}^{k}|f_{\alpha}(i)|.$

Similarly, the taxicab distance between the $j$th column profile and its
barycenter is

$distd_{Taxi}(\frac{P_{ij}}{P_{j+}},P_{i+})=\sum_{i=1}^{I}P_{i+}|\frac{P_{ij}}{P_{+j}P_{i+}}-1|=\sum_{i=1}^{I}|(\frac{P_{ij}}{P_{j+}}-P_{i+})|\
\leq\sum_{\alpha=1}^{k}|g_{\alpha}(j)|.$

The total raw data dispersion is

$weightedAve=\sum_{i,j}|P_{ij}-P_{i+}P_{+j}|\
=\sum_{i,j}P_{i+}P_{+j}|\frac{P_{ij}}{P_{+j}P_{i+}}-1|\ =$

$\ \ \ \ =\sum_{j=1}^{J}P_{+j}\
dist_{Taxi}(\frac{P_{ij}}{P_{+j}},P_{i+})=\sum_{i=1}^{I}P_{i+}\
dist_{Taxi}(\frac{P_{ij}}{P_{i+}},P_{+j})\leq\sum_{\alpha=1}^{k}\delta_{\alpha}.$

##  3 CA and TCA distortions

Distortion: Comparison of the distances on the raw data set and on the maps.
Given two metric spaces $(X,dist_{x})$ and $(Y,dist_{y})$, and an injective
mapping $\phi:X->Y,$ for two points $(u,v)\in X\times X$ and their mappings
$(\phi(u),\phi(v))\in Y\times Y$, our aim is the comparison of the distances
$dist_{x}(u,v)$ and $dist_{y}(\phi(u),\phi(v))$. We can have 3 cases:

Case 1: $dist_{x}(u,v)=dist_{y}(\phi(u),\phi(v))$; distortion is null, and the
mapping $\phi$ is called isometric embedding for the pair of points $(u,v)$.

Case 2: $dist_{x}(u,v)>dist_{y}(\phi(u),\phi(v))$; distortion is a
contraction, and the mapping $\phi$ is called contractive embedding for the
pair of points $(u,v)$.

Case 3: $dist_{x}(u,v)<dist_{y}(\phi(u),\phi(v))$; distortion is a stretching
(also named an expansion), and the mapping $\phi$ is called stretch embedding
for the pair of points $(u,v)$.

Let $d$ be the embedding dimension. For the next two subsections, we accept
the following:

Assumption: For $d=1,...,rank(p_{ij}-p_{i+}p_{+j})$,  $\sum_{\alpha=1}^{d}|$
$f_{\alpha}(i)|\ \neq 0.$

###  3.1 CA distortion

In CA: distortion is a CONTRACTION.

For $1\leq d<rank(p_{ij}-p_{i+}p_{+j})$

a)
$dist_{Benz}^{2}(\frac{P_{ij}}{P_{i+}},P_{+j})=\sum_{j=1}^{J}\frac{(\frac{P_{ij}}{P_{i+}}-P_{+j})^{2}}{P_{+j}}=\sum_{j=1}^{J}P_{+j}(\frac{P_{ij}}{P_{+j}P_{i+}}-1)^{2}\geq\sum_{\alpha=1}^{d}(f_{\alpha}(i))^{2}$

b) $weightedAve=\sum_{i=1}^{I}P_{i+}\
dist_{Benz}^{2}(\frac{P_{ij}}{P_{i+}},P_{+j})=\sum_{i,j}\frac{(p_{ij}-p_{i+}p_{+j})^{2}}{p_{i+}p_{+j}}>\sum_{\alpha=1}^{d}\delta_{\alpha}^{2}$

This can be re-expressed in the following way: There exists a constant
$c_{1}\in(0,1)$ such that

$c_{1}$
$dist_{Benz}^{2}(\frac{P_{ij}}{P_{i+}},P_{+j})\leq\sum_{\alpha=1}^{d}(f_{\alpha}(i))^{2}\leq
dist_{Benz}^{2}(\frac{P_{ij}}{P_{i+}},P_{+j})$

or

$c_{1}\leq\frac{\sum_{\alpha=1}^{d}(f_{\alpha}(i))^{2}}{dist_{Benz}^{2}(\frac{P_{ij}}{P_{i+}},P_{+j})}\leq
1$

where
$c_{1}=\min_{i}\frac{\sum_{\alpha=1}^{d}(f_{\alpha}(i))^{2}}{dist_{Benz}^{2}(\frac{P_{ij}}{P_{i+}},P_{+j})}$

