Bridging Qualitative, semi-Quantitative and Quantitative knowledge for modeling

Q3  (“Triple-Q”)

under PIKuliar Culture (ToPIK 7)[1]

 

Project leader:

Matthias K. B. Lüdeke, luedeke@pik-potsdam.de, 288 2578

Participants:

Klaus Eisenack, eisenack@pik-potsdam.de, 288 2625

Oliver Walkenhorst, walken@pik-potsdam.de, 288 2571

Further contributers:

Gerhard Petschel-Held (+ 9.9.2005)

Jürgen Kropp, kropp@pik-potsdam.de, 288 2526

Running time 1.1.2004 - 31.12.2006

 

Summary

 

In various realms of science quantitative mathematical modelling appeared as extremely fruitful explanatory and predictive tool (physics, meteorology, chemistry, etc). Some of the sciences relevant for Global Change research belong to this group, others, more qualitatively oriented sciences (political science, sociology, anthropology, etc), not. For PIK it is therefore a vital question how to include this important knowledge into integrated mathematical modelling activities. Q3 strives to develop methods to achieve this by a threefold strategy: improve existing concepts of qualitative modelling in close co-operation with applications in the qualitative sciences, clarify the role which formal deductive methods can play in these sciences and investigate, how the resulting qualitative mathematical models can best be coupled with well established quantitative models (e.g. the coupling of a quantitative climate model with a qualitative model of land use).

 

1                                                               Scientific background

 

Co-operation between qualitative and quantitative sciences is still sparse (Harvey and  Reed 1996). The possibilities which lie in a formalised (mathematical) deduction of results from systems of scientific assumptions are still unused in large parts of Global Change relevant research (political sciences, sociology, anthropology etc.). This is often based on the feeling that the framework of (dynamic) quantitative modelling is not appropriate (Haag and Matschonat 2002). A two-way strategy seems necessary to resolve this unproductive situation:

(A)   steps towards the qualitative sciences by offering an adequate and adapted concept of formalisation and outlining its scope and limits

(B)   steps towards integration by exploring ways to couple this formalised qualitative formulation with existing quantitative models

Such activities, encouraged by the SAB in 2002, already started at PIK with (A): In international projects on European urban development and land use in NE-Brazil, dynamic qualitative modelling is applied in close co-operation with "qualitative scientists". PIK-based activities on Syndrome-dynamics, resource dynamics and global governance also use this method. The now finishing methodological project QUIS provided an applicable algorithm for solving qualitative differential equations (QDEs). The current research on QDEs and qualitative reasoning shows an urgent demand on, i.a., the following issues:

·        epistemological "demarcation": which aspects of the qualitative sciences should become subject to formalised methods (Haag and Kaupenjohann 2001, Brajnik 1998, Eisenack et al. 2002)

·        extracting relevant information from the large solution sets of generalized dynamical systems (Bouwer and Bredeweg 2002, Eisenack and Petschel-Held 2002).

·        ways of further specifying the system in a still non-quantitative way to reduce the solution set (e.g. temporal order constraints, Clancy 1997, or qualitative rates if change, McIntosh 2003).

(B) is another large but unavoidable challenge. Here the main question is

·        how to couple QDEs with ODEs without running into the unresolved problems of integrating high dimensional differential inclusions (Berleant and Kuipers 1998, Chahma 2003)?

 

2                                                               Description of the work

2.1                                                               Aims and targets

The overall objective is to clarify in how far (and how) qualitative research (and its results) can be described by formal (mathematical) modelling and to supply algorithms (including their computer implementations) which allow for a practical realisation. From this objective together with the lacks in scientific knowledge identified above the following four aims for development and research emerge (increasingly challenging):

1.     Presently QDEs result in large qualitative behaviour graphs which sometimes hide interesting general dynamical properties (like irreversibility, bottlenecks for reaching sustainable regions etc.). Therefore - along the necessities occurring in practical applications of QDEs - algorithms as developed in the former QUIS project for extracting the relevant information from large graphs have to be implemented, improved and complemented by new ones.

2.     The actual QDE conceptualisation generates necessarily ambiguities in the resulting graphs. The development of additional non-quantitative constraints is of vital interest, since they can reduce these ambiguities. Research on such new constraints includes their interpretation in terms of their semantics (applicability to modellers) and their foundation in the theory of dynamical systems.

