=Paper=
{{Paper
|id=Vol-110/paper-9
|storemode=property
|title=Binary Factor Analysis with Help of Formal Concepts
|pdfUrl=https://ceur-ws.org/Vol-110/paper10.pdf
|volume=Vol-110
|dblpUrl=https://dblp.org/rec/conf/cla/KeprtS04
}}
==Binary Factor Analysis with Help of Formal Concepts==
https://ceur-ws.org/Vol-110/paper10.pdf
Binary
Binary Factor
Factor Analysis
Analysis
with
with Help of Formal Concepts
Help of Formal Concepts
Aleš Keprt, Václav Snášel
Aleš Keprt and Václav Snášel
Department of computer science, FEI, VŠB – Technical University Ostrava
Department
17. of Computer
listopadu 15, Science, FEI, VŠB – Technical
708 33 Ostrava-Poruba, Czech University
Republic Ostrava
17. listopadu 15, 708 33 Ostrava-Poruba,
Ales.Keprt@vsb.cz, Czech Republic
Vaclav.Snasel@vsb.cz
{ales.keprt, vaclav.snasel}@vsb.cz
Abstract. Binary factor analysis (BFA, also known as Boolean Factor
Analysis) is a nonhierarchical analysis of binary data, based on reduction
of binary space dimension. It allows us to find hidden relationships in
binary data, which can be used for data compression, data mining, or
intelligent data comparison for information retrieval. Unfortunately, we
can’t effectively use classical (i.e. non-binary) factor analysis methods
for binary data. In this article we show an approach based on utilizing
formal concept analysis to compute nonhierarchical BFA. Computation
of a concept lattice is a computationally expensive task too, still it helps
us to speed up the BFA computation.
Keywords: Binary factor analysis, boolean algebra, formal concepts
1 Introduction
Binary data are one of the basic stones of computers. In the ancient age of
computer science even non-binary information was forcibly transformed into and
stored in binary form, usually because of technical limitations. Today we can still
see binary data on a physical level, but not on a logical level. Instead, majority
of information is usually recorder and stored in its native or nearly native form,
which is non-binary in most cases. Sometimes, it goes even further - with the help
of fuzzy computing, even vague information can be processed successfully. From
one particular point of view, the vagueness can be understood as a counterpart
of the strictly binary data.
Data analysis and searching for important, but often hidden information isn’t
a new theme, it was already subject of statistics (statistology) long time before
informatics become an independent science. Although data analysis is in no way
new theme, it’s still very current one. From computer science point of view, it’s
important especially because of rapidly growing importance of Internet, or as a
consequence of general emphasis on economics and economical achievements.
In this point we realize, that with the current boom of new kinds of data
analysis and data mining, binary data is no more in the main focus. Although
people perceive the majority of real world quantities as non-binary, some in-
formation still has got binary nature. In poetic words: ”Binariness can occur.”
c V. Snášel, R. Bělohlávek (Eds.): CLA 2004, pp. 90–101, ISBN 80-248-0597-9.
VŠB – Technical University of Ostrava, Dept. of Computer Science, 2004.
Binary Factor Analysis with Help of Formal Concepts 91
We can use non-binary data analysis techniques for binary data as well, but
these techniques are usually based on linear algebra, approximation or finding
of global minima/maxima, and those don’t work well in the binary world. They
all fail because of the specific nature of binary data, which also requires specific
analytical methods.
Our goal is the research in the filed of binary factor analysis (BFA). BFA is
a nonlinear analysis of binary data, where neither classical linear algebra, nor
mathematical (functional) analysis can be used. In the past, it was repeatedly
experimentally proven that classical non-binary methods followed by the align-
ment of the results into binary or other dichotomous values give ill results. This
resulted in creation of some new methods, which use boolean (binary) algebra.
These binary methods were published in last 8 years, and are based on neural
networks, combinatorial searching, genetic algorithms, and transformation to the
problem of concept lattices. This paper focuses to the approach based on formal
concepts.