###  3.2 TCA distortion

In TCA: distortion is CONTRACTION or STRETCHING (also named EXPANSION). The
sign $?$ that we use below, means equal, less than or greater than.

a1) For $d=1$

$dist_{Taxi}(\frac{P_{ij}}{P_{i+}},P_{+j})=\sum_{j=1}^{J}|(\frac{P_{ij}}{P_{i+}}-P_{+j})|\
\geq|f_{1}(i)|\ $is a CONTRACTION

a2) For $2\leq d<rank(p_{ij}-p_{i+}p_{+j})$

$dist_{Taxi}(\frac{P_{ij}}{P_{+j}},P_{i+})=\sum_{j=1}^{J}|(\frac{P_{ij}}{P_{i+}}-P_{+j})|\
=\sum_{j=1}^{J}P_{+j}|\frac{P_{ij}}{P_{+j}P_{i+}}-1|\
?\sum_{\alpha=2}^{d}|f_{\alpha}(i)|\ \ $

b1) For $d=1$

$weightedAve=\sum_{j=1}^{J}P_{+j}\
dist_{Taxi}(\frac{P_{ij}}{P_{+j}},P_{i+})=\sum_{i,j}|P_{ij}-P_{i+}P_{+j}|\
>\delta_{1}$

b2) For $2\leq d<rank(p_{ij}-p_{i+}p_{+j})$

$weightedAve=\sum_{j=1}^{J}P_{+j}\
dist_{Taxi}(\frac{P_{ij}}{P_{+j}},P_{i+})=\sum_{i,j}|P_{ij}-P_{i+}P_{+j}|\ ?\
\sum_{\alpha=1}^{d}{}\delta_{\alpha}$

This can be re-expressed in the following way: There exist 2 constants
$c_{1}>0,$ $c_{2}>1$ and $c_{2}>c_{1}$, such that

$c_{1}$
$dist_{Taxi}(\frac{P_{ij}}{P_{+j}},P_{i+})\leq\sum_{\alpha=1}^{d}|f_{\alpha}(i)|\leq
c_{2}$ $dist_{Taxi}(\frac{P_{ij}}{P_{+j}},P_{i+})$

or

$c_{1}\leq\frac{\sum_{\alpha=1}^{d}|f_{\alpha}(i)|}{dist_{Taxi}(\frac{P_{ij}}{P_{+j}},P_{i+})}\leq
c_{2}$

where

$c_{1}=\min_{i}\frac{\sum_{\alpha=1}^{d}|f_{\alpha}(i)|}{dist_{Taxi}(\frac{P_{ij}}{P_{+j}},P_{i+})}$
and
$c_{2}=\max_{i}\frac{\sum_{\alpha=1}^{d}|f_{\alpha}(i)|}{dist_{Taxi}(\frac{P_{ij}}{P_{+j}},P_{i+})}$

Remark: We see that for the embedding dimension $d=1$, the distortion is a
CONTRACTION for both CA and TCA. While, for $d\geq 2$, the nature of
distortion can vary between CA and TCA.

We have:

Definition:

a) The upper bound for $TCA\ intrinsic\ dimension$ is:

$\dim
Upper_{intrinsic}=\arg\min_{d}(\sum_{\alpha=1}^{d}\delta_{\alpha}-|\sum_{i,j}|P_{ij}-P_{i+}P_{+j}|).$

b) The lower bound for $TCA\ intrinsic\ dimension$ is:

$\dim
Lower_{intrinsic}=\arg\min_{d}(\sum_{\alpha=1}^{d}\sum_{i,j}|P_{ij}-P_{i+}P_{+j}|\
-\delta_{\alpha}).$

Corollary: By b1) $\dim Upper_{intrinsic}\geq 2$ and $\dim
Lower_{intrinsic}\geq 1.$

##  4 Examples

We present some comparisons of the main results presented in the precedent
section. We apply three R packages: the ca package by Greenacre et al. (2022
), the ade4 package by Dray et al. (2023) and TaxicabCA by Allard and
Choulakian (2019).

