# Uncovering and displaying the coherent groups of rank data by exploratory
riffle shuffling

Vartan Choulakian and Jacques Allard  Université de Moncton, Canada  email:
<EMAIL_ADDRESS><EMAIL_ADDRESS>
(November 2020)

###### Abstract

Let $n$ respondents rank order $d$ items, and suppose that $d<<n$. Our main
task is to uncover and display the structure of the observed rank data by an
exploratory riffle shuffling procedure which sequentially decomposes the n
voters into a finite number of coherent groups plus a noisy group: where the
noisy group represents the outlier voters and each coherent group is composed
of a finite number of coherent clusters. We consider exploratory riffle
shuffling of a set of items to be equivalent to optimal two blocks seriation
of the items with crossing of some scores between the two blocks. A riffle
shuffled coherent cluster of voters within its coherent group is essentially
characterized by the following facts: a) Voters have identical first TCA
factor score, where TCA designates taxicab correspondence analysis, an L1
variant of correspondence analysis; b) Any preference is easily interpreted as
riffle shuffling of its items; c) The nature of different riffle shuffling of
items can be seen in the structure of the contingency table of the first-order
marginals constructed from the Borda scorings of the voters; d) The first TCA
factor scores of the items of a coherent cluster are interpreted as Borda
scale of the items. We also introduce a crossing index, which measures the
extent of crossing of scores of voters between the two blocks seriation of the
items. The novel approach is explained on the benchmarking SUSHI data set,
where we show that this data set has a very simple structure, which can also
be communicated in a tabular form.

Key words: Borda score and scale; exploratory riffle shuffle; coherent group;
coherent cluster; crossing index; taxicab correspondence analysis.

AMS 2010 subject classifications: 62H25, 62H30

##  1 Introduction

Ordering the elements of a set is a common decision making activity, such as,
voting for a political candidate, choosing a consumer product, etc. So there
is a huge literature concerning the analysis and interpretation of preference
data scattered in different disciplines. Often rank data is heterogenous: it
is composed of a finite mixture of components. The traditional methods of
finding mixture components of rank data are mostly based on parametric
probability models, distance or latent class models, and are useful for sparse
data and not for diffuse data.

Rank data are sparse if there are at most a small finite number of
permutations that capture the majority of the preferences; otherwise they are
diffuse. As a running example in this paper, we will consider the famous
benchmarking SUSHI data set enumerating $n=5000$ preferences of $d=10$ sushis,
see $\left[1\right]$. The SUSHI data set is diffuse, because there are at most
three counts for one observed permutation. It has been analyzed, among others
by $\left[2,3,4\right]$.

A second data set that we shall also analyze is the APA dataset of size
$n=5738$ by $d=5$, see $\left[5\right]$. APA data set is also considered as
non sparse by $\left[2\right]$, because all the 120 permutations exist with
positive probability.

For a general background on statistical methods for rank data, see the
excellent monograph by $\left[6\right]$ and the book $\left[7\right]$.

###  1.1 Riffle shuffle

The riffle shuffle, see $\left[8\right]$, is considered the most popular
method of card shuffling, in which one cuts a deck of $d$ cards (aka items)
into two piles of sizes $d_{1}$ and $d_{2}$, respectively, and successively
drops the cards, one by one, so that the piles are interleaved into one deck
again.

Let $V,$ named a voting profile, be a set of $n$ preferences on $d$ items.
Based on riffle shuffling ideas, $\left[2\right]$ proposed the notion of
riffled independence to model the joint probability distribution of observed
preferences of $V$. Independently, $\left[9\right]$ used it for visual
exploration of $V$, naming it two blocks partition of the Borda scored items
with crossing of some scores; this will be further developed here under the
following important

Assumption: $d<<n.$ This means that the sample size $n$ is quite large
compared to the number of items $d$.

SUSHI and APA data sets satisfy this assumption.

The most important first step in the application of a riffle shuffling
procedure is how to partition the $d$ items into two disjoint subsets. In the
probabilistic riffle shuffling approach of $\left[2\right]$, the partitioning
step is essentially done using some adhoc approach in the case of the SUSHI
data set or based on background second order information of the items in the
case of the APA data set. While in the exploratory riffle shuffling approach
of this paper an optimal partition is obtained by maximizing the cut norm of
row centered data, or equivalently by taxicab correspondence analysis of nega
coded data.

We compare the two formulations of riffle shuffle, probabilistic and
exploratory, in section 10.

###  1.2 Aim

Our aim is to explore and display a given voting profile $V$ by sequentially
partitioning it into $G$ coherent groups plus a noisy group; that is,

$V=\cup_{g=1}^{G}cohG(g)\cup noisyG,$ (1)

where $G$ represents the number of coherent groups and $cohG(g)$ is the $g$th
coherent group. Furthermore, each coherent group is partitioned into a finite
number of coherent clusters; that is,

$cohG(g)=\cup_{\alpha=1}^{c_{g}}cohC_{g}(\alpha)\text{ \ for }g=1,...,G,$ (2)

where $c_{g}$ represents the number of coherent clusters in the $g$th coherent
group. So the coherent clusters are the building blocks for the coherent
groups. We note the following facts:

Fact 1: The assumption $d<<n$ induces the new notion of coherency for the
clusters and consequently for the groups; it is a stronger characterization
than the notion of interpretability for groups as discussed in
$\left[9\right]$.

Fact 2: Each coherent group and its clusters have the same latent variable
summarized by the Borda scale.

Fact 3: Given that the proposed method sequentially peels the data like
Occam’s razor, the number of groups $G$ is calculated automatically.
Furthermore, outliers or uninformative voters belonging to the $noisyG$ are
easily tagged.

Fact 4: The approach is exploratory, visual, data analytic and is developed
within the framework of taxicab correspondence analysis (TCA). TCA is an L1
variant of correspondence analysis developed by $\left[10\right]$. TCA is a
dimension reduction technique similar to principal component analysis. In this
paper, we shall use only the first TCA factor scores of the items and of the
voters.

Two major advantages of our method are: First, we can easily identify
outliers. For the SUSHI data, our method tags 12.36% of the voters as
outliers, which form the noisy group. While no outliers in the SUSHI data have
been identified in $\left[3,\ 4\right]$. Second, it provides a tabular summary
of the preferences which compose a coherent group. For instance, consider the
first mixture component of the SUSHI data given in $\left[4\right]$, where the
modal ordering is almost the same as the Borda scale ordering of the ten
sushis in cohG(1) obtained by our method, see Table 14 in this paper. The
sample size of their first mixture component is 27.56 %, which is much smaller
than 48.36%, the sample size of our cohG(1), see Table 14. However, Table 13
of this paper provides a tabular-visual summary of the 2418 preferences which
form cohG(1). The visual summary describes different kinds of equivalent
similar riffle shufflings of the 2418 preferences, and it provides further
insight into the structure of the data. Such interesting visual summaries are
missing in $\left[3,\ 4\right]$.

###  1.3 Highlights of a coherent cluster

A coherent cluster of voters has interesting mathematical properties and is
essentially characterized by the following facts:

a) Voters have identical unique first TCA factor score.

b) Any voter preference is easily interpreted as a particular riffle shuffling
of its items.

c) The nature of riffle shuffling of the items can be observed in the
structure of the contingency table of the first-order marginals constructed
from the Borda scorings of the voters belonging to the coherent cluster.

d) The first TCA factor scores of the items of a coherent cluster are
interpreted as Borda scale of the items.

e) We also introduce the crossing index, which measures the extent of
interleaving or the crossing of scores of voters between two blocks seriation
of the items in a coherent cluster.

###  1.4 Organization

This paper has eleven sections and its contents are organized as follows:
Section 2 presents an overview of TCA; section 3 presents some preliminaries
on the Borda coding of the data and related tables and concepts; section 4
presents Theorem 1, which shows that the first principal dimension of TCA
clusters the voters into a finite number of clusters; section 5 discusses
coherent clusters and their mathematical properties; section 6 discusses
riffle shuffling in a coherent cluster; section 7 introduces the crossing
index; section 8 introduces the coherent groups; section 9 presents the
analysis of APA data set; section 10 presents a comparison of the two
formulations of riffle shuffle probabilistic and exploratory; and finally we
conclude in section 11.

All mathematical proofs are relegated to the appendix. Details of the
computation are shown only for the first coherent group of SUSHI data set.

##  2 An overview of taxicab correspondence analysis

Consider a $n\times p$ matrix $\mathbf{X}$ where $X_{ij}\geq 0.$ We have
$\sum_{j=1}^{p}\sum_{i=1}^{n}\mathbf{X}_{ij}=X_{\ast\ast}.$ Let
$\mathbf{P=X/}X_{\ast\ast}$ be the correspondence matrix associated to X; and
as usual, we define $p_{i\ast}=\sum_{j=1}^{p}p_{ij}$, $p_{\ast
j}=\sum_{i=1}^{n}p_{ij}$. Let $\mathbf{D}_{n}=Diag(p_{i\ast})$ a diagonal
matrix with diagonal elements $p_{i\ast}$. Similarly
$\mathbf{D}_{p}=Diag(p_{\ast j})$. Let $k=rank(\mathbf{P)}-1$.

In TCA the calculation of the dispersion measures $(\delta_{\alpha})$,
principal axes ($\mathbf{u}_{\alpha},\mathbf{v}_{\alpha}),$ principal basic
vectors $(\mathbf{a}_{\alpha},\mathbf{b}_{\alpha}),$ and principal factor
scores $(\mathbf{f}_{\alpha},\mathbf{g}_{\alpha})$ for $\alpha=1,...,k$ is
done in a stepwise manner. We put
$\mathbf{P}_{1}=(p_{ij}^{(1)}=p_{ij}-p_{i\ast}\ p_{\ast j})$. Let
$\mathbf{P_{\alpha}}$ be the residual correspondence matrix at the $\alpha$-th
iteration.

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

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

The $\alpha$-th principal axes are

$\mathbf{u}_{\alpha}\
=\arg\max_{\mathbf{u}\in\left\\{-1,+1\right\\}^{p}}\left|\left|\mathbf{P_{\alpha}u}\right|\right|_{1}\text{
\ \ and \ \ }\mathbf{v}_{\alpha}\
=\arg\max_{\mathbf{v}\in\left\\{-1,+1\right\\}^{n}}\left|\left|\mathbf{P_{\alpha}^{\prime}v}\right|\right|_{1}\text{,}$
(3)

and the $\alpha$-th basic principal vectors are

$\mathbf{a}_{\alpha}=\mathbf{P_{\alpha}u}_{\alpha}\text{ \ and \
}\mathbf{b}_{\alpha}=\mathbf{P_{\alpha}^{\prime}v}_{\alpha},$ (4)

and the $\alpha$-th principal factor scores are

$\mathbf{f}_{\alpha}=\mathbf{D}_{n}^{-1}\mathbf{a}_{\alpha}\text{ \ and \
}\mathbf{g}_{\alpha}=\mathbf{D}_{p}^{-1}\mathbf{b}_{\alpha};$ (5)

furthermore the following relations are also useful

$\mathbf{u}_{\alpha}=sgn(\mathbf{b}_{\alpha})=sgn(\mathbf{g}_{\alpha})\text{ \
and \ }\mathbf{v}_{\alpha}=sgn(\mathbf{a}_{\alpha})=sgn(\mathbf{f}_{\alpha}),$
(6)

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

$\begin{array}[]{cccc}\delta_{\alpha}&=&\left|\left|\mathbf{P_{\alpha}u}_{\alpha}\right|\right|_{1}=\left|\left|\mathbf{a}_{\alpha}\right|\right|_{1}=\mathbf{a}_{\alpha}^{\prime}\mathbf{v}_{\alpha}=\left|\left|\mathbf{D}_{n}\mathbf{f}_{\alpha}\right|\right|_{1}=\mathbf{u}_{\alpha}^{\prime}\mathbf{D}_{n}\mathbf{f}_{\alpha},&\\\
&=&\left|\left|\mathbf{P_{\alpha}^{\prime}v}_{\alpha}\right|\right|_{1}=\left|\left|\mathbf{b}_{\alpha}\right|\right|_{1}=\mathbf{b}_{\alpha}^{\prime}\mathbf{u}_{\alpha}=\left|\left|\mathbf{D}_{p}\mathbf{g}_{\alpha}\right|\right|_{1}=\mathbf{v}_{\alpha}^{\prime}\mathbf{D}_{p}\mathbf{g}_{\alpha}.&\end{array}$
(7)

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

$\mathbf{P_{\alpha+1}}=\mathbf{P_{\alpha}-D}_{n}\mathbf{f}_{\alpha}\mathbf{g}_{\alpha}^{{}^{\prime}}\mathbf{D}_{p}/\delta_{\alpha}.$
(8)

An interpretation of the term
$\mathbf{D}_{n}\mathbf{g}_{\alpha}\mathbf{f}_{\alpha}^{{}^{\prime}}\mathbf{D}_{p}/\delta_{\alpha}$
in (8) is that, it represents the best rank-1 approximation of the residual
correspondence matrix $\mathbf{P_{\alpha}}$, in the sense of taxicab norm.