2 Binary factor analysis
2.1 Problem definition
To describe the problem of Binary Factor Analysis (BFA) we can paraphrase
BMDP’s documentation (Bio-Medical Data Processing, see [1]).
BFA is a factor analysis of dichotomous (binary) data. This kind of analysis
differs from the classical factor analysis (see [18]) of binary valued data, even
though the goal and the model are symbolically similar. In other words, both
classical and binary analysis use symbolically the same notation, but their senses
are different.
The goal of BFA is to express p variables (x1 , x2 , . . . , xp ) by m factors
(f1 , f2 , . . . , fm ), where m p (m is considerably smaller than p). The model
can be written as
X=F A
x1,1 . . . x1,p f1,1 . . . f1,m a1,1 . . . a1,p
.. . . .. .. . . . . .. .
. . . = . . .. .. . ..
xn,1 . . . xn,p fn,1 . . . fn,m am,1 . . . am,p
where is binary matrix multiplication. For n cases, data matrix X, factor
scores matrix F, and factor loadings matrix A. The elements of all matrices are
valued 0 or 1 (i.e. binary).
2.2 Difference to classical factor analysis
Binary factor analysis uses boolean algebra, so matrices of factor scores and
loadings are both binary. See the following example: The result is 2 in classical
algebra
92 Aleš Keprt, Václav Snášel
1
1
0 = 1 · 1 + 1 · 1 + 0 · 0 + 1 · 0 = 2
[1 1 0 1] ·
0
but it’s 1 when using boolean algebra.
1
1
0 = 1 · 1 ⊕ 1 · 1 ⊕ 0 · 0 ⊕ 1 · 0 = 1
[1 1 0 1] ·
0
Sign ⊕ marks disjunction (logical sum), and sign · mars conjunction (logical
conjunction). Note that since we focus to binary values, the logical conjunction
is actually identical to the classic product.
In classical factor analysis, the score for each case, for a particular factor, is a
linear combination of all variables: variables with large loadings all contribute to
the score. In boolean factor analysis, a case has a score of one if it has a positive
response for any of the variables dominant in the factor (i.e. those not having
zero loadings) and zero otherwise.
2.3 Success and discrepancy
Obviously, not every X can be expressed as F A. The success of BFA is mea-
sured by comparing the observed binary responses (X) with those estimated by
multiplying the loadings and the scores (X̂ = F A). We count both positive
and negative discrepancies. Positive discrepancy is when the observed value (in
X) is one and the analysis (in X̂) estimates it to be zero, and reversely negative
discrepancy is when the observed value is zero and the analysis estimates it to
be one. Total count of discrepancies d is a suitable measure of difference between
observed values xi,j and calculated values x̂i,j .
p
n X
X
d= |x̂i,j − xi,j |
i=1 j=1
2.4 Terminology notes
Let’s summarize the terminology we use. Data to be analyzed are in matrix X.
Its columns xj represent variables, whereas its rows xi represent cases. The
factor analysis comes out from the generic thesis saying that variables, we can
observe, are just the effect of the factors, which are the real origin. (You can
find more details in [18].) So we focus on factors. We also try to keep number of
factors as low as possible, so we can say ”reducing variables to factors”.
The result is the pair of matrices. Matrix of factor scores F expresses the
input data by factors instead of variables. Matrix of factor loadings A defines
the relation between variables and factors, i.e. each row ai defines one particular
factor.
Binary Factor Analysis with Help of Formal Concepts 93
2.5 An example
As a basic example (see [1]) we consider a serological problem1 , where p tests are
performed on the blood of each of n subjects (by adding p reagents). The outcome
is described as positive (a value of one is assigned for the test in data matrix),
or negative (zero is assigned). In medical terms, the scores can be interpreted as
antigens (for each subject), and the loading as antibodies (for each test reagent).
