FUZZY INTERVAL NUMBER (FIN) TECHNIQUES FOR
CROSS LANGUAGE INFORMATION RETRIEVAL
Catherine Marinagi, Theodoros Alevizos, Vassilis G. Kaburlasos
Department of Industrial Informatics, T.E.I. of Kavala, Greece
Christos Skourlas
Department of Informatics, T.E.I. of Athens, Greece
Keywords: Fuzzy Interval Number (FIN), Information Retrieval (IR), Cross Language Information Retrieval (CLIR).
Abstract: A new method to handle problems of Information Retrieval (IR) and related applications is proposed. The
method is based on Fuzzy Interval Numbers (FINs) introduced in fuzzy system applications. Definition,
interpretation and a computation algorithm of FINs are presented. The frame of use FINs in IR is given. An
experiment showing the anticipated importance of these techniques in Cross Language Information
Retrieval (CLIR) is presented.
1 INTRODUCTION
Oard (Oard, 1997) classifies free (full) text Cross
Language Information Retrieval (CLIR) approaches
to corpus-based and knowledge-based approaches.
Knowledge-based approaches encompass dictionary-
based and ontology (thesaurus)-based approaches
while corpus-based approaches encompass parallel,
comparable and monolingual corpora. Dictionary-
based systems translate query terms one by one
using all the possible senses of the term. The main
drawbacks of this procedure are:
a) the lack of fully updated Machine Readable
Dictionaries (MRDs),
b) the ambiguity of the translations of terms results
in a 50% loss of precision (Davis, 1996).
Since the machine translation of the query is less
accurate than that of a full text, experiments have
been conducted with collections having machine
translations of all the collection texts to all
languages of interest. Such systems are really multi-
monolingual systems. Parallel and comparable
corpora systems are different: the parallel (or
comparable) corpora are used to “train” the system
and after that no translations are used for retrieval.
One such system, perhaps the most successful, is
based on Latent Semantic Indexing (LSI) (Dumais,
1996), (Berry, 1995). The main problem with this
approach is that it is not easy to find training parallel
corpora related to any collection.
Fuzzy (set) techniques were proposed for
Information Retrieval (IR) applications many years
ago (Radecki, 1979), (Kraft, 1993), mainly for
modeling.
Fuzzy Interval Numbers (FINs) were introduced by
Kaburlasos (Kaburlazos, 2004), (Petridis, 2003) in
fuzzy system applications.
A FIN may be interpreted as a conventional fuzzy
set; additional interpretations for a FIN are possible
including a statistical interpretation.
The special interest in these objects and associated
techniques for IR stems from their anticipated
capability to serve CLIR without the use of
dictionaries and translations.
The basic features of the method presented here are:
1) Documents are represented as FINs; a FIN
resembles a probability distribution.
2) The FIN representation of documents is based on
the use of the collection term frequency as the term
identifier.
3) The use of FIN distance instead of a similarity
measure.
There are indications, part of which is presented in
section 4 below, that a parallel corpora system can
be build using FIN techniques.
The structure of the remainder of this paper is as
follows: In section 2 a brief introduction to FINs and
other relevant concepts is given. In section 3 the
249
Marinagi C., Alevizos T., G. Kaburlasos V. and Skourlas C. (2006).
FUZZY INTERVAL NUMBER (FIN) TECHNIQUES FOR CROSS LANGUAGE INFORMATION RETRIEVAL.
In Proceedings of the Eighth International Conference on Enterprise Information Systems - AIDSS, pages 249-256
DOI: 10.5220/0002467002490256
Copyright
c
SciTePress
“conceptual” transition from document vectors to
document FINs is presented. Section 4 presents the
special interest of FINs in handling CLIR problems.
Conclusions and current work on the subject are
presented in section 5.
2 THEORETICAL
BACKGROUND
A. Generalized Intervals
A generalized interval of height h (∈ (0,1]) is a
mapping μ given by:
If x
1
< x
2
(positive generalized interval) then
elseif x
1
> x
2
(negative generalized interval) then
elseif x
1
= x
2
(trivial generalized interval) then
In this paper we use the more compact notation
[x
1
, x
2
]
h
instead of the μ notation.
The interpretation of a generalized interval depends
on an application; for instance if a feature is present
it could be indicated by a positive generalized
interval.
