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Statistical evaluation of rough set dependency analysis


TL;DR: This paper proposes to enhance RSDA by two simple statistical procedures, both based on randomization techniques, to evaluate the validity of prediction based on the approximation quality of attributes of rough set dependency analysis.
Abstract: Rough set data analysis (RSDA) has recently become a frequently studied symbolic method in data mining. Among other things, it is being used for the extraction of rules from databases; it is, however, not clear from within the methods of rough set analysis, whether the extracted rules are valid.In this paper, we suggest to enhance RSDA by two simple statistical procedures, both based on randomization techniques, to evaluate the validity of prediction based on the approximation quality of attributes of rough set dependency analysis. The first procedure tests the casualness of a prediction to ensure that the prediction is not based on only a few (casual) observations. The second procedure tests the conditional casualness of an attribute within a prediction rule.The procedures are applied to three data sets, originally published in the context of rough set analysis. We argue that several claims of these analyses need to be modified because of lacking validity, and that other possibly significant results were overlooked.

Summary (3 min read)

1 Introduction

  • Rough set analysis, an emerging technology in artificial intelligence (Pawlak et al. (1995)), has been compared with statistical models, see for example Wong et al. (1986), Krusi´nska et al. (1992a) or Krusińska et al. (1992b).
  • The methods will be applied to three different data sets: •.
  • The first set was published in Pawlak et al. (1986) and Słowi´nski & Słowiński (1990).
  • It utilizes rough set analysis to describe patients after highly selective vagotomy (HSV) for duodenal ulcer.
  • The authors show how statistical methodswithin rough set analysis highlight some of their results in a different way.

2 Rough set data analysis

  • Aninformation systemS = 〈U, Ω, Vq, f〉q∈Ω consists of 1.
  • Of particular interest in rough set dependency theory are those setsQ which use the least number of attributes, and still haveQ → P .
  • The intersection of all reducts ofP is called thecore ofP .
  • For eachR ⊆ Ω letPR be the partition ofU induced byθR. Define γQ(P ) = ∑ X∈PP |XθQ | |U | .(2.2) γQ(P ) is the relative frequency of the number of correctlyQ–classified elements with respect to the partition induced byP .
  • The larger the difference, the more important one regards the contribution ofq.

3.1 Casual dependencies

  • In the sequel the authors consider the case that a ruleQ → P was givenbeforeperforming the data analysis, and not obtained by optimizing the quality of approximation.
  • The latter needs additional treatment and will be discussed briefly in Section 3.5.
  • U} which preserves the cardinality of the classes.
  • Standard randomization techniques – for example Manly (1991), Chapter 1 – can now be applied to estimate this probability.
  • To decide whether the given rule is casual under the statistical assumption, the authors have to consider all 720 possible rules{σ(p), σ(q)} → d and their approximation qualities.

3.2 How the randomization procedure works

  • The proposed randomization test procedure is one way to model errors in terms of a statistical approach.
  • Because their approach is aimed to test the casualness of a rule system – and assume for a moment that this assumption really holds –, the assumption of representativeness is a problem of any analysis in most real life data bases.
  • Any observation within the other six classes ofθQ was randomly assigned to one of the three classes ofθP .
  • The percentage of the three rules – which is the true value of the approximation qualityγ – is varied by γ 0.0 0.1 0.2 0.3 Figure 1 shows the problem of granularity: GivenN = 10 observations and a true value ofγ = 0.0, the expectation of̂γ is about0.32; the granularity overshoot vanishes at aboutN = 40.
  • The power curves of an effectγ > 0.0 show that the randomization test has a reasonable power – at least in the chosen situation.

3.3 Computational considerations

  • It is well known that randomization is a rather expensive procedure, and one might have objections against this technique because of its cost in real life applications.
  • Iff(N ) is the time complexity for performing the computation of γ, the time complexity of the simulation based randomization procedure is1000f(N ).
  • If randomization is too costly for a data set, RSDA itself will not be applicable in this case.
  • Some simple short cuts such as a check whether the entropy of theQ partition is nearlog2(N ) may avoid superfluous computation.
  • For their re-analysis of the published data sets below it was not necessary to speed up the computations.

