=Paper=
{{Paper
|id=Vol-2123/paper10
|storemode=property
|title=A Data Analysis Application of Formal Independence Analysis
|pdfUrl=https://ceur-ws.org/Vol-2123/paper10.pdf
|volume=Vol-2123
|authors=Francisco J. Valverde-Albacete,Carmen Peláez-Moreno,Inma P. Cabrera, Pablo Cordero,Manuel Ojeda-Aciego
|dblpUrl=https://dblp.org/rec/conf/cla/Valverde-Albacete18
}}
==A Data Analysis Application of Formal Independence Analysis==
https://ceur-ws.org/Vol-2123/paper10.pdf
A Data Analysis Application of
Formal Independence Analysis?
Francisco J. Valverde-Albacete1 ?? , Carmen Peláez-Moreno1 , Inma P.
Cabrera2 , Pablo Cordero2 , and Manuel Ojeda-Aciego2
1
Depto. Teorı́a de Señal y Comunicaciones, Univ. Carlos III de Madrid, Madrid,
Spain, fva@tsc.uc3m.es carmen@tsc.uc3m.es
2
Dpt. Matemática Aplicada, Univ. de Málaga, Málaga, Spain
ipcabrera@uma.es pcordero@uma.es aciego@uma.es
Abstract. In this paper we present a new technique for the analysis of
data tables by means of Formal Independence Analysis (FIA). This is
an analogue of Formal Concept Analysis for the study of independence
relations in data, instead of hierarchical relations. A FIA of a context
produces, when possible, its block diagonalization by detecting pairs of
sets of objects and attributes that are not mutually incident, or tomoi,
that partition the context. In this paper we combine this technique with
the exploration of contexts with entries in a semifield to find independent
sets in contingency matrices. Specifically, we apply it to a number of
confusion matrices issued from cognitive experiments to find evidences
for the hypothesis of perceptual channels.
1 Introduction and Motivation
In this paper we derive a technique for data analysis from the recently introduced
Formal Independence Analysis, (FIA) [11]. This is an analysis technique for
formal contexts based on the description of certain pairs of subsets of objects and
attributes called tomoi, e.g. divisions, which are unrelated through the incidence.
We set out to demonstrate how these tomoi allow us to dissect the structure and
information of certain matrices.
Independent Perceptual Channels. Miller and Nicely [4] posited that
for certain human perceptual tasks—e.g. consonant perception—the underlying
structure of confusion matrices provide evidence of the existence of perceptual
channels associated with specific perceptual features. This work is aimed at pro-
viding a technique to make such channels evident with the goals and techniques
of Lattice Theory.
?
FJVA and CPM have been partially supported by the Spanish Government-MinECo
projects TEC2014-53390-P and TEC2017-84395-P. IPC and PC have been partially
supported by the Spanish Government-MinECo project TIN2017-89023-P; MOA
has been partially supported by the Spanish Government-MinECo project TIN2015-
70266-C2-P-1.
??
Corresponding author.
c paper author(s), 2018. Proceedings volume published and copyrighted by its editors.
Paper published in Dmitry I. Ignatov, Lhouari Nourine (Eds.): CLA 2018, pp.
117–128, Department of Computer Science, Palacký University Olomouc, 2018.
Copying permitted only for private and academic purposes.
118 Francisco J. Valverde-Albacete et al.
Specifically, consider the confusion matrix Cij describing the results of an
iterated classification experiment “when presented with stimulus i, and the (hu-
man) classifier answered response j.” If the hypothesis of independent channels
were true, we would expect this confusion matrix to be reordered by specific per-
mutations of its rows and columns into a block diagonal form, more specifically,
a squared block diagonal form. In this block-diagonal form, each block would
describe the confusions within a perceptual channel, while confusions outside
the channel would not be observed.
Reading Guide. In this paper we will use the recently developed FIA (Sec-
tion 2.1) to obtain a block-diagonal form for confusion matrices, that leads to the
independent virtual channel hypothesis of Miller and Nicely. This result actually
stems from the consideration of a disjoint union of subcontexts decomposition
technique already available from [2] that we relate to the notion of tomos and
boolean tomoi lattice (Section 2.2). Our main results are the theoretical tech-
nique (Section 3.1) and the actual analyses carried out in the Miller and Nicely
data (Section 3.2). We also provide a Discussion, a look into Further Work and
some Conclusions.