3.     The demand for coupling QDEs and ODEs is often expressed and a successful coupling technique would have a high potential for integrated assessment. Although this is a very challenging task for arbitrary dynamical systems with more than 4 dimensions, we propose pilot research on solution schemes for some specific types of systems.

4.     To weaken the borders existing between qualitative scientists and "modellers" a general clarification of the relation of qualitative sciences and qualitative modelling is necessary. This will shed light on the benefits and limitations of formal mathematical deduction methods in these sciences (this research theme was proposed in the last SAB comments).       

2.2                                                               Strategy

Due to the structure of the tasks encountered, the project will rely beside the continuous internal co-operation of the project staff on a seminar with the following functions:

(a)   Exchange between the QDE users at PIK* and Q3, where the concrete application problems are exchanged and collected

(b)   Dissemination the extensions to the actual and potential QDE users

(c)   Keeping in touch with the new developments in the qualitative reasoning community under participation of external experts

(d)   General discussion on bridging qualitative and quantitative science under participation of external experts

 

* LIFESTYLE; COMPROMISE; EURECA; INTERVUL; Syndromes/SPAREM; GLOGO; further PIK-internal co-operations are discussed

 

2.2.1       Aim 1

Function (a) will allow to specify the needs of the users with respect to the analysis of large behaviour graphs as a prerequisite for the problem guided development of post-processing tools. This development will be performed by using graph theory for the structural analysis and additional C++ and LEDA modules for the software implementation. Beside the identification of no-return-sets and lock-ins performed already in the QUIS project, systematic coupling and decomposition of QDEs will probably occur to be necessary.

2.2.2       Aim 2

In cases where even improved post-processing will not generate interpretable general results, the model formulation is simply to weak. A typical element of an ambiguity problem which cannot be resolved by post-processing is a so-called occurrence branching: when two variables are increasing towards important thresholds and we express only limited information about the speed of the associated processes in the model, we can not infer which variable reaches its threshold first. Thus, three cases are possible: variable 1 reaches the threshold first, or variable 2, or both at the same time. Via (a) possible additional model assumptions which would mathematically resolve these ambiguities can be discussed with respect to their semantics. Their foundation in the mathematical theory of dynamical systems will also be an important issue (e.g., higher derivatives for further characterisation of the QDEs).

Via (b) it will be ensured that during the whole project all interested users are informed on extensions developed in the project - even if they did not ask explicitly for them - thereby continuously improving the options for all users. Function (c) will ensure that external methodological developments will be perceived by inviting colleagues from the qualitative reasoning community to exchange actual progress in qualitative modelling. Here the contacts established under the MONET network will guarantee adequate resonance.

2.2.3       Aim 3

From a general mathematical point of view a coupled system of QDEs and ODEs is equivalent to a system of differential inclusions. It is often difficult to solve such a system numerically as computational complexity grows exponentially with the number of variables. But there is the realistic chance to develop algorithms which allow to treat larger systems (as necessary in global change research) because qualitative constraints and quantitative equations are separate modules with a limited number of exchange variables. Here it is a straightforward task to drive a qualitative module by the results of an ODE, while the opposite direction (which is the task in, e.g., the LIFESTYLE project) and the bi-directional coupling is rather challenging. Possible approaches are inverse methods (solve a quantitative guard-rail problem and filter the paths not consistent with the QDE) or related to hybrid systems theory (e.g. Schaft and Schumacher 2000). Here different qualitative states of the QDE-part (discrete) are associated with individual ODEs (continuous part). Despite these first ideas serious research on this field requires the co-operation with external mathematical expertise (e.g. from numerics). To establish this co-operation a 1-year pilot phase for aim 3 is planned, where a workshop with invited experts on the topic will be organized. This workshop will result in clarification of the most promising approach and a common proposal for funding. The objective is to start aim 3 research in 2005.

 

2.2.4       Aim 4

This is closely related to the seminar series (d). Here, three groups external experts are of interest: The first group are colleagues from the "qualitative sciences" which already deal with the implementation of formal methods in their discipline like, e.g, the sociologist Charles Ragin who introduced a Boolean algebra based method (QSA) into comparative social science studies. Other interesting experts of this category could be Andrew Bennett (political sciences) or Roland Scholz.