3.4 Conditional casual attributes

  • In rough set analysis, the decline of the approximation quality when omitting one attribute is usually used to determine whether an attribute within a minimal determining set is of high value for the prediction.
  • This approach does not take into account that the decline of approximation quality may be due to chance.
  • Assume that an additional attributer is conceptualized in three different ways: • A fine grained measurer1 using 8 categories, • A medium grained descriptionr2 using 4 categories.
  • Therefore the authors cannot trust the rules derived from the description{q, r1} → p, because the attributer1 is exchangeable with any random generated attributes = σ(r1).
  • Whereas the statistical evaluation of the additional predictive power of the three chosen attribute differs, the analysis of the decline of the approximation quality tells us nothingabout these differences.

3.5 Cross validation of learned dependencies

  • If rough set analysis is used to learn the best subset ofΩ to determineP , a simple randomization procedure is not sufficient, because it does not reflect the optimization of the learning procedure.
  • Within the test subset the same procedure can be used to validate the chosen attributes.
  • If the test procedure does not show a significant result, there are too few rules which can be used to predict the decision attributes from the learned attributes.
  • Note, that these rules need not be the same as those in the learning subset!.
  • If the additional attribute is conditional casual, the hypothesis that the rules in both sets of objects are identical should be kept.

4.1 Duodenal ulcer data

  • All data used in this paper are obtainable fromftp://luce.psycho.uni-osnabrueck.de/.
  • Pawlak et al. (1986) obtained – using rough set analysis – that the attribute setR, consisting of 3: Duration of disease 4: Complication 5: Basic HCI concentration 6: Basic Vol. of gastric juice 9: Stimulated HCI concentration 10: Stimulated Vol. of gastric juice suffices to predict attribute 12 (“Visick grading”).
  • The attribute set discussed in Pawlak et al. (1986) was based on a reduct searching procedure.
  • In order to discuss the cross validation procedure, the authors split the data set into 2 subsets containing 61 cases each.
  • Furthermore, the result suggests a reduction of the number of attributes withinR, because all attributes are conditional casual.

4.2 Earthquake data

  • In Teghem & Benjelloun (1992), the authors search for premonitory factors for earthquakes by emphasizing gas geochemistry.
  • The partition attribute (attribute 16) was the seismic activity on 155 days measured on the Richter scale.
  • The other attributes were radon concentration measured at 8 different locations (attributes 1-8) and 7 measures of climatic factors (attributes 9-15).
  • A problem with the information system was that it has an empty core with respect to attribute 16, and that an evaluation of some reducts turned out to be difficult.
  • The statistical evaluation of some of the information systems proposed by Teghem & Benjelloun (1992) gives us additional insights (Tab. 6).

4.3 Rough set analysis of Fisher’s Iris Data

  • Teghem & Charlet (1992) use the famous Iris data first published by Fisher (1936) to show the applicability of rough set dependency analysis for problems normally treated by discriminant analysis.
  • The setU consists of 150 flowers characterized by five attributes namely, 1. Petal length, 2. Petal width, 3. Sepal length, 4. Sepal width, and Table 7 validates the argument that only the attribute set{3, 4} should be used to predict the partition attribute.

5 Conclusion

  • Gathering evidence in procedures of Artificial Intelligence should not be based upon casual observations.
  • The authors approach shows how – in principle – a system using the rough set dependency analysis will defend itself against randomness.
  • The reanalysis of three published data sets shows that there is an urgent need for such a technique: Parts of the claimed results using the first two data sets are invalidated, some promising dependencies are overlooked and, as the authors show using data of Study 1, their proposed cross–validation technique offers a new horizon for the interpretation.
  • Concerning Study 3, the conclusions of the authors are validated.