2 Methods
2.1 Formal Independence Analysis
FIA was defined to complement the analysis of the information in formal contexts
carried out by FCA, originally in terms of the hierarchical relation of formal
concepts in terms of the inclusion between extents and intents. Instead, FIA
targets the relation of independence between sets of objects and attributes [11],
therefore called tomoi 3 The objects in the “extent” of a formal tomoi have no
relation with the attributes of the “intent” of the tomoi.
Theorem 1 (Basic theorem of formal independence analysis).
1. The context analysis phase: Given a formal context (G, M, I),
(a) The operators ·∼ : 2G → 2M and ·∼ : 2M → 2G
[
α∼ = M r I(g, ·) = {m ∈ M | g I\ m for all g ∈ α} (1)
g∈α
[
β∼ = G r I(·, m) = {g ∈ G | g I\ m for all m ∈ β} (2)
m∈β
form a right-Galois connection (·∼ , ·∼ ) : (2G , ⊆) (
*(2M , ⊆) whose formal
∼
tomoi are the pairs (α, β) such that α = β and α = β∼ .
3
From the Greek “tomos-tomoi”, division.
A Data Analysis Application of Formal Independence Analysis 119
(b) The set of formal tomoi A(G, M, I) with the relation
(α1 , β1 ) ≤ (α2 , β2 ) iff α1 ⊇ α2 iff β1 ⊆ β2
is a complete lattice, which is called the tomoi lattice of (G, M, I) and
denoted A(G, M, I), where infima and suprema are given by:
! !
^ [ \ ∼ _ \ ∼ [
(αt , βt ) = αt , βt (αt , βt ) = αt , βt
∼ ∼
t∈T t∈T t∈T t∈T t∈T t∈T
(c) The mappings γ : G → A(G, M, I) and µ : M → A(G, M, I)
∼ ∼ ∼
g 7→ γ(g) = ({g} ∼ , {g} ) m 7→ µ(m) = ({m}∼ , {m}∼ )
are such that γ(G) is infimum-dense in A(G, M, I) , µ(M ) is supremum-
dense in A(G, M, I).
2. The context synthesis phase: Given a complete lattice L = hL, ≤i
(a) L is isomorphic to4 A(G, M, I) if and only if there are mappings γ : G →
L and µ : M → L such that
– γ(G) is infimum-dense in L , µ(M ) is supremum-dense in L, and
– g I m is equivalent to γ(g) 6≥ µ(m) for all g ∈ G and all m ∈ M .
(b) In particular, L ∼
= A(L, L, 6≥) and, if L is finite, L ∼
= A(M (L), J(L), 6≥)
where M (L) and J(L) are the sets of meet- and join-irreducibles, respec-
tively, of L.
It is already known that the lattices of formal tomoi and concepts are deeply
related [13,7]. Recall that the contrary context to any (G, M, I) is the context
(M, G, I cd ), where the incidence has been transposed and inverted.
Proposition 1. The formal lattice of the contrary formal context is isomorphic
to the tomoi lattice:
A(G, M, I) ∼ = B(M, G, I cd )
2.2 Disjoint Context Sum and Adjoined Lattices
To set this scenario in a Formal Concept Analysis setting, recall from [2, Def-
inition 30] that the disjoint sum of two contexts K1 = (G1 , M1 , I1 ) and K2 =
(G2 , M2 , I2 ), with disjoint object and attribute sets is the context K1 ∪ K2 =
(G1 ∪ G2 , M1 ∪ M2 , I1 ∪ I2 ), and that the concept lattice of the total context is
the horizontal sum of the two concept lattices, that is, a union of the two lattices
which only overlap in the top and bottom elements
K = K1 ∪ K2 ⇐⇒ B(K1 ∪ K2 ) = B(K1 ) ∪ B(K2 ).
4
Read can be built as.
120 Francisco J. Valverde-Albacete et al.
This can be straightforwardly generalized to a finite number n of lattices,
n n
! n
[ [ [
K= Ki ⇐⇒ B Ki = B(Ki ). (3)
i=1 i=1 i=1
This is what we call in this paper an (explicit) block diagonal form for the
context, which results in a concept lattice of adjoined sublattices.
For this latter generalization, notice that each extent of K, except for the
extent G = ∪i Gi , is entirely contained in one of the sets Gi , and concept-lattice
dually for intents. So it makes sense to say that two non-extreme concepts are
orthogonal is they belong to different adjoined sublattices 5 .
The relationship between tomoi lattices and block decompositions is provided
by the following proposition.
Proposition 2. If A(G, M, I) ∼ = 2n then the context (G, M, I) has an explicit
block diagonal form.
