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Kamran Karimi (Ed.)
Proceedings of the Workshop on Causality and Causal
Discovery
In Conjunction with the Seventeenth Canadian Conference on
Artificial Intelligence (AI'2004) London, Ontario, Canada, 16 May
2004
Technical Report CS-2004-02 April 2004 Department of Computer
Science University of Regina Regina, Saskatchewan Canada S4S 0A2
ISSN 0828-3494 ISBN 0-7731-047-1
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ii
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iii
Preface
This volume contains papers selected for presentation at the
Workshop on Causality and Causal Discovery, in conjunction with the
Seventeenth Canadian Conference on Artificial Intelligence
(AI'2004), held in London, Ontario, Canada on 16 May 2004.
Causality and discovering causal relations are of interest because
they allow us to explain and control systems and phenomena. There
have been many debates on causality and whether it is possible to
discover causal relations automatically. Different approaches to
solving the problem of mining causality have been tried, such as
utilising conditional probability or temporal approaches.
Discussing, evaluating, and comparing these methods can add
perspective to the efforts of all the people involved in this
research area. The aim of this workshop is to bring researchers
from different backgrounds together to discuss the latest work
being done in this domain.
The occurrence of this workshop is the result of the joint
efforts of the authors, the programme committee members, and the
Canadian AI organisers. This volume would not have been possible
without the help of the members of the programme committee who
reviewed the papers attentively. The Canadian AI'2004 organisers,
General Chair, Kay Wiese (Simon Fraser University), Program
Co-Chairs Scott Goodwin and Ahmed Tawfik (both from the University
of Windsor), and Local Organiser Bob Mercer (University of Western
Ontario), supported the workshop from the beginning to the end.
Thanks to Weiming Shen for hosting the workshops at National
Research Council Canada (NRC) facilities and helping with the
co-ordination. The Department of Computer Science at the University
of Regina, and especially Howard Hamilton contributed their time
and resources towards the preparation of this volume. The efforts
of all the people not mentioned by name, who in any way helped in
making this workshop possible, are greatly appreciated.
April 2004 Kamran Karimi
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iv
Editor Kamran Karimi, University of Regina
Programme committee Cory Butz, University of Regina Eric
Neufeld, University of Saskatchewan Richard Scheines, Carnegie
Melon University Steven Sloman, Brown University
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v
Table of contents Jos Lehmann and Aldo Gangemi CAUSATIONT and
DOLCE …………………………………………………………………...…… 1 Bassem Sayrafi and Dirk Van
Gucht Inference Systems Derived from Additive Measures
…………………………………….……. 16 Kamran Karimi and Howard J. Hamilton From
Temporal Rules to One Dimensional Rules ……………………………………………... 30
Denver Dash Empirical Investigation of Equilibration-Manipulation
Commutability in Causal Models…….. 45
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CAUSATIONT and DOLCE
Jos Lehmann and Aldo Gangemi
Laboratory for Applied OntologyInstitute of Cognitive Science
and Technology
Italian National Research Councilhttp://www.loa-cnr.it/
Abstract. This paper offers an overview of CausatiOnt, a
semi-formalontology conceived as a basis for (automatic) legal
reasoning about cau-sation in fact. Moreover, a preliminary
axiomatization in DOLCE upperontology is provided of part of
CausatiOnt. This axiomatization is a steptoward making CausatiOnt,
or at least part of it, more rigorous and to-ward enabling the
automatic discovery of causal relations in the modelof a legal
case.
1 Introduction
In the context of a research in Artificial Intelligence and Law
(AI&Law), ex-tensively reported in [1] and, more concisely, in
[2], the problems posed by theautomation of legal responsibility
attribution are thoroughly analyzed and (par-tially) reduced to the
problems posed by automatic reasoning about causation.Based on such
reduction, the main contribution delivered by this research is
ananalytical subsumption hierarchy - an ontology, in Artificial
Intelligence (AI)terms - which semi-formally represents the
knowledge (i.e. the concepts and theconceptual relations) used in
the legal domain as the basis for reasoning aboutcausation. We call
such ontology CausatiOnt1.This paper offers a description of a work
in progress, which aims at axiomatiz-ing CausatiOnt within DOLCE
upper ontology [3]. This merging is being triedbecause, despite a
preliminary specification in Protégé-2000, CausatiOnt is stilltoo
complex for use in automatic reasoning, as it comprises knowledge
whichis, logically speaking, rather ambiguous. DOLCE, on the
contrary, has a wellfounded first order characterization [4], which
may help in making CausatiOntmore rigorous and, therefore,
potentially useful for the automatic discovery ofcausal relations
in the model of a legal case. We proceed as follows: section 2
dis-cusses the causal relation typically employed in legal
reasoning, causation in fact;section 3 presents the theoretical
basis and the class hierarchy of CausatiOnt;section 4 introduces
the preliminary results of the axiomatization of CausatiOntin
DOLCE; section 5 draws a conclusion.
1 From CAUSATIon ONTology.
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2 From legal responsibility to causation in fact
Legal Theory provides various arguments (see [2], section 1.1)
in favor of thefollowing legal theoretical position: reasoning
about the attribution of legal re-sponsibility to a person involved
in a case largely rests on causal reasoning. Froman AI&Law
perspective, this strongly suggests that the automation of legal
re-sponsibility attribution in one way or another requires the
automation of legalcausal reasoning. This may be achieved by
adopting, among other things, a suit-able ontology of causal
concepts, such as the one presented in sections 3 and 4of this
paper.Before presenting the ontology, we first spend some words on
the relation be-tween the notion of legal responsibility and the
underlying causal knowledge.This is meant to clarify the nature of
such knowledge and of the causal relationthat CausatiOnt is meant
to capture: causation in fact.Consider the following example, from
[5].
Example 1 (The Desert Traveler). A desert traveler T has two
enemies. Enemy1 poisons T’s canteen and Enemy 2, unaware of Enemy
1’s action, shoots andempties the canteen. A week later, T is found
dead and the two enemies confessto action and intention.