The set of all positive generalized intervals of
height h is denoted by
h
+
M , the set of all negative
generalized intervals by
h
−
M , the set of all trivial
generalized intervals by
h
0
M and the set of all
generalized intervals by M
h
=
h
−
M ∪
h
0
M
∪
h
+
M .
Two functions, that are going to be used in the
sequel, are defined:
Function support maps a generalized interval to the
corresponding conventional interval; support
([x
1
, x
2
]
h
) = [x
1
, x
2
] for positive, support([x
1
, x
2
]
h
) =
[x
2
, x
1
] for negative and support([x
1
, x
1
]
h
) = {x
1
} for
trivial generalized intervals.
Function sign: M
h
→ { –1, 0, +1 } maps a positive
generalized interval to +1, a negative generalized
interval to –1 and a trivial generalized interval to 0.
Now, we try to define a metric distance and an
inclusion measure function in the set (lattice) M
h
.
A relation ≤ in a set S is called partial ordering
relation if and only if it is:
1) x
≤ x (reflexive)
2) x
≤ y and y
≤ x imply x= y (antisymmetric)
3) x
≤ y and y
≤ z imply x
≤ z (transitive)
Therefore a partial order relation ≤ can be defined in
the set M
h
, h∈ (0,1]:
1) [a, b]
h
≤ [c, d]
h
⇔ support([a, b]
h
) ⊆
support ([c, d]
h
), for [a, b]
h
, [c, d]
h
∈
h
+
M
2) [a, b]
h
≤ [c, d]
h
⇔ support([c, d]
h
) ⊆
support([a, b]
h
), for [a, b]
h
, [c, d]
h
∈
h
−
M
3) [a, b]
h
≤ [c, d]
h
⇔ support([c, d]
h
) ∩
support([a, b]
h
) ≠ 0, for [a, b]
h
∈
h
−
M ,
[c, d]
h
∈
h
+
M
A partial ordering relation does not hold for all pairs
of generalized interval.
A lattice (L, ≤) is a partially ordered set and any two
elements have a unique greatest lower bound or
lattice meet (x ∧
L
y) and a unique least upper bound
or lattice join (x ∨
L
y).
A valuation v in a lattice L, defined as the area
“under” a generalized interval, is a real function
v:L → R which satisfies
v(x)+v(y)= v(x ∨
L
y) + v(x ∧
L
y), x,y ∈ L.
A valuation is called monotone if and only if x ≤ y
implies v(x) ≤ v(y) and positive if and only if x < y
implies v(x) < v(y) for x,y ∈ L.
A metric distance in a set S is a real function
d:S×S→ R which satisfies:
1) d(x,y) ≥ 0, x, y ∈ S
2) d(x,y) = 0 ⇔ x = y, x∈ S
3) d(x,y) = d(y,x) , x, y ∈ S (symmetry)
4) d(x,y) ≤ d(x,z) + d(z,y), x, y, z ∈ S
(triangle inequility)
Therefore a metric distance d:L×L→ R can be
defined in the lattice M
h
, h∈ (0,1] given by
d(x, y) = v(x ∨
L
y) - v(x ∧
L
y), x,y ∈ L.
A lattice is called complete when each of its subsets
has a least upper bound and a greatest lower bound.
In a complete lattice the positive valuation function
v can be used to define an inclusion measure
function k: L×L→ [0, 1] given by
v(u)
k(x, u) =
v(x ∨
L
u)
The lattice M
h
is not complete.
⎩
⎨
⎧
≤≤
=
otherwise,0
,
)(
21
],[
21
xxxh
x
h
xx
μ
⎩
⎨
⎧
≥≥−
=
otherwise,0
,
)(
21
],[
21
xxxh
x
h
xx
μ
⎩
⎨
⎧
=−
=
otherwise,0
},,{
)(
1
],[
21
xxhh
x
h
xx
μ
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250
Therefore we must define in a different way an
inclusion measure to quantify the degree of inclusion
of a lattice element into another one.
Definition
An inclusion measure σ in a non-complete lattice L
is a map σ: L×L→ [0, 1] such that for u, w, x ∈ L:
1) σ(x, x) = 1
2) u < w ⇒ σ(w, u) < 1
3) u ≤ w ⇒ σ(x, u) ≤ σ(x, w) (consistency property)
We have interchangeable used the notations σ(x, u),
σ(x ≤ u) because both the notations indicate a degree
of inclusion of x in u.