###  4.1 Colors of music data set

The data set is a contingency table of size $10\times 9$, found in Abdi and
Bera (2016); where $J=9$ kinds of music and $I=10$ colors. It is constructed
in the following way: ”A set of 12 children and 10 adults picked up the color
that best describes each of 9 pieces of music”. The sparsity of the data set
is 21.11% using the following R code:

sum(dataMatrix == 0)/length(dataMatrix)

[1] 0.2111111

Figures 1 and 2 display the CA and TCA maps; we note some important
differences. In the CA plot, the Black color dominates first principal
dimension. In Figure 2, the rows and the columns represent taxicab parabolas,
where the absolute value of the slope of the directrix is greater than 1; see
among others Gardner (1997, ch. 10), Wilson (2011) and Petrović et al. (2014).
The first principal dimension has very simple interpretation: From the left
side to right, the colors are ordered rom the most concentrated (Black and
Brown) to the most dispersed (purple and pink); see our comments on Meila and
Zhang (2023, subsection 5.1) in the introduction. The same interpretation
applies also to the types of music.

Table 1 displays some important CA raw data and embedded distance values of
the 10 colors:
$\sum_{j=1}^{J}\frac{(\frac{P_{ij}}{P_{i+}}-P_{+j})^{2}}{P_{+j}}=$
$dist_{Benz}^{2}(\frac{P_{ij}}{P_{i+}},P_{+j})$ is the Benzécri distance
between the $i$th color profile and its barycenter; while for the embedding
dimensions $d=1,2,3,$
$\sum_{\alpha=1}^{d}|f_{\alpha}(i)|^{2}=dist_{d}^{2}(\phi(\frac{P_{ij}}{P_{i+}}),\phi(P_{+j}))$
where $\phi=CA$ embedding on the first $d$ principal dimensions. We see that
$dist_{Benz}^{2}(\frac{P_{ij}}{P_{i+}},P_{+j})>dist_{d}^{2}(\phi(\frac{P_{ij}}{P_{i+}}),\phi(P_{+j})),$
so distortion values resulting from the CA embedddings on the first three
principal dimensions for the 10 rows (colors) represent contractions.

Table 1: CA distances of the 10 colors from their barycenter.
---
color$(i)$ | $dist_{Benz}^{2}(\frac{P_{ij}}{P_{i+}},P_{+j})$ | $|f_{1}(i)|^{2}$ | $\sum_{\alpha=1}^{2}|f_{\alpha}(i)|^{2}$ | $\sum_{\alpha=1}^{3}|f_{\alpha}(i)|^{2}$
Red | 0.2187 | $0.0007$ | $0.0904$ | $0.1054$
Orange | 0.3333 | $0.0984$ | $0.11521$ | $0.2099$
Yellow | 0.4544 | $0.1211$ | $0.1618$ | $0.2597$
Green | 0.4121 | $0.0019$ | $0.2423$ | $0.0490$
Blue | 0.5208 | $0.0067$ | $0.0490$ | $0.1001$
Purple | 0.7574 | $0.3828$ | $0.6082$ | $0.6104$
White | 1.3878 | $0.1074$ | $0.1106$ | $1.2299$
Black | 1.5363 | $1.4275$ | $1.5270$ | $1.5316$
Pink | 0.8750 | $0.3249$ | $0.4152$ | $0.5629$
Brown | 1.0204 | $0.0127$ | $1.0059$ | $1.0139$
weightedAve | 0.7462 | $0.2880$ | $0.48113$ | $0.6196$
cols&rows | $\sum_{i,j}\frac{(p_{ij}-p_{i+}p_{+j})^{2}}{p_{i+}p_{+j}}$ | $\delta_{1}^{2}$ | $\delta_{1}^{2}+\delta_{2}^{2}$ | $\sum_{\alpha=1}^{3}\delta_{\alpha}^{2}$