If a jury were asked to attribute the legal responsibility for
T’s death, it wouldprobably have to consider the following
additional information, which is leftimplicit in Example 1: T never
drank from the canteen, T was found dead bydehydration.Based on
such information, the jury would very probably come to an
unanimousdecision and indicate Enemy 2 as the responsible person
for T’s death. If askedwhy, the jury may answer: because Enemy 2
caused T’s death. If asked in whatsense Enemy 2 caused what he
caused, the jury would probably say that Enemy2’s action is a
counterfactual condition of T’s death, which makes it a cause.In
other words, had Enemy 2 not shot the canteen, T would still be
among us.But this is not true - it should be replied. Had Enemy 2
not shot the poisonedcanteen, T would have drunk from it and he
would not be among us anyway.Therefore, Enemy 2’s action is not a
counterfactual condition of T’s death. Is itstill its cause? - the
jury should be asked. Again its answer would probably beunanimous
and indicate Enemy 2’s action as the cause of T’s death in the
sensethat he is the most proximate cause of T’s death. If asked to
give a definitionof such proximity, the jurors would probably give
a temporal definition: Enemy2’s action is the latest cause of T’s
death. But, then again, it could be repliedthat from a strictly
physical point of view the heat of the Sun was definitely
atemporally more proximate cause than Enemy 2’s action.This “cat
and mouse game” with the jury could go on for a long time
becauseExample 1 is no real-life case. It is just a tricky and
underspecified combinationof circumstances devised by some smart
philosopher on some lazy day, with theexplicit purpose of fooling
imaginary juries. The example, though, does show thefollowing: a
“short circuit” in our causal understanding of a series of events
hasmajor consequences on our capacity to attribute (legal)
responsibility.
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[2] provides a legal theoretical bridge between the legal
concept of responsibilityand the causal notions that support its
attribution. Such bridge consists of fiveelements: first, the
distinction between causation in fact and legal causation; sec-ond,
the distinction between the ontological problems posed by causation
in factand the procedural problems posed by legal evidence and the
burden of proof;third, the definition of legal responsibility in
terms of liability and accountability;fourth, the definition of the
grounds for legal responsibility attribution, amongwhich causation
in fact; fifth, the definition of causation in fact. In the
followingwe briefly illustrate the first and the last of these
elements.The legal language makes a distinction between causation
in fact and legal cau-sation. On the one hand, the problem of
causation in fact is the problem ofunderstanding what actually
happened (i.e. what caused what) in a case. Suchfactual
interpretation is something legal experts usually take for granted
andmostly see as unproblematically achieved by common sense. In
Example 1 theconnection between the shooting of the canteen and T’s
death by dehydrationis an instance of causation in fact, because
Enemy 2 had the intention to killT, he believed that by shooting
the canteen T would die (rather than be savedfrom poisoning), he
shot the canteen, T died. On the contrary, the connectionbetween
the poisoning of the canteen and T’s death is not an instance of
causa-tion in fact, because T never drank from the canteen2. On the
other hand, legalcausation is the set of criteria that should be
applied either when a clear factualinterpretation of the case is
missing or when legal policy considerations shouldbe applied,
therefore adopting a causal interpretation that is different from
thefactual causal one. In Example 1, supposing that, after the
poisoning but beforethe shooting of the canteen, T had drunk from
it and supposing impossible toestablish the temporal priority
between the effects of poisoning and the effects ofdehydrating on
T’s body, the attribution of legal responsibility should be basedon
legal causation (for instance, by accepting that both Enemy 1’s and
Enemy2’s conducts legally caused T’s death).Now, how to give a
sufficiently general definition of causation in fact? There
arevarious traditional legal theoretical approaches to the problem
of giving this def-inition, most notably approaches based on the
notion of causal proximity or oncounterfactuals3. Traditional
approaches, though, suffer of a lack of an explicitaccount of the
elements of a case that a judicial authority should consider
whenassessing causation in fact. This jeopardizes consistency of
application of suchtests over large corpora of cases. In order to
overcome the common shortcomingof traditional approaches, Hart and
Honoré propose in [6] to base legal causalassessment on an
explicit definition of causation in fact, like the following
one.
Definition 1 (Causation in fact). Agent A causes an event e,
that mightinvolve agent B, if either of the following holds:
1. A starts some physical process that leads to e;2 Legally
speaking Enemy 1’s action may be considered just as an attempt at
murder-
ing T.3 Typical examples of counterfactual tests used in the
legal domain are the sine qua
non and the but for tests. For detailed overviews of these
approaches see [2] or [6].
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2. A provides reasons or draws attention to reasons which
influence the conductof B, who causes e;
3. A provides B with opportunities to cause e.4. All the
important negative variants of clauses 1, 2, 3
For what concerns Example 1 the causal connection between Enemy
2 shootingand T dying is non linear and may be considered either as
a case of the negativevariant of clause 1 above (Enemy 2’s conduct
prevents the physical process ofhydration which leads to T’s death
by dehydration) or as a case of clause 3 above(Enemy 2’s conduct
provides T with the opportunity of causing his own deathby
dehydration).In conclusion, Definition 1 carves a portion of causal
knowledge that is veryrelevant to AI&Law research.
3 An overview of CausatiOnt
In order to make Definition 1 more rigorous and possibly useful
to automaticclassification and/or interpretation, it should be
reconfigured along clear onto-logical lines and restructured by
means of a subsumption hierarchy, i.e. a socalled is-a hierarchy.
This is exactly the original purpose of CausatiOnt, the on-tology
presented in this section. It should be noticed that the
presentation ofCausatiOnt given here is rather theoretical. We only
occasionally exemplify theintuition behind each newly introduce
notion by referring to a subset of Example1 (namely: E1 = the
bullet is shot; E2 = the canteen is broken). But neitherin this
section nor in the following ones do we provide a complete model of
E1,E2 and of their causal connection, as this would require many
more pages thanavailable or a drastic cut in the theoretical
treatment of the introduced notions.
3.1 Philosophical preliminaries
The first and most obvious restructuring distinguishes in
Definition 1 four mainontological levels, corresponding to four
main types of causation, as usually de-scribed in the philosophical
literature: physical causation, agent causation, inter-personal
causation, negative causation4. Physical causation is described by
thefinal part of clause 1 of Definition 1, where the definition
mentions a physicalprocess that leads to an event. Agent causation
is described by the initial partof clause 1, where Definition 1
mentions an agent starting a physical process.The agreement around
cases of agent causation is not reached as easily as incases of
physical causation. This is due to the problem of detecting the
beliefs,
4 Distinguishing between varieties of causation is the pragmatic
answer of the phi-losophy of causation to the (temporary?) lack of
stable scientific theories of somefundamental phenomena. For
instance, without a stable neuropsychological solutionof the
mind-body problem, it is impossible to choose in a principled way
betweena reduction of agent causation to physical causation and a
reduction of physicalcausation to agent causation.