Kaburlazos has proved the following proposition:
Let the underlying positive valuation function
f: R→ R be a strictly increasing real function in R.
Then the real function v: M
h
→ R is given by
v([a, b]
h
) = sign ([a, b]
h
) c(h)
a
∫
b
[f(x) – f(a)] dx
where v is a positive valuation function in M
h
,
c: (0,1] → R
+
is a positive real function for
normalization. A metric distance in M
h
is given by:
d
h
(x, y) = v(x ∨ y) – v(x ∧ y)
As an example consider f(x)=x, c(h)=h.
Then the distance is given by:
d
h
([a, b]
h
,
[c, d]
h
)= h (|a-c|+|b-d|). As another
example, for f(x)= x
3
, h=1 and c(1)=0.5 we compute
the distance (between the intervals [-1, 1]
1
,
[2, 4]
1
)
d
h
([-1, 1]
1
,
[2, 4]
1
)= f(([-1, 1]
1
∨
[2, 4]
1
)
- f(([-1, 1]
1
∧
[2, 4]
1
) = 32.5 + 3.5 = 36.
The essential role of a positive valuation function v:
L
→
R is known to be a mapping from a lattice L of
semantics to the mathematical field R of real
numbers for carrying out computations.
B. Fuzzy Interval Numbers: Definition and
Interpretation
A positive Fuzzy Interval Number (FIN) is a
continuous function F: (0,1] →
h
+
M such that
h
1
≤ h
2
⇒ support(F (h
1
)) ⊇ support(F( h
2
)),
where 0 < h
1
≤ h
2
<1.
0
0.25
0.5
0.75
1
0 100 200 300 400
a
b
O
cd
x
h
Figure 1: An 86 value FIN. In the support(F(0.25)) =
[62,318] there are about 75% of the values
.
The set of all positive FINs is denoted by F
+
.
Similarly, trivial and negative FINs are defined.
Given a population (a vector) x = [x
1
,x
2
,…,x
N
] of
real numbers (measurements), sorted in ascending
order, a FIN can be computed by applying the
CALFIN algorithm given in the Appendix. In Fig.1
a FIN, calculated from a population of 86 values, is
shown. Given a FIN, any “cut” at a given height h
(∈ (0,1]) defines a generalized interval, denoted by
F(h). In Fig.1, F(0.25) is the generalized interval
[a,b]
0.25
represented by acdb.
A consequence of the CALFIN algorithm is the
following: Let F(1) = {m
1
}; approximately N/2 of
the values of x are smaller than m
1
and N/2 are
greater than m
1
. Let F(0.5) = [p
1/2
,q
1/2
]
0.5
;
approximately N/4 of the values of x lie in [p
1/2
,m
1
]
and N/4 in [m
1
,q
1/2
]. In more general terms: for any
h ∈ (0,1] approximately 100(1 - h)% of the N values
of x are within support(F(h)).
C. FIN Metrics
Let m
h
: R → R
+
be a positive real function – a
mass function – for h ∈ (0,1] (could be independent
of h) and f
h
(x) =
∫
x
h
dttm
0
)(
.
Obviously, f
h
is strictly increasing. The real function
v
h
:
h
+
M → R, given by v
h
([a,b]
h
) = f
h
(b) – f
h
(a)
is a positive valuation function in the set of positive
generalized intervals of height h.
d
h
([a,b]
h
,[c,d]
h
) =
[f
h
(a∨c) – f
h
(a∧c)] + [f
h
(b∨d) – f
h
(b∧d)],
where a∧c= min{a,c} and a∨c= max{a,c}, is a
metric distance between the two generalized
intervals [a,b]
h
and [c,d]
h
. In Fig.2 an interpretation
of a, b, c, d in the case of two FINs is shown.
0
0.25
0.5
0.75
1
0 100 200 300 400 500
ctf(mod)
h
a’ c’ b’ d’
a c b d
Figure 2: Two FINs F
1
and F
2
(representing two
documents of the CACM test collection). The points a, b,
c, d used to define d
h
(F
1
(h),F
2
(h)) = d
h
([a,b]
h
,[c,d]
h
).