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Statistical Evaluation of Rough Set Dependency Analysis
Ivo Düntsch
1
School of Information and Software Engineering
University of Ulster
Newtownabbey, BT 37 0QB, N.Ireland
I.Duentsch@ulst.ac.uk
Günther Gediga
1
FB Psychologie / Methodenlehre
Universität Osnabck
49069 Osnabrück, Germany
gg@Luce.Psycho.Uni-Osnabrueck.DE
and
Institut für semantische Informationsverarbeitung
Universität Osnabck
December 12, 1996
1
Equal authorship implied

Summary
Rough set data analysis (RSDA) has recently become a frequently studied symbolic method in data
mining. Among other things, it is being used for the extraction of rules from databases; it is, however,
not clear from within the methods of rough set analysis, whether the extracted rules are valid.
In this paper, we suggest to enhance RSDA by two simple statistical procedures, both based on ran-
domization techniques, to evaluate the validity of prediction based on the approximation quality of
attributes of rough set dependency analysis. The first procedure tests the casualness of a prediction
to ensure that the prediction is not based on only a few (casual) observations. The second procedure
tests the conditional casualness of an attribute within a prediction rule.
The procedures are applied to three data sets, originally publishedin the context of rough set analysis.
We argue that several claims of these analyses need to be modified because of lacking validity, and
that other possibly significant results were overlooked.
Keywords: Rough sets, dependency analysis, statistical evaluation, validation, randomization test

1 Introduction
Rough set analysis, an emerging technology in artificial intelligence (Pawlak et al. (1995)), has been
compared with statistical models, see for example Wong et al. (1986), Krusi´nska et al. (1992a) or
Krusi´nska et al. (1992b). One area of application of rough set theory is the extraction of rules from
databases; these rules then are sometimes claimed tobe usefulfor future decisionmaking or prediction
of events. However, if such a rule is based on only a few observations, its usefulness for prediction is
arguable (see also Krusi´nska et al. (1992a), p 253 in this context).
The aim of this paper is to employ statistical methods which are compatible with the rough set phi-
losophy to evaluate the “prediction quality” of rough set dependency analysis. The methods will be
applied to three different data sets:
The rst set was publishedin Pawlak et al. (1986) and Słowi´nski & Słowi´nski (1990). It utilizes
rough set analysisto describe patientsafter highlyselectivevagotomy (HSV) for duodenalulcer.
The statistical validity of the conclusions will be discussed.
The second example is the discussion of earthquake data published by Teghem & Charlet
(1992). The main reason why we use this example is that it demonstrates the applicability of
our approach in the situation when the prediction success is perfect in terms of rough analysis.
The third example is used by Teghem & Benjelloun (1992) to compare statistical and rough set
methods. We show how statistical methods within rough set analysis highlight some of their
results in a different way.
2 Rough set data analysis
A major area of application of rough set theory is the study of dependencies among attributes of
information systems. An information system S = hU, ,V
q
,fi
q
consists of
1. A set U of objects,
2. A nite set of attributes,
3. For each q asetV
q
of attribute values,
4. An information function f : U × V
def
=
S
qQ
V
q
with f(x, q) V
q
for all x U, q .
We think of the descriptor f(x, q) as the value which object x takes at attribute q.
With each Q we associate an equivalence relation θ
Q
on U by
x y (θ
Q
)
def
⇐⇒ f(x, q)=f(y, q) for all q Q.
If x U ,thenθ
Q
x is the equivalence class of θ
Q
containing x.
1