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desires and intentions of the agent that starts the physical
process. Things be-come even more complex when considering
interpersonal causation, described byclauses 2 and 3. One might be
tempted to consider interpersonal causation justas a subcase of
agent causation, where the psychological state of an agent exertsa
causal influence on another agent. Things are not that simple,
though. Thecausal influence that an agent may exercise on someone
else may be physical innature or psychological or a combination of
the two. Finally, the most elusivecase of causation is negative
causation. Definition 1 refers to negative causationin clause 4 as
to all the important negative variants of the preceding clauses. It
isontologically very difficult, almost paradoxical, to accept the
general idea thatsomething that does not exist can cause anything.
For reasons of space we cannot analyze the subtleties of this
fascinating problem here.In [2] definitions are given for physical
and agent causation within the widerstructure of CausatiOnt and
some analytical material is provided on interper-sonal and negative
causation, which are both left as research objectives. In thispaper
we limit the scope of the presentation of CausatiOnt to the
knowledgeneeded for defining physical causation (shown in figure
1). In other words, wepresent only the knowledge needed for
assessing causal relations between events,without considering
actions.Before starting with the detailed presentation of the class
hierarchy shown infigure 1, the following general philosophical
biases of CausatiOnt with respectto physical causation should be
highlighted:
Cognitivism CausatiOnt is based on the assumption that causal
relations areneither purely ontological nor purely epistemological.
Therefore, the repre-sentation of causal knowledge cannot be
limited to the ontological elementsof causal relations (i.e. the
entities). It must be extended to the epistemo-logical elements
(i.e. the categories) and to the phenomenological relationsbetween
them (i.e. the dimensions). This extension might seem as a
nonparsimonious scientific practice. But it gives us some room to
explain whatin causal reasoning pertains to us as observing
entities and what pertainsto the world as observed entity.
Furthermore, by not limiting ourselves toontology we provide a
clear way of distinguishing semantically similar terms(e.g.,
matter, a category; mass, a dimension; object, an entity). In a
similarfashion, we are able to adopt the distinction defined in [7]
between causality(a category, representing general causal
principles) and causation (a reifiedrelation, i.e. an entity,
representing particular causal relations). All this willfurther be
explained in section 3.2.
Singularism According to singularism, physical causation relates
events, i.e.particular changes of the world located in space and
time5 [8].
Functionalism Functionalism [9], [10], [11] may be seen as the
continuation ofsingularism by other means. The main difference from
singularism is thatfunctionalism seeks sharper tools than the
notion of change for detecting
5 Ducasse would for instance say that the cause of the
particular change E2 is E1 ifE1 alone occurred in the immediate
environment of E2 immediately before. This, ofcourse, begs the
question - what is the definition of ‘immediate environment’?
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physical causation. The various functionalist views proposed so
far try toreduce the notion of causation to physical notions, such
as energy or mo-mentum transfer between physical processes, in
accordance to contemporaryPhysics6.
Formalism According to CausatiOnt, like according to most
treatments ofcausal relations, physical causation has the formal
properties of transitiv-ity, asymmetry and non reflexivity.
Energy
Category of Existence
isa
Matter
isa
Change
isa
Space
isa
Process
Physical Entity
isa
Object
isa
Dimension
Noesis
isa
Category
isa
Entity
isa
Quality
Category of Experience
isa
Time
isa
Quantity
isa
isa isa
Causation
Physical Causation
isa
Occurrence
isa
Event
isa
isa isa
Causality
isa
Fig. 1. General hierarchy of CausatiOnt
3.2 CausatiOnt
We present here the class hierarchy shown, at different levels
of detail, in figures1 and 2. This hierarchy is an image of a
preliminary specification of Causa-tiOnt in Protégé-2000, a
fairly liberal knowledge representation tool, based onthe classical
is-a relation. Protégé-2000’s liberalism includes the possibility
ofdistinguishing among the following data types in an ontology.
Class, i.e. a set of(prototypical) individuals (so called
instances). A class has a name, that uniquelyidentifies it and,
possibly, a number of slots that intensionally describe it; it
isrelated by is-a relations to its subclasses and by i-o (instance
of) relations to itsinstances. Slot, i.e. a (user defined) binary
relation between the instances of aclass and the instances of
another class, or a literal (symbolic or numeric). System
6 For instance, a functionalist would consider a relation
between E1 and E2 as causal,if the actual physical intersection
between E1 and E2 involves exchange of a con-served quantity (e.g.
energy). Such exchange may be seen as a criterion for
furtherspecification of the ‘immediate environment’ used by
singularists
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class, i.e. a class that has classes as its instances (i.e. a
metaclass). The creationof system classes is usually used in order
to expand Protégé-2000’s knowledgemodel because classes and slots
are all instances of system class. Constraint, i.e.an assertion
that restricts the domain and the range of slots.Protégé-2000
variegated data types allow to represent knowledge that pertainsto,
at least, three logical orders (instances, classes, system
classes). Such spec-ifications may then be subject to further
specification in order to fully expressthem at the first order. In
the rest of this section we provide exactly the firstliberal
specification of CausatiOnt. For each introduced notion we provide
asynthetic natural language definition, some comments and the
indication of howthe notion is implemented in Protégé-2000. Next
section provides indications ofhow CausatiOnt has been imported
into DOLCE, in order to axiomatize it in asemantically well founded
model.
Definition 2 (Noesis). Noesis is the psychological counterpart
of experience(i.e. perception, learning and reasoning).
The notion of noesis has a rather long philosophical tradition,
which dates backto Greek Philosophy. As far as we are concerned, we
adopt here the notion ofnoesis in its broadest cognitive sense. We
consider all the experiences of an in-dividual human being to be
physical phenomena. On the one hand, perceptualexperiences (e.g.
perceiving the form of the canteen) are the result of the
inter-action between the physical world (i.e. light) and an
individual’s sensory system(e.g. his optic nerve and other parts of
his brain). On the other hand, intellec-tual experiences (e.g.
thinking about the notion of form) occur in the brain, i.e.they too
are physical phenomena. Besides their physical nature, though,
bothperceptual and intellectual experiences generally seem to have
a psychologicalcounterpart, i.e. a part of which the individual is
aware (i.e. the form of the can-teen, in the example of perceptual
experiences, and the notion of form, in theexample of intellectual
experiences). Any such psychological counterpart of anexperience is
noesis. Noesis is represented in Protégé-2000 as a standard
class,with no slots.
Definition 3 (Category). Category is knowledge-related (i.e.
epistemological)noesis.