In CA and TCA, the principal factor scores are centered; that is,

$\sum_{i=1}^{n}f_{\alpha}(i)p_{i\ast}=0=\sum_{j=1}^{p}g_{\alpha}(j)p_{\ast
j}\text{ \ \ \ for \ \ }\alpha=1,...,k.$ (9)

The reconstitution formula in TCA and CA is

$p_{ij}=p_{i.}p_{.j}\left[1+\sum_{\alpha=1}^{k}f_{\alpha}(i)g_{\alpha}(j)/\delta_{\alpha}\right].$
(10)

In TCA, the calculation of the principal component weights,
$\mathbf{u}_{\alpha}$ and $\mathbf{v}_{\alpha},$ and the principal factor
scores, $\mathbf{g}_{\alpha}$ and $\mathbf{f}_{\alpha},$ can be accomplished
by two algorithms. The first one is based on complete enumeration based on
equation (3). The second one is based on iterating the transition formulae
(4,5,6). This is an ascent algorithm; that is, it increases the value of the
objective function at each iteration, see $\left[11\right]$. 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.

The TCA map is obtained by plotting $(\mathbf{g}_{1},\mathbf{g}_{2})$ or
$(\mathbf{f}_{1},\mathbf{f}_{2}).$

##  3 Preliminaries

In this section we review a) The Borda scoring of a voting profile V into R
and the Borda scale; b) Contingency table of the first order marginals of R;
c) The coded tables Rdouble and R${}_{nega}.$

###  3.1 Borda scorings and Borda scale

Let $A=\\{a_{1},a_{2},\ldots,a_{d}\\}$ denote a set of $d$
alternatives/candidates/items, and $V$ a set of $n$ voters/individuals/judges.
In this paper we consider the linear orderings/rankings/preferences, in which
all $d$ objects are rank-ordered according to their levels of desirability by
the $n$ voters. We denote a linear order by a sequence
$\mathbf{s}=(a_{k_{1}}\succ a_{k_{2}}\succ\ldots\succ a_{k_{d}})$, where
$a_{k_{1}}\succ a_{k_{2}}$ means that the alternative $a_{k_{1}}$ is preferred
to the alternative $a_{k_{2}}.$ The Borda scoring of $\mathbf{s}$, see
$\left[12\right],$ is the vector $b(\mathbf{s)}$ where to the element
$a_{k_{j}}$the score of $(d-j)$ is assigned, because $a_{k_{j}}$ is preferred
to $(d-j)$ other alternatives; or equivalently it is the $j$th most preferred
alternative. Let $\mathbf{R=(}r_{ij})$ be the matrix having $n$ rows and $d$
columns, where $r_{ij}$ designates the Borda score of the $i$th voter’s
preference of the $j$th alternative. We note that the $i$th row of
$\mathbf{R}$ will be an element of $S_{d}$ the set of permutations of the
elements of the set $\left\\{0,1,2,...,d-1\right\\}.$ A toy example of
$\mathbf{R}$ is presented in Table 1 for $n=4$ and $d=3$.

The Borda scale of the elements of $A$ is
$\mathbf{\beta}=\mathbf{1}_{n}^{\prime}\mathbf{R}/n,$ where $\mathbf{1}_{n}$
is a column vector of $1$’s having $n$ coordinates. The Borda scale
seriates/orders the $d$ items of the set $A$ according to their average
scores: $\mathbf{\beta}(j)>\mathbf{\beta}(i)$ means item $j$ is preferred to
item $i$, and $\mathbf{\beta}(j)=\mathbf{\beta}(i)$ means both items
$(a_{i},a_{j})$ are equally preferred. In the toy example of Table 1, the
Borda scale seriates $\\{A,B\\}\succ C$.

Similarly, we define the reverse Borda score of $\mathbf{s}$ to be the vector
$\overline{b}$($\mathbf{s)}$, which assigns to the element $a_{k_{j}}$the
score of $(j-1).$ We denote
$\overline{\mathbf{R}}\mathbf{=(}\overline{r}_{ij})$ to be the matrix having
$n$ rows and $d$ columns, where $\overline{r}_{ij}$ designates the reverse
Borda score of the $i$th judge’s nonpreference of the $j$th alternative. The
reverse Borda scale of the $d$ items is
$\overline{\mathbf{\beta}}=\mathbf{1}_{n}^{\prime}\overline{\mathbf{R}}/n.$

We note that

$\mathbf{R+}\overline{\mathbf{R}}=(d-1)\mathbf{1}_{n}\mathbf{1}_{d}^{\prime}$

and

$\mathbf{\beta+}\overline{\mathbf{\beta}}=(d-1)\mathbf{1}_{d}^{\prime}.$

Table 1: Toy example with $n=4$ preferences of $d=3$ items.
---
| $\mathbf{R}$ |  |  |  | $\overline{\mathbf{R}}$ |
$A\succ B\succ C$ | 0 | 1 | 2 | 2 | 1 | 0
$A\succ C\succ B$ | 1 | 0 | 2 | 1 | 2 | 0
$B\succ A\succ C$ | 0 | 2 | 1 | 2 | 0 | 1
$B\succ C\succ A$ | 1 | 2 | 0 | 1 | 0 | 2
Borda scale $\mathbf{\beta}$ | 0.5 | 1.25 | 1.25 |  |  |
$\text{reverse Borda scale}\overline{\text{ }\mathbf{\beta}}$ |  |  |  | 1.5 | 0.75 | 0.75
nega $n\overline{\mathbf{\beta}}$ |  |  |  | 6 | 3 | 3

###  3.2 Contingency table of first-order marginals

The contingency table of first order marginals of an observed voting profile
$V$ on $d$ items is a square $d\times d$ matrix M, where
$\mathbf{M(}i,j\mathbf{)}$ stores the number of times that item $j$ has Borda
score $i$ for $i=0,...,d-1,$ see $\left[6,\ \text{p.17}\right]$. Table 2
displays the matrix M for the toy example $\mathbf{R}$ displayed in Table 1.
We note the following facts:

a) It has uniform row and column marginals equal to the sample size.

b) We can compute the Borda scale $\mathbf{\beta}$ from it.

c) It reveals the nature of crossing of scores attributed to the items for a
given binary partition of the items. For the toy example, consider the
partition $\left\\{C\right\\}$ and $\left\\{B,A\right\\}$ with attributed
scores of $\left\\{0\right\\}$ and $\left\\{1,2\right\\}$ respectively (this
is the first step in a riffle shuffle). Then the highlighted cells (marked in
bold) in Table 2 show that there are two crossing of scores, permutation
(transposition) of the scores 0 and 1, between the sets $\left\\{C\right\\}$
and $\left\\{B,A\right\\}$, (this is the second step in a random shuffle).
Furthermore the third row of Table 2 shows that the score 2 is equally
attributed to both items of the set $\left\\{B,A\right\\}$ and it never
crossed to $\left\\{C\right\\}$.

Table 2: The matrix of first-order marginals of R.
---
| $C$ | $B$ | $A$ | row sum
$0$ | 2 | 1 | 1 | 4
$1$ | 2 | 1 | 1 | 4
$2$ | 0 | 2 | 2 | 4
column sum | 4 | 4 | 4 |
Borda scale $\mathbf{\beta}$ | 0.5 | 1.25 | 1.25 |

###  3.3 Coded tables $\mathbf{R}_{double}$ and $\mathbf{R}_{nega}$

Our methodological approach is based on Benzécri’s platform, see $\left[13,\
p.1113\right],$ that we quote: “ the main problem inductive statistics has to
face is to build tables that, through appropriate coding and eventual
supplementation, give to the available data such a shape that the analysis is
able to extract from it the answer to any question that we are allowed to
ask”. Italics are ours.

There are three elements in Benzécri’s platform: a) coding, a kind of pre-
processing of data, will be discussed in the following paragraph; b) eventual
supplementation consists in applying TCA and not correspondence analysis (CA),
because in the CA case we do not have a result similar to Theorem 1; c)
question that we are allowed to ask is to explore and visualize rank data.

Within the CA framework, there are two codings of rank data
$\mathbf{R}_{double}$ and $\mathbf{R}_{nega.}$.

####  3.3.1 $\mathbf{R}_{double}$

The first one is the doubled table of size $(2n)\times d$

$\mathbf{R}_{double}=(_{\overline{\mathbf{R}}}^{\mathbf{R}})$

proposed independently by $\left[14,\ 15\right]$, where they showed that CA of
$\mathbf{R}_{double}$ is equivalent to the dual scaling of Nishisato coding of
rank data, see $\left[16\right]$. CA of $\mathbf{R}_{double}$ is equivalent to
CA of its first residual correspondence matrix

$\mathbf{P}_{double}^{1}=\frac{1}{t}(_{-(r_{ij}-\frac{d-1}{2})}^{(r_{ij}-\frac{d-1}{2})}),$

where $t=nd(d-1)$. The structure of $\mathbf{P}_{double}^{1}$ shows that each
row is centered as in Carroll’s multidimensional preference analysis
procedure, MDPREF, exposed in Alvo and Yu (2014, p.15). In TCA the objective
function to maximize is a combinatorial problem, see equation (3); and the
first iteration in TCA of $\mathbf{R}_{double}$ corresponds to computing

$\begin{array}[]{cccc}\delta_{1}^{double}&=&\max_{\mathbf{v\in}\left\\{-1,1\right\\}^{n}}||(\mathbf{v}^{t}\
|\ \mathbf{-v}^{t})\mathbf{P}_{double}^{1}||_{1}&\\\
&=&\max_{\mathbf{v\in}\left\\{-1,1\right\\}^{n}}\frac{2}{t}\sum_{j=1}^{d}|\sum_{i=1}^{n}(r_{ij}-\frac{d-1}{2})v_{i}|\text{.}&\end{array}$

####  3.3.2 $\mathbf{R}_{nega}$

In the second approach, we summarize $\overline{\mathbf{R}}$ by its column
total; that is, we create a row named $\mathbf{nega=}$
$n\overline{\mathbf{\beta}}=\mathbf{1}_{n}^{\prime}\overline{\mathbf{R}},$
then we vertically concatenate $\mathbf{nega}$ to $\mathbf{R}$, thus obtaining

$\mathbf{R}_{nega}=(_{\mathbf{nega}}^{\mathbf{R}})$

of size $(n+1)\times d.$

$\left[17\right]$ discussed the relationship between TCA of
$\mathbf{R}_{double}$ and TCA of $\mathbf{R}_{nega}$: TCA of
$\mathbf{R}_{nega}$ can be considered as constrained TCA of
$\mathbf{R}_{double}$, because we are constraining the vector
$\mathbf{-v}^{t}=\mathbf{-1}_{n}^{t}$ in (11); that is, the objective function
to maximize corresponds to computing

$\begin{array}[]{cccc}\delta_{1}&=&\max_{\mathbf{v\in}\left\\{-1,1\right\\}^{n}}||(\mathbf{v}^{t}\
|\ \mathbf{-1}_{n}^{t})\mathbf{P}_{double}^{1}||_{1}&\\\
&=&\max_{\mathbf{v\in}\left\\{-1,1\right\\}^{n}}||(\mathbf{v}^{t}\ \
\mathbf{-}1)\mathbf{P}_{nega}^{1}||_{1}\\\
&=&\max_{\mathbf{v\in}\left\\{-1,1\right\\}^{n}}\frac{1}{t}\sum_{j=1}^{d}|\sum_{i=1}^{n}(r_{ij}-\frac{d-1}{2})(v_{i}+1)|\text{.}&\end{array}$

So, we see that if in (11) the optimal value of $\mathbf{v}=\mathbf{1}_{n}$,
then $\delta_{1}^{double}=\delta_{1},$ otherwise
$\delta_{1}^{double}>\delta_{1}$.

Let

$\mathbf{v}_{1}=\arg\max_{\mathbf{v\in}\left\\{-1,1\right\\}^{n}}\frac{1}{t}\sum_{j=1}^{d}|\sum_{i=1}^{n}(r_{ij}-\frac{d-1}{2})(v_{i}+1)|.$

Define the set of indices $I_{+}=\left\\{i|v_{1i}=1\right\\}$ and
$I_{-}=\left\\{i|v_{1i}=-1\right\\},$ where $\mathbf{v}_{1}=(v_{1i}).$ Then

$\delta_{1}=\frac{2}{t}\sum_{j=1}^{d}|\sum_{i\in
I_{+}}(r_{ij}-\frac{d-1}{2})|$ (13)

shows that the summation in (13) is restricted to the subset of assessors that
belong to $I_{+}$. The subset $I_{+}$ indexes the voters having the same
direction in their votes. Given that we are uniquely interested in the first
TCA dimension, all the necessary information is encapsulated in $I_{+}$, as
discussed in $\left[17,\ 9\right]$ using other arguments.