Table 2 displays some important TCA raw data and embedded distance values of
the 10 colors: $\sum_{j=1}^{J}|\frac{P_{ij}}{P_{i+}}-P_{+j}|=$
$dist_{Taxi}(\frac{P_{ij}}{P_{i+}},P_{+j})$ is the Taxicab distance between
the $i$th color profile and its barycenter; while for $d=1,2,3,$
$\sum_{\alpha=1}^{d}|f_{\alpha}(i)|\
=dist_{d}(\phi(\frac{P_{ij}}{P_{i+}}),\phi(P_{+j}))$ where $\phi=TCA$
embedding on the first $d$ principal dimensions. We see that for $d=1,\
dist_{Taxi}(\frac{P_{ij}}{P_{i+}},P_{+j})>dist_{1}(\phi(\frac{P_{ij}}{P_{i+}}),\phi(P_{+j})),$
so distortion values resulting from the TCA embedddings on the first principal
dimension for the 10 rows (colors) represent contractions; while for $d=2$,
$dist_{Taxi}(\frac{P_{ij}}{P_{i+}},P_{+j})?dist_{2}(\phi(\frac{P_{ij}}{P_{i+}}),\phi(P_{+j})),$
so distortion values resulting from the TCA embedddings on the first two
principal dimensions for the 10 rows (colors) represent contractions or
stretchings (in bold colors). Finally, for $d=3$,
$dist_{Taxi}(\frac{P_{ij}}{P_{i+}},P_{+j})<dist_{3}(\phi(\frac{P_{ij}}{P_{i+}}),\phi(P_{+j})),$
so distortion values resulting from the TCA embedddings on the first three
principal dimensions for the 10 rows (colors) represent stretchings (in bold
colors).

Furthermore, $TCA\ intrinsic\ dimension$ is $\dim_{intrinsic}=2.$

Table 2: TCA distances of the 10 colors from their barycenter.
---
music$(i)$ | $dist_{Taxi}(\frac{P_{ij}}{P_{+j}},P_{i+})$ | $|f_{1}(i)|$ | $\sum_{\alpha=1}^{2}|f_{\alpha}(i)|$ | $\sum_{\alpha=1}^{3}|f_{\alpha}(i)|$
Red | 0.4444 | $0.2222$ | $0.3456$ | 0.4671
Orange | 0.4444 | $0.3333$ | 0.5371 | 0.7394
Yellow | 0.5688 | $0.4088$ | 0.7152 | 0.9890
Green | 0.5411 | $0.1546$ | $0.3564$ | 0.7083
Blue | 0.5965 | $0.1637$ | $0.5354$ | 0.7459
Purple | 0.8034 | $0.7350$ | 0.8141 | 0.9773
White | 1.0476 | $0.1746$ | $0.7239$ | 1.5496
Black | 1.0038 | $0.6973$ | 1.4940 | 1.9288
Pink | 0.7777 | $0.5555$ | $0.6974$ | 1.0278
Brown | 0.9206 | $0.6349$ | 1.1779 | 1.5834
weightedAve | 0.7048 | $0.4063$ | 0.7644 | 1.0890
cols&rows | $\sum_{i,j}|P_{ij}-P_{i+}P_{+j}|$ | $\delta_{1}$ | $\delta_{1}+\delta_{2}$ | $\sum_{\alpha=1}^{3}\delta_{\alpha}$

###  4.2 Aravo ecological abundance data set

We consider the Aravo data set N of size $75\times 82$ found in the R package
ade4 by Dray et al.(2023). This dataset describes the distribution of 82
species of Alpine plants in 75 sites. The percentage of zero counts in N is
$59.29\%$.

Figures 3 and 4 display the CA and TCA maps; both represent parabolas.
However, in Figure 4, the rows represent taxicab parabola, where the absolute
value of the slope of the directrix is less than 1. Figures 2 and 4 represent
the two kinds of Taxicab parabolas as discussed by, see among others Gardner
(1997, ch. 10), Wilson (2011) and Petrović et al. (2014).

we have: $\delta_{1}=0.627$, $\delta_{1}+\delta_{2}=1.085,\
\delta_{1}+\delta_{2}+\delta_{3}=1.463$ and
$\sum_{i,j}|P_{ij}-P_{i+}P_{+j}|=1.249;$ so $\dim Upper_{intrinsic}=3$ and
$\dim Lower_{intrinsic}=2;$ given the parabola, we deduce that TCA
$\dim_{intrinsic}=2.$

###  4.3 Rodent ecological abundance data

We consider the rodent data set N of size $28\times 9$ found in the R package
TaxicabCA by Allard and Choulakian (2019), taken from Quinn and Keough (2002).
This is an abundance data set of 9 species of rodents in 28 cities in
California. The percentage of zero counts in N is $66.27\%$. Choulakian (2017)
analyzed it by comparing the CA and TCA maps; using Benzécri’s criterion,
Choulakian (2021) showed that it has quasi 2-blocks diagonal structure (or
partly decomposable); furthermore, Choulakian (2023) analyzed it by Goodman’s
marginal-free correspondence analysis.

