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Gaussian Distribution

Gaussian Distribution. Casti’s Questions. 1. What is a model? 2. What is a “good” model? 3. How do we represent a process (P) as a formal system (F)? 4. How can we compare two models, F 1 and F 2 , of the same P? 5. When is a relation, R, between observables considered a “law”?

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Gaussian Distribution

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  1. Gaussian Distribution

  2. Casti’s Questions • 1. What is a model? • 2. What is a “good” model? • 3. How do we represent a process (P) as a formal system (F)? • 4. How can we compare two models, F1 and F2, of the same P? • 5. When is a relation, R, between observables considered a “law”? • 6. How do we identify key observables to formulate a model? • 7. When can two systems that behave similarly be considered as models of each other?

  3. STEPS IN MODEL BUILDING • IDENTIFICATION: WHAT’S THE QUESTION? • ASSUMPTIONS: WHAT’S IMPORTANT; WHAT’S NOT? • CONSTRUCTION: MATHEMATICAL FORMULATION • ANALYSIS: SOLUTIONS • INTERPRETATION: WHAT DOES IT MEAN? • VALIDATION: DOES IT ACCORD WITH KNOWN DATA? • IMPLEMENTATION: CAN IT PREDICT NEW DATA?

  4. SOME EMPIRICAL VARIABLES ENTERING MODELS • RESPONSE RATE • IRTS • BOUTS • LATENCY • CHANGEOVERS • DWELL TIMES • CORRECTS and ERRORS • REINFORCER RATES, QUANTITIES, QUALITIES • STIMULUS PROPERTIES AND ARRANGEMENTS • MOTIVATIONAL CONDITIONS

  5. THEORETICAL VARIABLES • PROBABILITY OF A RESPONSE • REFLEX RESERVE • RESPONSE STRENGTH • MOMENTUM • INCENTIVE • VALUE • ASSOCIATIVE STRENGTH • STATES • COSTS • ENERGY RESERVE

  6. THE STRUCTURE OF MODELS STATIC vs. DYNAMIC HERRNSTEIN’S HYPERBOLA vs. FEEDBACK FUNCTIONS DISCRETE vs. CONTINUOUS RESCORLA-WAGNER vs. LINEAR SYSTEMS DETERMINISTIC vs. STOCHASTIC HYPERBOLIC DISCOUNTING vs. PROBABILITY MACHINES

  7. OTHER FEATURES OF MODELS • LINEAR vs. NONLINEAR • MEMORY vs. NON-MEMORY • COMPUTATIONAL • OPTIMALITY ASSUMPTIONS AND DETERMINATIONS • CURVE FITTING AND MODEL BUILDING • DELAY • MODELS BASED ON OTHER MODELS • MOLAR vs. MOLECULAR

  8. SOME PROBLEMS ADDRESSED • SCHEDULES OF REINFORCEMENT • CHOICE • DELAY AND SELF-CONTROL • BEHAVIOR DYNAMICS • STIMULUS CONTROL • TIMING • BEHAVIORAL ECOLOGY • BEHAVIORAL ECONOMICS • MOTIVATION • MEMORY

  9. SOME MATHEMATICAL METHODS • FUNCTION PROPERTIES AND CURVE FITTING • DIFFERENTIAL AND INTEGRAL CALCULUS • CALCULUS OF VARIATIONS • DIFFERENTIAL AND DIFFERENCE EQUATIONS • COMPUTATIONAL MODELING • STOCHASTIC METHODS

  10. From Davison & McCarthy (1988)

  11. From Davison & McCarthy (1988)

  12. MATCHING LAW Herrnstein R1 / (R1+R2) = r1 / (r1+r2) Baum R1 / R2 = b (r1 / r2)a

  13. FUNCTIONAL PROPERTIES AND CURVE FITTING • What is the “real” delay function? Vt = V0 / (1 + Kt) Vt = V0/(1 + Kt)s Vt = V0/(M + Kts) Vt = V0/(M + ts) Vt = V0 exp(-Mt)

  14. WHAT GOOD IS THE CALCULUS? • Finding rates of change. (e.g., Think of slopes in a cumulative record.) • A derivative of function at a point is the slope of the function at that point. If the slope (derivative) is zero at a point, that point is a maxima or minima (may be local). • Finding areas (integration)---the opposite of finding derivatives. • Leads to the study of differential equations and dynamical systems.

