Copyright | Copyright (c) 2009-2017 David Sorokin <david.sorokin@gmail.com> |
---|---|

License | BSD3 |

Maintainer | David Sorokin <david.sorokin@gmail.com> |

Stability | experimental |

Safe Haskell | None |

Language | Haskell2010 |

Tested with: GHC 8.0.1

This module defines integrals and other functions of System Dynamics.

## Synopsis

- (.==.) :: Eq a => Dynamics a -> Dynamics a -> Dynamics Bool
- (./=.) :: Eq a => Dynamics a -> Dynamics a -> Dynamics Bool
- (.<.) :: Ord a => Dynamics a -> Dynamics a -> Dynamics Bool
- (.>=.) :: Ord a => Dynamics a -> Dynamics a -> Dynamics Bool
- (.>.) :: Ord a => Dynamics a -> Dynamics a -> Dynamics Bool
- (.<=.) :: Ord a => Dynamics a -> Dynamics a -> Dynamics Bool
- maxDynamics :: Ord a => Dynamics a -> Dynamics a -> Dynamics a
- minDynamics :: Ord a => Dynamics a -> Dynamics a -> Dynamics a
- ifDynamics :: Dynamics Bool -> Dynamics a -> Dynamics a -> Dynamics a
- integ :: Dynamics Double -> Dynamics Double -> Simulation (Dynamics Double)
- integEither :: Dynamics (Either Double Double) -> Dynamics Double -> Simulation (Dynamics Double)
- smoothI :: Dynamics Double -> Dynamics Double -> Dynamics Double -> Simulation (Dynamics Double)
- smooth :: Dynamics Double -> Dynamics Double -> Simulation (Dynamics Double)
- smooth3I :: Dynamics Double -> Dynamics Double -> Dynamics Double -> Simulation (Dynamics Double)
- smooth3 :: Dynamics Double -> Dynamics Double -> Simulation (Dynamics Double)
- smoothNI :: Dynamics Double -> Dynamics Double -> Int -> Dynamics Double -> Simulation (Dynamics Double)
- smoothN :: Dynamics Double -> Dynamics Double -> Int -> Simulation (Dynamics Double)
- delay1I :: Dynamics Double -> Dynamics Double -> Dynamics Double -> Simulation (Dynamics Double)
- delay1 :: Dynamics Double -> Dynamics Double -> Simulation (Dynamics Double)
- delay3I :: Dynamics Double -> Dynamics Double -> Dynamics Double -> Simulation (Dynamics Double)
- delay3 :: Dynamics Double -> Dynamics Double -> Simulation (Dynamics Double)
- delayNI :: Dynamics Double -> Dynamics Double -> Int -> Dynamics Double -> Simulation (Dynamics Double)
- delayN :: Dynamics Double -> Dynamics Double -> Int -> Simulation (Dynamics Double)
- forecast :: Dynamics Double -> Dynamics Double -> Dynamics Double -> Simulation (Dynamics Double)
- trend :: Dynamics Double -> Dynamics Double -> Dynamics Double -> Simulation (Dynamics Double)
- diffsum :: (Num a, Unboxed a) => Dynamics a -> Dynamics a -> Simulation (Dynamics a)
- diffsumEither :: (Num a, Unboxed a) => Dynamics (Either a a) -> Dynamics a -> Simulation (Dynamics a)
- lookupDynamics :: Dynamics Double -> Array Int (Double, Double) -> Dynamics Double
- lookupStepwiseDynamics :: Dynamics Double -> Array Int (Double, Double) -> Dynamics Double
- delay :: Dynamics a -> Dynamics Double -> Dynamics a
- delayI :: Dynamics a -> Dynamics Double -> Dynamics a -> Simulation (Dynamics a)
- delayByDT :: Dynamics a -> Dynamics Int -> Dynamics a
- delayIByDT :: Dynamics a -> Dynamics Int -> Dynamics a -> Simulation (Dynamics a)
- step :: Dynamics Double -> Dynamics Double -> Dynamics Double
- pulse :: Dynamics Double -> Dynamics Double -> Dynamics Double
- pulseP :: Dynamics Double -> Dynamics Double -> Dynamics Double -> Dynamics Double
- ramp :: Dynamics Double -> Dynamics Double -> Dynamics Double -> Dynamics Double
- npv :: Dynamics Double -> Dynamics Double -> Dynamics Double -> Dynamics Double -> Simulation (Dynamics Double)
- npve :: Dynamics Double -> Dynamics Double -> Dynamics Double -> Dynamics Double -> Simulation (Dynamics Double)

# Equality and Ordering

ifDynamics :: Dynamics Bool -> Dynamics a -> Dynamics a -> Dynamics a Source #

Implement the if-then-else operator.