Furthermore, $\delta_{1}$ in (13) equals four times the cut norm of
$\mathbf{R}_{centered}(i,j)=\frac{1}{t}(r_{ij}-\frac{d-1}{2}),$ where the cut
norm is defined to be

$\displaystyle\left|\left|\mathbf{R}_{centered}\right|\right|_{cut}$
$\displaystyle=$ $\displaystyle\max_{S,T}\frac{1}{t}\sum_{j\in S}\sum_{i\in
T}(r_{ij}-\frac{d-1}{2})$ $\displaystyle=$ $\displaystyle\frac{1}{t}\sum_{j\in
S_{+}}\sum_{i\in I_{+}}(r_{ij}-\frac{d-1}{2})$ $\displaystyle=$
$\displaystyle\delta_{1}/4,$

where $S\subseteq\left\\{1,...,d\right\\}$ and $T\subseteq I;$ it shows that
the subsets $I_{+}$ and $S_{+}$ are positively associated, for further details
see for instance, $\left[18,\ 19\right]$.

In the sequel, we will consider only the application of TCA to
$\mathbf{R}_{nega}$.

##  4 First TCA voter factor scores of Rnega

We show the results on the SUSHI data set enumerating $n=5000$ preferences of
$d=10$ sushis, see $\left[1\right]$. Even though, our interest concerns only
the first TCA voter factor scores of a voting profile $V_{1},$ it is a common
practice in CA circles to present the principal map of the row and column
projections.

Figures 1 and 2 display the principal maps obtained from CA and TCA of
$R_{nega}$ of the SUSHI data denoted by $V_{1}$. We observe that, TCA clusters
the voters into a finite number of discrete patterns, while CA does not: This
is the main reason that we prefer the use of TCA to the use of the classical
well known dimension reduction technique CA.

We have the following theorem concerning the first TCA principal factor scores
of the voters belonging to a profile $V_{1}$, $f_{1}(i)$ for $i=1,...,n$,
where the first principal axis partitions the $d$ items into $d_{1}$ and
$d_{2}$ parts such that $d=d_{1}+d_{2}.$

(a) Figure 1:CA map of SUSHI rank data

(b) Figure 2: TCA map of SUSHI rank data

Theorem 1

a) The maximum number of distinct clusters of the $n$ voters belonging to
$V_{1}$ on the first TCA principal axis (distinct $f_{1}(i\mathbf{)}$ values
for $i\mathbf{\in}V_{1})$ is $d_{1}d_{2}+1.$

b) The maximum value that $f_{1}(i\mathbf{)}$ can attain is
$2\frac{d_{1}d_{2}}{d(d-1)}.$

c) The minimum value that $f_{1}(i\mathbf{)}$ can attain is
$-2\frac{d_{1}d_{2}}{d(d-1)}.$

d) If the number of distinct clusters is maximum, $d_{1}d_{2}+1$, then the gap
between two contiguous $f_{1}(i\mathbf{)}$ values is $\frac{4}{d(d-1)}.\vskip
12.0pt plus 4.0pt minus 4.0pt$

Remark 1

a) We fix $f_{1}(nega)<0$ to eliminate the sign indeterminacy of the first
bilinear term in (10).

b) We partition $V_{1}$ into $d_{1}d_{2}+1$ clusters,
$V_{1}=\cup_{\alpha=1}^{d_{1}d_{2}+1}V_{1,\alpha}$, where the voters of the
$\alpha$th cluster are characterized by their first TCA factor score; that is,
$V_{1,\alpha}=\left\\{i\in
V_{1}\mathbf{:}f_{1}^{V_{1}}(i\mathbf{)=}2\frac{d_{1}d_{2}}{d(d-1)}-(\alpha-1)\frac{4}{d(d-1)}\right\\}$
for $\alpha=1,...,d_{1}d_{2}+1$.

Example 1: In Figure 2, $d_{1}=4$ and $d_{2}=6,$ and we observe

Fact 1: by Theorem 1a, 5000 preferences are clustered into $d_{1}d_{2}+1=25$
clusters on the first TCA principal axis.

Fact 2: by Theorem 1b, the maximum value of $f_{1}(i\mathbf{)=}$
$48/90\mathbf{.}$

Fact 3: by Theorem 1c, the minimum value of $f_{1}(i\mathbf{)=}$
$-48/90\mathbf{.}$

Fact 4: by Theorem 1d, the gap separating two contiguous clusters of voters on
the first TCA principal axis is $4/90.\vskip 12.0pt plus 4.0pt minus 4.0pt$

A cluster of voters defined in Remark 1b, $V_{1,\alpha}$ for
$\alpha=1,...,d_{1}d_{2}+1,$ can be classified as coherent or incoherent. And
this will be discussed in the next section.

##  5 Coherent cluster

The following definition characterizes a coherent cluster.

Definition 1 (Coherency of a cluster of voters $V_{1,\alpha}$ for
$\alpha=1,...,d_{1}d_{2}+1$)

A cluster of voters $V_{1,\alpha}$ $\subseteq V_{1}$ is coherent if
$f_{1}^{V_{1,\alpha}}(v\mathbf{)=}2\frac{d_{1}d_{2}}{d(d-1)}-(\alpha-1)\frac{4}{d(d-1)}$
for all $v\mathbf{\in}V_{1,\alpha},$ where $f_{1}^{V_{1,\alpha}}(i\mathbf{)}$
is the first TCA factor score of the voter $i\mathbf{\in}V_{1,\alpha}$
obtained from TCA of subprofile $V_{1,\alpha}.\vskip 12.0pt plus 4.0pt minus
4.0pt$

Remark 2:

a) It is important to distinguish between $f_{1}^{V_{1}}(i\mathbf{)}$ for
$i=1,...,|V_{1}|$ where $n=|V_{1}|,$ and $f_{1}^{V_{1,\alpha}}(i\mathbf{)}$
for $i=1,...,|V_{1,\alpha}|,$ where $|V_{1,\alpha}|$ represents the sample
size of the cluster $|V_{1,\alpha}|.$

b) Definition 1 implies that a cluster $V_{1,\alpha}$ is coherent when for all
voters $i\mathbf{\in}V_{1,\alpha}$ the first TCA factor score
$f_{1}^{V_{1,\alpha}}(i\mathbf{)}$ does not depend on the voter $i$, but it
depends on $(\alpha,d_{1},d_{2}).\vskip 12.0pt plus 4.0pt minus 4.0pt$

Corollary 1: It follows from Remark 1a and equation (13) that, a necessary
condition, but not sufficient, for a cluster $V_{1,\alpha}$ to be coherent is
that its first TCA factor score obtained from TCA of $V_{1}$ is strictly
positive; that is, $0<f_{1}^{V_{1}}(i)$ for $i\in V_{1,\alpha}.$

(a) Figure 3 : TCA map of $\mbox{V}_{1,1}$.

(b) Figure 4 : TCA map of $\mbox{V}_{1,2}$.

(c) Figure 5 : TCA map of $\mbox{V}_{1,3}$.

(d) Figure 6 : TCA map of $\mbox{V}_{1,4}$.

(e) Figure 7 : TCA map of $\mbox{V}_{1,5}$.

(f) Figure 8 : TCA map of $\mbox{V}_{1,6}$.

(g) Figure 9 : TCA map of $\mbox{V}_{1,7}$.

(h) Figure 10: TCA map of $\mbox{V}_{1,8}$.

Example 2: Figures 3 through 9 show the coherency of the clusters of voters
$V_{1,\alpha}$ for $\alpha=1,...,7,$ where dots represent clusters of voters;
while Figure 10 shows the incoherence of the cluster $V_{1,8}.$ Further, the
first three columns of Table 3 display the mathematical formulation of the 7
coherent clusters $cohC_{1}(\alpha)=V_{1,\alpha}$ for $\alpha=1,...,7$ as
defined in Remark 1b and their sample sizes $|V_{1,\alpha}|.$

Table 3: Characteristics of $cohC_{1}(\alpha)=$ $V_{1,\alpha}$ of SUSHI data.
---
$\alpha$ | $|V_{1,\alpha}|$ |  description of $V_{1,\alpha}$ | $\delta_{1}(V_{1,\alpha})$ | $T_{v\in V_{1,\alpha}}(\tau_{J_{1}}(S_{1}))$ | $Cross(V_{1,\alpha})$
$1$ | $314$ | $\left\\{i\mathbf{:}f_{1}^{V_{1}}(i\mathbf{)}=48/90\right\\}$ | $48/90$ | $6$ | $0$
$2$ | $235$ | $\left\\{i\mathbf{:}f_{1}^{V_{1}}(i\mathbf{)}=44/90\right\\}$ | $44/90$ | $7$ | $1/12$
$3$ | $326$ | $\left\\{i\mathbf{:}f_{1}^{V_{1}}(i\mathbf{)}=40/90\right\\}$ | $40/90$ | $8$ | $2/12$
$4$ | $315$ | $\left\\{i\mathbf{:}f_{1}^{V_{1}}(i\mathbf{)}=36/90\right\\}$ | $36/90$ | $9$ | $3/12$
$5$ | $452$ | $\left\\{i\mathbf{:}f_{1}^{V_{1}}(i\mathbf{)}=32/90\right\\}$ | $32/90$ | $10$ | $4/12$
$6$ | $375$ | $\left\\{i\mathbf{:}f_{1}^{V_{1}}(i\mathbf{)}=28/90\right\\}$ | $28/90$ | $11$ | $5/12$
$7$ | $401$ | $\left\\{i\mathbf{:}f_{1}^{V_{1}}(i\mathbf{)}=24/90\right\\}$ | $24/90$ | $12$ | $6/12$

Proposition 1: For a voting profile $V$, $\delta_{1}(V)\geq|f_{1}(nega)|$,
where $\delta_{1}(V)$ is the first TCA dispersion value obtained from TCA of
$V,$ and $f_{1}(nega)$ is the first TCA factor score of the row $nega$.

The equality in Proposition 1 is attained only for coherent clusters as shown
in the following result.

Proposition 2: The first TCA dispersion value of a coherent cluster
$cohC_{1}(\alpha)$ satisfies

$\displaystyle\delta_{1}(cohC_{1}(\alpha))$ $\displaystyle=$
$\displaystyle|f_{1}^{V_{1,\alpha}}(nega)|.$ $\displaystyle\mathbf{=}$
$\displaystyle 2\frac{d_{1}d_{2}}{d(d-1)}-(\alpha-1)\frac{4}{d(d-1)}$

Example 3: propostion 2 can be observed by looking at the columns 3 and 4 of
Table 3 which concern the 7 coherent clusters $cohC_{1}(\alpha)=V_{1,\alpha}$
for $\alpha=1,...,7$. While for the incoherent cluster $V_{1,8}$ with sample
size of $|V_{1,8}|=335,$ we observe:
$V_{1,8}=\left\\{i\mathbf{:}f_{1}^{V_{1}}(i\mathbf{)}=20/90=0.222\right\\},$
and by Proposition 1, $\delta_{1}(V_{1,8})=0.2354$ $>2/9.$ This means that the
335 voters belonging to $V_{1,8}$ form a cluster within the whole sample of
5000 voters, but separated as 335 voters they do not form a coherent cluster.

###  5.1 Interpretability of a coherent cluster

The following result shows that for coherent clusters, the first TCA dimension
can be interpreted as Borda scaled factor.

Proposition 3: The first TCA column factor score of the item $j,$ $g_{1}(j),$
is an affine function of the Borda scale $\beta(j);$ that is,
$g_{1}(j)=\frac{2}{d-1}\beta(j)-1$ for $j=1,...,d.$ Or
$corr(\mathbf{g}_{1},\mathbf{\beta})=1.\vskip 12.0pt plus 4.0pt minus 4.0pt$

Remark 3:

The first TCA principal factor score of item $j$ for $j=1,...,d$ is bounded:
$-1\leq g_{1}(j)\leq 1,$ because $0\leq\beta(j)\leq d-1.\vskip 12.0pt plus
4.0pt minus 4.0pt$

Example 4: Table 4 displays the Borda scales of the items, sushis, in the
seven coherent clusters $cohC_{1}(\alpha)=V_{1,\alpha}$ for $\alpha=1,...,7.$
To identify the sushi type, one has to refer to Figure 2; for instance, $j10$
corresponds to $10cucumber$ $roll$ in Figure 2. We observe the following main
fact: For each of the seven coherent clusters, the first TCA principal axis
produced the same binary partition of the items:
$J_{1}=\left\\{j10,j7,j4,j9\right\\}$ characterized by $4.5>\beta(j_{1})$ for
$j_{1}\in J_{1}$, and $J_{2}=\left\\{j3,j1,j2,j6,j5,j8\right\\}$ characterized
by $\beta(j_{1})$ $>4.5$ for $j_{2}\in J_{2}.$ The six sushis in $J_{2}$ have
Borda scales above average score of $4.5=(0+9)/2$; while the four sushis in
$J_{1}$ have Borda scales below average score of $4.5.\vskip 12.0pt plus 4.0pt
minus 4.0pt\ $

Table 4: Borda scales of the 10 sushis in the seven coherent clusters.
---
Borda scale | items
| j10 | j7 | j4 | j9 | j3 | j1 | j2 | j6 | j5 | j8
$\mathbf{\beta}(cohC_{1}(1))$ | 0.66 | 1.31 | 1.87 | 2.16 | 5.55 | 5.78 | 6.03 | 6.58 | 7.31 | 7.52
$\mathbf{\beta}(cohC_{1}(2))$ | 0.69 | 1.29 | 2.44 | 2.59 | 5.47 | 5.43 | 5.50 | 6.35 | 7.38 | 7.86
$\mathbf{\beta}(cohC_{1}(3))$ | 0.65 | 1.60 | 3.04 | 2.71 | 5.25 | 5.25 | 5.39 | 6.26 | 7.17 | 7.68
$\mathbf{\beta}(cohC_{1}(4))$ | 0.83 | 1.79 | 3.10 | 3.28 | 5.30 | 4.74 | 5.22 | 6.34 | 6.76 | 7.64
$\mathbf{\beta}(cohC_{1}(5))$ | 1.12 | 2.02 | 3.26 | 3.60 | 5.70 | 4.74 | 5.27 | 5.75 | 5.99 | 7.60
$\mathbf{\beta}(cohC_{1}(6))$ | 1.12 | 2.33 | 3.62 | 3.93 | 5.68 | 4.98 | 5.21 | 5.33 | 5.25 | 7.56
$\mathbf{\beta}(cohC_{1}(7))$ | 1.42 | 2.74 | 3.84 | 4.00 | 5.45 | 4.70 | 5.02 | 5.26 | 5.20 | 7.38

Now we ask the question what are the differences among the seven coherent
clusters? The answer is riffle shuffling of the scores of the items, which we
discuss next.