The benefit for Q3 will be twofold: the horizon of possible approaches to the formalisation of qualitative sciences will be enlarged, thereby generating new ideas for QDE extensions (aim 1 and 2) and, secondly, experiences on the acceptance and usage of formal deduction methods can be collected on a broader scale as a basis for assessing chances and limitations of formalisation. The second group comprises qualitative scientists which were confronted with formal methods, reporting on their view of these exercises. Here contacts exist to a large pool of experts from ongoing or former integrated modelling projects. The third group are epistomologists dealing with the methodology of science. The task of the project group will be to summarise, compare and systematise the presented positions. The result will allow for an improved integration of qualitative sciences into integrated modelling projects reducing present misunderstandings and friction.

 

Selected Publications:

Klaus Eisenack, Matthias Lüdeke, Gerhard Petschel-Held, Jürgen Scheffran and Jürgen Kropp (2006):
Qualitative Modeling Techniques to Assess Patterns of Global Change
in J. Scheffran and J. Kropp (ed.): Advanced Methods for Decision Making and Risk Management in Sustainability Science. Nova Science Publishers, New York. Forthcoming.
download

 

Klaus Eisenack (2005):
Model Ensembles for Natural Resource Management
PhD Thesis, Free University Berlin.
download

 

Klaus Eisenack and Gerhard Petschel-Held (2002):
Graph Theoreticel Analysis of Qualitative Models in Sustainability Science
in N. Agell and J. A. Ortega (ed.): Working Papers of 16th Workshop on Qualitative Reasoning, 53-60. Edición Digial@tres, Sevilla.
download

 

 

 

 

 

3                                                               Results of the former QUIS project (1/2002-12/2003)

 

Q3 is partially the continuation of the former QUIS project. During the last three years methodological improvements for the analysis of large qualitative models were made, which are closely related to the conceptual development for its use for sustainability issues. In parallel, the qualitative simulation software has been substantially enhanced for its use in other PIK projects and for external partners.

Usually, the solution of a qualitative differential equation (QDE) consists of a large number of so-called qualitative states, each representing a possible combination of trends of the relevant variables. These states are linked by transitions indicating possible changes in time. As QDEs are generalized dynamical systems in the sense that they can be interpreted as a whole class of ODEs, these transitions are, in general, not unique. Thus, the main research focussed at revealing robust features in this complex structure:

·        The identification of clusters of states which are meaningful to the modeller. Three types of clusters have proved to be valuable: lock-ins, no-return-sets and GCBs. No-return-sets are operational modes of a system which can not be re-entered once they are left, and are therefore related to irreversible processes. Contrary, GCBs are modes which can persist for long times although much fluctuations are observed on the first glance. Lock-ins have shown to be of high relevance for the syndrom approach. They denote a situation where the system cannot get out of ‘vicious circle’ (or ‘angelic circle’). These concepts have been carefully defined and their inter-relationships where investigated (Eisenack and Petschel-Held 2002). Algorithms for their detection have been developed and integrated into the modelling software.

·        Combining QDEs with viability theory. Here, the modeller is looking for management strategies which can guarantee that the systems is kept between some specified guardrails. Viability theory shows that under some very general preconditions it is sufficient to investigate the systems dynamics near these guardrails. By restricting the solution of a QDE to this region of the phase space, the resulting graph is much simpler and qualitative management strategies can be derived (Eisenack 2003).

Further research and development work was performed in the following directions:

·        The coupling QDEs and ODEs: This was kept on an explorative level, since the complexity of this task became clear and the project resources were not sufficient for its solution. Problems occur both on the theoretical and the computational level. The latter is associated with the numerics of differential inclusions, where only slow progress is made in the scientific comunitiy. In the new Q3 project a more detailed pilot phase on this topic is planned with the aim to find external partners to obtain a critical mass on expertise.

·        Software development started nearly from scratch, since only a very in-transparent, slow simulator implemented in LISP and a very basic kernel in C was available. A re-implementation in C++ resulted in a computational speed-up of about factor 100. By using LEDA, a software library for graph theoretical algorithms, important post-processing options could be integrated to the more user-friendly application. This includes further simplification methods and graphical output routines which make the simulation results easier to interpret (e.g. projection, clustering, zooming). However, this work is not completely finished, since some relevant and theoretically understood functions (e.g. coupling of qualitative models) are not fully implemented yet. The user-interface is currently improved in collaboration with the projects where the software is applied. Stopping the activity in this phase would result in a substantial loss of former effort, as this part of the work is nearly completed and merely a technical matter.

