1. Juan A. Ortega, Jesus Torres, Rafael M. Gasca, Departamento de Lenguajes y Sistemas Informáticos University of Seville (Spain) A new methodology for analysis of semiqualitative dynamic models with constraints A new methodology for analysis of semiqualitative dynamic models with constraints

2. n Model that evolves in the time n Qualitative and quantitative knowledge n Constraints + Objectives Semiqualitative model with constraints Semiqualitative model with constraints Study its temporal evolution Obtain its behaviour patterns

3. n Two interconnected tank system Objectives p r1r1 r2r2 Evolve in the time t0t0 x2x2 x1x1 t1t1 t2t2 t3t3 tftf

4. n Two interconnected tanks system Objectives Qualitative and quantitative knowledge - p is a moderadately positive influent - x 1,x 2 contain a slightly positive quantity of liquid at the initial time p r1r1 r2r2 x2x2 x1x1 0.4 x 2 0.6 x 2 g1g1 r 1 = g 1 ( x 1 – x 2 ) h1h1 5 8 y0y0 x0x0 ++ 0 0 r 2 = h 1 ( x 2 )

5. n Two interconnected tank system Objectives p r1r1 r2r2 x2x2 x1x1 Constraints - Height of the tanks is moderately positive

6. n Two interconnected tank system Objectives p r1r1 r2r2 x2x2 x1x1 Evolve in the time Qualitative and quantita- tive knowledge Constraints Semiqualitative model with constraints Semiqualitative model with constraints Study its temporal evolution Obtain its behavior patterns

7. n Two interconnected tanks system –Study its temporal evolution Objectives p r1r1 r2r2 x2x2 x1x1 If always the system reaches a stable equilibrium If it is reached an equilibrium where x 1 < x 2 If sometime the height of a tank is overflowed If sometime x 1 < x 2

8. n Two interconnected tanks system –Obtain its behaviour patterns Objectives p r1r1 r2r2 x2x2 x1x1 Depending on the influent p: –“a tank is overflowed” –“a tank is no overflowed and always x 1 >x 2 “ –“a tank is no overflowed and sometime x 1 <x 2 ” if p > 0.4 then a tank is overflowed if p > 0.1 & p < 0.4 then a tank is no overflowed & always x 1 >x 2 if p < 0.1 then a tank is no overflowed & sometime x 1 <x 2

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

10. Semiqualitative methodology Dynamic System Labelled Database Classification Queries Learning Transformation techniques Stochastic techniques Quantitative Models M F Semiqualitative Model S Trajectory Database Quantitative simulation T Modelling Answers System Behaviour

11. n A formalism to incorporate qualitative knowledge –qualitative operators and labels –envelope functions –qualitative continuous functions n This methodology allows us to study all the states of a dynamic system: stationary and transient states. n 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

12. Semiqualitative models  (x,x,y,q,t), x(t 0 ) = x 0,  0 (q,x 0 )  variables, parameters,...  numbers and intervals  arithmetic operators and functions  qualitative knowledge  qualitative operators and labels  envelope functions  qualitative continuous functions  x: state variables  x: derivative of x  q: parameters  y: auxiliary variables   : constraints dx dt 

13. n Qualitative operators –Every operator is defined by means of a real interval I op. –This interval is given by the experts –Unary qualitative operators U(e) Every qualitative variable has its own unary operators defined U x = {VN x, MN x, LN x, A0 x, LP x, MP x, VP x } –Binary qualitative operators B(e 1,e 2 ) They are applied between two qualitative magnitudes B = {=, , , «, , ~, , »} Qualitative knowledge Qualitative operators

14. n 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 x I y g g y=g(x),  x  I g(x)  g(x)

15. Qualitative knowledge Qualitative continuous functions n 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  {P 1, s 1, P 2,..., s k-1, P k } with P i =( d i, e i ), s i  { +, -, 0 } h  {(– , +  ),–,(x 0,0), –,(x 1,y 0 ),+,(x 2,0),+,(0,y 1 ),+,(x 3,y 2 ), –,(x 4,0),–,(+ ,–  )} x0x0 x1x1 x2x2 x3x3 x4x4 y2y2 y1y1 y0y0 –– ++ h 0

