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Slide1: 

Juan A. Ortega, Jesus Torres, Rafael M. Gasca, Departamento de Lenguajes y Sistemas Informáticos University of Seville (Spain)

Objectives: 

Model that evolves in the time Qualitative and quantitative knowledge Constraints Objectives

Objectives: 

Two interconnected tank system Objectives Evolve in the time t0

Objectives: 

Two interconnected tanks system Objectives Qualitative and quantitative knowledge - p is a moderadately positive influent - x1,x2 contain a slightly positive quantity of liquid at the initial time

Objectives: 

Two interconnected tank system Objectives Constraints - Height of the tanks is moderately positive

Objectives: 

Two interconnected tank system Objectives Evolve in the time Qualitative and quantita- tive knowledge Constraints

Objectives: 

Two interconnected tanks system Study its temporal evolution Objectives If always the system reaches a stable equilibrium If it is reached an equilibrium where x1 < x2 If sometime the height of a tank is overflowed If sometime x1 < x2

Objectives: 

Two interconnected tanks system Obtain its behaviour patterns Objectives Depending on the influent p: “a tank is overflowed” “a tank is no overflowed and always x1>x2“ “a tank is no overflowed and sometime x1<x2”

Outline: 

Outline Semiqualitative methodology Semiqualitative models Qualitative knowledge Generation of trajectories database Query/classification language Theoretical study of the conclusions Application to a logistic growth model with a delay Conclusions and further work

Semiqualitative methodology: 

Semiqualitative methodology Dynamic System Classification Queries Learning Transformation techniques Stochastic techniques Semiqualitative Model S Trajectory Database Quantitative simulation T Modelling Answers System Behaviour

Semiqualitative methodology: 

A formalism to incorporate qualitative knowledge qualitative operators and labels envelope functions qualitative continuous functions This methodology allows us to study all the states of a dynamic system: stationary and transient states. Main idea: “A semiqualitative model is transformed into a family of quantitative models. Every quantitative model has a different quantitative behaviour, however, they may have similar quantitative behaviours” Semiqualitative methodology

Semiqualitative models: 

Semiqualitative models (x,x,y,q,t), x(t0) = x0 , 0 (q,x0 ) variables, parameters, ... numbers and intervals arithmetic operators and functions qualitative knowledge qualitative operators and labels envelope functions qualitative continuous functions •

Qualitative knowledgeQualitative operators: 

Qualitative operators Every operator is defined by means of a real interval Iop. This interval is given by the experts Unary qualitative operators U(e) Every qualitative variable has its own unary operators defined Ux = {VNx , MNx , LNx , A0x , LPx , MPx , VPx } Binary qualitative operators B(e1,e2) They are applied between two qualitative magnitudes B = {=,  ,  , «, , ~<, , ~>, , »} Qualitative knowledge Qualitative operators

Qualitative knowledgeEnvelope functions: 

A envelope function represents the family of functions included between a upper function g and a lower one g into a domain I. Qualitative knowledge Envelope functions y=g(x), <g(x), g(x), I> x I • g(x)  g(x)

Qualitative knowledgeQualitative continuous functions: 

Qualitative knowledge Qualitative continuous functions A qualitative continuous function represents a constraint in-volving the values of y and x according to the properties of h y=h(x) h  {P1, s1, P2, ..., sk-1, Pk} with Pi =( di, ei ), si  { +, -, 0 }

Transformation techniques : 

Semiqualitative model S Family of quantitative models F Transformation techniques Transformation rules •

Generation of trajectories database: 

Database generation T T:={ } for i=1 to N M := Choose Model (F) r := Quantitative Simulation (M) T := T  r Choose Model (F) for every interval parameter and qualitative variable p  F v:=Choose Value (Domain (p)) substitute p by v in M for every function h  F H:=Choose H (h) substitute h by H in M Generation of trajectories database r1 rn T • • •

Query/classification language: 

Abstract Syntax Query/classification language

Query/classification language: 

Abstract Syntax Query/classification language

Query/classification language: 

If always the system reaches a stable equilibrium  rT EQ If it is reached an equilibrium where x1 < x2  rT EQ  (always (t ~ tF  x1<x2)) If sometime x1 < x2  rT sometime x1< x2 Query/classification language

Application to a logistic growth model with a delay: 

It is very common to find growth processes in which an initial phase of exponential growth is followed by another phase of approaching to a saturation value asymptotically They abound in natural, social and socio-technical systems: evolution of bacteria, mineral extraction economic development world population growth Application to a logistic growth model with a delay

Application to a logistic growth model with a delay: 

Let S be a semiqualitative model of these systems where a delay has been added. Its differential equations are Application to a logistic growth model with a delay

Application to a logistic growth model with a delay: 

We would like to know if an equilibrium is always reached to know if there is logistic growth equilibrium to know if all the trajectories reach the decay equilibrium without oscillations to classify the database in accordance with the behaviours of the system Applying the proposed methodology is obtained a time-series database Application to a logistic growth model with a delay

Application to a logistic growth model with a delay: 

Queries Application to a logistic growth model with a delay

Application to a logistic growth model with a delay: 

Application to a logistic growth model with a delay

Application to a logistic growth model with a delay: 

X/t Recovered equilibrium Extinction Retarded catastrophe Application to a logistic growth model with a delay

Conclusions and further work: 

A new methodology has been presented in order to automates the analysis of dynamic systems with qualitative and quantitative knowledge The methodology applied a transformation process, stochastic techniques and quantitative simulation. Quantitative simulations are stored into a database and a query/classification language has been defined In the future the language will be enrich with operators for comparing trajectories, and for comparing regions of the same trajectory. Clustering algorithms will be applied in other to obtain automatically the behaviours of the systems Dynamic systems with explicit constraints and with multiple scales of time are also one of our future points of interest Conclusions and further work