A category is a kind of noesis, which cannot be
(philosophically) reduced to anyother kinds. It must therefore be
postulated. Categories form the intellectualbackground of our
noetic experience of the world (i.e. of our perception, learningand
reasoning about the world). Even though categories play a crucial
role innoesis, we are hardly aware of them in our experience. When
perceiving, learn-ing or reasoning we are not fully aware of the
categories that are supportingour effort. For instance, when
reasoning about (i.e. having an intellectual expe-rience of) or
perceiving (i.e. having a physical experience of) an entity (e.g.
anobject, say, the bullet or the canteen), a number of categories
(e.g. matter andquantity) make our experience possible, even though
they are not immediatelypresent to our mind and/or to our sensory
system. Categories are, therefore,here understood as in (Kantian)
Epistemology: as the basic notions on which
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our (intellectual and perceptual) experience builds up7. Our
intent is to use cat-egories as purely descriptive notions that
clarify the intuitive meaning of theterms that are used in
reasoning about entities (which we call the dimensions,see below).
As shown in figure 1 we distinguish between two main groups
ofcategories: the categories of existence and the categories of
experience. The op-position between these two types of categories
is the epistemological equivalentof the opposition, within noesis,
between entity (or Ontology) and category (orEpistemology). In
other words, just like in noesis, where we distinguish
existence(the entity) from knowledge (the category), in category we
distinguish betweenthe knowledge of what exists (category of
existence) from the knowledge of themodes of knowledge (category of
experience). These second categories describehow we know what
exists (or, rather, how we know the categories of
existence).Categories of existence encompass notions such as space,
matter, energy, change,causality; whereas category of experience
encompass notions such as quantity,quality and time8. Categories
are all represented in Protégé-2000 as subclassesof noesis, with
no slots.Two categories of existence that deserve some attention
here are change andcausality. On the one hand, we postulate change
as a separate category fromtime following the philosophical
position [13] according to which change must beassumed as distinct
from time in order for objects to keep their identity throughthe
occurrence of events (i.e. temporal individuals) that change them.
Further-more, following [7] we propose to distinguish causality
from causation and to seethe former as a kind of change. In other
words we propose to see causality as anur -element of our knowledge
of what exists: causality is a piece of our knowledgeof how what
exists can change. For instance, in Example 1 there is a
causalityrelation between, on the one hand, the shooting of the
bullet or the poisoning ofthe canteen (possible causes) and, on the
other hand, the death of the traveler(possible effect). But there
is a relation of causation only between the shootingof the bullet
(actual cause) and the death of the traveler (actual effect).
Wetherefore propose to see causality as the epistemological
counterpart of an on-tological dependence. In other words, the
build up of experience by means ofcausality requires the concurrent
presence of certain categories of existence. Forinstance, we
propose here to adopt the following ontological dependence
betweencategories of existence as the standard notion of causality:
energy cannot existwithout matter, matter cannot exist without
space.
Definition 4 (Dimension). Dimension is experience-related (i.e.
phenomeno-logical) noesis. A dimension relates two categories.
7 We want to avoid to use here the expression a priori for
describing the status ofcategories. As a matter of fact, under a
noetical perspective nothing is a priori andone may see categories
as the result of evolution, both of individuals and of species.
8 The main philosophical rationale behind having time as a
category of experience isthe idea that when we talk about time we
do not connote an entity or a naturaldimension that exists with
independence of what we are as (human) observers. Thefoundation of
the notion of time rests on the biology of the observer [12].
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The cognitive build up provided by the categories allows
dimensions to emerge.The standard example of a dimension is mass.
By experience, all physical ob-jects have a mass, which is the
quantity of matter they comprise. We never have,though, a concrete
experience of either matter or quantity as such. Therefore,we must
assume their existence as categories, rather than as entities, and
em-ploy them in the definition of the notion of mass. In other
words, the concretenotion of mass relates the epistemological to
the ontological part of our noeticexperience. We experience objects
(ontology) as having mass (phenomenology),which relates two
categories: matter and quantity (epistemology). In the defini-tions
of dimensions, we associate categories to one another with the
expression‘experienced by means of’. This is to underline the fact
that the definition ofdimensions in terms of categories is not an
ontological but a phenomenologicaldefinition. We therefore say, for
instance, that mass is matter experienced bymeans of quantity
(rather than mass is a quantity of matter), where the experi-ence
of matter by means of quantity is a purely intellectual one, as
both matterand quantity are categories, not entities. Furthermore,
it should be noticed thatwe use the expression ‘experienced by
means of’ also in the definition of entitiesin terms of dimensions.
In this case, the expression ‘experienced by means of’refers to the
perceptual (rather than the intellectual) experience of an entity
(e.g.an object) through a dimension (e.g. mass).The following
dimensions have been defined: volume (i.e., space experienced
bymeans of quantity), form (i.e. space experienced by means of
quality), location(i.e., space experienced by means of time); mass
(i.e., matter experienced bymeans of quantity), material (i.e.,
matter experienced by means of quality), state(i.e., matter
experienced by means of time); work (i.e., energy experienced
bymeans of quantity), energy-form (i.e., energy experienced by
means of quality),power (i.e., energy experienced by means of
time); direction (change experiencedby means of quantity),
transition (change experienced by means of quality), pe-riod
(change experienced by means of time).All dimensions are
represented in Protégé-2000 as instances of the class dimen-sion.
This, in turn, is a subclass both of noesis and of standard slot,
which isa type of system class. In other words, the instances of
the class dimension areparticular kinds of slots, which by
definition associate a category of existencewith a category of
experience.
Definition 5 (Entity). Entity is existence-related (i.e.
ontological) noesis.
The notion of entity indicates something that exists separately
from other thingsand has a clear identity. In Example 1 everything
is an entity. Entity is repre-sented in Protégé-2000 as a
subclass of noesis with no slots.
Definition 6 (Physical entity). Physical entity is an entity
experienced bymeans of one or more of the following dimensions:
volume, form, location, mass,material, state, work, energy-form,
power, direction, transition, period.
Physical entity is represented in Protégé-2000 as a subclass
of entity with noslots.
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Physical Causation
CausationCause Instance EventEffect Instance Event
isa
Entity
Physical Entity
isa
Occurrence
isa
ProcessTransition Instance Dimension
Direction StringEnergy-Form String
Power IntegerWork IntegerPeriod Integer
isa
ObjectLocation StringForm String
Material StringState Integer
Volume IntegerMass Integer
isa
EventSubject Instance Object
Occurence-of Instance Process
isa isa
Fig. 2. Entities in CausatiOnt
Definition 7 (Object). Object is a physical entity which is
experienced bymeans of all of the following dimensions: volume,
form, location, mass, ma-terial, state.