Intuitively, x y (θ
Q
) if the objects x and y are indiscernible with respect to the values of their
attributes from Q. If X U,thenthe lower approximation of X by Q
X
θ
Q
=
[
{θ
Q
x : θ
Q
x X}
is the set of all correctly classified elements of X with respect toθ
Q
, i.e. with the information available
from the attributes given in Q.
Suppose that P, Q . We say that P is dependent on Q written as Q P if every class of θ
P
is a union of classes of θ
Q
. In other words, the classification of U induced by θ
P
can be expressed by
the classification induced by θ
Q
.
In order to simplify notation we shall in the sequel usually write Q p instead of Q →{p} and θ
p
instead of θ
{p}
.
Each dependency Q P leads to a set of rules as follows: Suppose that Q
def
= {q
0
,...,q
n
},and
P
def
= {p
0
,...,p
k
}. For each set {t
0
,...,t
n
} where t
i
V
q
i
there is a uniquely determined set
{s
0
,...,s
k
} with s
i
V
p
i
such that
(x U)[f (x, q
0
)=t
0
···∧f(x, q
n
)=t
n
) (f(x, p
0
)=s
0
···∧f(x, p
k
)=s
k
)].(2.1)
Of particular interest in rough set dependency theory are those sets Q which use the least number of
attributes, and still have Q P . A set with this property called a minimal determining set for P .In
other words, a set Q is minimal determining for P ,ifQ P ,andR 6→ P for all R
(
Q.
If such Q is a subset of P we call Q a reduct of P. It is not hard to see, that each P has a reduct,
though this need not be unique. The intersection of all reducts of P is called the core of P .UnlessP
has only one reduct, the core of P is not itself a reduct.
For each R let P
R
be the partition of U induced by θ
R
.Dene
γ
Q
(P )=
P
X∈P
P
|X
θ
Q
|
|U |
.(2.2)
γ
Q
(P ) is the relative frequency of the number of correctly Q–classified elements with respect to
the partition induced by P . It is usually interpreted in rough set analysis as a measurement of the
prediction success of a set of inference rules based on value combinations of Q and value combinations
of P of the form given in (2.1). The prediction success is perfect, if γ
Q
(P )=1; in this case, Q P .
Suppose that Q is a reduct of P ,sothatQ P ,andQ \{q}6P for any q Q. In rough
set theory, the impact of attribute q on the fact that Q P is usually measured by the drop of the
approximation function γ from 1 to γ
Q\{q}
(P ): The larger the difference, the more important one
regards the contribution of q. We shall show below that this interpretation needs to be taken with care
in some cases, and additional statistical evidence may be needed.
2

3 Casual rules and randomization analysis
3.1 Casual dependencies
In the sequel we consider the case that a rule Q P was given before performing the data analysis,
and not obtained by optimizing the quality of approximation. The latter needs additional treatment
andwillbediscussedbrieyinSection3.5.
Suppose that θ
Q
is the identity relation id
U
on U. Then, θ
Q
θ
P
for all P ,i.e.Q P for
all P . Furthermore, each class of θ
Q
consists of exactly one element, and therefore, any rule
Q P is based on exactly one observation. We call such a rule deterministic casual.
If a rule is not deterministic casual, it nevertheless may be based on a few observationsonly, and thus,
its prediction quality could be limited; such rules may be called casual. Therefore, the need arises for
a statistical procedure which tests the casualness of a rule based on mechanisms of rough set analysis.
Assume that theinformation system is the realization of a randomprocessin which the attribute values
of Q and P are realized independently of each other. If no additional information is present, it may be
assumed that the attribute value combinations within Q and P are fixed and the matching of the Q, P
combinations is drawn at random.
Let σ be a permutation of U ,andQ . We define a new information function f
σ(Q)
by
f
σ(Q)
(x, r)
def
=
f(σ(x),r)), if r Q,
f(x, r), otherwise,
and let γ
σ(Q)
(P ) be the approximation of the prediction of P by Q in the new information system.
Note that the structure of the equivalence relation θ
σ(Q)
determined by Q in the revised system is the
same as that of the original θ
Q
. In other words, there is a bijective mapping
τ : {θ
σ(Q)
x : x U}→{θ
Q
x : x U}
which preserves the cardinality of the classes. In particular, if θ
Q
is the identity on U ,soisθ
σ(Q)
.It
follows that for a rule Q p with θ
Q
= id
U
,wehaveγ
σ(Q)
(p)=1as well for all permutations σ of
U.
The distribution of the prediction success is given by the set
R
P,Q
def
= { γ
σ(Q)
(P ):σ a permutation of U }.
Let H be the null hypothesis;we have to estimate the position of the observed approximation quality
γ
obs
def
= γ
Q
(P ) in the set R
P,Q
, i.e. to estimate the probability p(γ
R
γ
obs
|H). Standard ran-
domization techniques for example Manly (1991), Chapter 1 can now be applied to estimate this
probability.
If p(γ
R
γ
obs
|H) is low conventionally in the upper 5% region –, the assumption of randomness
can be rejected, otherwise, if
p(γ
R
γ
obs
|H) > 0.05,
we call the rule (random) casual.
3