Proof (Sketch). By Theorem 1 (item 2.b) the context (G, M, I) can be trans-
formed into another one whose object-concepts are the meet-irreducible elements
and whose attribute-concepts are the join-irreducible elements and, hence, be-
cause of the isomorphism with 2n , they are the co-atoms and the atoms, re-
spectively. Moreover, they are complementary pairs of one object-tomos and one
attribute-tomos.
As consequence, it is possible to reorganize the tabular expression of (G, M, I)
in such a way that we obtain a block diagonal form. t
u
3 Results
3.1 Theoretical Analysis.
The purpose of proving the existence of independent channels for different per-
cepts can be achieved by reducing a confusion matrix to a block diagonal form.
But, confusion matrices are not binary incidences and may not be subject to a
simple process of block diagonalization. Instead, we may look for an approximate
block-diagonal block, that retains the main structure of the confusions.
We can motivate this approximation in the following way:
– A perfect classifier would obtain a diagonal matrix of counts. This has been
proven in terms of information-theoretic arguments in [9], for instance.
– But in most cases what we can hope for is a diagonally dominant matrix,
that is not even symmetrical. For instance, the heatmap of the symmetrized
confusion matrix for the M&N data for −6dB, to the left of Fig. 1, shows
such a shape. Even its symmetrical part of CS has a corresponding structure
that is far from being block-diagonal, e.g. center of Fig. 1.
5
The basis for this definition is, of course, the embedding of extents and intents as
vectors in semimodules over an idempotent semifield which allows us to define a dot
product between extents, resp. intents. [10]. Note that in idempotent semimodules,
which are zero-sum free, null dot-products can only occur for vectors of disjoint
support, and this is precisely the case at hand.
A Data Analysis Application of Formal Independence Analysis 121
Fig. 1: (Color online) Heatmaps of the count confusion matrix in M&N for a
SNR of −6dB. Left: count matrix; C center: symmetrized count matrix CS ;
right: antisymmetrical residue CA .
– Using structural analysis from an adequately transformed matrix M = f (CA )
we could use the paradigm of Landscapes-of-Knowledge (LoK) [14] extended
to multi-valued contexts [8] to explore the sequence of boolean incidences
I(ϕ)ij = Mij Q ϕ where ϕ ranges in the values of the original matrix:
• Choosing I(ϕ)ij = Mij ≥ ϕ uses the min-plus structural analysis, while
• Choosing I(ϕ)ij = Mij ≤ ϕ uses the max-plus structural analysis.
– The criterion for finding a “correct” value for ϕ is to ensure that the I(ϕ)
has a tomoi lattice that is boolean. A proxy criterion for this is to select and
inspect only those ϕ whose number of formal tomoi is a power of 2. Note
that after obtaining the appropriate ϕ by Proposition 2 we would have the
block-decomposition.
In the following section, we check the feasibility of this scheme on the Miller
and Nicely data.
3.2 FIA Exploration of Confusion Matrices
Data Description. In this paper we will use the data from the Miller and
Nicely study to show examples of phenomena and test the proposed data anal-
ysis procedures. These are the confusion data of a consonant perception task,
and we will refer to it as the M&N data. Specifically they are six different con-
fusion matrices of 16 entries for different Signal-to-Noise Ratios (SNR) in dB of
{−18, −12, −6, 0, 6, 12} obtained in a (human) speech recognition task for the
consonants listed in Table 1. The stimuli where balanced, but the responses may
be unbalanced due to non-symmetrical confusion effects.
Data Preprocessing. Due to the symmetry inherent in the confusion task,
since the category of the responses was the same as that of the stimuli, we
extracted the symmetric component of each confusion matrix. This was done
by obtaining from each matrix C its symmetric component CS = (C + C t )/2.
122 Francisco J. Valverde-Albacete et al.
Table 1: Ordering of the consonants used in the confusion matrices analyzed
(from [4]).
symbol p , t , k , f , th , s , sh , b , d ,R g , v , dh , z , zh , m , n
phone /p/, /t/, /k/, /f/, /θ /, /s/, / /, /b/, /d/, /g/, /v/, /ð/, /z/, /zh/, /m/, /n/
The antisymmetric component CA = (C − C t )/2 can then be interpreted as a
residue. For the M&N confusion matrix at −6dB these two components can be
seen in Fig. 1.
