This paper introduces Temporal Concept Analysis (TCA) as the theory of temporal phenomena described with tools of Formal Concept Analysis (FCA). The basic notion in TCA is that of a Conceptual Time System introduced by the author [42,45] such that state spaces and phase spaces can be defined as concept lattices which clearly represent the meaning of states with respect to the chosen time description. In this paper two closely related types of conceptual time systems are introduced, namely 'conceptual time systems with parts' and 'conceptual time systems with actual objects'. The latter yields a generalization of classical systems with many similar subsystems or particles. Some industrial and scientific applications of conceptual movies of subsystems (particles) are presented.
For the investigation of temporal phenomena many theories have been developed. A
nice overview over the history of the development of formal representations of 'time'
can be found in [
For any two of these theories the system descriptions are different, and Lin [
Clearly, a general definition of 'state' can be given only within a general system description – and that should cover the most simple and useful system descriptions namely protocols of statements or measurements taken at some 'real system'. While arbitrary statements about a 'real system' are formally represented by n-ary relations measurements and their values at a certain 'object' or at a certain 'point of time' are usually represented in data tables, hence by a ternary relation interpreted as 'the measurement taken at a certain point of time yields a certain value'.
The practical relevance of data tables for knowledge representation in general and the
theoretical importance of lattices introduced by Birkhoff [
Besides that 'data – law polarity' the representation of granularity plays an important
role. It seems to be obvious that the choice of a coarser granularity in the data yields
fewer states. Therefore, to define 'states' a granularity tool should be introduced in the
formal description of a system. In Conceptual Time Systems the granularity tool is
Conceptual Scaling Theory which yields flexible possibilities for the generation of
mental frames. These frames are constructed as concept lattices of formal contexts
(called scales) which are used to describe the contextual meaning of the values of
many-valued contexts ([
Conceptual Scaling Theory is closely related to other granularity describing theories,
as for example Fuzzy Theory (Zadeh [
Now, we shortly discuss the formal representation of 'time' in Conceptual Time
Systems. There are many possibilities to represent 'time' in many-valued contexts. A
simple and often used way is to collect all the data for a given 'time point' in a row of
a data table. But what is a 'time point'? Should we start with a set of 'time points' used
as formal objects of a many-valued context? Should we introduce a linear ordering on
this set? Should we introduce intervals of a linearly ordered set of time points? How
to represent the non-linear ordering of the intervals? Should we introduce time
measurements as many-valued attributes of a many-valued context? What is the role of
other measurements in space or other domains?
To give a short answer: we introduce a set G whose elements are called 'time
granules' or 'time objects'. They play the role of 'elementary pieces of time' or 'time
instants' like 'morning' and they are contextually described by two many-valued
contexts T and C on the same set G of time granules. The first many-valued context T,
called the 'time part' describes the time granules by time measurements, for example,
the time granule 'morning' begins at 9 and ends at 12 o'clock. The second
manyvalued context C, called the 'event part' describes all other measurements taken at the
'real system'. Each many-valued attribute in the time part and the event part has a
conceptual scale describing the granularity we wish to use for its values.
That simple and general framework of a Conceptual Time System together with
Conceptual Scaling Theory [
The following example serves for several purposes: At first we use it to demonstrate (for readers not familiar with Formal Concept Analysis) a formal context, its concept lattice, object concepts, attribute concepts and line diagrams. Then we use it as the derived context of a conceptual time system, for the introduction of 'conceptual time systems with parts', 'object construction', and 'conceptual time systems with actual objects'.
The following Table 1 is a cross-table of a formal context K = (G,M,I) with 4 objects
and 7 attributes. It is understood as a short protocol of some observations of stars.