See [14] for more on these terms.
2.6 Application to text documents
BFA can be also used to analyse a collection of text documents. In that case the
data matrix X is built up of a collection of text documents D represented as p-
dimensional binary vectors di , i ∈ 1, 2, . . . , n. Columns of X represent particular
words. Particular cell xi,j equals to one when document i contains word j, and
zero otherwise. In other words, data matrix X is built in a very intuitive way.
It should be noted that some kind of smart (i.e. semantic) preprocessing
could be made in order to let the analysis make more sense. For example we
usually want to take world and worlds as the same word. Although the binary
factor analysis has no problems with finding this kind similarities itself, it is
computationally very expensive, so any kind of preprocessing which can decrease
the size of input data matrix X is very useful. We can also use WordNet, or
thesaurus to combine synonyms. For additional details see [5].
3 The goal of exact binary factor analysis
In classic factor analysis, we don’t even try to find 100% perfect solution, be-
cause it’s simply impossible. Fortunately, there are many techniques that give a
good suboptimal solution (see [18]). Unfortunately, these classic factor analysis
techniques are not directly applicable to our special binary conditions. While
classic techniques are based on the system of correlations and approximations,
these terms can be hardly used in binary world. Although it is possible to apply
classic (i.e. non-boolean non-binary) factor analysis to binary data, if we really
focus to BFA with restriction to boolean arithmetic, we must advance another
way.
You can find the basic BFA solver in BMDP – Bio-Medical Data Processing
software package (see [1]). Unfortunately, BMDP became a commercial product,
so the source code of this software package isn’t available to the public, and even
the BFA solver itself isn’t available anymore. Yet worse, there are suspicions
saying that BMDP’s utility is useless, as it actually just guesses the F and A
matrices, and then only explores the similar matrices, so it only finds local
minimum of the vector error function.
1
Serologic test is a blood test to detect the presence of antibodies against microor-
ganism. See serology entry in [14].
94 Aleš Keprt, Václav Snášel
One interesting suboptimal BFA method comes from Húsek, Frolov et al.
(see [16], [7], [6], [2], [8]). It is based on a Hopfield-like neural network, so it finds
a suboptimal solution. The main advantage of this method is that it can analyse
very large data sets, which can’t be simply processed by exact BFA methods.
Although the mentioned neural network based solver is promising, we actu-
ally didn’t have any one really exact method, which could be used to proof the
other (suboptimal) BFA solvers. So we started to work on it.
4 Blind search based solver
The very basic algorithm blindly searches among all possible combinations of F
and A. This is obviously 100% exact, but also extremely computational expen-
sive, which makes this kind of solver in its basic implementation simply unusable.
To be more exact, we can express the limits of blind search solver in units
of n, p and m. Since we need to express matrix X as the product of matrices
F A, which are n × m and m × p in size, we need to try on all combinations
of m · (n + p) bits. And this is very limiting, even when trying to find only
3 factors from 10 × 10 data set (m = 3, n = 10, p = 10), we end up with
computational complexity of 2m·(n+p) = 260 , which is quite behind the scope of
current computers.
Although the blind search algorithm can be optimized (see [8],[11]), it’s still
quite unusable in real world.
5 Concept lattices
Another method of solving BFA problem is based on concept lattices. This sec-
tion gives minimum necessary introduction to concept lattices, and especially
formal concepts, which are the key part of the algorithm.
Definition 1 (Formal context, objects, attributes).
Triple (X, Y, R), where X and Y are sets, and R is a binary relation R ⊆ X ×Y ,
is called formal context. Elements of X are called objects, and elements of
Y are called attributes. We say ”object A has attribute B”, just when A ⊆ X,
B ⊆ Y and (A, B) ∈ R. t
u
Definition 2 (Derivation operators).