##  6 Exploratory riffle shuffling

$\left[8\right]$ is the seminal reference on riffle shuffling of cards.
$\left[2\right]$ generalized the notion of independence of two subsets of
items to riffled independence to uncover the structure of rank data. Within
the framework of data analysis of preferences, exploratory riffle shuffling
can be described in the following way. We have two sets: $J$ a set of $d$
distinct items and $S$ a set of $d$ Borda scores. We partition both sets into
two disjoint subsets of sizes $d_{1}$ and $d_{2}=d-d_{1};$ that is,
$J=J_{1}\cup J_{2}$ with $J_{1}=\left\\{j_{1},j_{2},...,j_{d_{1}}\right\\}$
and $S=S_{1}\cup S_{2}$ with $S_{1}=\left\\{0,1,...,d_{1}-1\right\\}.$ Riffle
shuffling consists of two steps. In the first step, we attribute the scores of
$S_{1}$ to $J_{1}$ and the scores of $S_{2}$ to $J_{2}.$ In the second step,
we permute some scores attributed to $J_{1}$ with the same number of scores
attributed to $J_{2}.$ The second step can be mathematically described as an
application of a permutation $\tau$, such that
$\tau_{J}(S_{1},S_{2})=(\tau_{J_{1}}(S_{1}),\tau_{J_{2}}(S_{2})).$ We
interpret $\tau_{J_{1}}(S_{1})$ as the set of scores attributed to $J_{1},$
and $\tau_{J_{2}}(S_{2})$ as the set of scores attributed to $J_{2}.$

Example 5: Table 5 displays a toy example with $n=7$ voters’ Borda scorings of
$d=10$ items with $J_{1}=\left\\{a,b,c,d\right\\}$ and
$J_{2}=\left\\{e,f,g,h,i,j\right\\}.$ We observe the following: a) The first
four voters have only done the first step in a riffle shuffle: each one of
them has attributed the scores in $S_{1}=\left\\{0,1,2,3\right\\}$ to the
items in $J_{1}$ and the scores in $S_{2}=\left\\{4,5,6,7,8,9\right\\}$ to the
items in $J_{2}.$ This can be described as
$\tau_{J}(S_{1},S_{2})=(S_{1},S_{2});$ that is the permutation $\tau$ is the
identity permutation; so there is no crossing of scores between $J_{1}$ and
$J_{2}$. b) Voters 5, 6 and 7 have done both steps in a riffle shuffle. Voters
5 and 6 have permuted score 3 with 5, so we have
$\tau_{J_{1}}(S_{1})=\left\\{0,1,2,\mathbf{5}\right\\}$ and
$\tau_{J_{2}}(S_{2})=\left\\{4,\mathbf{3},6,7\mathbf{,}8,9\right\\}$. Voter 7
has permuted the scores $\left\\{\mathbf{2,3}\right\\}$ with
$\left\\{\mathbf{4,5}\right\\}$, so we have
$\tau_{J_{1}}(S_{1})=\left\\{0,1,\mathbf{4},\mathbf{5}\right\\}$ and
$\tau_{J_{2}}(S_{2})=\left\\{\mathbf{2},\mathbf{3},6,7\mathbf{,}8,9\right\\}$.

Further, we note by $|\tau_{J_{1}}(S_{1})|$ the number of voters who have done
the riffle shuffle $(\tau_{J_{1}}(S_{1}),\tau_{J_{2}}(S_{2}))$. So
$|\tau_{J_{1}}(S_{1})=\left\\{0,1,2,3\right\\}|=4,$
$|\left\\{0,1,2,\mathbf{5}\right\\}|=2$ and
$|\left\\{0,1,\mathbf{4},\mathbf{5}\right\\}|=1.$ The permuted scores between
the two blocks of items is in bold in Table 5.

Table 5: Borda scorings of 10 items by 7 voters.
---
voter | items
| a | b | c | d | e | f | g | h | i | j
1 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9
2 | 0 | 2 | 3 | 1 | 6 | 4 | 5 | 8 | 7 | 9
3 | 3 | 2 | 1 | 0 | 5 | 6 | 4 | 9 | 7 | 8
4 | 2 | 1 | 0 | 3 | 8 | 7 | 9 | 4 | 5 | 6
5 | 0 | 1 | 2 | 5 | 4 | 3 | 6 | 7 | 8 | 9
6 | 1 | 2 | 5 | 0 | 3 | 6 | 4 | 9 | 7 | 8
7 | 0 | 4 | 5 | 1 | 6 | 8 | 9 | 2 | 7 | 3

Remark 4: A useful observation that we get from Example 5 is that we can
concentrate our study either on $J_{1}$ or on $J_{2}:$ For if we know
$\tau_{J_{1}}(S_{1})$, the scores attributed to $J_{1},$ we can deduce
$\tau_{J_{2}}(S_{2})$, the scores attributed to $J_{2}$ because of mutual
exclusivity constraints ensuring that any two items, say $a$ and $b,$ never
map to the same rank by a voter.

A simple measure of magnitude of $(d_{1},d_{2})$ riffle shuffling of a voter
$i$ is the sum of its Borda scores attributed to the items in $J_{1};$ that
is,

$T_{i}(\tau_{J_{1}}(S_{1}))=\sum_{j\in J_{1}}r_{ij},$

where $r_{ij}$ is the Borda score attributed to item $j$ by voter $i$. In
Table 5, for the first four voters, $T_{i}(\tau_{J_{1}}(S_{1}))=6$ for
$i=1,...,4,$ which is the minimum attainable sum of scores; it implies that
for these voters there is no crossing of scores between the two blocks $J_{1}$
and $J_{2}$. While for voters 5 and 6, $T_{i}(\tau_{J_{1}}(S_{1}))=8$ for
$i=5,6;$ for voter 7, $T_{7}(\tau_{J_{1}}(S_{1}))=10$. These values show that
the crossing of scores between the two blocks $J_{1}$ and $J_{2}$ of voters 5
and 6 are at a lower level than the crossing of scores for voter 7.

For relatively small sample sizes, it is easy to enumerate the different types
of $(d_{1},d_{2})$ riffle shuffles. For relatively large sample sizes, we use
the contingency table of first-order marginals, that we discuss next.

###  6.1 Types of $(d_{1},d_{2})$ riffle shufflings in a coherent cluster

The contingency table of first order marginals of an observed voting profile
$V$ on $d$ items is a square $d\times d$ matrix M, where
$\mathbf{M(}i,j\mathbf{)}$ stores the number of times that item $j$ has Borda
score $i$ for $i=0,...,d-1,$ see subsection 3.2. It helps us to observe types
of $(d_{1},d_{2})$ riffle shufflings in a coherent cluster as we explain in
Example 6.

Table 6: M${}_{1,1},$ contingency table of first-order marginals of
$cohC_{1}(1).$
---
Borda | items
scores | j10 | j7 | j4 | j9 | j3 | j1 | j2 | j6 | j5 | j8 | sum
0 | 174 | 92 | 37 | 11 | 0 | 0 | 0 | 0 | 0 | 0 | 314
1 | 88 | 88 | 76 | 62 | 0 | 0 | 0 | 0 | 0 | 0 | 314
2 | 38 | 78 | 91 | 107 | 0 | 0 | 0 | 0 | 0 | 0 | 314
3 | 14 | 56 | 110 | 134 | 0 | 0 | 0 | 0 | 0 | 0 | 314
4 | 0 | 0 | 0 | 0 | 92 | 78 | 73 | 38 | 21 | 12 | 314
5 | 0 | 0 | 0 | 0 | 95 | 77 | 59 | 42 | 23 | 18 | 314
6 | 0 | 0 | 0 | 0 | 47 | 63 | 70 | 65 | 37 | 32 | 314
7 | 0 | 0 | 0 | 0 | 35 | 49 | 45 | 72 | 68 | 45 | 314
8 | 0 | 0 | 0 | 0 | 32 | 27 | 32 | 62 | 87 | 74 | 314
9 | 0 | 0 | 0 | 0 | 13 | 20 | 35 | 35 | 78 | 133 | 314
$\beta$ | 0.66 | 1.31 | 1.87 | 2.16 | 5.55 | 5.78 | 6.03 | 6.58 |  7.31 | 7.52 |

Table 7: M${}_{1,2},$ contingency table of first-order marginals of
$cohC_{1}(2).$
---
Borda | items
scores | j10 | j7 | j4 | j9 | j3 | j1 | j2 | j6 | j5 | j8 | sum
0 | 127 | 70 | 32 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 235
1 | 69 | 82 | 38 | 46 | 0 | 0 | 0 | 0 | 0 | 0 | 235
2 | 32 | 56 | 62 | 85 | 0 | 0 | 0 | 0 | 0 | 0 | 235
3 | 0 | 0 | 0 | 0 | 55 | 59 | 74 | 29 | 15 | 3 |  235
4 | 7 | 27 | 103 | 98 | 0 | 0 | 0 | 0 | 0 | 0 | 235
5 | 0 | 0 | 0 | 0 | 68 | 60 | 42 | 41 | 11 | 13 | 235
6 | 0 | 0 | 0 | 0 | 49 | 53 | 35 | 48 | 32 | 18 | 235
7 | 0 | 0 | 0 | 0 | 26 | 35 | 42 | 48 | 40 | 44 | 235
8 | 0 | 0 | 0 | 0 | 28 | 15 | 22 | 44 | 70 | 56 | 235
9 | 0 | 0 | 0 | 0 | 9 | 13 | 20 | 25 | 67 | 101 | 235
$\beta$ | 0.69 | 1.29 | 2.44 | 2.59 | 5.47 | 5.43 | 5.50 | 6.35 |  7.38 | 7.86 |

Table 8: M${}_{1,3},$ contingency table of first-order marginals of
$cohC_{1}(3).$
---
Borda | items
scores | j10 | j7 | j4 | j9 | j3 | j1 | j2 | j6 | j5 | j8 | sum
0 | 182 | 97 | 33 | 14 | 0 | 0 | 0 | 0 | 0 | 0 | 326
1 | 104 | 100 | 46 | 76 | 0 | 0 | 0 | 0 | 0 | 0 | 326
2 | 19 | 41 | 41 | 70 | 40 | 37 | 46 | 17 | 12 | 3 | 326
3 | 16 | 35 | 53 | 51 | 39 | 48 | 43 | 22 | 13 | 6 | 326
4 | 3 | 29 | 62 | 61 | 40 | 41 | 43 | 32 | 9 | 6 | 326
5 | 2 | 24 | 91 | 54 | 39 | 43 | 23 | 24 | 16 | 10 | 326
6 | 0 | 0 | 0 | 0 | 70 | 65 | 51 | 60 | 45 | 35 | 326
7 | 0 | 0 | 0 | 0 | 53 | 36 | 52 | 74 | 56 | 55 | 326
8 | 0 | 0 | 0 | 0 | 35 | 33 | 33 | 57 | 80 | 88 | 326
9 | 0 | 0 | 0 | 0 | 10 | 23 | 35 | 40 | 95 | 123 | 326
$\beta$ | 0.65 | 1.60 | 3.04 | 2.71 | 5.25 | 5.25 | 5.39 | 6.26 |  7.17 | 7.68 |