In Example 1 objects are the bullet and the canteen. Object is
represented inProtégé-2000 as a subclass of entity with slots
(its dimensions).
Definition 8 (Process). Process is a physical entity experienced
by means ofall of the following dimensions: work, energy-form,
power, direction, transition,period.
In Example 1 being shot and being broken are processes. Process
is representedin Protégé-2000 as a subclass of entity with slots
(its dimensions).
Definition 9 (Occurrence). Occurrence is a reified relation
between objects,processes and/or occurrences.
Occurrence is represented in Protégé-2000 as a subclass of
entity with no slots.
Definition 10 (Event). Event is an occurrence of a process (the
occurrenceof) which changes the value of a dimension of an object
(the subject).
In Example 1 an example of event is the trigger being
pulled.Finally, the notion of causation may be defined.
Definition 11 (Causation). Causation is an occurrence of two
events, thecause and the effect.
Definition 11 is the counterpart within CausatiOnt of definition
1. It is very broadand it is needed as a definitional node in the
ontology. In other words, all the
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clauses that provide the sufficient conditions for more
restrictive (and thereforemore interesting) causal relations are
provided in the definitions subsumed byDefinition 11. This does not
mean that the relation introduced in Definition 11is
indistinguishable from simple sequencing of events. Definition 11
introduces atype of occurrence. This has, of course, a rather
strong implication: by definitionall reified relations between
events are causal relations.
Definition 12 (Physical causation). Physical causation is
causation betweenan event E1, which is an occurrence of a physical
process P1 (the occurrence of)involving an object O1 (the subject),
and event E2, which is an occurrence of aphysical process P2 (the
occurrence of) involving an object O2 (the subject). Arelation of
physical causation holds between E1, the cause, and E2, the effect,
ifthe following conditions are met:
1. O1 and O2 are not the same object, according to the adopted
identity criterionfor objects.Comment: the subjects must be truly
distinguished objects.
2. P1 and P2 are not the same process, according to the adopted
identity crite-rion for processes.Comment: an event cannot cause
itself. By this clause we adopt the viewthat causation is a non
reflexive relation.
3. P1’s period precedes P2’s period.Comment: the cause
temporally precedes the effect. Even for processes thatare
temporally distributed (i.e. continuous) the causing process starts
be-fore the caused one. By this clause we adopt the view that
causation is atemporally asymmetric relation.
4. P1’s energy-form is the same as P2’s energy-form or E2 is
reducible to eventsE2,1. . . E2,n such that:(a) E2,1. . . E2,n are
occurrences of processes P2,1. . . P2,n, which all have the
same energy form of P1.(b) E2,1. . . E2,n have as their subjects
objects O2,1. . .O2,n, which are the
grains of O2, according to the adopted structural
constraints.Comment: in the interaction between two objects energy
is transferred ortransformed. In this latter case, the
transformation of energy should be re-ducible to a transfer of
energy between the cause and the events occurringto the structural
components of the object of the effect (its grains accordingto a
chosen granularity).
5. P1’s direction is the same as P2’s direction or P1’s power is
greater or equalto P2’s power or P1’s work is greater or equal to
P2’s work.Comment: this clause accounts for the fact that usually
changes of one signcause changes of the same sign (i.e. an increase
can usually only be caused byan increase and a decrease by a
decrease). If this condition cannot be tested(which might be the
case when lack of information makes it impossible toestablish the
directions of either P1 or P2) or if it is not satisfied, one
maywant to use the principle of the dispersion of energy in order
to distinguishthe cause from the effect.
-
6. The category of existence of P2’s transition can not exist
without the categoryof existence of P1’s transition, according to
the adopted causality constraint.Comment: changes in O1’s
dimensions can only affect those dimensions of O2that are
ontologically dependent on the dimensions changed in O1,
accordingto the adopted causality constraint between categories of
existence.
It should be added that we take physical causation to be a
transitive relation.Definition 12 is represented in Protégé-2000
as a subclass of causation with slots.The conditions listed in the
definition should be implemented as a series of con-straints.The
information given on E1 and E2 so far may be used by the reader for
anintuitive testing of clauses 1, 2, 3, 6 of definition 12. Clauses
4 and 5 are moredifficult to test, not only for what concerns the
information given here on E1 andE2, but in general for any two
couples of non repeatable events. In conclusion,the most important
characteristic of definition 12 is its use of a controlled
vo-cabulary, which defines terms that pertain to three distinct
philosophical levels:epistemology, phenomenology and ontology. Such
modularity makes it possibleto define causation by means of several
types of traditionally distinct criteriaemployed within the same
one definition: formalism (clauses 1, 2), singularismand
functionalism (clauses 3, 4, 5), cognitivism (clause 6).
4 Preliminary axiomatization of CausatiOnt in DOLCE
DOLCE (Descriptive Ontology for Linguistic and Cognitive
Engineering) is anontology of particulars, as shown in the top
class of Figure 3. DOLCE is basedon a fundamental distinction
between four types of entities: Endurants, Perdu-rants, Qualities
and Abstract entities. Endurants are wholly present (i.e., alltheir
proper parts are present) at any time they are present. Endurants
roughlycorrespond to objects in CausatiOnt. Perdurants, on the
other hand, just extendin time by accumulating different temporal
parts, so that, at any time they arepresent, they are only
partially present, in the sense that some of their propertemporal
parts (e.g., their previous or future phases) may be not present.
