Copyright	Copyright (c) 2009-2014, David Sorokin <david.sorokin@gmail.com>
License	BSD3
Maintainer	David Sorokin <david.sorokin@gmail.com>
Stability	experimental
Safe Haskell	Safe-Inferred
Language	Haskell2010

Simulation.Aivika.Trans.SystemDynamics

Contents

Equality and Ordering
Ordinary Differential Equations
Difference Equations
Table Functions
Discrete Functions
Financial Functions

Description

Tested with: GHC 7.8.3

This module defines integrals and other functions of System Dynamics.

Synopsis

Equality and Ordering

(.==.) :: (MonadComp m, Eq a) => Dynamics m a -> Dynamics m a -> Dynamics m Bool Source

Compare for equality.

(./=.) :: (MonadComp m, Eq a) => Dynamics m a -> Dynamics m a -> Dynamics m Bool Source

Compare for inequality.

(.<.) :: (MonadComp m, Ord a) => Dynamics m a -> Dynamics m a -> Dynamics m Bool Source

Compare for ordering.

(.>=.) :: (MonadComp m, Ord a) => Dynamics m a -> Dynamics m a -> Dynamics m Bool Source

Compare for ordering.

(.>.) :: (MonadComp m, Ord a) => Dynamics m a -> Dynamics m a -> Dynamics m Bool Source

Compare for ordering.

(.<=.) :: (MonadComp m, Ord a) => Dynamics m a -> Dynamics m a -> Dynamics m Bool Source

Compare for ordering.

maxDynamics :: (MonadComp m, Ord a) => Dynamics m a -> Dynamics m a -> Dynamics m a Source

Return the maximum.

minDynamics :: (MonadComp m, Ord a) => Dynamics m a -> Dynamics m a -> Dynamics m a Source

Return the minimum.

ifDynamics :: MonadComp m => Dynamics m Bool -> Dynamics m a -> Dynamics m a -> Dynamics m a Source

Implement the if-then-else operator.

Ordinary Differential Equations

integ Source

Arguments

:: (MonadComp m, MonadFix m)
=> Dynamics m Double	the derivative
-> Dynamics m Double	the initial value
-> Simulation m (Dynamics m Double)	the integral

Return an integral with the specified derivative and initial value.

To create a loopback, you should use the recursive do-notation. It allows defining the differential equations unordered as in mathematics:

model = 
  mdo a <- integ (- ka * a) 100
      b <- integ (ka * a - kb * b) 0
      c <- integ (kb * b) 0
      let ka = 1
          kb = 1
      runDynamicsInStopTime $ sequence [a, b, c]

integEither Source

Arguments

:: (MonadComp m, MonadFix m)
=> Dynamics m (Either Double Double)	either set a new `Left` integral value, or use a `Right` derivative
-> Dynamics m Double	the initial value
-> Simulation m (Dynamics m Double)

Like integ but allows either setting a new Left integral value, or integrating using the Right derivative directly within computation.

This function always uses Euler's method.

smoothI Source

Arguments

:: (MonadComp m, MonadFix m)
=> Dynamics m Double	the value to smooth over time
-> Dynamics m Double	time
-> Dynamics m Double	the initial value
-> Simulation m (Dynamics m Double)	the first order exponential smooth

Return the first order exponential smooth.

To create a loopback, you should use the recursive do-notation with help of which the function itself is defined:

smoothI x t i =
  mdo y <- integ ((x - y) / t) i
      return y

smooth Source

Arguments

:: (MonadComp m, MonadFix m)
=> Dynamics m Double	the value to smooth over time
-> Dynamics m Double	time
-> Simulation m (Dynamics m Double)	the first order exponential smooth

Return the first order exponential smooth.

This is a simplified version of the smoothI function without specifing the initial value.

smooth3I Source

Arguments

:: (MonadComp m, MonadFix m)
=> Dynamics m Double	the value to smooth over time
-> Dynamics m Double	time
-> Dynamics m Double	the initial value
-> Simulation m (Dynamics m Double)	the third order exponential smooth

Return the third order exponential smooth.