For subsets A ⊆ X and B ⊆ Y , we define
A↑ = {b ∈ B | ∀a ∈ A : (a, b) ∈ R}
B ↓ = {a ∈ A | ∀b ∈ B : (a, b) ∈ R}
t
u
In other words, A↑ is the set of attributes common to all objects of A, and
similarly B ↓ is the set of all objects, which have all attributes of B.
Binary Factor Analysis with Help of Formal Concepts 95
Note: We just defined two operators ↑ and ↓ :
↑
: P (X) → P (Y )
↓
: P (Y ) → P (X)
where P (X) and P (Y ) are sets of all subsets of X and Y respectively.
Definition 3 (Formal concept).
Let (X, Y, R) be a formal context. Then pair (A, B), where A ⊆ X, B ⊆ Y ,
A↑ = B and B ↓ = A, is called formal concept of (X, Y, R).
Set A is called extent of (A, B), and set B is called intent of (A, B). t
u
Definition 4 (Concept ordering).
Let (A1 , B1 ) and (A2 , B2 ) be formal concepts. Then (A1 , B1 ) is called subconcept
of (A2 , B2 ), just when A1 ⊆ A2 (which is equivalent to B1 ⊇ B1 ). We write
(A1 , B1 ) ≤ (A2 , B2 ). We can also say that (A2 , B2 ) is superconcept of (A1 , B1 ).
t
u
In this article we just need to know the basics of concepts and their meaning.
For more detailed, descriptive, and well understandable introduction to Formal
Concept Analysis and Concept Lattices, see [3], [12] or [15].
6 BFA using formal concepts
If we want to speed up the simple blind-search algorithm, we can try to find
some factor candidates, instead of checking out all possible bit-combinations.
The technique which can help us significantly is Formal Concept Analysis (FCA,
see [12]). FCA is based on concept lattices, but we actually work with formal
concepts only, so the theory we need is quite simple.
6.1 The strategy
We can still use some good parts of the blind-search program (matrix optimiza-
tions, optimized bitwise operations using boolean algebra, etc.), but instead of
checking out all possible bit combinations, we work with concepts as the factor
candidates. In addition, we can adopt some strategy optimizations (as discussed
above) to concepts, so the final algorithm is quite fast; its strength actually relies
on the concept-building algorithm we use.
So the BFA algorithm is then as follows:
1. Compute all concepts of X. (We use a standalone program based on Lindig’s
algorithm.)
2. Import the list of concepts, and optimize it, so it correspond to our optimized
data matrix X. (This is simple. We just merge objects and attributes the same
way, as we merged duplicate rows and columns of X respectively.)
96 Aleš Keprt, Václav Snášel
3. Remove all concepts with too many one’s. (The number of one’s per factor
is one of our starting constraints.)
4. Use the remaining concepts as the factor candidates, and find the best m-
element subset (according to discrepancy formulae).
This way we can find the BFA solution quite fast, compared to the blind
search algorithm. Although the algorithm described here looks quite simple2 ,
there is a couple of things, we must be aware of.
6.2 More details
The most important FCA consequence is that 100% correct BFA solution can
always be found among all subsets of concepts. This is very important, because
it is the main guarantee of the correctness of the concept based BFA solver.
(This is Keprt’s theorem, firstly published in [11].).
Other important feature of FCA based concepts is that they never directly
generate any negative discrepancy. It is a direct consequence of FCA qualities,
and affects the semantic sense of the result. As we discussed above (and see also
[1]), negative discrepancy is a case when F A gives 1 when it should be 0. From
semantic point of view, this (the negative discrepancy) is commonly unwanted
phenomenon. In consequence, the fact that there’s no negative discrepancy in
the concepts, may have negative impact on the result, but the reality is usually
right opposite. (Compare this to the quick sort phenomenon.)
The absence of negative discrepancies coming from concepts applies to A
matrix only. It doesn’t apply to F matrix, we still can use any suitable values
for it. In consequence, we always start with concepts not generating negative
discrepancy, which are semantically better, and end up with best suitable factor
scores F, which give the lowest discrepancy. So it seems to be quite good feature.