Perdu-rants roughly correspond to processes in CausatiOnt. DOLCE’s
third branch isQuality. Qualities can be seen as the basic entities
we can perceive or measure:shapes, colors, sizes, sounds, smells,
as well as weights, lengths, electrical charges,etc. Qualities may
be clustered in quality types. The term ‘quality’ is often usedas a
synonymous of ‘property’, but this is not the case in DOLCE:
qualities areparticulars, properties are universals. Qualities
inhere to entities: every entity(including qualities themselves)
comes with certain qualities, which exist as longas the entity
exists. DOLCE’s qualities are not comparable to CausatiOnt’s
di-mensions, because the latter are not entities. DOLCE
distinguishes between aquality (e.g., the capacity of the canteen
in Example 1), and its value (e.g., 1liter). Values are Abstracts,
called qualia in DOLCE, and describe the positionof an individual
quality within a certain conceptual space, called here
qualityspace. Such quality spaces are subsumed by the fourth branch
of DOLCE, i.e.abstract entities, and they are called Regions. So
when we say that two canteens
-
Particular
Perdurant
isa
Abstract
isa
Quality
isa
Endurant
isa
Event
isa
Stative
isa
Physical Object
Physical Endurant
isa
State
isa
Process
isa
Abstract Region
Region
isa
Physical Region
isa
isa
Physical Quality
isa isa
Fig. 3. General hierarchy of DOLCE
have (exactly) the same capacity, in DOLCE we mean that their
capacity quali-ties, which are distinct entities, have the same
position in the measure-for-fluidsspace, that is they have the same
capacity quale. This distinction between qual-ities and qualia is
inspired by the so-called trope theory. Its intuitive rationale
ismainly due to the fact that natural language - in certain
constructs - often seemsto make a similar distinction. Each quality
type has an associated quality spacewith a specific structure. For
example, lengths are usually associated to a metriclinear space,
and colors to a topological 2D space etc. For a full specification
andformal characterization of DOLCE refer to [4]9. Our first effort
in axiomatizingCausatiOnt in DOLCE10 has been directed at importing
CausatiOnt’s epistemo-logical and phenomenological branches into
DOLCE. As shown in figure 4 andin the following set of definitions,
categories are Abstract regions (definitions1-10). By (11) we have
defined CausantiOnt’s relation ExperiencedByMeansOfin terms of
DOLCE’s relation ExactLocation, which generically locates any
typeof particular in a region. In (12-13) we have hooked up
categories and DOLCE’squalities, by means of DOLCE’s relation
QLocation, which relates qualities toregions. In (14) we have
defined the ontological constraint for causality. Finallyin (15) we
give an example of how dimensions should be defined in DOLCE asa
relation between a particular and a region.
Categoryc(x) → AbstractRegion(x) (1)Categoryc(x) ≡
(2)CategoryOfExistencec(x) ∨∨CategoryOfExperiencec(x)
9 Available on
http://wonderweb.semanticweb.org/deliverables/D18.shtml10 In order
to avoid confusion with DOLCE’s original predicates, in the
following all
the predicates introduced in DOLCE from CausatiOnt are
distinguished by the su-perscript c.
-
Physical Endurant
Category
Category of Existence
isa
Category of Experience
isa
Abstract Region
isa
ExperiencedByMeansOf
Region
isa
Quality
InherentIn
MainCategory
QLocation
Fig. 4. Import of CausatiOnt into DOLCE
CategoryOfExistencec(spacec) (3)CategoryOfExistencec(matterc)
(4)CategoryOfExistencec(energyc) (5)CategoryOfExistencec(changec)
(6)CategoryOfExperiencec(quantityc)
(7)CategoryOfExperiencec(qualityc) (8)CategoryOfExperiencec(timec)
(9)ExactLocation(changec, causalityc) (10)ExperiencedByMeansOfc(x,
y) =def (11)CategoryOfExistencec(x) ∧ CategoryOfExperiencec(y)
∧∧ExactLocation(x, y)
HasCategoryc(x, y) =def (12)Quality(x) ∧ Categoryc(y)
∧QLocation(x, y)
MainCategoryc(x, y, z) =def (13)Quality(z) ∧HasCategoryc(z, x)
∧HasCategoryc(z, y) ∧∧ExperiencedByMeansOfc(x, y)
CausalityOrderc(x, y, z, w) =def (14)Quality(z) ∧Quality(w)
∧∧∃x∗MainCategoryc(x, x∗, z) ∧
-
∧∃y∗MainCategoryc(y, y∗, w) ∧∧(x = spacec → (y = spacec ∨ y =
matterc ∨ y = energyc)∧(x = matterc → (y = matterc ∨ y =
energyc)∧(x = energyc → (y = energyc))
V olumec(x, y) =def (15)PhysicalEndurant(x) ∧ ∃zInherentIn(z, x)
∧QLocation(z, y) ∧∧MainCategoryc(spacec, quantityc, z)
5 Conclusion
Based on axioms (1-15) further research efforts will be directed
at defining therelation of causation in DOLCE by means of a
representation paradigm calledDescriptions and Situations, which
extends DOLCE and is now under devel-opment. Once this definitional
phase is complete, an implementation of the re-sulting knowledge
structure will be attempted. All this is aimed at creating
theconceptual basis of a tool for automatic testing, relative to
Definition 1, of (legal)models of causation in fact.
References
[1] Lehmann, J.: Causation in Artificial Intelligence and Law -
A modelling approach.PhD thesis, University of Amsterdam - Faculty
of Law - Department of ComputerScience and Law (2003)
[2] Lehmann, J., Breuker, J., Brouwer, B.: Causation in
ai&law (to appear). AI andLaw (2004)
[3] Gangemi, A., Guarino, N., C., Oltramari, A., Schneider, L.:
Sweetening ontologieswith dolce. In: Proceedings of EKAW 2002:
166-181. (2002)
[4] Masolo, C., Borgo, S., Gangemi, A., Guarino, N., Oltramari,
A.: Wonderweb de-liverable d18 - final report. Technical report,
National Research Council - Instituteof Cognitive Science and
Technology (2003)
[5] Pearl, J.: Causality. Cambridge University Press (2000)[6]
Hart, H., Honore, T.: Causation in the Law. Oxford University Press
(1985)[7] Hulswit, M.: A semeiotic account of causation - The
cement of the Universe from
a Peircean perspective. PhD thesis, Katholieke Universiteit
Nijmegen (1998)[8] Ducasse, C.: On the nature and observability of
the causal relation. Journal of
Philosophy 23 57-68 (1926)[9] Russell, B.: Human Knowledge.
Simon and Schuster (1948)
[10] Salmon, W.: Scientific Explanation and the Causal Structure
of the World. Prince-ton: Princeton University Press (1984)
[11] Dowe, P.: Causality and conserved quantities: A reply to
salmon. Philosophy ofScience 62, 321-333 (1995)
[12] Maturana, H.: The nature of time.
http://www.inteco.cl/biology/nature.htm(1995)
[13] Lombard, L.: Event - A metaphysical study. Routledge and
Kegan Paul (1986)
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-
From Temporal Rules to One Dimensional Rules
Kamran Karimi and Howard J. Hamilton
Department of Computer Science University of Regina
Regina, Saskatchewan CANADA S4S 0A2
{karimi, hamilton}@cs.uregina.ca
Abstract. In this paper we propose a new algorithm, called
1DIMERS (One Dimensional Investigation Method for Enregistered
Record Sequences), to mine rules in any data of sequential nature,
temporal or spatial. We assume that each record in the sequence is
at the same temporal or spatial distance from others, and we do not
constrain the rules to follow any monotonic direction, meaning that
the rules can involve condition attributes in previous and next
records relative to the decision attribute. Removing the conceptual
temporal limitations makes 1DIMERS a generalised form of TIMERS
(Temporal Investigation Method for Enregistered Record Sequences).