Citations
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TL;DR: The algorithm for feature selection is based on an application of a rough set method to the result of principal components analysis (PCA) used for feature projection and reduction.
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763 citations


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Journal ArticleDOI
Ivo Düntsch1, Günther GedigaInstitutions (1)
TL;DR: This work presents three model selection criteria, using information theoretic entropy in the spirit of the minimum description length principle, based on the principle of indifference combined with the maximum entropy principle, thus keeping external model assumptions to a minimum.
Abstract: The main statistics used in rough set data analysis, the approximation quality, is of limited value when there is a choice of competing models for predicting a decision variable. In keeping within the rough set philosophy of non-invasive data analysis, we present three model selection criteria, using information theoretic entropy in the spirit of the minimum description length principle. Our main procedure is based on the principle of indifference combined with the maximum entropy principle, thus keeping external model assumptions to a minimum. The applicability of the proposed method is demonstrated by a comparison of its error rates with results of C4.5, using 14 published data sets.

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Book ChapterDOI
Jarosław Stepaniuk1Institutions (1)
01 Dec 2000-
Abstract: The amount of electronic data available is growing very fast and this explosive growth in databases has generated a need for new techniques and tools that can intelligently and automatically extract implicit, previously unknown, hidden and potentially useful information and knowledge from these data These tools and techniques are the subject of the field of Knowledge Discovery in Databases In this Chapter we discuss selected rough set based solutions to two main knowledge discovery problems, namely the description problem and the classification (prediction) problem

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01 Jan 1993-
TL;DR: This article presents bootstrap methods for estimation, using simple arguments, with Minitab macros for implementing these methods, as well as some examples of how these methods could be used for estimation purposes.
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  • ...Teghem & Charlet (1992) use the famous Iris data first published by Fisher (1936) to show the applicability of rough set dependency analysis for problems normally treated by discriminant analysis....

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TL;DR: This approach seems to be of fundamental importance to artificial intelligence (AI) and cognitive sciences, especially in the areas of machine learning, knowledge acquisition, decision analysis, knowledge discovery from databases, expert systems, decision support systems, inductive reasoning, and pattern recognition.
Abstract: Rough set theory, introduced by Zdzislaw Pawlak in the early 1980s [11, 12], is a new mathematical tool to deal with vagueness and uncertainty. This approach seems to be of fundamental importance to artificial intelligence (AI) and cognitive sciences, especially in the areas of machine learning, knowledge acquisition, decision analysis, knowledge discovery from databases, expert systems, decision support systems, inductive reasoning, and pattern recognition.

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TL;DR: The writer really shows how the simple words can maximize how the impression of this book is uttered directly for the readers.
Abstract: Every word to utter from the writer involves the element of this life. The writer really shows how the simple words can maximize how the impression of this book is uttered directly for the readers. Even you have known about the content of randomization tests so much, you can easily do it for your better connection. In delivering the presence of the book concept, you can find out the boo site here.

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