  15. OPTIMALITY EXAMPLE Staddon’s response-cost model of a random ratio, RR n: C(r) = (n / r) + Q r. n: ratio requirement r: response rate (n / r): average inter-food interval Q: cost/response/unit time. Q r: cost proportional to rate. Question: For a given ratio requirement, what response rate is the least costly?

  16. STADDON RATIO MODEL(CON’D) • The one free parameter is Q. • To find minimal cost, find dC/dr and set equal to 0. dC/dr = (-n / r -2) + Q = 0. Thus, r = √n/Q.

  17. HOW FAST SHOULD A PIGEON EAT? • ASSUMPTION: REDUCTION OF FITNESS RELATED TO LEVEL OF DEPRIVATION AND THE RATE OF CHANGE OF DEPRIVATION.

  18. Euler Condition (i.e. pigeon should feed at a gradually decreasing rate reducing deprivation exponentially.)

  19. DIFFERENTIAL EQUATIONS AND DYNAMICS THE COURSE OF TRUE LOVE Suppose William is in love with Zelda, but Zelda is a fickle lover. The more William loves her, the more she dislikes him---but when he loses interest in her, her feelings for him warm up. On the other hand, William reacts to her: When she loves him, his love for her grows and when she loses interest, he also loses interest. Let w(t) = William’s feelings for Zelda at time t. Let z(t) = Zelda’s feelings for William at time t. • dw/dt = az w(0) = α • dz/dt = -bw z(0) = β

  20. Interacting Contingencies • Predator-Prey • Competition • Mutualism/Symbiosis All involve non-linear processes. Can we model these in the operant laboratory through arranging interactive contingencies?

  21. Example: Competition Imagine a two-compartment pigeon chamber with a partition. Each pigeon (P1 & P2) pecks for food under a schedule which delivers food according to a feed-back function where the rate of reinforcement is some positive function of the rate of response. (e.g.,VI t, VR n or some combination). The rate of responding of, say, P1 controls the rate of reinforcement for P2 and vice-versa by changing the schedule parameter for the other bird. For example, as P1’s rate increases, this leads to a decrease in P2’s rate of reinforcement, leading to decreased rate for P2, leading to a decreased rate for P1, etc.

  22. COMPETITION MODEL Pigeon 1 (VI t1) Pigeon 2 (VI t2) R1 increases R2 decreases  t2 increases t1 increases (r2 decreases) and (r1 decreases) and vice versa. vice versa.

  23. Some Assumptions and Parameters 1. Rate of responding is a positive function of rate of reinforcement (r). 2.In the absence of a competitor, reinforcement rate (r) will increase logistically in proportion to a1 or a2 to asymptote at b1 or b2,respectively. 3. The parameters c12 and c21 specify the intensity of competition in the reduction of reinforcement rate contributed by an increase in response rate by each pigeon on the other.

  24. Competitive System Predicted behavior depends on parameter values, most notably the level of competition (c12 & c21). Assuming all other parameters are equal for the two birds, if both c12 and c21 < 0, then a stable state with each bird settling for less than maximum possible reinforcement rate. Otherwise, one or the other will predominate.

  25. A sphere moving through a medium with retarding force a function of initial velocity, V, reaches a terminal velocity VT.

  26. ASSUME A SPHERICAL PIGEON Falling body in a viscous medium model of behavioral momentum. 1’ 1 m1 m2 v0 v0 v0 m2 2 2 m1 vT2 vT1

  27. Solution

  28. m1 = 2 x m2 m1 v m2 t

  29. MARBLE IN A DOUBLE WELLDynamic model of choice d2x / dt2 + a (dx / dt) – bx + cx2 +dx3 = A cos(2πft)

  30. HERRNSTEIN’S HYPERBOLA

  31. SECOND-ORDER,SECOND-DEGREE DIFFERENTIAL EQUATION DOES THIS HAVE A CLOSED-FORMED SOLUTION?

  32. YES! HERRNSTEIN’S HYPERBOLA

  33. Substitute these (4) and (5) values in (3) gives:

  34. What could this God-awful equation mean? I don’t know! Can we simplify the problem?

  35. To simplify, invert Herrnstein’s Hyperbola and follow same procedure to yield a linear ODE:

  36. This ODE arises in simple problems in electromagnetics and thermodynamics by application of Laplace’s Equation: And describes, for example, how the temperature will change between two concentric spheres held at different temperatures—depends on their RELATIVE values—like Herrnstein’s r and ro.

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