We leave the interpretation of the distortion values in Tables 3 and 4 to the
reader.

However we see that: $TCA\ intrinsic\ dimension$ is $\dim_{intrinsic}=2.$

Table 3: CA distances of the 9 rodents from their barycenter.
---
rodent$(j)$ | $\sum_{i=1}^{I}\frac{(\frac{P_{ij}}{P_{+j}}-P_{i+})^{2}}{P_{i+}}$ | $|g_{1}(j)|^{2}$ | $\sum_{\alpha=1}^{2}|g_{\alpha}(j)|^{2}$ | $\sum_{\alpha=1}^{3}|g_{\alpha}(j)|^{2}$
1 | 34.91 | $6.79$ | $34.86$ | $34.87$
2 | 5.85 | $5.24$ | $5.84$ | $5.85$
3 | 0.245 | $0.09$ | $0.10$ | $0.12$
4 | 1.60 | $0.17$ | $0.17$ | $1.49$
5 | 0.49 | $0.10$ | $0.10$ | $0.11$
6 | 0.60 | $0.07$ | $0.08$ | $0.36$
7 | 3.00 | $0.20$ | $0.20$ | $2.04$
8 | 2.50 | $0.08$ | $0.09$ | $0.10$
9 | 1.63 | $0.15$ | $0.15$ | $0.35$
weightedAve | 1.719 | $0.746$ | $1.205$ | $1.493$
cols&rows | $\sum_{i,j}\frac{(p_{ij}-p_{i+}p_{+j})^{2}}{p_{i+}p_{+j}}$ | $\delta_{1}^{2}$ | $\delta_{1}^{2}+\delta_{2}^{2}$ | $\sum_{\alpha=1}^{3}\delta_{\alpha}^{2}$

Table 4: TCA distances of the 9 rodents from their barycenter.
---
rodent$(j)$ | $\sum_{i=1}^{I}|(\frac{P_{ij}}{P_{+j}}-P_{i+})|$ | $|g_{1}(j)|$ | $\sum_{\alpha=1}^{2}|g_{\alpha}(j)|$ | $\sum_{\alpha=1}^{3}|g_{\alpha}(j)|$
1 | 1.888 | $0.803$ | $1.595$ | 2.522
2 | 1.408 | $0.876$ | 1.819 | 2.483
3 | 0.390 | $0.306$ | 0.489 | 0.682
4 | 1.15 | $0.848$ | 1.605 | 2.134
5 | 0.542 | $0.149$ | $0.341$ | $0.514$
6 | 0.547 | $0.426$ | 0.814 | 1.124
7 | 1.439 | $1.232$ | 2.110 | 2.689
8 | 1.141 | $0.558$ | 1.621 | 2.425
9 | 1.212 | $0.482$ | $0.527$ | 1.324
weightedAve | 0.705 | $0.478$ | 0.900 | 1.248
cols&rows | $\sum_{i,j}|P_{ij}-P_{i+}P_{+j}|$ | $\delta_{1}$ | $\delta_{1}+\delta_{2}$ | $\sum_{\alpha=1}^{3}\delta_{\alpha}$

###  4.4 Sacred books data set

The data set represents an extremely sparse contingency table of textual data
(a bag of words) found in Sah and Fokoue (2019). It is of size $I\times
J=590\times 8265$, where $I=590$ represents fragments of chapters of 8 sacred
books and $J=8265$ represents the number of distinct words. The real
percentage of zero cells is 98.65%. It is studied and visualized quite in
detail by (12+1) dimension reduction methods in Ma, Sun and Zou (2022,2023).
Choulakian and Allard (2023) visualized this data set by the joint use of CA
and TCA methods. From Table 1 in Choulakian and Allard (2023), we have the
first 4 principal values: $\delta_{1}=0.669$,
$\delta_{2}=0.437,\delta_{3}=0.431,\delta_{4}=0.391$ and
$\sum_{i,j}|P_{ij}-P_{i+}P_{+j}|=1.748976;$ so $\dim Upper_{intrinsic}=4$ and
$\dim Lower_{intrinsic}=3.$