Table 9: M${}_{1,4},$ contingency table of first-order marginals of
$cohC_{1}(4).$
---
Borda | items
scores | j10 | j7 | j4 | j9 | j3 | j1 | j2 | j6 | j5 | j8 | sum
0 | 164 | 93 | 44 | 14 | 0 | 0 | 0 | 0 | 0 | 0 | 315
1 | 78 | 71 | 30 | 36 | 10 | 31 | 32 | 9 | 16 | 2 | 315
2 | 44 | 53 | 49 | 50 | 32 | 39 | 27 | 10 | 8 | 3 | 315
3 | 22 | 52 | 58 | 87 | 24 | 20 | 24 | 15 | 7 | 6 | 315
4 | 5 | 17 | 35 | 43 | 51 | 61 | 41 | 25 | 23 | 14 | 315
5 | 1 | 11 | 61 | 46 | 43 | 42 | 34 | 35 | 26 | 16 | 315
6 | 1 | 18 | 38 | 39 | 52 | 37 | 37 | 49 | 28 | 16 | 315
7 | 0 | 0 | 0 | 0 | 49 | 44 | 51 | 61 | 54 | 56 | 315
8 | 0 | 0 | 0 | 0 | 37 | 28 | 47 | 72 | 69 | 52 | 315
9 | 0 | 0 | 0 | 0 | 17 | 13 | 22 | 39 | 84 | 140 | 315
$\beta$ | 0.83 | 1.79 | 3.10 | 3.28 | 5.30 | 4.74 | 5.22 | 6.34 |  6.76 | 7.64 |

Table 10: M${}_{1,5},$ contingency table of first-order marginals of
$cohC_{1}(5).$
---
Borda | items
scores | j10 | j7 | j4 | j9 | j3 | j1 | j2 | j6 | j5 | j8 | sum
0 | 188 | 99 | 36 | 10 | 6 | 25 | 30 | 22 | 34 | 2 | 452
1 | 132 | 109 | 69 | 57 | 12 | 30 | 21 | 13 | 9 | 0 | 452
2 | 69 | 88 | 59 | 67 | 28 | 46 | 40 | 28 | 20 | 7 | 452
3 | 39 | 72 | 85 | 92 | 34 | 44 | 21 | 31 | 25 | 9 | 452
4 | 12 | 35 | 76 | 81 | 50 | 57 | 53 | 36 | 38 | 14 | 452
5 | 6 | 29 | 63 | 72 | 63 | 64 | 53 | 41 | 40 | 21 | 452
6 | 3 | 11 | 34 | 36 | 71 | 68 | 64 | 75 | 45 | 45 | 452
7 | 3 | 9 | 30 | 37 | 87 | 45 | 62 | 73 | 57 | 49 | 452
8 | 0 | 0 | 0 | 0 | 71 | 41 | 47 | 72 | 95 | 126 | 452
9 | 0 | 0 | 0 | 0 | 30 | 32 | 61 | 61 | 89 | 179 | 452
$\beta$ | 1.12 | 2.02 | 3.26 | 3.60 | 5.70 | 4.74 | 5.27 | 5.75 |  5.99 | 7.60 |

Table 11: M${}_{1,6},$ contingency table of first-order marginals of
$cohC_{1}(6).$
---
Borda | items
scores | j10 | j7 | j4 | j9 | j3 | j1 | j2 | j6 | j5 | j8 | sum
0 | 151 | 81 | 31 | 14 | 8 | 14 | 19 | 18 | 39 | 0 | 375
1 | 112 | 79 | 44 | 33 | 12 | 21 | 26 | 25 | 19 | 4 | 375
2 | 66 | 72 | 52 | 63 | 16 | 24 | 29 | 22 | 28 | 3 | 375
3 | 26 | 52 | 68 | 68 | 22 | 45 | 31 | 29 | 25 | 9 | 375
4 | 8 | 26 | 42 | 37 | 52 | 67 | 41 | 45 | 45 | 12 | 375
5 | 8 | 27 | 56 | 61 | 44 | 49 | 42 | 36 | 28 | 24 | 375
6 | 3 | 21 | 36 | 52 | 64 | 42 | 49 | 50 | 29 | 29 | 375
7 | 0 | 7 | 25 | 31 | 70 | 43 | 44 | 59 | 45 | 51 | 375
8 | 1 | 10 | 21 | 16 | 66 | 33 | 46 | 49 | 45 | 88 | 375
9 | 0 | 0 | 0 | 0 | 21 | 37 | 48 | 42 | 72 | 155 | 375
$\beta$ | 1.12 | 2.33 | 3.62 | 3.93 | 5.68 | 4.98 | 5.21 | 5.33 |  5.25 | 7.56 |

Table 12: M${}_{1,7},$ contingency table of first-order marginals of
$cohC_{1}(7).$
---
Borda | items
scores | j10 | j7 | j4 | j9 | j3 | j1 | j2 | j6 | j5 | j8 | sum
0 | 129 | 65 | 46 | 14 | 11 | 24 | 35 | 23 | 52 | 2 | 401
1 | 122 | 77 | 53 | 35 | 14 | 28 | 24 | 19 | 25 | 4 | 401
2 | 74 | 69 | 50 | 51 | 24 | 41 | 36 | 31 | 19 | 6 | 401
3 | 36 | 51 | 31 | 66 | 44 | 48 | 39 | 46 | 30 | 10 | 401
4 | 24 | 50 | 40 | 71 | 51 | 45 | 38 | 37 | 27 | 18 | 401
5 | 7 | 45 | 49 | 56 | 43 | 53 | 39 | 48 | 32 | 29 | 401
6 | 5 | 23 | 73 | 68 | 42 | 50 | 42 | 33 | 40 | 25 | 401
7 | 3 | 10 | 31 | 28 | 85 | 39 | 43 | 54 | 51 | 57 | 401
8 | 1 | 3 | 17 | 5 | 58 | 46 | 47 | 65 | 57 | 102 | 401
9 | 0 | 8 | 11 | 7 | 29 | 27 | 58 | 45 | 68 | 148 | 401
$\beta$ | 1.42 | 2.74 | 3.84 | 4.00 | 5.45 | 4.70 | 5.02 | 5.26 |  5.20 | 7.38 |

Example 6: Tables 6 to 12 display M1,α for $\alpha=1,...,7,$ the contingency
tables of first order marginals of the seven coherent clusters of the SUSHI
data, respectively. We observe the following:

Each one of them reveals the nature of the riffle shuffles of its coherent
cluster, which are summarized in Table 13. The number of observed $(4,6)$
blocks of scores for the seven coherent clusters,
($\tau_{J_{1}}(S_{1}),\tau_{J_{2}}(S_{2})),$ is only 27 in Table 13 out of the
possible total number of $10!/(4!6!)=210$. The counts of the observed $(4,6)$
blocks do not seem to be uniformly distributed in Table 13. Furthermore, we
observe that as $\alpha$ increases from 1 to 7, the magnitude of riffle
shuffles, $T_{v}(\tau_{J_{1}}(S_{1})),$ increases in the coherent clusters
from 6 to 12. Integers in bold in Table 13 are the shuffled-crossed scores.

Table 13: Types of riffle shuffles in the 7 coherent clusters of SUSHI data.
---
$cohC_{1}(\alpha)$ | scores given to | sum of |  | $cohC_{1}(\alpha)$ | scores given to | sum of |
| $\left\\{j10,j7,j4,j9\right\\}$ | scores | count |  | $\left\\{j10,j7,j4,j9\right\\}$ | scores | count
$cohC_{1}(1)$ | $\left\\{0,1,2,3\right\\}$ | 6 | 314 | $cohC_{1}(6)$ | $\left\\{0,1,2,\mathbf{8}\right\\}$ | 11 | 48
$cohC_{1}(2)$ | $\left\\{0,1,2,\mathbf{4}\right\\}$ | 7 | 235 |  | $\left\\{0,1,\mathbf{7},3\right\\}$ | 11 | 63
$cohC_{1}(3)$ | $\left\\{0,1,2,\mathbf{5}\right\\}$ | 8 | 171 |  | $\left\\{0,\mathbf{6},2,3\right\\}$ | 11 | 53
| $\left\\{0,1,\mathbf{4},3\right\\}$ | 8 | 155 |  | $\left\\{\mathbf{5},1,2,3\right\\}$ | 11 | 98
$cohC_{1}(4)$ | $\left\\{0,1,2,\mathbf{6}\right\\}$ | 9 | 96 |  | $\left\\{0,1,\mathbf{4,6}\right\\}$ | 11 | 59
| $\left\\{0,1,\mathbf{5},3\right\\}$ | 9 | 119 |  | $\left\\{0,\mathbf{4},2,\mathbf{5}\right\\}$ | 11 | 54
| $\left\\{0,\mathbf{4},2,3\right\\}$ | 9 | 100 | $cohC_{1}(7)$ | $\left\\{0,1,2,\mathbf{9}\right\\}$ | 12 | 26
$cohC_{1}(5)$ | $\left\\{0,1,2,\mathbf{7}\right\\}$ | 10 | 79 |  | $\left\\{0,1,\mathbf{8},3\right\\}$ | 12 | 26
| $\left\\{0,1,\mathbf{6},3\right\\}$ | 10 | 84 |  | $\left\\{0,\mathbf{7},2,3\right\\}$ | 12 | 33
| $\left\\{0,\mathbf{5},2,3\right\\}$ | 10 | 85 |  | $\left\\{\mathbf{6},1,2,3\right\\}$ | 12 | 43
| $\left\\{\mathbf{4},1,2,3\right\\}$ | 10 | 119 |  | $\left\\{0,\mathbf{4,5},3\right\\}$ | 12 | 38
| $\left\\{0,1,\mathbf{4,5}\right\\}$ | 10 | 85 |  | $\left\\{0,\mathbf{4},2,\mathbf{6}\right\\}$ | 12 | 39
|  |  |  |  | $\left\\{0,1,\mathbf{4},\mathbf{7}\right\\}$ | 12 | 49
|  |  |  |  | $\left\\{0,1,\mathbf{5,6}\right\\}$ | 12 | 82
|  |  |  |  | $\left\\{\mathbf{4},1,2,\mathbf{5}\right\\}$ | 12 | 65

The counts in Table 13 are calculated from M1,α for $\alpha=1,...,7,$ by
reasoning on the permutation of scores between the sets $S_{1}$ and $S_{2}$.
Here are the details, where $J_{1}=\left\\{j10,j7,j4,j9\right\\}$.

a)$\ cohC_{1}(1)$

$|\left\\{0,1,2,3\right\\}|=314,$ which is the number of $0$s attributed to
$J_{1}$ in M${}_{1,1}.\ $Among the M1,α for $\alpha=1,...,7$, note that M1,1
is the only contingency table of first-order marginals which is block
diagonal.

b) $cohC_{1}(2)$

$|\left\\{0,1,2,4\right\\}|=235,$ which is the number of $4$s attributed to
$J_{1}$ in M${}_{1,2}.$

c) $cohC_{1}(3)$

$|\left\\{0,1,2,5\right\\}|=171,$ which is the number of $5$s attributed to
$J_{1}\ $in M${}_{1,3}.$

$|\left\\{0,1,\mathbf{4},3\right\\}|=155,$ which is the number of $4$s
attributed to $J_{1}$ in M${}_{1,3}.$

d) $cohC_{1}(4)$

$|\left\\{0,1,2,\mathbf{6}\right\\}|=96,$ which is the number of $6$s
attributed to $J_{1}$ in M${}_{1,4}.$

$|\left\\{0,1,\mathbf{5},3\right\\}|=119,$ which is the number of $5$s
attributed to $J_{1}$ in M${}_{1,4}.$

$|\left\\{0,\mathbf{4},2,3\right\\}|=100,$ which is the number of $4$s
attributed to $J_{1}$ in M${}_{1,4}.$

e) $cohC_{1}(5)$

$|\left\\{0,1,2,\mathbf{7}\right\\}|=79,$ which is the number of $7$s
attributed to $J_{1}$ in M${}_{1,5}.$

$|\left\\{0,1,\mathbf{6},3\right\\}|=84,$ which is the number of $6$s
attributed to $J_{1}$ in M${}_{1,5}.$

$|\left\\{0,\mathbf{5},2,3\right\\}|=85,$ which is the number of $1$s not
attributed to $J_{1}$ in M${}_{1,5}.$

$|\left\\{0,1,\mathbf{4},\mathbf{5}\right\\}|+|\left\\{0,\mathbf{5},2,3\right\\}|=170,$
which is the total number of $5$s attributed to $J_{1}\ $in
$\mathbf{M}_{1,5};$ so
$|\left\\{0,1,\mathbf{4},\mathbf{5}\right\\}|=170-85=85.$

$|\left\\{\mathbf{4},1,2,3\right\\}|=119,$ which is the number of $0$s not
attributed to $J_{1}\ $in $\mathbf{M}_{1,5}.$

f) $cohC_{1}(6)$

$|\left\\{0,1,2,\mathbf{8}\right\\}|=48,$ which is the number of $8$s
attributed to $J_{1}\ $in $\mathbf{M}_{1,6}.$

$|\left\\{0,1,\mathbf{7},3\right\\}|=63,$ which is the number of $7$s
attributed to $J_{1}\ $in $\mathbf{M}_{1,6}.$

$|\left\\{\mathbf{5},1,2,3\right\\}|=98,$ which is the number of $0$s not
attributed to $J_{1}\ $in $\mathbf{M}_{1,6}.$

$|\left\\{0,\mathbf{4},2,\mathbf{5}\right\\}|=152-98=54,$ where $152$ is the
total number of $5$s attributed to $J_{1}\ $in $\mathbf{M}_{1,6}.$

$|\left\\{0,1,\mathbf{4},\mathbf{6}\right\\}|=113-54=59,$ where $113$ is the
total number of $4$s attributed to $J_{1}\ $in $\mathbf{M}_{1,6}.$

$|\left\\{0,\mathbf{6},2,3\right\\}|=112-59=53,$ where $112$ is the total
number of $6$s attributed to $J_{1}\ $in $\mathbf{M}_{1,6}.$

g) $cohC_{1}(7)$

$|\left\\{0,1,2,\mathbf{9}\right\\}|=26,$ which is the number of $9$s
attributed to $J_{1}\ $in $\mathbf{M}_{1,7}.$

$|\left\\{0,1,\mathbf{8},3\right\\}|=26,$ which is the number of $8$s
attributed to $J_{1}\ $in $\mathbf{M}_{1,7}.$

For the remaining counts, we have to solve the following system of 7 linear
equations, where, $u=|\left\\{0,\mathbf{7},2,3\right\\}|$,
$t=|\left\\{0,\mathbf{4},\mathbf{5},3\right\\}|$,
$s=|\left\\{0,\mathbf{4},2,\mathbf{6}\right\\}|$,
$w=|\left\\{0,1,\mathbf{4,7}\right\\}|$,
$z=|\left\\{0,1,\mathbf{5,6}\right\\}|$,
$x=|\left\\{\mathbf{6},1,2,3\right\\}|$, and
$y=|\left\\{\mathbf{4},1,2,5\right\\}|$.