Reading example: Observation 1 has the attributes 'Monday', 'morning', 'east' and
'luminous'. The attributes 'east' and 'morning' have the same attribute concept, namely
the concept (A,B) with the extent A = {1,3} and the intent B = {east, morning,
luminous}. The two object concepts of 1 and 3 are sub-concepts of (A,B). For an
introduction to FCA the reader is referred to [
The following Table 2 describes in some sense the 'same' situation as Table 1. Indeed, Table 1 is the derived context of the conceptual time system given by Table 2 and the scales in Table3. There are four 'time objects' or 'time granules', labeled 1,2,3,4, interpreted as 'pieces of time' during which some measurements took place. For each time object g there is a time description of g (in the second and third column) and an event description (in the fourth and fifth column). In the general definition of a conceptual time system such tables are described by two many-valued contexts on the same set G of 'time objects'. By definition a conceptual time system contains also for each of its many-valued attributes a scale. In this example the scales are given by the following Table 3 (using some abbreviations): Mo Tu
Mo ×
Tu × mo ev mo × ev × east west west × lum lum ×
To get a flavour of the rich possibilities of conceptual scaling the reader is referred to
Wolff [
At first we give a short description of Conceptual System Theory developed by the
author [
Let G be an arbitrary set and T := ((G, M, W, IT), (Sm | m ∈ M)) and C := ((G, E, V, I), (Se | e ∈ E )) scaled many-valued contexts (on the same object set G). Then the pair (T, C) is called a conceptual time system on G.
T is called the time part and C the event part of (T, C).
The elements of the set G are called time granules or time objects, interpreted as pieces of time like 'morning' which are described by time measurements, formally represented by the many-valued attributes of the many-valued context (G, M, W, IT). The scales Sm of the time measurements m ∈ M describe the granularity of this conceptual 'language' about the values of the time measurements. The many-valued context (G, E, V, I) of the event part C is interpreted as the measurement table of the events of that system. The scales Se of the events e ∈ E describe the chosen granularity for the values of the event measurements.
In a conceptual time system (T, C) we would like to say 'The system is at time granule g in state s(g)'. This is introduced in the following definition where the object concepts of the derived context KC are defined as the states. As to the time part we call the object concepts of the derived context KT the time states or time granule concepts.
Let (T, C) be a conceptual time system and KT and KC the derived contexts of T and C. For each time granule g we define the state s(g) of (T, C ) at time granule g by s(g) := γC(g) := the object concept of g in KC and the time state or time granule concept t(g) of (T, C ) at time granule g by t(g) := γT(g) := the object concept of g in K . T The set S(T, C):= {s(g) | g ∈ G } is called the state space of (T, C). We say that the system (T, C) is at time granule g in the state s ∈ S(T, C) iff s = s(g). This definition yields the 'partition meaning' of states, namely, that the set G of time granules is partitioned by the extents of the states, or, equivalently, that a system is at each time granule in exactly one state. Clearly, for any two time granules g and h, s(g) = s(h) iff for each event e the values e(g) and e(h) have the same scale attributes and therefore the same object concept in the scale Se.
For the following discussion we need the notion of a subsystem of a conceptual time system and an embedding of the concepts of a subsystem into the concept lattice of the given system.
Let C = ((G, E, V, I), (Se | e ∈ E )) be a scaled many-valued context; for an arbitrary
subset R ⊆ E let IR := {(g,e,v) ∈ I | e ∈ R}. Then
C(R) := ((G, R, V, IR), (Se | e ∈ R )) is called the R-part of C or a subsystem of C.
Analogously we define the 'R-part of a formal context (G, M, I)' (where R ⊆ M);
([
Let K := (G, M, I) be a formal context, and R ⊆ M, and K(R) the R-part of K. Then the mapping φR which maps each concept (A,B) of K(R) onto the concept φR((A,B)) := (A,A↑) of K with the same extent, but possibly a larger intent A↑ in K is called the part embedding from B(K(R)) into B(K).
The part embedding φR is a meet-preserving order-embedding ([
C
In classical Mathematical System Theory [
Let (T, C) be a conceptual time system on G and KT and KC the derived contexts of T and C.