To create a loopback, you should use the recursive do-notation with help of which the function itself is defined:

smooth3I x t i =
  mdo y  <- integ ((s2 - y) / t') i
      s2 <- integ ((s1 - s2) / t') i
      s1 <- integ ((x - s1) / t') i
      let t' = t / 3.0
      return y

smooth3 Source

Arguments

:: (MonadComp m, MonadFix m)
=> Dynamics m Double	the value to smooth over time
-> Dynamics m Double	time
-> Simulation m (Dynamics m Double)	the third order exponential smooth

Return the third order exponential smooth.

This is a simplified version of the smooth3I function without specifying the initial value.

smoothNI Source

Arguments

:: (MonadComp m, MonadFix m)
=> Dynamics m Double	the value to smooth over time
-> Dynamics m Double	time
-> Int	the order
-> Dynamics m Double	the initial value
-> Simulation m (Dynamics m Double)	the n'th order exponential smooth

Return the n'th order exponential smooth.

The result is not discrete in that sense that it may change within the integration time interval depending on the integration method used. Probably, you should apply the discreteDynamics function to the result if you want to achieve an effect when the value is not changed within the time interval, which is used sometimes.

smoothN Source

Arguments

:: (MonadComp m, MonadFix m)
=> Dynamics m Double	the value to smooth over time
-> Dynamics m Double	time
-> Int	the order
-> Simulation m (Dynamics m Double)	the n'th order exponential smooth

Return the n'th order exponential smooth.

This is a simplified version of the smoothNI function without specifying the initial value.

delay1I Source

Arguments

:: (MonadComp m, MonadFix m)
=> Dynamics m Double	the value to conserve
-> Dynamics m Double	time
-> Dynamics m Double	the initial value
-> Simulation m (Dynamics m Double)	the first order exponential delay

Return the first order exponential delay.

To create a loopback, you should use the recursive do-notation with help of which the function itself is defined:

delay1I x t i =
  mdo y <- integ (x - y / t) (i * t)
      return $ y / t

delay1 Source

Arguments

:: (MonadComp m, MonadFix m)
=> Dynamics m Double	the value to conserve
-> Dynamics m Double	time
-> Simulation m (Dynamics m Double)	the first order exponential delay

Return the first order exponential delay.

This is a simplified version of the delay1I function without specifying the initial value.

delay3I Source

Arguments

:: (MonadComp m, MonadFix m)
=> Dynamics m Double	the value to conserve
-> Dynamics m Double	time
-> Dynamics m Double	the initial value
-> Simulation m (Dynamics m Double)	the third order exponential delay

Return the third order exponential delay.

delay3 Source

Arguments

:: (MonadComp m, MonadFix m)
=> Dynamics m Double	the value to conserve
-> Dynamics m Double	time
-> Simulation m (Dynamics m Double)	the third order exponential delay

Return the third order exponential delay.

This is a simplified version of the delay3I function without specifying the initial value.

delayNI Source

Arguments

:: (MonadComp m, MonadFix m)
=> Dynamics m Double	the value to conserve
-> Dynamics m Double	time
-> Int	the order
-> Dynamics m Double	the initial value
-> Simulation m (Dynamics m Double)	the n'th order exponential delay

Return the n'th order exponential delay.

delayN Source

Arguments

:: (MonadComp m, MonadFix m)
=> Dynamics m Double	the value to conserve
-> Dynamics m Double	time
-> Int	the order
-> Simulation m (Dynamics m Double)	the n'th order exponential delay

Return the n'th order exponential delay.

This is a simplified version of the delayNI function without specifying the initial value.

forecast Source

Arguments

:: (MonadComp m, MonadFix m)
=> Dynamics m Double	the value to forecast
-> Dynamics m Double	the average time
-> Dynamics m Double	the time horizon
-> Simulation m (Dynamics m Double)	the forecast

Return the forecast.

The function has the following definition:

forecast x at hz =
  do y <- smooth x at
     return $ x * (1.0 + (x / y - 1.0) / at * hz)

trend Source

Arguments

:: (MonadComp m, MonadFix m)
=> Dynamics m Double	the value for which the trend is calculated
-> Dynamics m Double	the average time
-> Dynamics m Double	the initial value
-> Simulation m (Dynamics m Double)	the fractional change rate

Return the trend.