TIMERS merges consequent records together, and then finds causal or
acausal relationships among the variables in the merged records.
The kind of rules discovered by TIMERS has also been called
sequential rules. In general the passage of time is limited to one
direction, which has been used in our previous work to distinguish
between causality and acausality. Since in principle it is possible
to move back and forth along a sequence, with general sequential
data we can no longer intuitively speak of causality and acausality
based on a direction. As a result, in 1DIMERS we substitute the
terms "causality" and "acausality" with "forward-predictive" and
"backward-predictive," respectively. 1DIMERS and TIMERS may each be
applicable to a different problem depending on the user's choice,
and we give examples of each program's applicability. In previous
work we have been using C4.5 as the classifier for creating
temporal rules. Here we employ CART as well, which has the ability
to regress as well as classify, and show that the results are
independent of the underlying rule discovery program.
1. Introduction
Given a sequence of records, the problem we are considering is
finding rules for predicting the value of a decision attribute that
appears in each record. The traditional approach is to look for a
relationship among the decision attribute and other attributes
within the same record. One example rule would be [If(a = 6) then
(decision = false)]. This method may not produce good results if
there is an inter-record relationship among the attributes. Bounded
by temporal constraints, in such a case we usually expect only the
previous records to affect the current decision attribute, but here
we investigate the possibility that the decision attribute's value
is determined by attributes not only in previous records, but in
next records, or both previous and next records. One example rule
in this context would be: [If{(acurrent-1 = 2) AND (bcurrent+1 =
2)} then (decisioncurrent = false)].
-
.
The attributes are now qualified with their position relative to
the position of the decision attribute. In this example,
"current-1" could be read as "previous," and "current+1" could be
read as "next."
Relying on data records that appear one after the other in a
sequence from a single source (so they are related), sets our
approach apart from methods such as [16] than do not consider any
ordering among the input records. In this paper we use the term
"one-dimensional" instead of "sequential" to avoid confusion with
common terminology. Though the data we deal with is sequential in
nature, it need not obey any temporal order. Also, the attributes
in each record in the sequence can represent any number of
dimensions, temporal or spatial. "One-dimensional" here refers to
the fact that there is a single temporal or spatial ordering among
the records. Also, a sequence implies an unbreakable order, while
in this paper we present rules that go back or forth (or both) in
the sequence to predict the value of a decision attribute. This
usage may not be suitable for real-time execution of temporal rules
(we can't give a verdict until sometime in the future). We expect
them to be of value in cases where predictive power is of more
importance, or we are processing stored temporal data where at any
given instant the future and past are available. Alternatively we
may be processing spatial data, where future and past are simply
substituted by notions of nearby locations (neighbourhood), and
considered available. Lifting the restriction of following an
inherent order in rules opens the door to new methods of analysing
data. Other than that, there are hints that in spite of the
intuitive appeal of finding patterns and rules in temporal
sequences of data such as time series in a fixed temporal
direction, in some cases the results may not be useful [6].
Sequential data and sequential rules have been studied before
[1, 4, 20]. For example, in [4] the authors provide a genetic
algorithm solution to the problem of detecting rules that manoeuvre
a plane that is being chased by a missile in a two dimensional
space. Discrete attributes such as speed, direction of the missile,
turning rate of the plane, etc. are measured during 20 time steps.
It is assumed that after 20 steps the missile will stop the chase.
The rules discovered in that paper form part of a plan, and the
genetic algorithm changes parts of the plan to make them better
suitable to solving the problem. Since the aim of the plans is to
prevent a hit, the system is developed to produce rules that come
up with evasive actions. The rules are then used in a simulator to
measure their effectiveness. Time is obviously the sequencing
factor in this example. In this paper we provide one example of
sequential data that resembles this application in the sense that
it consists of 15 measurements made after a failure is detected in
a robot. The observations are then used to classify failure types.
However in general we are interested in predicting the value of an
attribute that is included in each record. "Being hit," or
"failure" does not appear in any of the records, while an attribute
such as soil temperature can be measured at regular intervals along
with other related variables. Other examples in this paper address
the problem of predicting the value of such an attribute.
The remainder of the paper is organised as follows. Section 2
provides background for our work and also presents intuitive
examples of the concepts used in the paper. Section 3 formally
presents the 1DIMERS (One Dimensional Investigation Method for
Enregistered Record Sequences) algorithm, which is a generalised
version of TIMERS (Temporal Investigation Method for Enregistered
Record Sequences). In Section 4 we present the
-
.
results of experiments with TIMERS, showing its effectiveness in
solving temporal problems. The results of running TIMERS on a Robot
learning problem, involving fixed number of relevant records (w = n
= 15), are presented. It is a classification problem, with discrete
values for the decision attribute. Other experiments in Section 4
show that TIMERS is effective for solving regression problems,
where the decision attribute is continuous, and provide the results
of running CART on two datasets. C4.5's results are provided for
comparison purposes. 1DIMERS overlaps with TIMERS in its method,
and for comparison's sake its results are provided in each case
after trying TIMERS. Section 5 looks at another application domain
that closely resembles the temporal domain, and that is spatial
sequential data, and shows that the same techniques are effective
there. TIMERS and 1DIMERS' results are presented and compared.
Section 6 concludes the paper.
2. Background
Our previous work focuses on discovering temporal rules [7, 8]
that allow us to predict what happens next, given previous
observations. A temporal rule is a rule that involves time, i.e.
the condition attributes appear at different times than the
decision attribute. We divide temporal rules into two possible
categories, causal, and acausal. A causal relation is one that
involves attributes in the past affecting the decision attribute in
the future [19]. The past affecting the future is the normal
direction of time, and provides our definition of causality with an
intuitive sense. We also consider the case where the future
observations affect the past. We consider future affecting the past
as a sign of acausality [15], or temporal co-occurrence. In an
acausal relationship, the attributes just happen to be observed
together over time, while none is causing the other. In this case
there may be a hidden cause that has escaped our observation. Other
than causality and acausality, the third possibility is that the
relationship between the condition attributes and the decision
attribute is instantaneous, meaning that value of the decision
attribute is best determined by the condition attributes at the
same time.