6.3 Implementation issues
It’s clear that the data matrix X is usually quite large, and makes the finding
of the formal concepts the main issue. Currently we use the standalone CL
(concept lattice) builder. It is optimized for finding concept lattices, but that’s
not right what we need. In the future, we should consider adopting some kind
of CL building algorithm directly into BFA solver. This will save a lot of time
when working with large data sets, because we don’t need to know the concept
hierarchy.
We don’t even need to know all the formal concepts, because the starting
constraints limit the maximum number of one’s in a factor, which is directly
applicable to CL building.
2
Everything’s simple, when you know it.
Binary Factor Analysis with Help of Formal Concepts 97
7 Experiments - results and computation times
Here we present three experiments using typical data sets taken from other
papers discussing BFA (see [17], [16], [7], [6], [2], [8]). We focus not only to
results (i.e. discrepancy of the solutions), but also to the computation times.
BFA computation is always a time consuming task, so faster algorithms are
generally preferred.
7.1 Data sets p3 and p2
Data set p3 is a representative of the simplest data. It is a sparse matrix, 100×100
bits in size. This data matrix can be expressed with just 5 factors, what leads to
searching for two 500-bit matrices. Due to the sparseness of the p3 data set, the
computation complexity is quite low, and allows us to use simple blind search.
Data set p2 is the same in size, but this one is not a sparse data set. Its
theoretical complexity based on its size is the same as the complexity of p3, but
the real complexity makes it impossible to be solved with simple blind search.
In other words, our new algorithm based on formal concepts is the only way to
compute exact BFA on p2 data set.
Table 1. Computation times
data set factors one’s time (m:s) discrepancy notes
p3.txt 5 2–4 61:36 0 375 combinations
p3.txt 5 3 0:12 0 120 combinations
p3.txt 5 1–10 0:00 0 8/10 concepts
p2.txt 2 6 11:44 743 54264 combinations
p2.txt 5 1–10 0:07 0 80/111 concepts
p2.txt 5 6–8 0:00 0 30/111 concepts
The results are shown in table 1. Data set p3 is rather simple, its factor
loadings (particular rows of A) all contain just 3 one’s. The first row in the table
shows that it takes over 61 minutes to find these factors, when we search among
all combinations with 2, 3 or 4 one’s per factor. If we knew that there are just 3
one’s per factor, we could specify it as a constraint, and got the result in just 12
seconds (see table 1, row 2). Indeed we usually don’t know it in real situations.
Third row shows that when using formal concepts, we can find all factors in
just 0 seconds, even when we search all possible combinations with 1 to 10 one’s
per factor. You can see the concept lattice in figure 1, with factors expressively
circled.
Data set p2 is much more complex, because it is created from factors con-
taining 6 one’s each. In this case the blind-search algorithm was able to find just
2 factors. It took almost 12 minutes, and discrepancy was 743. In addition, the
two found factors are wrong, which is not a surprise according to the fact that
there are actually 5 factors, and factors can’t be searched individually. It was
98 Aleš Keprt, Václav Snášel
0a
0
100o
1a
77
77o
1a 3a 2a
36 66 84
36o 22o 42o
3a 3a 3a 3a
69 39 57 69
23o 13o 19o 23o
30a
0
0o
Fig. 1. Concept lattice of p3 data set.
not possible to find more factors using blind-search algorithm. Estimated times
for computing 3 to 5 factors with the same constraints (limiting number of one’s
per factor to 6) are shown in table 2. It shows that it would take up to 3.5×109
years to find all factors.