In addition to research and development the following activities were pursued in QUIS:

·        The training of and exchange with qualitative modellers in other PIK projects, mainly SYNAPSE and GLOGO, and external partners (e.g. from ICBM, Oldenburg; RIVM, Bilthoven).

·        Supervision of several students for practical training, and one student for a diploma thesis.

Dissemenation activities comprised presentations and publications:

·        QR-01: 16th International Workshop on Qualitative Reasoning, San Antonio, Texas, May 2001.

·        QR-02: 17th International Workshop on Qualitative Reasoning, Sitges, Spain, June 2002.

·        Workshop on connectionist and structural complexity of dynamical networks, Paris, March 2003.

·        Jahrestagung of the German Physical Society, Dresden, March 2003.

·        Moldenhauer et al. (1999)

·        Petschel-Held and Lüdeke (2001)

·        Eisenack and Petschel-Held (2002)

·        Eisenack et al. (2002)

·        Eisenack (2003)

 

4                                                               References

Aubin J-P (1991): Viability Theory, Birkhäuser.

Berleant D and Kuipers B J (1998): Qualitative and quantitative simulation: Bridging the gap, Artifical Intelligence 95(2): 215-255.

Bower A and Bredeweg B (2002): Aggregation of Qualitative Simulations for Explanation, Working Papers of 16th Workshop on Qualitative Reasoning, 11-19. Download at http://www.upc.es/web/QR2002/Principal.htm

Brajnik G and Lines M (1998): Qualitative modeling and simulation of socio-economic phenomena, Journal of Artificial Societies and Social Simulation 1(1).

Chahma I A (2003): Set-Valued discrete approximation of state-constrained differential inclusions, Bayreuther Mathematische Schriften 67, 3-162.

Clancy D J (1997): Solving complexity and ambiguity problems within qualitative simulation, dissertation at the University of Texas at Austin.

Eisenack K and Petschel-Held G (2002): Graph Theoreticel Analysis of Qualitative Models in Sustainability Science, Working Papers of 16th International Workshop on Qualitative Reasoning, 53-60. Download at http://www.upc.es/web/QR2002/Principal.htm

Eisenack K, Moldenhauer O and Reusswig F (2002): Möglichkeiten und Grenzen qualitativer und semiqualitativer Modellierung von Natur-Gesellschafts-Interaktionen. In: Sozial-Ökologische Forschung, Balzer I and Wächter M (eds.), ökom verlag, 377-389.

Eisenack K (2003): Qualitative Viability Analysis of a Bio-Socio-Economic System. Working Papers of 17th International Workshop on Qualitative Reasoning, in press.

McIntosh B S (2003): Qualitative modelling with imprecise ecological knowledge: a framework for simulation, Environmental Modelling and Software 18: 295-307.

Harvey DL, Reed M (1996): Social Science as the Study of Complex Systems. In: Chaos Theory in the Social Sciences, Douglas Kiel, L and Elliot E., Ann Arbor, 295-345.

Haag D, Kaupenjohann, M (2001): Parameters, prediction, post-normal science and the precautionary principle . Ecological Modelling 144, 45-60.

Haag D, Matschonat, G (2002): Paradigmen zur Repäsentation und zum Management komplexer Systeme. In: Sozial-Ökologische Forschung, Balzer, I and Wächter, M, München, 409-427.

Moldenhauer O, Bruckner T and Petschel-Held G (1999): The use of semi-qualitative reasoning and probability distributions in assessing possible behaviors of a socio-economic system. In: Computational Intelligence for Modelling, Control and Automation, Mohammadian M (ed.), IOS Press, 410-416.

Petschel-Held G and Lüdeke MKB (2001): Integration of case studies on global change by means of qualitative differential equations. Integrated Assessment 2 (3), 123-138.

Schaft A v d and Schumacher H (2000): An Introduction to Hybrid Dynamical Systems, Springer.


[1] Interdisciplinary research is at the heart of PIK tasks including the integration across the scientific cultures of natural and social sciences. Q3 attempts to provide mathematical methods for this integration and to clarify to what degree the theories in qualitative sciences can be formalized and thereby incorporated into integrated mathematical models. Both parts of the project are at the front of research and PIK is the optimal place to perform them as methodological development is in close interaction with applications. ToPIK 7 could profit from becoming a focal point of this research field.