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in this paper , we investigate the problem of mining numerical data .
this problem arises in many practical situations , e.g. analysis of gene and transcriptomic data in biology , soil characteristics and land occupation in agronomy , demographic data in economics , temperatures in climate analysis , etc .
we introduce an original framework for mining numerical data based on advances in itemset mining and in formal concept analysis ( fca ,  @xcite ) , respectively condensed representations of itemsets and pattern structures in fca  @xcite .
the mining of frequent itemsets in binary data , considering a set of objects and a set of associated attributes or items , is studied for a long time and usually involves the so - called `` pattern flooding '' problem @xcite .
a way of dealing with pattern flooding is to search for equivalence classes of itemsets , i.e. itemsets shared by the same set of objects ( or having the same image ) . for an equivalence class , there is one maximal itemset , which corresponds to a `` closed set '' , and possibly several minimal elements corresponding to `` generators '' ( or `` key itemsets '' ) . from these elements , families of association rules
can be extracted .
these itemsets are also related to fca , where a concept lattice is built from a binary context and where formal concepts are closed sets of objects and attributes .
the present work is rooted both in fca and pattern mining with the objective of extracting interval patterns from numerical data .
our approach is based on `` pattern structures '' where complex descriptions can be associated with objects . in  @xcite
we introduced closed interval patterns in the context of gene expression data mining .
intuitively , an interval pattern is a vector of intervals , each dimension corresponding to a range of values of a given attribute ; it is closed when composed of the smallest intervals characterizing a same set of objects .    in the present paper , we complete and extend this first attempt . considering numerical data , some general characteristics of equivalence classes remain , e.g. one maximal element which is a closed pattern and possibly several generators which are minimal patterns w.r.t . a subsumption relation defined on patterns .
we show that directly extracting patterns data from numerical is more efficient using pattern structures than working on binary data with associated scaling procedures .
we also provide a semantics to interval patterns in the euclidean space , design and experiment algorithms to extract frequent closed interval patterns and their generators .
the problem of mining patterns in numerical data is usually referred as quantitative itemset / association rule mining  @xcite .
generally , an appropriate discretization splits attribute ranges into intervals maximizing some interest functions , e.g. support , confidence . however
, none of these works covers the notion of equivalence classes , closed patterns , and generators , and this is one of the originality of the present paper .
the plan of the paper is as follows .
firstly , we introduce the problem of mining numerical data and interval patterns .
then , we recall the basics of fca and interordinal scaling .
we pose a number of questions that we propose to answer using our framework of interval pattern structures dealing with numerical data .
we then detail two original algorithms for extracting closed interval patterns and their generators .
these algorithms are evaluated in the last section on real - world data .
finally , we end the paper in discussing related work and giving perspectives to the present research work . as a complement , an extended version of this paper
is given in  @xcite , providing algorithms pseudo - code and a longer discussion on the usefulness of interval patterns in classification problems and privacy preserving data - mining .
we propose a definition of interval patterns for numerical data .
intuitively , each object of a numerical dataset corresponds to a vector of numbers , where each dimension stands for an attribute .
accordingly , an interval pattern is a vector of intervals , and each dimension describes the range of a numerical attribute .
we only consider finite intervals and that the set of attributes / dimensions is assumed to bed ( canonically ) ordered .
a numerical dataset is given by a set of objects @xmath0 , a set of numerical attributes @xmath1 , where the range of @xmath2 is a finite set noted @xmath3 .
@xmath4 means that @xmath5 is the value of attribute @xmath6 for object @xmath7 .    .a
numerical dataset .
[ cols="<,<,<,<",options="header " , ]     in both cases , using binary data is better when the minimal support is high ( e.g. 90% ) . for low supports , a critical issue , our algorithms deliver better execution times .
most importantly , the global redundancy effect discards the use of binary data , e.g. only @xmath8 of all fisg are actually fipg in dataset bl .
finally , the algorithm _ minintchangeg - t _ outperforms _ minintchangeg - h_. _ minintchangeg - t _ however needs more memory since it stores each closed set of objects as a word in the trie , and to each word the list of associated fipg .
it is very interesting to analyse the compression ability of closed interval patterns and generators .
for that , we compare in each dataset the number of those patterns w.r.t . to all possible interval patterns .
it gives the ratio of closed ( generators ) in the whole search space . in both cases ,
ratio varies between @xmath9 and @xmath10 .
this means that the volume of useful interval patterns , either closed or generators , is very low w.r.t .
the set of all possible interval patterns , justifying our interest in equivalence classes for interval patterns .
we discussed the important problem of pattern discovery in numerical data with a new and original formalization of interval patterns .
the classical fca / itemset - mining settings are adapted accordingly : from a closure operator naturally rise the notions of equivalence classes , closed and generator patterns , and we designed corresponding algorithms .
an appropriate semantics of interval patterns shows from a theoretical ( redundancy ) and practical ( computation times ) points of view that mining equivalent binary data ( encoding all possible intervals ) is not acceptable .
this is due to the fact that interval patterns are provided with a stronger partial ordering than is - itemsets ( classical set inclusion ) , hence pattern structures yield significantly less generators w.r.t .
their semantics .    dealing with interval patterns has applications in computational geometry , machine learning and data - mining , e.g.  @xcite and references therein .
it is indeed highly related to the actual problem of ( maximal ) @xmath11-boxes which corresponds to interval patterns ( generators ) with support @xmath11 . when @xmath12 , it corresponds to largest empty subspaces of the data .
our contribution to this field is the characterization of smaller subsets ( closed and generators ) .    in data - mining ,
closed patterns and their generators are crucial for extracting valid and informative association rules  @xcite , while generators can be preferable to closed patterns following the minimum descriptions length principle for so - called itemset - based classifiers  @xcite .
how these notions can be shifted to interval patterns is an original perspective of research rising questions concerning missing values , fault - tolerant patterns , and interestingness measures that are critical issues even in classical itemset mining : although the compression ability of closed interval patterns and generators is spectacular , the number of patterns remains too high for large datasets .
however , bringing the problem of numerical pattern mining into well known settings in favor of these perspectives of research .
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j.  li , h.  li , l.  wong , j.  pei , and g.  dong .
minimum description length principle : generators are preferable to closed patterns . in _ innovative applications of artificial intelligence conference_.
aaai press , 2006 .
we investigate the problem of mining numerical data with formal concept analysis . 
 the usual way is to use a scaling procedure transforming numerical attributes into binary ones leading either to a loss of information or of efficiency , in particular w.r.t . 
 the volume of extracted patterns . by contrast 
 , we propose to directly work on numerical data in a more precise and efficient way . 
 for that , the notions of closed patterns , generators and equivalent classes are revisited in the numerical context . 
 moreover , two algorithms are proposed and tested in an evaluation involving real - world data , showing the quality of the present approach .