FUZZY INTERVAL NUMBER (FIN) TECHNIQUES FOR FUZZY INTERVAL NUMBER (FIN) TECHNIQUES FOR
251
Given two positive FINs F
1
and F
2
,
d(F
1
,F
2
)=
∫
1
0
21
))(),(( hdhFhFdc
h
where c is an user-defined positive constant, is a
metric distance (for a proof see (Kaburlazos, 2004))
3 USING FINS TO REPRESENT
DOCUMENTS
In the Vector Space Model (Salton, 1983) for
Information Retrieval, a text document is
represented by a vector in a space of many
dimensions, one for each different term in the
collection. In the simplest case, the components of
each vector are the frequencies of the corresponding
terms in the document:
Doc
k
= ( f
k1
, f
k2
, … f
kn
)
f
kj
stands for the frequency of occurrence of term t
j
in document Doc
k
. In Fig.3 one such vector is shown
as a histogram.
0
1
2
3
4
0 50 100 150
term
tf
Figure 3: A document vector as a histogram. Each value
on the term axis represents a term (stem), e.g. “51” stands
for “industri” and “104” for “research”
.
Let ctf
j
be the total frequency of occurrence of
term t
j
in the whole collection. Then ctf
j
is equal to
∑
k
kj
tf
The collection term frequencies are going to be used
as term identifiers. In order to ensure the uniqueness
of the identifiers a multiple of a small ε is added to
the ctfs when needed. In Fig.4 the new form of the
document vector histogram is shown.
Figure 4: The document vector histogram in (modified) ctf
abscissae. Each value on the ctf (mod) axis represents a
term (stem), e.g. “1.24775” stands for “industri” and
“2.09009” for “research”
.
The next step is to break the “high bars” to multiple
pieces of height 1, placed side by side, separated by
some ε′ < ε; this is shown in Fig.5. Now, the original
histogram has been transformed to a “density
graph”.
0
1
0123456
ctf(mod)
Figure 5: The “density graph” representation of the
document of Fig. 1.
The abscissae vector is exactly the “number
population” from which the document FIN (Fig.6) is
computed from by the CALFIN algorithm described
in the Appendix.
0
0.25
0.5
0.75
1
012345
ctf(mod)
h
Figure 6: The document FIN along with the median bars.
The numbers of terms on the left and right sides of the bar
with height 1 are equal
.
0
1
2
3
4
0123456
ctf(mod)
tf
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252
The FIN distance is used instead of the similarity
measure between documents: the smaller the
distance the more similar the documents. This
a) means that for each query a FIN must be
calculated and
b) imposes a serious requirement on the
queries: they must have many terms. As
can be seen in the CALFIN algorithm
(appendix), at least two terms are needed to
calculate a FIN. Moreover, experience
shows that many term documents (queries)
give “better” fins.
4 FINS FOR CLIR
A. The Hypothesis
Consider a document set {doc(l)
1
, doc(l)
2
, …,
doc(l)
n
}, all in language l. Suppose that for each
document doc(l)
k
there exist translations doc(j)
k
to
the languages j = 1, …, m. So we have a
multilingual document collection:
{ 1≤ j ≤ m, 1≤ k ≤ n , doc(j)
k
}
Assume that:
1. The stopword lists of all languages are
translations of each other (partially unrealistic).
2. All the translations are done “1 word to 1
word”, i.e. we have no phrasal translations of
words (highly unrealistic (Ballesteros, 1997)).
3. There is no different polysemy between any two
languages (highly unrealistic (Ballesteros,
1998)).
Under these assumptions: Consider a term in
language l, t(l)
j
. Let tf(l)
jk
be the term frequency in
doc(l)
k
and ctf(l)
j
the total frequency of the term in
the collection. The following equalities hold:
tf(1)
jk
= tf(2)
jk
= … = tf(m)
jk
ctf(1)
j
= ctf(2)
j
= … = ctf(m)
j
If docF(l)
k
is the FIN representing doc(l)
k
then the
distances of the translated document FINs will be
approximately 0:
A
1
: d(docF(l
1
)
k
,docF(l
2
)
k
) ≈ 0
The distances can be nullified exactly with the use of
a dictionary.
Moreover, let qryF(l) be the FIN of a query
submitted to a FIN-based IR System that manages
the collection. Then:
A
2
: d(qryF(l),docF(l)
k
) ≈ d(qryF(l),docF(j)
k
),
j ∈ 1..m
This means that Cross Language Information
Retrieval is achievable without the use of
dictionaries.
