Qualitative and Quantitative Simulation: Bridging the gap

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Qualitative and Quantitative Simulation Bridging
the gap

• Daniel Berleant , Benjamin Kuipers

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Introduction

• Aim is to combine numerical simulation and
  qualitative simulation.
• Numerical Simulation can not infer infinite
  values at all.
• Qualitative simulation can not infer which
  qualitative behavior will be te one to occur in a
  given instance.

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Introduction (Cont)

• Q2 works on simulations that contain only few
  time points -where significant Qualitative events
  occur- limiting the quantitative inferences.
• Q3 works on the tree of qualitative simulation
  trajectories produced by Q2.
• It only introduces step size refinement to allow
  better inferences both qualitative and
  quantitative simulation.

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Step Size Refinement

• With few time points in Q2 a variable may change
  significantly when going from one time point to
  the next.
• As a result numerical inferences would be weak.
• Q3 adds new time points reducing the step size
  between them so that change in a variable could
  be controlled in a narrower range.

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Step Size Refinement Phases

• Phase I Generating Simulation Trace
• Phase II Progressive Refinement

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Phase I Generating Simulation Trace

• Qualitative Simulation
• Propogate quantitative Information
• Iterate

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(1)Qualitative Simulation

• Q2 is called as a subroutine and each behaviour
  is represented as a constraint network.
• Constraint Network relates all model variables
  at the time value of each qualitative state in
  the simulation.

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(2)Propogate Quantitative Information

• Make use of quantitative information.
• If an interval for one variable becomes narrower
  it may effect the other variables connected to it
  with a constraint model.
• 4 kinds of constraints can appear.

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(2) Types of Constraints - 1

• Arithmetic constraints

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(2) Types of Constraints - 2

• Greater and less than constraints among
  different qualitative landmark values.
• Eg
• topHeight gt bottomHeight
• topHeight ? 100,200 , bottomHeight ? 0,8)
• As a result bottomHeight ? 0,200

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(2) Types of Constraints - 3

• Monotonicity Constraint

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(2) Types of Constraints - 4

• Mean Value Constraint

Interval Extension
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Phase II Progressive Refinement

• In phase one quantitatively annotated qualitative
  simualtion is done.
• Addition of new time points to the existing ones
  in order to reduce the step size is the
  refinement process.

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Step Size Refinement Algorithm

• Given finite number ordered T0,T1,..Tn time
  points of Ti with intervals (lb(Ti),ub(Ti))
  representing the uncertainity of the interval.
• 1) Locate a gap

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Step Size Refinement Algorithm(2)

• 2) Interpolate a state Insert new state Ta with
  zero-width interval t,t and initialize the
  other variables.
• 3)Propagate interval bounds Creation of new
  states with new inrtervals for variables may
  narrow the other intervals at the other time
  points by propagating throughout the network.

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Step Size Refinement Algorithm

• What if there is no proper gap?
• Try to create one by decreasing the upper bound
  of the previous time point and increasing the
  lower bound of the next time point.
• Two Algorithms
• Target Interval Splitting
• Behaviour Splitting

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Target Interval Splitting(TIS)

• Split the intervals randomly, test each of them,
  if an inconsistency appears for any of the
  intervals, rule out the inconsistent interval.

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Behavior Splitting

• It copies a qualitative behavior , only changes
  the interval belonging to it, instead of taking
  the original interval, a separate sub-interval is
  used.
• For each copy , a propogation is done through
  out the network if any inconsistency occurs ,
  rule out the interval.
• When splitting for infinite values arbitrary
  high number can be used such as 106.

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Gap Creation

• Target Interval splitting should be used before
  Behavior Splitting since copying qualitative
  behaviors for sub-intervals will cause high
  computational complexity in subsequent simualtion
  refinement.
• If TIS and BS are not enough, use another model
  variable other than TIME which has a gap.

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Correctness

• Correctness implies that any interval for a
  variable contains all possible values.
• However bounds may include extraneous values for
  following reasons
• Excess width eg X-X 1,2 1,2 -1,1
• Impossible values between possible values

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Correctness (Cont)

• Q3 can be accepted correct since it does not
  narrow down the intervals too much.
• But for full accounting of correctness of Q3, we
  have to assume that machine arithmetic does not
  introduce incorrectness through round-off errors.
• COMMON LISP language is used.

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Convergence

• For numerical predictions convergence means
  improving point predictions, for interval
  simulations it means narrowing interval all the
  way to correct point predictions.
• Q3 is accepted as converging even if the step
  size is very small going to zero since

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Convergence (Cont)

• The theorem shows convergence where implies
  amount of uncertainity at time point b,
  implies uncertainity in the initial conditions
  and it is equal to zero.
• As according to the theorem where h ?
  0,h0.

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Stability

• Stability means change in the starting values by
  a fixed amount produces a bounded change in
  numerical solution.
• Stability constraint accepted as
• Step size refinement effects positively even
  when initial conditions and model parameters are
  not completely specified via inetrvals
• More precise inital conditions result into more
  precise predictions.

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Termination

• Q3s aim is to increase the lower bound or
  decrease the upper bound during constraint
  propagation.It keeps doing this only if the bound
  will change by a proportion of its value greater
  than some constant e.

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Conclusion

• Q3 provides much better predictions compared to
  Q2 since it employes strengths of both
  qualitative and interval reasoning algorithms.
• From qualitative simulation the guarantee that
  all qualitative behaviors will be found.
• From interval simulation the guarantee that
  the trajectory of any real system conforming to
  an incompletely specified model is enclosed by
  one of the predicted behaviors.

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Conlusion

• As step size goes to ZERO , stability is
  provided.
• As step size converges , uncertainity converges
  and this provides the convergence property.
• From both qualitative and interval simulation
• The ability to express and make predictions
  from partial knowledge.

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Conclusion

• Q3 depends on step size refinement and
  propagation of interval labels.
• It is both pragmatic and theoretical.
• Pragmatic since it demonstrates an effective
  method of obtaining better quantitaive bounds on
  semi-quantitative simulation.
• Theoratical since it guarantees correctness,
  convergence and stability.