Table 2. Estimated computation times
data set factors one’s estimated time
p2.txt 3 6 440 days
p2.txt 4 6 65700 years
p2.txt 5 6 3.5×109 years
As you can see at the bottom of table 1, we can find all 5 factors of p2 easily
in just 7 seconds, searching among candidates containing 1 to 10 one’s. The time
can be reduced to 0 seconds once again, if we reduce searching to the range of
6 to 8 one’s per factor. You can see the concept lattice in figure 2, with factors
marked as well. As you can see, the factors are non-overlapping, i.e. they are not
connected to each other. Note that in general factors can arbitrarily overlap.
7.2 Berry’s data set
Berry’s data set is an example of real data. It is a set of text documents describing
occurrences of 18 words in 14 documents, i.e. X is 14 × 18 bits in size. We can’t
even try to find the factors with simple blind search, because it would take years
to find just a few of them. On the other side, we can find 10 or more factors
with help of formal concepts.
Binary Factor Analysis with Help of Formal Concepts 99
2a
200
100o
3a 3a 3a 3a 3a
258 264 258 282 279
86o 88o 86o 94o 93o
4a 4a 4a 4a 4a 4a 4a 4a 4a 4a
296 288 320 316 328 296 320 324 316 348
74o 72o 80o 79o 82o 74o 80o 81o 79o 87o
6a 5a 5a 5a 5a 5a 5a 5a 7a 5a 5a 5a 7a
372 340 300 325 335 330 365 340 378 335 375 365 469
62o 68o 60o 65o 67o 66o 73o 68o 54o 67o 75o 73o 67o
7a 7a 7a 7a 6a 8a 6a 6a 8a 6a 6a 8a 8a 8a 8a 6a 8a
336 350 392 385 330 496 324 318 320 366 354 384 424 376 440 366 424
48o 50o 56o 55o 55o 62o 54o 53o 40o 61o 59o 48o 53o 47o 55o 61o 53o
8a 8a 8a 8a 8a 8a 9a 7a 7a 9a 9a 9a 9a 7a 9a 9a 9a 9a
288 352 328 336 344 392 432 301 343 432 495 306 297 329 369 351 369 369
36o 44o 41o 42o 43o 49o 48o 43o 49o 48o 55o 34o 33o 47o 41o 39o 41o 41o
9a 9a 12a 9a 11a 9a 9a 10a 11a 10a 10a 8a 12a 12a 10a 10a 10a 10a 12a
279 270 408 261 253 315 333 340 396 410 230 296 336 420 410 290 270 270 408
31o 30o 34o 29o 23o 35o 37o 34o 36o 41o 23o 37o 28o 35o 41o 29o 27o 27o 34o
10a 13a 10a 10a 12a 13a 12a 12a 12a 13a 12a 13a 11a 13a 13a 11a 13a
190 260 250 230 204 351 192 264 288 182 324 273 187 273 273 187 260
19o 20o 25o 23o 17o 27o 16o 22o 24o 14o 27o 21o 17o 21o 21o 17o 20o
15a 13a 16a 13a 16a 16a 15a 15a 16a 14a
195 78 112 156 160 224 105 105 224 140
13o 6o 7o 12o 10o 14o 7o 7o 14o 10o
30a
0
0o
Fig. 2. Concept lattice of p2 data set.
This data set is interesting in that it needs 14 factors to be successfully
expressed as F A. You can see all factors in figure 3, with all 14 factors circled.
Fig. 3. Concept lattice of Berry’s data set.
100 Aleš Keprt, Václav Snášel
8 Conclusion
This paper presented an idea of exploiting theory of formal concept analysis to
do a nonhierarchical analysis of binary data, namely binary factor analysis. The
experiments revealed that this approach is a big step forward from the usual
simple blind search.
Still, we don’t understand this approach as a final step. It is quite fast on
small data sets, but the complexity quickly increases as data set becomes larger.
Formal concepts-based algorithm can be also used as a reference algorithm for
testing the promising neural network-based solver (see [16], [7], [6], [2], [8]). For
sure, the future work will more focus on the possibilities of exploiting formal
concepts and concept lattices for BFA.
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