B. The Experiment
The experiment aims to test the aforementioned
statements A
1
and A
2
in the “real world”.
A small document collection comprising 3 short
documents in english and their greek translations
(the documents originate from EU databases) was
used. The documents were slightly modified in order
to improve their compliance with the hypotheses of
part A. The term content of the documents is shown
in Table 1. Apparently it was not easy to avoid
phrasal translations and terms with different
polysemy.
Table 1: The term content of the documents.
Total Number of Terms
Number of Distinct
Terms
english greek English greek
doc1 69 69 67 58
doc2 83 83 50 51
doc3 79 86 65 71
Total Nt
en
=231 Nt
gr
=238 151 154
The FINs of the documents were calculated taking
all terms without exceptions. The FINs of the
english documents are shown in Fig.7. The steep
ascent of the left side of the curves is due to the
inclusion of all the terms appearing only once in the
collection.
In Fig.8 the FINs of the greek and english
versions of document 3 are shown. They are almost
identical except for the right “tail” of the greek FIN.
This is a result of different polysemy.
0
0.25
0.5
0.75
1
01234567
ctf(mod)
h
doc1
doc2
doc3
Figure 7: The FINs of the english documents.
FUZZY INTERVAL NUMBER (FIN) TECHNIQUES FOR FUZZY INTERVAL NUMBER (FIN) TECHNIQUES FOR
253
0
0.25
0.5
0.75
1
01234567
ctf(mod)
eng
gre
h
Figure 8: The FINs of the english and greek versions of
document 3
.
For the distance calculations a bell-shaped mass
function is selected:
m
h
(t) =
2
2
2
)max(
⎟
⎠
⎞
⎜
⎝
⎛
−+
+
ctf
tA
h
β
α
The positive real numbers A, α, β, are parameters;
max(ctf) is the maximum value of all the collection
term frequencies.
That kind of mass function degrades the
contribution of to the FIN distance of the terms with
ctf = 1. The same degradation applies to the high ctf
terms. So, the contribution to the distance of the
“tail” to the right of the greek FIN in Fig.8 is
degraded but the same applies in general to all high
frequency terms even those that do not have high
document frequency.
The verification of A1 and A2 is translated as
follows:
A
1
: d(docF(e)
1
,docF(g)
1
), d(docF(e)
2
,docF(g)
2
) and
d(docF(e)
3
,docF(g)
3
) are significantly smaller that
other distances between FINs.
A2: Instead of query documents the FINs of the
documents of the collection are used. So instead of
d( qryF(l),docF(l)
k
) ≈ d(qryF(l),docF(j)
k
), j ∈ 1..m
the following is examined:
d(docF(l)
m
,docF(l)
n
) ≈ d(docF(l
1
)
m
,docF(l
2
)
n
), m,
n = 1, 2, 3, m ≠ n, l, l
1
, l
2
= e, g
In Table 2 the FIN distances of all pairs of
documents in the collection are shown. As expected
the smallest are the distances between the greek and
english versions of the same documents.
If d
1e1g
= d( doc(e)1F, doc(g)1F ) = 0.03515 then
d( doc(e)2F, doc(g)2F ) ≈ 1.2 d
1e1g
and
d( doc(e)2F, doc(g)2F ) ≈ 2.06 d
1e1g
Apart from these, the smallest distance is
d( doc(e)1F, doc(e)3F ) = 0.27195 ≈ 7.7 d
1e1g
. That
is: the largest distance between two versions of the
same document is about 3.8 times smaller than the
smallest distance between versions of different
documents. Moreover:
d(docF(l
1
)
m
,docF(l
2
)
n
) ≤ 1.15 d(docF(l
3
)
m
,docF(l
4
)
n
),
m, n = 1, 2, 3, m ≠ n, l
1
, l
2
, l
3
, l
4
= e, g
That is: the distance between any two versions of the
same document is at most 1.15 times larger than the
distance of any two other versions of the same
document.
In conclusion:
1) The distances of different versions (languages)
of the same document are considerably smaller
than others.
2) The distances of two different documents are
almost the same irrespective of the language of
the documents.