The function has the following definition:

trend x at i =
  do y <- smoothI x at (x / (1.0 + i * at))
     return $ (x / y - 1.0) / at

Difference Equations

diffsum Source

Arguments

:: (MonadComp m, MonadFix m, Unboxed m a, Num a)
=> Dynamics m a	the difference
-> Dynamics m a	the initial value
-> Simulation m (Dynamics m a)	the sum

Retun the sum for the difference equation. It is like an integral returned by the integ function, only now the difference is used instead of derivative.

As usual, to create a loopback, you should use the recursive do-notation.

diffsumEither Source

Arguments

:: (MonadComp m, MonadFix m, Unboxed m a, Num a)
=> Dynamics m (Either a a)	either set the `Left` value for the sum, or add the `Right` difference to the sum
-> Dynamics m a	the initial value
-> Simulation m (Dynamics m a)	the sum

Like diffsum but allows either setting a new Left sum value, or adding the Right difference.

Table Functions

lookupDynamics :: MonadComp m => Dynamics m Double -> Array Int (Double, Double) -> Dynamics m Double Source

Lookup x in a table of pairs (x, y) using linear interpolation.

lookupStepwiseDynamics :: MonadComp m => Dynamics m Double -> Array Int (Double, Double) -> Dynamics m Double Source

Lookup x in a table of pairs (x, y) using stepwise function.

Discrete Functions

delay Source

Arguments

:: MonadComp m
=> Dynamics m a	the value to delay
-> Dynamics m Double	the lag time
-> Dynamics m a	the delayed value

Return the delayed value using the specified lag time.

delayI Source

Arguments

:: MonadComp m
=> Dynamics m a	the value to delay
-> Dynamics m Double	the lag time
-> Dynamics m a	the initial value
-> Simulation m (Dynamics m a)	the delayed value

Return the delayed value using the specified lag time and initial value. Because of the latter, it allows creating a loop back.

step Source

Arguments

:: MonadComp m
=> Dynamics m Double	the height
-> Dynamics m Double	the step time
-> Dynamics m Double

Computation that returns 0 until the step time and then returns the specified height.

pulse Source

Arguments

:: MonadComp m
=> Dynamics m Double	the time start
-> Dynamics m Double	the interval width
-> Dynamics m Double

Computation that returns 1, starting at the time start, and lasting for the interval width; 0 is returned at all other times.

pulseP Source

Arguments

:: MonadComp m
=> Dynamics m Double	the time start
-> Dynamics m Double	the interval width
-> Dynamics m Double	the time period
-> Dynamics m Double

Computation that returns 1, starting at the time start, and lasting for the interval width and then repeats this pattern with the specified period; 0 is returned at all other times.

ramp Source

Arguments

:: MonadComp m
=> Dynamics m Double	the slope parameter
-> Dynamics m Double	the time start
-> Dynamics m Double	the end time
-> Dynamics m Double

Computation that returns 0 until the specified time start and then slopes upward until the end time and then holds constant.

Financial Functions

npv Source

Arguments

:: (MonadComp m, MonadFix m)
=> Dynamics m Double	the stream
-> Dynamics m Double	the discount rate
-> Dynamics m Double	the initial value
-> Dynamics m Double	factor
-> Simulation m (Dynamics m Double)	the Net Present Value (NPV)

Return the Net Present Value (NPV) of the stream computed using the specified discount rate, the initial value and some factor (usually 1).

It is defined in the following way:

npv stream rate init factor =
  mdo let dt' = liftParameter dt
      df <- integ (- df * rate) 1
      accum <- integ (stream * df) init
      return $ (accum + dt' * stream * df) * factor

npve Source

Arguments

:: (MonadComp m, MonadFix m)
=> Dynamics m Double	the stream
-> Dynamics m Double	the discount rate
-> Dynamics m Double	the initial value
-> Dynamics m Double	factor
-> Simulation m (Dynamics m Double)	the Net Present Value End (NPVE)

Return the Net Present Value End of period (NPVE) of the stream computed using the specified discount rate, the initial value and some factor.

It is defined in the following way:

npve stream rate init factor =
  mdo let dt' = liftParameter dt
      df <- integ (- df * rate / (1 + rate * dt')) (1 / (1 + rate * dt'))
      accum <- integ (stream * df) init
      return $ (accum + dt' * stream * df) * factor