$x+y=147,\ $which is the number of $0$s not attributed to $J_{1}\ $in
$\mathbf{M}_{1,7}.$

$u+w=72,\ $which is the number of $7$s attributed to $J_{1}\ $in
$\mathbf{M}_{1,7}.$

$s+z+x=169,\ $which is the number of $6$s attributed to $J_{1}\ $in
$\mathbf{M}_{1,7}.$

$t+z+y=157,\ $which is the number of $5$s attributed to $J_{1}\ $in
$\mathbf{M}_{1,7}.$

$t+s+w+y=185,\ $which is the number of $4$s attributed to $J_{1}\ $in
$\mathbf{M}_{1,7}.$

$u+t+x=158,\ $which is the number of $3$s attributed to $J_{1}\ $in
$\mathbf{M}_{1,7}.$

$u+s+x+y=218,\ $which is the number of $2$s attributed to $J_{1}\ $in
$\mathbf{M}_{1,7}.$

##  7 Crossing index

The following $(d_{1},d_{2})$ crossing index is based on the internal
dispersion of a voting profile.

Definition 3: For a voting profile $V$ we define its crossing index to be

$\displaystyle Cross(V)$ $\displaystyle=$ $\displaystyle
1-\frac{\delta_{1}(V_{d_{1},d_{2}})}{\max_{V}\delta_{1}(V_{d_{1},d_{2}})},$
$\displaystyle=$ $\displaystyle
1-\frac{\delta_{1}(V_{d_{1},d_{2}})}{2\frac{d_{1}d_{2}}{d(d-1)}}\ \ \text{by
Proposition 2.}$

where $\delta_{1}(V_{d_{1},d_{2}})$ is the first taxicab dispersion obtained
from TCA of $V$ and $(d_{1},d_{2})$ represents the optimal TCA binary
partition of the $d$ items of $V$ such that $d=d_{1}+d_{2}.\vskip 12.0pt plus
4.0pt minus 4.0pt$

Proposition 4: The crossing index of a coherent cluster is

$Cross(cohC(\alpha))=\frac{2(\alpha-1)}{d_{1}d_{2}}.$

Example 7: The last column in Table 3 contains the values of the crossing
indices of the seven coherent clusters of the first iteration of SUSHI data.
We observe: a) $Cross(cohC_{1}(1))=0$, because the structure of its matrix of
first order marginals, M${}_{1,1},$ is block diagonal; which means that the
permutation $\tau$ is the identical permutation, so there are no crossing of
scores between the two subsets of items $J_{1}$ and $J_{2}$ in $cohC_{1}(1).$
b) $Cross(cohC_{1}(\alpha))$ for $\alpha=1,...,7$ is a uniformly increasing
function of $\alpha,$ similar in spirit to the $T_{v}(\tau_{J_{1}}(S_{1}))$
statistic. c) For the incoherent cluster $V_{1,8}$, we have:
$\delta_{1}(V_{1,8})=0.2354$ given in Example 3; and $d_{1}=d_{2}=5$ from
Figure 10. So
$Cross(V_{1,8})=1-\frac{0.2354}{2(5)(5)/(10(9))}=1-0.4237=0.5763.$

##  8 Coherent group

Our aim is to explore a given voting profile $V$ by uncovering its coherent
mixture groups, see equation (1); that is, $V=\cup_{g=1}^{G}cohG(g)\cup
noisyG$, where $G$ represents the number of coherent groups and $cohG(g)$ is
the $g$th coherent group. The computation is done by an iterative procedure in
$n_{G}$ steps for $n_{G}\geq G$ that we describe:

For $g=1$; let $V_{1}=V;$ compute $cohG(1)$ from $V_{1},$ then partition
$V_{1}=V_{2}\cup cohG(1);$

For $g=2$; compute $cohG(2)$ from $V_{2},$ then partition $V_{2}=V_{3}\cup
cohG(2);$

By continuing the above procedure, after $n_{G}$ steps, we get
$V=\cup_{g=1}^{n_{G}}cohG(g).\vskip 12.0pt plus 4.0pt minus 4.0pt$

However, some of the higher ordered coherent groups may have relatively small
sample sizes; so by considering these as outliers, we lump them together thus
forming the noisy group denoted by $noisyG$ in equation (1).

Let us recall the definition of a coherent group given in equation 2

$cohG(g)=\cup_{\alpha=1}^{c_{g}}cohC_{g}(\alpha)\text{ \ for }g=1,...,G;$

that is, a coherent group is the union of its coherent clusters. This implies
that the sample size of $cohG(g)$ equals the sum of the sample sizes of its
coherent clusters

$|cohG(g)|\ =\sum_{\alpha=1}^{c_{g}}|cohC_{g}(\alpha)|.$

As an example, for the SUSHI data, from the 2nd column of Table 3 we can
compute the sample size of the first coherent group

$\displaystyle|cohG(1)|$ $\displaystyle=$
$\displaystyle\sum_{\alpha=1}^{c_{g}=7}|cohC_{1}(\alpha)|$ $\displaystyle=$
$\displaystyle 2418.$

Furthermore, $cohG(1)$ is composed of 27 observed riffle shuffles summarized
in Table 13, which provides quite a detailed view of its inner structure.

The next result shows important characteristics of a coherent group inherited
from its coherent clusters.

Theorem 2: ( Properties of a coherent group $cohG(g)$)

a) The first principal column factor score $\mathbf{g}_{1}$ of the $d$ items
in a coherent group is the weighted average of the first principal column
factor score $\mathbf{g}_{1}$ of the $d$ items of its coherent clusters; that
is,

$\displaystyle g_{1}(j$ $\displaystyle\in$ $\displaystyle
cohG(g))=\sum_{\alpha=1}^{c_{g}}\frac{|cohC_{g}(\alpha)|}{|cohG(g)|}g_{1}(j\in
cohC_{g}(\alpha))\text{\ \ for }j=1,...,d.$ $\displaystyle=$
$\displaystyle\frac{2}{d-1}\sum_{\alpha=1}^{c_{g}}\frac{|cohC_{g}(\alpha)|}{|cohG(g)|}\beta(j\in
cohC_{g}(\alpha))-1\text{ \ \ by Proposition 3.}$

And $corr(\mathbf{g}_{1}(cohG(g),\mathbf{\beta}(cohG(g))=1.$

b) The first TCA dispersion value of a coherent group is the weighted average
of the first TCA dispersion values of its coherent clusters; that is,

$\delta_{1}(cohG(g))=\sum_{\alpha=1}^{c_{g}}\frac{|cohC_{g}(\alpha)|}{|cohG(g)|}\delta_{1}(cohC_{g}(\alpha)).$

c) The crossing index of a coherent group is the weighted average of the
crossing indices of its coherent clusters; that is,

$Cross(cohG(g))=\sum_{\alpha=1}^{c_{g}}\frac{|cohC_{g}(\alpha)|}{|cohG(g)|}Cross(cohC_{g}(\alpha)).$

Example 8: Table 14 summarizes the first four coherent groups of SUSHI data,
which emerged after 5 iterations. For $g=1$, we get
$cohG(1)=\cup_{\alpha=1}^{c_{1}=7}cohC_{1}(\alpha);$ that is, the first
coherent group of voters, the majority, is composed of 48.36% of the sample
with crossing index of 27.3%. Standard errors of the Borda scale of the items
in $cohG(1)\ $in Table 14 are:

$(0.046,0.051,0.042,0.042,0.053,0.047,0.037,0.034,0.037,0.025).$

We can discern the following grouped seriation (bucket ranking) of the items

$j8\succ j5\succ j6\succ\left\\{j3,j2\right\\}\succ
j1\succ\left\\{j9,j4\right\\}\succ\left\\{j7\right\\}\succ\left\\{j10\right\\}.$

The groupings are based on the standard 95% confidence intervals of the Borda
scale of the items.

The 2nd coherent group $cohG(2),$ summarized by its Borda scales in Table 14,
is made up of eight coherent clusters; it is composed of 19.0% of the sample
with crossing index of 35.38%. The voters in this coherent group disapprove
$\left\\{uni(seaurchin),sake(salmonroe)\right\\},$ which are considered more
”daring sushis”.

The third coherent group $cohG(3),$ summarized by its Borda scales in Table
14, is made up of eight coherent clusters; it is composed of 13.24% of the
sample with crossing index of 27.3%. The voters in this coherent group prefer
the three types of tuna sushis with sea urchin sushis.

The fourth coherent group $cohG(4),$ summarized by its Borda scales in Table
14, is made up of eight coherent clusters; it is composed of 6.94% of the
sample with crossing index of 35.27%. The voters disapprove the three types of
tuna sushis.

Remark 6:

a) Note that the number of preferred sushis in $cohG(1)$ and $cohG(2)$ are
six; that is $|J_{2}|=6.$ While the number of preferred sushis in $cohG(3)$
and $cohG(4)$ are four.

b) The four coherent groups summarized in Table 14 can also be described as
two bipolar latent factors: By noting that the only major difference between
the first two coherent groups is that (5\. uni (sea urchin), 6. sake (salmon
roe)) are swapped with (7\. tamago (egg), 4. ika (squid)). While the only
major difference between the third and fourth coherent groups is that the
three tunas are swapped with (4\. ika (squid), 5. uni (sea urchin), 1. ebi
(shrimp)).

c) We consider the fifth group as noisy (outliers not shown) composed of
12.36% of the remaining sample: it contains
$cohG(5)=\cup_{\alpha=1}^{2}cohC_{5}(\alpha)$ whose sample size is $38$, a
very small number. For the sake of completeness we also provide the sample
sizes of its two coherent clusters $|cohC_{5}(1)|=22$ and $|cohC_{5}(2)|=16$.

Table 14: The first four coherent groups of SUSHI data and related statistics.
---
$\mathbf{cohG(1)=\cup}_{\alpha=1}^{7}\mathbf{cohC}_{1}\mathbf{(\alpha)}$ | $\mathbf{\beta}$ | $\mathbf{cohG(2)=\cup}_{\alpha=1}^{8}\mathbf{cohC}_{2}\mathbf{(\alpha)}$ | $\mathbf{\beta}$
8\. toro (fatty tuna) | $\mathbf{7.62}$ | 8\. toro (fatty tuna) | $\mathbf{6.15}$
5\. uni (sea urchin) | $\mathbf{6.31}$ | 2\. anago (sea eel) | $\mathbf{5.97}$
6\. sake (salmon roe) | $\mathbf{5.92}$ | 1\. ebi (shrimp) | $\mathbf{5.92}$
3\. maguro (tuna) | $\mathbf{5.49}$ | 7\. tamago (egg) | $\mathbf{5.76}$
2\. anago (sea eel) | $\mathbf{5.35}$ | 3\. maguro (tuna) | $\mathbf{5.55}$
1\. ebi (shrimp) | $\mathbf{5.04}$ | 4\. ika (squid) | $\mathbf{5.41}$
9\. tekka-maki (tuna roll) | 3.27 | 9\. tekka-maki (tuna roll) | 3.80
4\. ika (squid) | 3.10 | 10\. kappa-maki (cucumber roll) | 2.56
7\. tamago (egg) | 1.94 | 6\. sake (salmon roe) | 2.45
10\. kappa-maki (cucumber roll) | 0.97 | 5\. uni (sea urchin) | 1.44
$Cross(cohG(1))=27.3\%$ |  | $Cross(cohG(2))=35.38\%$ |
$|cohG(1)|=2418\ (48.36\%)$ |  | $|cohG(2)|=955\ (19.10\%)$ |
|  |  |
$\mathbf{cohG(3)=\cup}_{\alpha=1}^{8}\mathbf{cohC}_{3}\mathbf{(\alpha)}$ | $\mathbf{\beta}$ | $\mathbf{cohG(4)=\cup}_{\alpha=1}^{8}\mathbf{cohC}_{4}\mathbf{(\alpha)}$ | $\mathbf{\beta}$
8\. toro (fatty tuna) | $\mathbf{7.31}$ | 4\. ika (squid) | $\mathbf{6.67}$
6\. sake (salmon roe) | $\mathbf{6.62}$ | 5\. uni (sea urchin) | $\mathbf{6.50}$
3\. maguro (tuna) | $\mathbf{6.30}$ | 6\. sake (salmon roe) | $\mathbf{6.43}$
9\. tekka-maki (tuna roll) | $\mathbf{6.00}$ | 1\. ebi (shrimp) | $\mathbf{6.16}$
7\. tamago (egg) | 3.76 | 8\. toro (fatty tuna) | 3.69
4\. ika (squid) | 3.41 | 7\. tamago (egg) | 3.39
2\. anago (sea eel) | 3.00 | 2\. anago (sea eel) | 3.21
1\. ebi (shrimp) | 2.92 | 9\. tekka-maki (tuna roll) | 3.14
10\. kappa-maki (cucumber roll) | 2.86 | 10\. kappa-maki (cucumber roll) |  2.99
5\. uni (sea urchin) | 2.80 | 3\. maguro (tuna) | 2.80
$Cross(cohG(3))=31.37\%$ |  | $Cross(cohG(4))=35.27\%$ |
$|cohG(3)|\ =662\ (13.24\%)$ |  | $|cohG(4)|\ =347\ (6.94\%)$ |