The data were preprocessed to obtain both the Pointwise Mutual Information
(MI) and the Weighted Pointwise Mutual information (WPMI) as shown in
Fig. 2. Although prior work suggested that WPMI lends itself to more clear
Fig. 2: (Color online) Heatmaps of the confusion matrix in M&N for a SNR
of −6dB for different preprocessing. Left: pointwise mutual information. Right:
weighted pointwise mutual information.
analyses, for the purpose of finding independent blocks in the matrix, we found—
on using both types of data preprocessing—MI to retain more details about
confusions that define the blocks, e.g. between elements that share (unknown)
features
R motivating the confusion, for instance the voiceless fricatives /s/ vs.
/ /.
Data Analysis. We carried out min-plus exploratory analysis in the MI-
transformed confusion matrix above by thresholding for each ϕ in increasing
order and generated a sequence of K = 105 (binary) formal contexts K(ϕk ) =
(G, M, I(ϕk )), k ∈ [1, . . . K].
For each of these contexts, we calculated the number of formal tomoi for each
thresholded I(ϕ) by actually working out the formal concepts of the contrary
A Data Analysis Application of Formal Independence Analysis 123
context K(ϕ)cd = (M, G, \I t ). We do not explore at ϕ = −∞ which entails a
trivial full-incidence and a count of one tomoi.
The graph of these counts in base-2 logarithm, shown in Fig. 3.a, allows us
to define three regions:
(a) Tomoi count of I(ϕ) vs. ϕ(dB)
(b) I(ϕ) at ϕ = −1.299010 (c) I(ϕ) at ϕ = 1.539160
Fig. 3: (Color online) Number of formal tomoi vs ϕ for I(ϕ) and heatmaps for
two highlighted ϕ.
124 Francisco J. Valverde-Albacete et al.
– An initial segment where the threshold is too lax and we see essentially
few blocks and a number of “noise” tomoi, where our assumption, viz. that
there are virtual channels, does not hold.
In the example being analyzed, this is the range (−5.2, 1.53), to the left of the
leftmost vertical line in Fig. 3.a . To ascertain the shapes of the thresholded
we present an instance for where ϕ ≈ −1.23 and |K(ϕ)| = 30 focused on by
the leftmost circle. In the heatmap of Fig. 3.b we can see and inkling of three
different blocks, but since they are not complete, a number of “noisy” tomoi
appear, making the tomoi lattice drift away from 23 . Figure 4.a shows this
non-boolean tomoi lattice whose incidence is that of Fig. 3.b
– A middle segment where we start seeing many blocks, and consequently
the number of tomoi |A(G, M, I(ϕ))| falls exactly into one of the powers of
2, where our assumption holds.
In the example, this is ϕ ∈ [1.53, 2.07] between the vertical lines in Fig. 3.a
comprising the ramp where the cardinalities range from 29 to 214 tomoi.
This is the case, for instance, of ϕ ≈ −1.53, |K(ϕ)| = 29 , signaled as the
rightmost red circle. We can see the 9-block incidence in Fig. 3.c, while
Fig. 4.b shows the boolean lattice K(ϕ) ∼ = 29 . For reference, the (average)
mutual information for this matrix, M I−6dB = 1.80 falls within this range,
and would generate the tomoi lattice isomorphic to 210 .
– A final segment where the threshold is too stringent and we no longer see
a block diagonal form.
This is the least interesting zone for us. In the example it appears as a
descending slope in the range ϕ ∈ (2.07, 3.61) of Fig. 3.a.
We checked whether this behavior was analogous for all confusion matrices
by analyzing the rest of the matrices at different SNR. The following are the
main trends of analysis:
– We could only obtain boolean tomoi lattices considering all stimuli for those
confusion matrices with SNR of {18, 12, 6, 0, −6}. The matrices at SNR ∈
{−12, −18} were too noisy and some elements in the diagonal were less stable
than elements off the diagonal, hence they disappeared on early exploration.
– In all of the instances where in some range of MI values the exploration
procedure obtained boolean tomoi lattices, the average MI for the whole
matrix, that is in the standard definition of mutual information, actually
belonged in the range where the hypothesis held. Most of the times, this MI
was close to the value for values of ϕ that obtained the boolean tomoi lattice
of highest cardinality.
– The highest SNR in the confusion matrix being analyzed, the higher num-
ber of blocks in I(ϕ). This is congruent with the supposition that high SNR
situations allow us to distinguish individual phones better and it is there-
fore more difficult to obtain evidence of the perceptual channels through
confusions.
Extracting Perceptual Channels. The tomoi provide the basis for obtaining
the perceptual channels on top of boolean tomoi lattices, since for every object-
A Data Analysis Application of Formal Independence Analysis 125
tomoi, a meet-irreducible, its complement is an attribute tomoi, hence a join-
irreducible. By the properties of complementary tomoi, the crossed extents and
intents, define the blocks in the block diagonalization.