In the second phase of the experiment one more
english language document (doc(e)4) is inserted in
the collection but not its greek version. The number
of english terms is increased by ΔNt
en
= 58. Now the
distances are changed:
d(docF(l
1
)
m
,docF(l
2
)
n
) ≤ 1.32 d(docF(l
3
)
m
,docF(l
4
)
n
),
m, n = 1, 2, 3, m ≠ n, l
1
, l
2
, l
3
, l
4
= e, g
To rebalance the collection characteristics we
renormalize the greek FIN absissae multiplying
χ
j
by:
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
Δ
+
en
en
j
r
NT
NT
ctf
x
k
)max(
1
Table 1
FIN distances of the documents of the
collection
Doc(e)1 doc(e)2 doc(e)3 doc(g)1 doc(g)2 doc(g)3
doc(e)1 0.00000 0.34044 0.27195
0.03515
0.32658 0.29084
doc(e)2 0.34044 0.00000 0.60588 0.31314
0.07230
0.61259
doc(e)3 0.27195 0.60588 0.00000 0.29938 0.59363
0.04258
doc(g)1
0.03515
0.31314 0.29938 0.00000 0.30065 0.31147
doc(g)2 0.32658
0.07230
0.59363 0.30065 0.00000 0.60035
doc(g)3 0.29084 0.61259
0.04258
0.31147 0.60035 0.00000
Table 2: FIN distances of the documents of the collection.
ICEIS 2006 - ARTIFICIAL INTELLIGENCE AND DECISION SUPPORT SYSTEMS
254
After that:
d(docF(l
1
)
m
,docF(l
2
)
n
)
≤
d(docF(l
3
)
m
,docF(l
4
)
n
),
m, n = 1, 2, 3, m
≠
n, l
1
, l
2
, l
3
, l
4
= e, g
5 CONCLUSIONS AND FUTURE
WORK
FIN techniques seem promising for IR and related
applications; the prospect of CLIR without
dictionaries is very intriguing. Nevertheless there are
quite enough topics to be considered carefully.
These techniques, for monolingual and cross
language IR, work with long documents and queries
that can give “good” FINs. Unfortunately, this is not
the case with the queries submitted to Internet search
engines; these queries very often have just a couple
of terms (Kobayashi, 2000). The FIN techniques can
be more successful in problems of document
classification where documents with many terms –
and “better” FINs– must be handled.
The quality of a FIN does not depend on number
of terms only; it must be considered with the mass
function for the distance calculation. In the previous
paragraph a “soft” degradation of the contribution to
the distance of terms with ctf = 1, has been
attempted through the mass function. A better idea
would be probably to ignore completely these terms
in the FIN computation. In FIN construction, term
document frequency (df) must be taken into account
as well.
The bell-shaped mass function seems to be a
reasonable one but other ideas should be considered
in conjunction with FIN computation.
Last but not least the renormalization scheme: in
any multilingual collection the numbers of terms in
different languages are random and a solid and
flexible re-balancing scheme is needed, which is not
independent of the FIN construction method and the
distance calculation (mass function).
The optimal determination of the parameters A, α,
β and k
r
is part of the system training process using
parallel corpora.
At present, experiments are been conducted along
these lines mainly with two of the standard
monolingual collections, namely CACM and WSJ.
These collections are of interest because they have
relatively longer queries.
On the other hand, a trilingual – greek / english /
french – test collection is been built for CLIR
experimentation. The parallel corpora are created by
translation by hand (although there is some
mechanical help). Experiments are conducted and
records of performance are kept during various
stages of the parallel corpora creation.
ACKNOWLEDGMENTS
This work was co-funded by 75% from the E.U. and
25% from the Greek Government under the
framework of the Education and Initial Vocational
Training Program – Archimedes.
REFERENCES
Ballesteros, L. and W. B. Croft, “Phrasal Translation and
Query Expansion Techniques for Cross-Language
Information Retrieval” in the Proceedings of the 20th
International ACM SIGIR Conference on Research
and Development in Information Retrieval (SIGIR-
97), pp. 84-91, 1997.
Ballesteros, L. and W. B. Croft, “Resolving Ambiguity for
Crosslanguage Retrieval” in the Proceedings of the
21st International ACM SIGIR Conference on
Research and Development in Information Retrieval
(SIGIR-98), pp. 64-71, 1998.