###  4.5 Saporta-Tambrea data set

Saporta and Tambrea (1993, II.3 example) present a data table of size
$19\times 13$, where the counts represent the number of times each of 1000
respondents associates an item ( among 19) to 13 brands of diet butters. Due
to multiple answers, the total number of counts $n=21900$. There are no zero
counts in the table. By applying the Malinvaud test, they found the $CA\
intrinsic\ dimension=2.$

Using TCA, we have: $\delta_{1}=0.05386276$,
$\delta_{1}+\delta_{2}=0.08952342,\
\delta_{1}+\delta_{2}+\delta_{3}=0.12062140$ and
$\sum_{i,j}|P_{ij}-P_{i+}P_{+j}|=0.08858070;$ so $TCA\ intrinsic\ dimension$
is $\dim_{intrinsic}=2.$

###  4.6 Food Of The World data set

This data set is a contingency table of size $26\times 68,$ where 26 world
cuisines are described by their $68$ cooking ingredients; it is available
online at

https://github.com/HerveAbdi/data4PCCAR/tree/master/data

The percentage of zero cells is 21.15%.

CA and TCA maps (not shown) are quite similar. Using TCA, we have:
$\delta_{1}=0.4083893$,
$\delta_{1}+\delta_{2}=0.6428507,\delta_{1}+\delta_{2}+\delta_{3}=0.8548540$
and $\sum_{i,j}|P_{ij}-P_{i+}P_{+j}|\ =0.5907166;$ so $TCA\ intrinsic\
dimension$ is $\dim_{intrinsic}=2.$

##  5 Conclusion

In dimension reduction methods for visualization problems, our impression from
observations is that EUCLIDEAN GEOMETRY is quite RIGID, because of the
ORTHOGONALITY property; so embedding distortions are contractions. While
TAXICAB GEOMETRY is quite FLEXIBLE, because orthogonality property is replaced
by the CONJUGACY property, thus embedding distortions are contractions or
stretchings. To be more precise as we stated in section 2:

In CA, the variance ordered factor scores are orthogonal:

$0=\sum_{i=1}^{I}f_{\alpha}(i)f_{\beta}(i)p_{i+}=\sum_{j=1}^{J}g_{\alpha}(j)g_{\beta}(j)p_{+j}$
for $\alpha\neq\beta.$

In TCA, the dispersion ordered factor scores are conjugate:

$0=\sum_{i=1}^{I}f_{\alpha}(i)\
sign(f_{\beta}(i))p_{i+}=\sum_{j=1}^{J}g_{\alpha}(j)\
sign(g_{\beta}(j))p_{+j}$ for $\alpha>\beta.$

To see clearly the flexibility of the Taxicab approach we provide two simple
examples.

Example 1: We consider two vectors v${}_{1}=(_{2}^{1})$ and
v${}_{2}=(_{1}^{2}).$ It is IMPOSSIBLE to find a vector v $\mathbf{=\
}(_{y}^{x})$ $\neq\mathbf{0}$ which is orthogonal to both vectors v1 and v2;
that is, vtv${}_{1}\ =\ $vtv${}_{2}=0$. While it is POSSIBLE to find a vector
v that is conjugate to both vectors v1 and v2: for instance v $=$
$(_{-3}^{3});$ then vtsign(v${}_{1})=\ $vtsign(v${}_{2})=0$.

Example 2: The famous recreational mathematician Gardner (1997, ch. 10)
provides many examples of Taxicab plane gometry, which distinguishes it from
the Euclidean plane geometry, and shows its flexibility. We provided an
example of data analysis displayed in Figure 2. A second simple example
discussed by Gardner is: ”A taxicab scalene triangle with corners A, B, and C
and sides of 14, 8, and 6 is shown at the left in Figure 65. The sides of taxi
polygons must of course be taxi paths, and the paths that make up a polygon of
specified dimensions may vary in shape but not in length. Observe how the
triangle in the illustration violates the Euclidean theorem that the sum of
any two sides of a triangle must be greater than the third side. In this case
the sum of two sides equals the third: 6 + 8 equals 14.”

As stated in Example 2, Taxicab plane geometry is considered recreational
mathematics and a curiosity for high-school students. But we think, it is a
new open rich area of research for data analysis worth pursuing.

Acknowledgements: Choulakian’s research has been supported by NSERC of Canada.

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