The apposition KT|KC of the derived contexts is called the phase context of (T, C). The concept lattice B(KT|KC) is called the phase space of (T, C). For any 'general time granule' (A,B) ∈ B(KT) and any 'general state' (C,D) ∈ B(KC) we say, that the system (T, C) is during the general time granule (A,B) always in the general state (C,D) iff A ⊆ C (which is equivalent to φT(A,B) ≤ φC(C,D) in B(KT|KC), where φT and φC are the part embeddings of B(KT) and B(KC) into B(KT|KC)). And we say, that the system (T, C) is during the general time granule (A,B) sometimes [never] in the general state (C,D) iff A ∩ C ≠ ∅ [resp. A ∩ C = ∅].
In Table 1 the formal context K is the apposition of KT (with 4 time attributes) and KC (with 3 event attributes). For time granule 1 the pair (t(1), s(1)) is the 'first' phase of this system, where t(1) = ({1}, {Monday, morning}) and s(1) = ({1,3}, {east, luminous}). By the previous definition (T, C) is during time granule 1 always in the state s(1) since φT(t(1)) = ({1}, {Monday, morning, east, luminous}) ≤ ({1,3}, {morning, east, luminous}) = φC(s(1)) in B(K) = B(KT|KC).
If we interpret the observations in the previous example as being caused by a single star, then this star could be nicely represented by the attribute concept of 'luminous'. The extent of this concept describes the time when this star was 'discovered' by the observation of its 'permanent' attribute 'luminous'. This star looks like a 'planet' since 'it moves' in the state space.
If we interpret the same observations as being caused by two stars, namely the 'morning star' which 'always' (Monday and Tuesday) occurs in the morning in the east, and the 'evening star' which 'always' occurs in the evening in the west, then these two stars look like fixed stars. The 'morning star' can be nicely represented as the formal concept ({1,3}, {morning, east, luminous}). Clearly, we could explain these observations also by four stars which might be represented as the object concepts of the four time granules.
This example shows that Conceptual Time Systems might be useful to understand from a philosophical point of view the role of objects, its construction from temporal phenomena and therefore its intimate relationship to our notion of time. For further discussions I call this relationship the 'object – time duality'. Is there any meaningful connection to the 'particle – wave duality' or the complementarity of energy and time?
Clearly, since time granules are used as formal objects in conceptual time systems the introduction of 'usual' objects like 'morning star' leads to the problem of the formal representation of them. Let us assume that we have accepted two objects 'morning star' and 'evening star'; then we might introduce these two stars as subsystems of a system of some stars. The following definition describes such conceptual time systems with (some selected ) subsystems or parts.
In applications a conceptual time system with parts might describe not only its specified subsystems but also the relations among these subsystems, for example the flow of information among several participants of a computer network.
Let (T, C) be a conceptual time system, T = ((G, M, W, IT), (Sm | m ∈ M)), C = ((G, E, V, I), (Se | e ∈ E )) and P some index set. If for each p ∈ P Tp is the M – p part of T for some Mp ⊆ M and Cp is the Ep –part of C for some Ep ⊆ E then (T, C, ((Tp, Cp) | p ∈ P)) is called a conceptual time system with parts from P. observ.
day Monday Monday Tuesday Tuesday day time morning evening morning evening
morning star space brightness east luminous
/ / east luminous / /
evening star space brightness
/ / west luminous
/ / west luminous In Table 4 the time part of the 'morning star' is described by the columns 1,2,3 and the event part by the columns 1,4,5.
The corresponding scales are chosen as nominal scales for 'day' and 'day time' and as the following scales (scaling also the missing value "/") for the columns 4,5,6,7 :
The set P of parts is chosen as {morning star, evening star}, their time parts are chosen as T, their event parts are described by columns 4, 5 respectively 6, 7 (together with the corresponding scales).
In many applications objects are introduced as something which is relatively stable, but sometimes objects are changing quite rapidly with time. Then we usually combine such an object with a 'piece of time' as for example 'my father in his youth' or the 'morning star at Monday morning'. If we like to use the parts as new objects, then it is very natural to combine a part p and a time granule g to a pair (p,g). This is done in the following example. observation (ms,1) (ms,2) (ms,3) (ms,4) (es,1) (es,2) (es,3) (es,4)
day Monday Monday Tuesday Tuesday Monday Monday Tuesday Tuesday day time morning evening morning evening morning evening morning evening space east
/ east / / west
/ west brightness luminous
/ luminous / / luminous
/
luminous
In Table 6 the information of Table 4 (where the parts are graphically represented
using apposition of their tables) is now represented using subposition of their tables.