We proposed the TIMERS method to detect a causal or acausal
relation among temporal sequences of data [11, 13, 14]. TIMERS
provides a set of tests and guidelines, for judging the nature of a
relationship. It is partly performed by software, and partly by the
domain expert who is analysing the data. Following this algorithm,
we generate classification rules from the data, using an operation
called flattening. Flattening merges consecutive records in the
normal, forward, direction of time (for the causality test) or the
backward direction of time (for the acausality test). The number of
records merged is determined by a time window w, and represents our
guess as to how many records may be involved in an inter-record
relationship. The quality of the rules, determined by their
training or predictive accuracy, allows us to judge the data as
containing a causal or acausal relation. TIMERS performs three
tests: One without flattening, to test the instantaneous
hypothesis, and two others to determine the temporal
characteristics of the data. The order to consider goes from
instantaneous, to acausal, to causal. So if the results of an
instantaneous test is about the same or better than the other two,
then we declare the relationship among the decision and condition
attributes to be instantaneous. Otherwise if the results of the
acausality test is about the same, or better than the causality
test, then we
-
.
declare the relationship as acausal. Otherwise the relationship
is causal. This order implies that when dealing with temporal
relationships, the tendency is to declare it as acausal. More
explanation is provided in [13], where an algorithm for flattening
data in both forward and backward directions is provided.
In a sequential spatial dataset it is reasonable to assume that
there may be connections between records at the neighbouring
positions, before and after the current record. In this paper we
introduce the sliding position flattening method which includes
forward and backward flattening as special cases. The principle
behind the sliding position method is that both previous and next
records can be influential in determining the current value of the
decision attribute. In a temporal domain this means considering
both past and future observations. With any fixed window size w,
the new flattening algorithm first places the current decision
attribute at position one, and uses the next w-1 records to predict
its value. This corresponds to a backward flattening in TIMERS,
where future values are used to predict the past. Then the current
attribute is set at position 2, and the previous record (position
one) and the next w-2 records are used for prediction. This case
has no correspondence in our previous algorithm in [13]. This
movement of the current position continues and at the end it is set
to w, and the previous w-1 records are used for prediction. This
corresponds to forward flattening in TIMERS.
As an example consider four temporally consecutive records, each
with four fields: , , , . Suppose we are interested in predicting
the value of the last (Boolean) variable. Using a window of size 3,
we can merge them as in Table 1. The decision attribute is
indicated in bold characters. When it comes to the record involving
the decision attribute, we do not consider any condition attributes
in the same record as the decision [13]. The Record.value notation
in Table 1 means that we are only including the decision attribute.
For example, would contain , where false is the decision attribute
in R3. This is to make sure that minimum amount of data is shared
between the original record and the flattened record.
Instantaneous. w = 1 (original data)
Forward (Causality). w = 3
Backward (Acausality). w = 3
Sliding position. w = 3
R1 = R2 = R3 = R4 =
Table 1. Results of flattening using the forward, backward, and
sliding position methods
Normal flattening (vs. sliding position flattening) with a
window size of w reduces the number of records by w-1. It is
possible that not all the data in a dataset follow each other
temporally, but only every n records. For example, every two
records were generated one after the other, but there is no
relationship between the first record and the third record, or any
other record. In this case we consider the window size w to be the
same as n, and
-
.
perform flattening so that every consecutive n records are
merged into one, and thus the number of flattened records is
divided by n. Sliding position flattening increases the number of
records.
TIMERS and 1DIMERS perform a series of pre-processing
operations, notably flattening, then provide the processed data to
another software to generate rules or trees and evaluate them. A
post-processing phase can then follow, in which the data are
presented to the user in a sequentially meaningful way. We have
tried C4.5 [17] before, and because of its availability of source
code, have been able to integrate it into our TimeSleuth software
[9, 12]. TimeSleuth performs all processing before and after
running C4.5 and hence partially implements both the TIMERS and
1DIMERS algorithms. Results of comparing TimeSleuth with other
causality miners appear in [14]. Being a classifier, to apply data
with continuous variables to C4.5, one has to perform
discretisation on the decision attribute, which is not always
reasonable with continuous data. In this paper we use another
package called CART [2] which can classify as well as perform
regression. We evaluate our method with CART as the underlying
rule-discoverer, and we show that it performs with little basic
variation with different rule-discovery programs.
To use CART we had to perform many of the pre- and
post-processing operations "by hand," i.e., using tools that were
not integrated into CART. We performed flattening with TimeSleuth,
and then deleted the current-time attributes using Mcrosoft Excel.
The results for CART were not presented in a temporally valid way
since CART, like most other data mining and machine learning
algorithms, does not consider any order among its input records. So
in the output a variable from the future could precede a variable
from the past, for example.
3. The 1DIMERS Algorithm
Modern physics has established time and space as a unity, where
one is inconceivable without the other. However, time remains an
anomaly because unlike the spatial dimensions, it seems that one
cannot move back in time, although experiments have shown that at
the particle level, this is in fact possible [5]. Discovering
temporal associations that predict the future, based on past
observations, is possible, and one can conceptually use the same
idea for one-dimensional space as well. Our previous work has used
the distinction between moving back and forth in time as the basis
of distinguishing causality on one side, and acausality (or
temporal co-occurrence) on the other. Since we do not consider an
explicit representation of time as necessary (time is implicitly
present in the order of the records), a measure such as length can
be substituted for time.
Consider the problem of drilling a well. The well can be
regarded as a one-dimensional entity. As the drill is making its
way through the ground, new points are explored and registered.
When we stop, we have a series of records that follow each other
along the line. While it seems that the data was produced in a
certain temporal order, one could argue that if the drilling were
started from the opposite side, then we would be encountering the
points from the reverse direction of time. It makes perfect sense
to analyse the drilling data in any direction of time, with the
results being valid in both cases.
-
.
Of course now it is not possible to talk about cause and effect
because what happens to precede something in one direction, will be
following it in the opposite direction.
1DIMERS is an evolution of TIMERS. It changes the terminology
from a temporal domain to a spatial domain and provides a more
general flattening method, as intuitively described in Section 2.
In 1DIMERS an instantaneous rule becomes a punctual rule (happens
on a point instead of at an instant). A causal rule becomes a
forward-predictive rule. An acausal rule becomes a
backward-predicitve rule. The intuitive distinction of causal vs.
acausal rules does not exist here. "Forward" and "backward" in
1DIMERS simply refer to the original direction of the data. The
lack of distinction between the two possible directions of movement
on a line side steps some conceptual problems and debates about
causality [3]. In 1DIMERS two new categories are added to
one-dimensi