##  9 APA data set

The 1980 American Psychological Association (APA) presidential election had
five candidates: $\left\\{A,C\right\\}$ were research psychologists,
$\left\\{D,E\right\\}$ were clinical psychologists and $B$ was a community
psychologist. In this election, voters ranked the five candidates in order of
preference. Among the 15449 votes, 5738 votes ranked all five candidates. We
consider the data set which records the 5738 complete votes; it is available
in $\left[20,\ p.96\right]$ and $\left[5,\ Table\ 1\right]$. The winner was
candidate $C$.

(a) Figure 11 : TCA map of $\mbox{Coh}_{1}C_{1}$ of APA data

(b) Figure 12: TCA map of $\mbox{Coh}_{1}C_{2}$ of APA data

Table 15 compares the results obtained by our method and the best distance-
based mixture model given in $\left[21\right]$. Distance-based models have two
parameters, a central modal ranking and a precision parameter. The precision
parameter measures the peakedness of the distribution. $\left[21\right]$ found
that the Cayley distance produced better results than the Kendall and Spearman
distances using BIC (Bayesian information criterion) and ICL (integrated
complete likelihood) criteria. Parts a and b of Table 15, are reproduced from
$\left[21,\ \text{Tables 4 and 5}\right]$.

Part c of Table 15 summarizes the results of our approach, where we only
describe the first four coherent groups: We find only the first two coherent
groups as meaningfully interpretable based on the a priori knowledge of the
candidates. Voters in $cohG(1)$, with sample size of 31%, prefer the research
oriented psychologists $\left\\{A,C\right\\}$ over the rest. Voters in
$cohG(2)$, with sample size of 23.7%, prefer the clinical psychologists
$\left\\{D,E\right\\}$ over the rest. We interpret $cohG(3)$ and $cohG(4)$ as
mixed B with 14.23% and 12.% of the voters, respectively. Additionally, there
is a $noisyG$ making up 19.1% of the sample, which comprises $cohG(5)$
displayed in Table 15.

$\left[5\right]$ discussed this data set quite in detail; surprisingly, our
results confirm his observations: a) There are two groups of candidates,
$\left\\{A,C\right\\}$ and $\left\\{D,E\right\\}.$ The voters line up behind
one group or the other ; b) The APA divides into academicians and clinicians
who are on uneasy terms. Voters seem to choose one type or the other, and then
choose within; but the group effect predominates ; c) Candidate $B$ seems to
fall in the middle, perhaps closer to $D$ and $E$.

The following important observation emerges from the comparison of results in
Table 15. We have two distinct concepts of groups for rank data, categorical
and latent variable based. To see this, consider groups 3 and 4 in part a of
Table 15: Group 3 is based on the modal category $B\succ C\succ A\succ D\succ
E$ and group 4 is based on the modal category $B\succ C\succ A\succ E\succ D.$
The only difference between these two modal categories is the permutation of
the least ranked two clinical psychologist candidates $\left\\{D,E\right\\};$
this difference is not important and does not appear in our approach, which is
a latent variable approach.

Table 15: A summary of results derived from three methods of analysis of
---
APA election data. Parts a) and b) are from Murphy and Martin (2003).
a) Parameters of the best mixture model selected, Cayley-based, using BIC
Group | sample% | modal orderings | precision
$1$ | $42$ | $D\succ B\succ E\succ C\succ A$ | $0.16$
$2$ | $31$ | $C\succ D\succ E\succ A\succ B$ | $0.79$
$3$ | $12$ | $B\succ C\succ A\succ D\succ E$ | $1.52$
$4$ | $8$ | $B\succ C\succ A\succ E\succ D$ | $1.81$
$5$ | $7$ | $B\succ D\succ A\succ E\succ C$ | $1.72$
b) Parameters of the best mixture model selected, Cayley-based, using ICL
Group | sample% | modal ordering | precision
$1$ | $100$ | $B\succ C\succ A\succ E\succ D$ | $0.25$
c) The first five coherent groups, each composed of two coherent clusters.
Group | sample% | $\beta(C)$ | $\beta(A)$ | $\beta(B)$ | $\beta(E)$ | $\beta(D)$ | $Cross$
cohG(1) Research | $31.0$ | $\mathbf{3.55}$ | $\mathbf{3.15}$ | $1.31$ | $1.15$ | $0.85$ | $10.22\%$
cohG(2) Clinical | $23.7$ | $0.83$ | $1.28$ | $1.28$ | $\mathbf{3.31}$ | $\mathbf{3.30}$ | $12.90\%$
cohG(3) mixed B | $14.2$ | $0.66$ | $\mathbf{2.70}$ | $\mathbf{2.96}$ | $0.71$ | $\mathbf{2.97}$ | $12.45\%$
cohG(4) mixed B | $12.0$ | $\mathbf{2.85}$ | $0.77$ | $\mathbf{2.86}$ | $\mathbf{2.80}$ | $0.72$ | $10.22\%$
cohG(5) outlier | $8.6$ | $0.96$ | $\mathbf{3.30}$ | $1.31$ | $\mathbf{3.40}$ | $1.00$ | $9.88\%$

###  9.1 Description

The eight coherent clusters of the first four coherent groups can simply be
described as:

$coh_{1}C(1):T_{v}(\tau_{J_{2}}(S_{2})=\tau_{\left\\{A,C\right\\}}\left\\{3,4\right\\}=\left\\{3,4\right\\})=7$
for $v=1,...,1233.$

$coh_{1}C(2):T_{v}(\tau_{J_{2}}(S_{2})=\tau_{\left\\{A,C\right\\}}\left\\{3,4\right\\}=\left\\{\mathbf{2},4\right\\})=6$
for $v=1,...,545.$

$coh_{2}C(1):T_{v}(\tau_{J_{2}}(S_{2})=\tau_{\left\\{D,E\right\\}}\left\\{3,4\right\\}=\left\\{3,4\right\\})=7$
for $v=1,...,834.$

$coh_{2}C(2):T_{v}(\tau_{J_{2}}(S_{2})=\tau_{\left\\{D,E\right\\}}\left\\{3,4\right\\}=\left\\{\mathbf{2},4\right\\})=6$
for $v=1,...,526.$

$coh_{3}C(1):T_{v}(\tau_{J_{1}}(S_{1})=\tau_{\left\\{C,E\right\\}}\left\\{0,1\right\\}=\left\\{0,1\right\\})=1$
for $v=1,...,512.$

$coh_{3}C(2):T_{v}(\tau_{J_{1}}(S_{1})=\tau_{\left\\{C,E\right\\}}\left\\{0,1\right\\}=\left\\{0,\mathbf{2}\right\\})=2$
for $v=1,...,305.$

$coh_{4}C(1):T_{v}(\tau_{J_{1}}(S_{1})=\tau_{\left\\{A,D\right\\}}\left\\{0,1\right\\}=\left\\{0,1\right\\})=1$
for $v=1,...,350.$

$coh_{4}C(2):T_{v}(\tau_{J_{1}}(S_{1})=\tau_{\left\\{A,D\right\\}}\left\\{0,1\right\\}=\left\\{0,\mathbf{2}\right\\})=2$
for $v=1,...,338.$

In this case, we can also visualize all the orderings belonging to a coherent
group: Figures 11 and 12 display all the preferences belonging to the two
coherent clusters of the first coherent group. The label $CAEBD162$ in Figure
11 should be interpreted as the preference $C\succ A\succ E\succ B\succ D$
repeated 162 times.

##  10 Riffle independence model

Riffle independence is a nonparametric probabilistic modelling method of
preferences developed by $\left[2\right]$, which generalizes the independence
model. It can be described in the following way:

(a) Partition the set $J$ of $d$ distinct items into two disjoint subsets
$J_{1}$ of size $d_{1}$ and $J_{2}$ of size $d_{2}$. Then generate an ordering
of items within each subset according to a certain ranking model. This implies
that any ordering of the $d$ items can be written as a direct product of two
disconnected orderings; which in its turn implies the independence of the two
subsets $J_{1}$ and $J_{2}$. So the model complexity of this step is of order
$d_{1}!+d_{2}!.$

(b) Interleave the two independent orderings for these two subsets using a
riffle shuffle to form a combined ordering. An interleaving is a binary
mapping from the set of orderings to $\left\\{J_{1},J_{2}\right\\}$. The model
complexity of this step is of order $d!/(d_{1}!d_{2}!).$ The interleaving step
generates the riffled independence of the two subsets $J_{1}$ and $J_{2}$.

So the combined model complexity of both steps is
$d_{1}!+d_{2}!+d!/(d_{1}!d_{2}!)$ which is much smaller than
$d!=(d_{1}+d_{2})!$.

For example, consider an ordering of the items in the set
$J=\left\\{A,B,C,D,E,F\right\\}$ from its two subsets
$J_{1}=\left\\{A,C\right\\}$ and $J_{2}=\left\\{B,D,E,F\right\\}.\ $In the
first step, relative orderings of the items in $J_{1}$ and $J_{2}$ are drawn
independently. Suppose we obtain the relative ordering $\varphi(J_{1})=(C\succ
A)$ in $J_{1},$ and the relative ordering $\varphi(J_{2})=(B\succ D\succ
F\succ E)$ in $J_{2}.$ Then, in the second step, the two relative orderings
are combined by interleaving the items in the two subsets. For instance, if
the interleaving process is
$\omega(J_{1},J_{2})=(J_{1},J_{2},J_{2},J_{1},J_{2},J_{2})$, where the
relative ordering of the items in each subset remains unchanged, the combined
ordering is then determined by the composition

$\displaystyle\omega(J_{1},J_{2})\ast(\varphi(J_{1}),\varphi(J_{2}))$
$\displaystyle=$ $\displaystyle(C\succ B\succ D\succ A\succ F\succ E)$
$\displaystyle=$ $\displaystyle\varphi(J).$

Given the two subsets $J_{1}$ and $J_{2}$ with their orderings
$\varphi(J_{1})$ and $\varphi(J_{2})$ and interleaving $\omega(J_{1},J_{2})$
generated from models with probability distributions $f_{J_{1}},$ $g_{J_{2}}$
and $m_{\omega}$, respectively, the probability of observed ordering under the
riffle independence model is

$P(\varphi(J))=m_{\omega}(\omega(J_{1},J_{2}))f_{J_{1}}(\varphi(J_{1}))g_{J_{2}}(\varphi(J_{2}).$

There are two formulations of riffle shuffle for rank data in statistics:
probabilistic and exploratory. In the riffled independence model, the set of
items is partitioned recursively, while in the exploratory approach the set of
voters is partitioned recursively.

##  11 Conclusion

The main contribution of this paper is the introduction of an exploratory
riffle shuffling procedure to reveal and display the structure of diffuse rank
data for large sample sizes. The new notion of a coherent cluster, that we
developed, is simply based on the geometric notion of taxicab projection of
points on the first TCA axis globally and locally; furthermore, it has nice
mathematical properties. Coherent clusters of a coherent group represent the
same latent variable opposing preferred items to disliked items, and can
easily be interpreted and displayed.

Like Occam’s razor, step by step, our procedure peels the essential structural
layers (coherent groups) of rank data.

Our method was able to discover some other aspects of the rank data, such as
outliers or small groups, which are eclipsed or masked by well established
methods, such as distance or random utility based methods. The major reasons
for this is that in random utility based methods the multivariate nature of a
preference is reduced to binary preferences (paired comparisons), and in
Mallows distance related methods distances between any two preferences are
bounded.