16. n Semiqualitative model S n Family of quantitative models F Transformation techniques  (x,x,y,q,t),x(t ) = x,  (q,x ) 0000 Transformation rules x=f(x,y,p,t), x(t 0 ) = x 0, p  I p, x 0  I 0

17. n Database generation T T:={ } for i=1 to N M := Choose Model (F) r := Quantitative Simulation (M) T := T  r n 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 r1r1 rnrn T

18. n Abstract Syntax Query/classification language Queries

19. n Abstract Syntax Query/classification language Classification

20. If always the system reaches a stable equilibrium  r  T  EQ If it is reached an equilibrium where x 1 < x 2  r  T  EQ  (always (t ~ t F  x 1 <x 2 )) If sometime x 1 < x 2  r  T  sometime x 1 < x 2 If always the system reaches a stable equilibrium  r  T  EQ If it is reached an equilibrium where x 1 < x 2  r  T  EQ  (always (t ~ t F  x 1 <x 2 )) If sometime x 1 < x 2  r  T  sometime x 1 < x 2 p r1r1 r2r2 x2x2 x1x1 true falsetrue Query/classification language

21. n 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 n They abound in natural, social and socio-technical systems: –evolution of bacteria, –mineral extraction –economic development –world population growth t Logistic growth Decay and extinction Application to a logistic growth model with a delay t Exponential growth Asymptotic behaviour

22. n Let S be a semiqualitative model of these systems where a delay has been added. Its differential equations are x = (n h 1 (y) – m) x, y = delay  (x), x >0, h 1  {(– , –  ),+,(x 0,0),+,(0,1),+,(x 1,y 0 ), –,(1,0),–,(+ ,–  )}  x 0  [LP x,MP x ],  [MP , VP  ], LP x (m), LP x (n) 00 y0y0 h 1 1 x0x0 x1x1 1 –– –– ++ 0 Application to a logistic growth model with a delay

23. n 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 n Applying the proposed methodology is obtained a time-series database Application to a logistic growth model with a delay

24. n Queries If an equilibrium is always reached  r  T  EQ If an equilibrium is always reached  r  T  EQ True, therefore there are no limit cycles If there is a logistic growth equilibrium  r  T  EQ  always (t ~ t F  x  0) If there is a logistic growth equilibrium  r  T  EQ  always (t ~ t F  x  0) True (1 st behaviour pattern) If the decay equilibrium is reached without oscillations  r  T  EQ  always (t ~ t F  x  0 )  (length([ x  0],{x})  0) If the decay equilibrium is reached without oscillations  r  T  EQ  always (t ~ t F  x  0 )  (length([ x  0],{x})  0) False, there are two ways to reach this equilibrium, with and without oscillations (2 nd y 3 rd behaviour patterns ) Application to a logistic growth model with a delay

25. n All time-series were classified with a label n The obtained conclusions are in accordance when a mathema- tical reasoning is carried out n Behaviour patterns [r, EQ  length([x  0],{x})>0  always (t ~ t F  x  0)]  recoved equil. [r, EQ  length([x  0],{x})>0  always (t ~ t F  x  0)]  ret. catast. [r, EQ  length([x  0],{x})  0  always (t ~ t F  x  0)]  extinction [r, EQ  length([x  0],{x})>0  always (t ~ t F  x  0)]  recoved equil. [r, EQ  length([x  0],{x})>0  always (t ~ t F  x  0)]  ret. catast. [r, EQ  length([x  0],{x})  0  always (t ~ t F  x  0)]  extinction Application to a logistic growth model with a delay

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

27. n A new methodology has been presented in order to automates the analysis of dynamic systems with qualitative and quantitative knowledge n The methodology applied a transformation process, stochastic techniques and quantitative simulation. n Quantitative simulations are stored into a database and a query/classification language has been defined n 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