(For a definition of apposition and subposition the reader is referred to [
Let P and G be sets and Π ⊆ P×G , then a conceptual time system (T,C) on Π is called a conceptual time system with actual objects from P×G. P is called the set of objects or particles, G the set of time points and Π the set of actual objects (which is the set of time granules of (T,C)).
Clearly, each conceptual time system with actual objects can be transformed (in many ways) into a conceptual time system with parts. While in conceptual time systems with parts for each time granule g the intent of g (in the derived context) contains all the information given in all parts, 'this information' is distributed over the intents of all actual objects (p,g). The details of these constructions are not explained here.
Conceptual time systems with actual objects can be used to represent the development
of many objects in a common phase space or – omitting the time part – in a common
state space. Classical examples are a swarm of birds or a physical system of particles.
The author has applied this method in cooperation with a computer firm in the
production of wafers based on process data of a reactor; it was applied in a chemical
plant for the visualization of a chemical process in a distillation column [
In this application an air-conditioning plant in the pharmaceutical industry was investigated to control the behavior of the plant. Therefore several temperature time series were collected. The usual visual inspection of these discrete data represented in 'curves' finds some phases which occur nearly every day, for example the phase that during the afternoon the temperature outside and inside is relatively high. To represent such phases systematically we constructed several conceptual time systems with different scales based on the same many-valued context describing the original temperature measurements taken at each hour of the first six months (=182 days) of the year 2000 using 4368 = 182 x 24 hours as time objects. Each hour is described in the time part by three attributes 'month', 'day', and 'hour' where the values of 'hour' range from 0 to 23. Clearly, the temperature measurements form the event part.
To demonstrate the method we take from these data only the first three days and only two temperatures, namely the outside temperature and the temperature in one of the production rooms. For the 72 = 3 x 24 hours we use labels, for example '100' for the midnight hour of the first day, and '323' for the last hour of the third day. The line diagram in Figure 2 shows the phase space of a conceptual time system using an ordinal scale with five concepts for the attribute 'hour' and two ordinal scales each with three concepts for the temperatures inside and outside.
The process starts at midnight of the first day (in the phase of time granule 100) where the temperatures outside and inside are low. The system changes to another phase at time granule 108, but remains in the same state. After time granule 110 the system changes to a state with a medium inside temperature. The change to the phase of 112 shows that the transitions between two phases clearly need not follow the lines of the line diagram. Omitting the 'hour'-attributes we get the projection of this phase space onto the state space (described by the six points in the upper-left part) which can be embedded into a 3 x 3 –grid spanned by the two 3-chains for the temperatures. A typical phase space implication: hour:>=16 ⇒ Temp-outside:<=6.0.
In the following example a conceptual time system with actual objects is described.
We represent a therapeutic process of an anorectic young woman over a period of
about three years [
The actual state of Formal Concept Analysis and Conceptual System Theory seems to be a promising starting point for a new discipline in Conceptual Knowledge Processing namely Temporal Concept Analysis understood as the theory of temporal phenomena described with tools of Formal Concept Analysis. On the basis of a clear definition of states and phases as formal concepts a general description of temporal phenomena in all areas of science can be developed.
One of the next steps will be a conceptual transition theory such that transitions in
automata theory [
For the application of Temporal Concept Analysis programs have to be developed
which represent complex temporal systems in easily understandable visualizations.
One of the most promising possibilities are conceptual movies introduced by the
author [
I like to thank Prof. Walter Sedelow for his encouragement and his help during the development of this research area and for his suggestion to introduce the name 'Temporal Concept Analysis'.
Thanks also to the firm NAVICON GmbH Frankfurt for the permission for using their excellent programs on Conceptual Knowledge Processing in their NAVICON DECISION SUITE.