We presented a new index, $Cross$, that quantifies the extent of crossing of
scores between the optimal binary partition of the items that resulted from
TCA. The crossing index of a group is based on the first taxicab dispersion
measure: it takes values between 0 and 100%, so it is easily interpretable.

The proposed approach can easily be generalized to the analysis of rankings
with ties and partial rankings.

The package TaxicabCA written in R available on CRAN can be used to do the
calculations.

Acknowledgement: Choulakian’s research has been supported by NSERC grant
(RGPIN-2017-05092) of Canada.

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Appendix

Let $\mathbf{R}=(r_{ij})$ for $i=1,...,n$ and $j=1,...,d$ represent the Borda
scorings for preferences, where $r_{ij}$ takes values $0,...,d-1.$ Similarly,
let $\overline{\mathbf{R}}$ represent the reverse Borda scorings, whose column
sums are the cordinates of the row named $\mathbf{nega=}$
$n\overline{\mathbf{\beta}}=\mathbf{1}_{n}^{\prime}\overline{\mathbf{R}}.$ We
consider the application of TCA to the data set

$\mathbf{R}_{nega}=(_{\mathbf{nega}}^{\mathbf{R}})$

of size $(n+1)\times d.$ So let

$\mathbf{P}=\mathbf{R}_{nega}/t$

be the correspondence table associated with $\mathbf{R}_{nega},$ where
$t=2n\sum_{j=0}^{d-1}=nd(d-1).$ We have

$\begin{array}[]{cccc}p_{i\ast}&=&\frac{1}{2n}\;\;\;\text{for}\quad
i=1,...,n&\end{array}$ (14)
$\begin{array}[]{cccc}&=&\frac{1}{2}\;\;\;\text{for}\quad i=n+1,&\end{array}$
(15)

and

$p_{\ast j}=\frac{1}{d}\text{\ \ \ \ \ for\ \ \ }j=1,...,d.$ (16)

The first residuel correspondence matrix will be

$\begin{array}[]{ccccc}p_{ij}^{(1)}&=&p_{ij}-p_{i\ast}p_{\ast
j}&\qquad\qquad\qquad(17)\\\
&=&\frac{r_{ij}}{t}-\frac{1}{2n}.\frac{1}{d}\text{\ \ \ \ for\ \ \ \
}i=1,...,n&\qquad\qquad\qquad(18)\\\
&=&\frac{\mathbf{nega}_{j}}{t}-\frac{1}{2}.\frac{1}{d}\ \ \ \ \text{for\ \ \ \
}i=n+1.&\qquad\qquad\qquad(19)\end{array}$

Consider the nontrivial binary partition of the set
$S=\left\\{0,1,...,d-1\right\\}$ into $S=S_{1}\cup S_{2},$ where
$|S_{1}|=d_{1},$ $|S_{2}|=d_{2}$ and $d=d_{1}+d_{2}.$ To eliminate the sign
indeterminacy in the first TCA principal axis, we fix
$\mathbf{v}_{1}(nega)=\mathbf{v}_{1}(n+1)=-1;$ and we designate by $S_{1}$ the
set of item indices such that the first TCA principal axis coordinates are
negative, that is, $\mathbf{u}_{1}(j)=-1$ for $j\in S_{1}.$ It follows that
$\mathbf{u}_{1}(j)=1$ for $j\in S_{2}$.

Now we have by (4) for $i=1,...,n$

$\begin{array}[]{llll}a_{i1}&=&\sum_{j=1}^{d}\mathbf{u}_{1}(j)p_{ij}^{(1)}\\\
&=&\sum_{j\in S_{1}}\mathbf{u}_{1}(j)p_{ij}^{(1)}+\sum_{j\in
S_{2}}\mathbf{u}_{1}(j)p_{ij}^{(1)}&\\\ &=&-\sum_{j\in
S_{1}}p_{ij}^{(1)}+\sum_{j\in S_{2}}p_{ij}^{(1)}\\\ &=&-2\sum_{j\in
S_{1}}p_{ij}^{(1)}\ \ \ \ \text{by\ \ }(17)&\\\ &=&-2\sum_{j\in
S_{1}}(\frac{r_{ij}}{t}-\frac{1}{2n}.\frac{1}{d})\text{\ \ \ by\ \ (18)}&\\\
&=&\frac{d_{1}}{nd}-\frac{2}{t}\sum_{j\in S_{1}}r_{ij};&\end{array}$ (20)

and from which we deduce by (5) for $i=1,...,n$

$\begin{array}[]{lll}f_{i1}&=&\frac{a_{i1}}{p_{i\ast}}\\\
&=&\frac{2d_{1}}{d}-\frac{4}{d(d-1)}\sum_{j\in S_{1}}r_{ij}.\end{array}$ (21)

We have the following Theorem concerning the first TCA principal factor scores
of respondents $f_{i1}$ for $i=1,...,n.$

Theorem 1:

a) The maximum number of distinct clusters of $n$ respondents on the first TCA
principal axis (distinct $f_{i1}$ values$)$ is $d_{1}d_{2}+1.$

Proof: We consider the two extreme cases of $S_{1}$ and calculate the
summation term in (21):

For $S_{1}=\left\\{0,1,...,d_{1}-1\right\\}$,$\ \sum_{j\in
S_{1}}r_{ij}=\sum_{j=0}^{d_{1}-1}j=\frac{d_{1}(d_{1}-1)}{2}.$

For $S_{1}=\left\\{d-d_{1},1,...,d-1\right\\},$ $\sum_{j\in
S_{1}}r_{ij}=\sum_{j=d-d_{1}}^{d-1}j=\sum_{j=d_{2}}^{d-1}j=\frac{d_{1}(d_{2}+d-1)}{2}$.

It follows that

$\frac{d_{1}(d_{1}-1)}{2}\leq\sum_{j\in
S_{1}}r_{ij}\leq\frac{d_{1}(d_{2}+d-1)}{2};$

so $\sum_{j\in S_{1}}r_{ij}$ can take at most
$\frac{d_{1}(d_{2}+d-1)}{2}-\frac{d_{1}(d_{1}-1)}{2}+1=d_{1}d_{2}+1$ values.

b) The maximum value that $f_{i1}$ can attain is $2\frac{d_{1}d_{2}}{d(d-1)}.$

Proof: From (21) and Part a, it follows that the maximum value that $f_{i1}$
can attain is
$(\frac{2d_{1}}{d}-\frac{4}{d(d-1)}\frac{d_{1}(d_{1}-1)}{2})=2\frac{d_{1}d_{2}}{d(d-1)}.$

c) The minimum value that $f_{i1}$ can attain is
$-2\frac{d_{1}d_{2}}{d(d-1)}.$

Proof: From (21) and Part a, it follows that the minimum value that $f_{i1}$
can attain is
$(\frac{2d_{1}}{d}-\frac{4}{d(d-1)}\frac{d_{1}(d_{2}+d-1)}{2})=-2\frac{d_{1}d_{2}}{d(d-1)}.$

d) If the number of distinct clusters is maximum, $d_{1}d_{2}+1$, then the gap
between two contiguous $f_{i1}$ values is $\frac{4}{d(d-1)}.$

Proof: Suppose that the number of distinct clusters is maximum,
$d_{1}d_{2}+1$. We consider the first TCA factor score
$f_{i1}=\frac{2d_{1}}{d}-\frac{4}{d(d-1)}\sum_{j\in S_{1}}r_{ij}$ which is
different in value from the two extreme values $\pm
2\frac{d_{1}d_{2}}{d(d-1)}.$ Then
$f_{i_{1}1}=\frac{2d_{1}}{d}-\frac{4}{d(d-1)}(-1+\sum_{j\in S_{1}}r_{ij})$
will be the contiguous higher value to $f_{i1};$ and similarly
$f_{i_{2}1}=\frac{2d_{1}}{d}-\frac{4}{d(d-1)}(1+\sum_{j\in S_{1}}r_{ij})$ will
be the contiguous lower value to $f_{i1};$ and the required result follows.

Proposition 1: For a voting profile $V$,
$\delta_{1}\geq|f_{1}(\mathbf{nega})|$.

Proof: Let $\mathbf{a}_{1}=(_{a_{1}(nega)}^{\mathbf{a}_{11}}).$We need the
following three observations.

First, it is well known that $\mathbf{a}_{1}$ is centered by (5) and (9),

$\displaystyle\mathbf{1}_{n+1}^{\prime}\mathbf{a}_{1}$ $\displaystyle=$
$\displaystyle 0,$ $\displaystyle=$
$\displaystyle\mathbf{1}_{n}^{\prime}\mathbf{a}_{11}+a_{1}(nega);$

from which we get,

$|\mathbf{1}_{n}^{\prime}\ \mathbf{a}_{11}|=|a_{1}(nega)|.$ (22)

Second, by triangle inequality of the L1 norm we have

$||\mathbf{a}_{11}||_{1}\geq|\mathbf{1}_{n}^{\prime}\mathbf{a}_{11}|.$ (23)

Third, the marginal relative frequency of the nega row is $p_{nega\ast}=1/2$
by (15) , and $f_{i1}=a_{i1}/p_{i\ast}$ for $i=1,...,n+1$ by (5); so we have

$f_{1}(nega)=2a_{1}(nega).$ (24)

Now we have by (7)

$\begin{array}[]{llll}\delta_{1}&=&||\mathbf{a}_{1}\mathbf{||}_{1}\\\
&=&||\mathbf{a}_{11}||_{1}+|a_{1}(nega)|\\\
&\geq&|\mathbf{1}_{n}^{\prime}\mathbf{a}_{11}|+|a_{1}(nega)|\text{\ \ \ by \ \
(23)}\\\ &=&2|a_{1}(nega)|\ \ \ \ \text{by\ \ \ (22)}\\\
&=&|f_{1}(nega)|\text{\ \ \ \ by\ \ (24)}\end{array}$

Propostion 2: Let $cohC_{m}(\alpha)=V_{m,\alpha}$ be the $\alpha$th coherent
cluster of the $m$th coherent group characterized by
$f_{1}^{V_{m,\alpha}}(\mathbf{\sigma)=}f_{\alpha}^{V_{m}}$ for all
$\mathbf{\sigma\in}cohC_{m}(\alpha)$. Then
$\delta_{1}=f_{\alpha}^{V_{m}}=-f_{1}(\mathbf{nega}).$

Proof: By Definition 1 of the coherency of the cluster $V_{m,\alpha},$ we have
$0<f_{1}^{V_{m,\alpha}}(i)=f_{\alpha}^{V_{m}}$ for
$i=1,...,|cohC_{m}(\alpha)|$; by (5) it follows that $0<a_{i1}=$
$f_{\alpha}^{V_{m}}/n$ for $i=1,...,|cohC_{m}(\alpha)|$ ; so (25) becomes
equality,
$||\mathbf{a}_{11}||_{1}=\sum_{i=1}^{n}a_{i1}=|\mathbf{1}_{n}^{\prime}\mathbf{a}_{11}|$,
and the required result follows.

Proposition 3 is a corollary to the following general result

Theorem 3: If the first TCA principal axis of the columns of
$\mathbf{R}_{nega}$ is $\mathbf{v}_{1}=(_{-1}^{\mathbf{1}_{n}})$, then

the first principal column factor score $\mathbf{g}_{1}$ of the $d$ items is
an affine function of the Borda scale $\mathbf{\beta};$ that is,
$g_{1}(j)=\frac{2}{d-1}\beta(j)-1$ or $corr(\mathbf{g}_{1},\mathbf{\beta})=1.$

Proof: Suppose that $\mathbf{v}_{1}=(_{-1}^{\mathbf{1}_{n}});$ then by (4) for
$j=1,...,d$

$\displaystyle b_{1}(j)$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{n+1}v_{1}(i)p_{ij}^{(1)}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{n}p_{ij}^{(1)}-p_{(n+1)j}^{(1)}$ $\displaystyle=$
$\displaystyle 2\sum_{i=1}^{n}p_{ij}^{(1)}\text{ \ \ by\ \ \ (17)}$
$\displaystyle=$ $\displaystyle 2\sum_{i=1}^{n}(p_{ij}-p_{i\ast}p_{\ast j})$
$\displaystyle=$ $\displaystyle 2\sum_{i=1}^{n}r_{ij}/t-p_{\ast j}\ \ \
\text{by\ \ (14)}$ $\displaystyle=$ $\displaystyle 2n\beta(j)/t-p_{\ast j}$

Thus by (5) for $j=1,...,d$

$\displaystyle g_{1}(j)$ $\displaystyle=$ $\displaystyle b_{1}(j)/p_{\ast j}$
$\displaystyle=$ $\displaystyle\frac{2n\beta(j)/t-p_{\ast j}}{p_{\ast j}}$
$\displaystyle=$ $\displaystyle\frac{2\beta(j)}{d-1}-1.$

Proposition 4: The crossing index of a coherent cluster is

$Cross(cohC(\alpha))=\frac{2(\alpha-1)}{d_{1}d_{2}}.$

Proof: Easily shown by using Definition 3 and Proposition 2.

The proof of Theorem 2a easily follows from Theorem 3. The proof of Theorem 2b
is similar to the proof of Propostion 1. The proof of Theorem 2c is similar to
the proof of Propostion 4.