To see this, consider Table 2 of object-tomoi extents and their complementary
tomoi intents—the attribute tomoi—to be used to build the block-diagonal form
of (3). We see how, modulo a permutation, they constitute a refinement of the
perceptual channels that Miller and Nicely proposed [4].
Table 2: Paired table of meet- and join-irreducibles of K(ϕ) in Fig. 4.b .
Object and attribute subsets object-tomoi extent complementary attribute-tomoi intent
(G1 , M1 ) {p, t, k} {p, t, k}
(G2 , M2 ) {f, θ} {f, θ}
(G3 , M3 ) {s}
R {s}
R
(G4 , M4 ) { } { }
(G5 , M5 ) {b} {b}
(G6 , M6 ) {d, g} {d, g}
(G7 , M7 ) {v, ð} {v, ð}
(G8 , M8 ) {z, zh} {z, zh}
(G9 , M9 ) {m, n} {m, n}
Discussion and Further Work. Note that the problem we address in this
paper was already approached in [12], but not solved satisfactorily, and we believe
FIA provides a principled approach to the study of independent blocks within
matrices.
In fact, FIA seems to detect much finer perceptual channels than the origi-
nal paper suggests, perhaps because of the granularity of the perceptual features
used there (see below). In order to obtain a rougher partition of phones to sup-
port Miller and Nicely’s hypothesis, we have tried to analyze a balanced mixture
of all the confusion matrices. But FIA has proven too strong for this unrealistic
type of noise: the absence of confusions at 12dB dominates the behavior of the
mixture, and the confusions from those behaviors at −18dB and −12dB are lost.
Recall that it is precisely from the confusions where we obtain the evidence for
the perceptual channels, so clearly a more nuanced approach to such mixture
would be needed.
Although we have provided a data-induced procedure to obtain perceptual
channels from confusion matrices, this is only a first step in actually obtaining the
experimental channels. In particular, we have not investigated justifying those
channels in terms of perceptual characteristics. While Miller and Nicely pro-
posed an encoding of phones based on traditional categorical phonetic features,
modern studies favor the consideration of numeric features. In our opinion this
necessarily entails considering idempotent semimodule models of such spaces [6]
and would lead to higher-value hypotheses. This is left for future work.
126 Francisco J. Valverde-Albacete et al.
By no means is ours the only attempt at block-diagonalizing matrices over
idempotent semifields. In fact, such process is important for the calculation of the
Moore-Penrose inverse of a matrix over an idempotent semifield [5]. The main
difference with out work is that we are trying to approximate the block-diagonal
form in the presence of an implicit noise.
Yet a more general version of the problem is that of Cell Formation (CF)
in Group Technology, in the field of Manufacturing [1], because it involves the
block diagonalization of rectangular matrices. FIA is not restricted to squared
matrices, but our application and interpretation indeed are because of the fact
that confusion matrices are usually square. CF therefore opens up as an open
research and application avenue for FIA.
Finally, further work is necessary to ascertain the relationship of lattices of
formal tomoi to lattices of formal concepts, as well as to find out whether these
are the only information lenses available for formal contexts, or how to measure
the “quality” of the tomoi, in an effort similar to that shown for triadic analysis
and triclustering in [3]. Our next main aim, though, is to incorporate these
techniques in the over-arching exploratory data analysis framework first laid out
in full in [8].
4 Conclusions
We have introduced a new technique to analyze data tables based on the newly
proposed Formal Independence Analysis. The purpose of the technique is to ob-
tain tomoi,—pairs of sets of objects and attributes unrelated through an incident
relation— and their complements in the lattice of tomoi, which define as many
partitions of the sets of objects and attributes. These tomoi will then be used to
define a block-diagonal form for the incidence.
We apply the technique to the diagonally-dominant incidences of confusion
matrices. By a process of exploration we select special thresholds that obtain
boolean tomoi lattices. In these lattices we obtain the meet-irreducible object-
tomoi and their complements, the join-irreducible attribute-tomoi, that define
diagonal blocks on the looked-for incidence.
These diagonal blocks can be interpreted as virtual channels that transmit
different types of information in the spirit of some classical perceptual experi-
ments, e.g. Miller and Nicely’s.
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(a) Tomoi lattice for ϕ = −1.299010
(b) Tomoi lattice for ϕ = 1.539160
Fig. 4: (Color online) Tomoi lattices for the two incidences of Fig. 3