Berry, M. and P. Young, “Using latent semantic indexing
for multi-language information retrieval” Computers
and the Humanities, vol. 29, no 6, pp. 413-429, 1995.
Davis, M., “New experiments in cross-language text
retrieval at NMSU’s Computing Research Lab” in D.
K. Harman, ed., The Fifth Text Retrieval Conference
(TREC-5), NIST, 1996.
Dumais, S. T., T.K. Landauer, M.L. Littman, “Automatic
cross-linguistic information retrieval using latent
semantic indexing” in G. Grefenstette, ed., Working
Notes of the Workshop on Cross-Linguistic
Information Retrieval. ACM SIGIR.
Kaburlasos, V.G., “Fuzzy Interval Numbers (FINs):
Lattice Theoretic Tools for Improving Prediction of
Sugar Production from Populations of Measurements,”
IEEE Trans. on Man, Machine and Cybernetics – Part
B, vol. 34, no 2, pp. 1017-1030, 2004.
Kobayashi, M. and K. Takeda, “Information Retrieval on
the Web,” ACM Computing Surveys, 32, 2 (2000), pp
144-173.
Kraft, D.H. and D.A. Buell, “Fuzzy Set and Generalized
Boolean Retrieval Systems” in Readings in Fuzzy Sets
for Intelligent Systems, D. Dubius, H.Prade, R.R.
Yager (eds) 1993.
Oard, D.W., “Alternative Approaches for Cross-Language
Text Retrieval” in Cross-Language Text and Speech
Retrieval, AAAI Technical Report SS-97-05.
Available at
http://www.clis.umd.edu/dlrg/filter/sss/papers/
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Petridis, V. and V.G. Kaburlasos, “FINkNN: A Fuzzy
Interval Number k-Near-est Neighbor Classifier for
prediction of sugar production from populations of
samples,” Journal of Machine Learning Research, vol.
4 (Apr), pp. 17-37, 2003 (can be downloaded from
http://www.jmlr.org).
Radecki,T, “Fuzzy Set Theoretical Approach to Document
Retrieval” in Information Processing and
Management, v.15, Pergammon-Press 1979.
Salton, G. and M.J. McGill, Introduction to Modern
Information Retrieval, McGraw-Hill, 1983.
APPENDIX – FIN COMPUTATION
Consider a vector of real numbers x = [x
1
,x
2
,…,x
N
]
such that x
1
≤ x
2
≤ … ≤ x
N
. A FIN can be
constructed according to the following algorithm
CALFIN where dim(x) denotes the dimension of
vector x, e.g. dim([2,-1])= 2, dim([-3,4,0,-1,7])= 5,
etc.
The median(x) of a vector x = [x
1
,x
2
,…,x
N
] is
defined to be a number such that half of the N
numbers x
1
,x
2
,…,x
N
are smaller than median(x) and
the other half are larger than median(x); for instance,
the median([x
1
,x
2
,x
3
]) with x
1
< x
2
< x
3
equals x
2
,
whereas the median([x
1
,x
2
,x
3
,x
4
]) with x
1
< x
2
< x
3
<
x
4
was computed here as median([x
1
,x
2
,x
3
,x
4
])=
(x
2
+ x
3
)/2.
Algorithm CALFIN
1. Let x be a vector of real numbers.
2. Order incrementally the numbers in vector
x.
3. Initially vector pts is empty.
4. function calfin(x) {
5. while (dim(x) ≠ 1)
6. medi:= median(x)
7. insert medi in vector pts
8. x_left:= elements in vector x less-than
number median(x)
9. x_right:= elements in vector x larger-than
number median(x)
10. calfin(x_left)
11. calfin(x_right)
12. endwhile
13. } //function calfin(x)
14. Sort vector pts incrementally.
15. Store in vector val, dim(pts)/2 numbers
from 0 up to 1 in steps of 2/dim(pts)
followed by another dim(pts)/2 numbers
from 1 down to 0 in steps of 2/dim(pts).
The above procedure is repeated recursively log
2
N
times, until “half vectors” are computed including a
single number; the latter numbers are, by definition,
median numbers. The computed median values are
stored (sorted) in vector pts whose entries constitute
the abscissae of a positive FIN’s membership
function; the corresponding ordinate values are
computed in vector val. Note that algorithm
CALFIN produces a positive FIN with a
membership function μ(x) such that μ(x)=1 for
exactly one number x.
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