# Ordinary Differential Equations

:: Dynamics Double | the derivative |

-> Dynamics Double | the initial value |

-> Simulation (Dynamics 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 :: Simulation [Double] 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]

:: Dynamics Double | the value to smooth over time |

-> Dynamics Double | time |

-> Dynamics Double | the initial value |

-> Simulation (Dynamics 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

:: Dynamics Double | the value to smooth over time |

-> Dynamics Double | time |

-> Simulation (Dynamics 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.

:: Dynamics Double | the value to smooth over time |

-> Dynamics Double | time |

-> Dynamics Double | the initial value |

-> Simulation (Dynamics 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

:: Dynamics Double | the value to smooth over time |

-> Dynamics Double | time |

-> Simulation (Dynamics 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.

:: Dynamics Double | the value to smooth over time |

-> Dynamics Double | time |

-> Int | the order |

-> Dynamics Double | the initial value |

-> Simulation (Dynamics 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.

:: Dynamics Double | the value to smooth over time |

-> Dynamics Double | time |

-> Int | the order |

-> Simulation (Dynamics 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.

:: Dynamics Double | the value to conserve |

-> Dynamics Double | time |

-> Dynamics Double | the initial value |

-> Simulation (Dynamics 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

:: Dynamics Double | the value to conserve |

-> Dynamics Double | time |

-> Simulation (Dynamics 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.

:: Dynamics Double | the value to conserve |

-> Dynamics Double | time |

-> Dynamics Double | the initial value |

-> Simulation (Dynamics Double) | the third order exponential delay |

Return the third order exponential delay.

:: Dynamics Double | the value to conserve |

-> Dynamics Double | time |

-> Simulation (Dynamics 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.

:: Dynamics Double | the value to conserve |

-> Dynamics Double | time |

-> Int | the order |

-> Dynamics Double | the initial value |

-> Simulation (Dynamics Double) | the n'th order exponential delay |

Return the n'th order exponential delay.

:: Dynamics Double | the value to conserve |

-> Dynamics Double | time |

-> Int | the order |

-> Simulation (Dynamics 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.

:: Dynamics Double | the value to forecast |

-> Dynamics Double | the average time |

-> Dynamics Double | the time horizon |

-> Simulation (Dynamics 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)

:: Dynamics Double | the value for which the trend is calculated |

-> Dynamics Double | the average time |

-> Dynamics Double | the initial value |

-> Simulation (Dynamics 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

:: (Num a, Unboxed a) | |

=> Dynamics a | the difference |

-> Dynamics a | the initial value |

-> Simulation (Dynamics 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.

# Table Functions

lookupDynamics :: Dynamics Double -> Array Int (Double, Double) -> Dynamics Double Source #

Lookup `x`

in a table of pairs `(x, y)`

using linear interpolation.

lookupStepwiseDynamics :: Dynamics Double -> Array Int (Double, Double) -> Dynamics Double Source #

Lookup `x`

in a table of pairs `(x, y)`

using stepwise function.

# Discrete Functions

Return the delayed value using the specified lag time.
This function is less accurate than `delayByDT`

.

:: Dynamics a | the value to delay |

-> Dynamics Double | the lag time |

-> Dynamics a | the initial value |

-> Simulation (Dynamics 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.
This function is less accurate than `delayIByDT`

.

:: Dynamics a | the value to delay |

-> Dynamics Int | the delay as a multiplication of the corresponding number and the integration time step |

-> Dynamics a | the delayed value |

Return the delayed value by the specified positive number of integration time steps used for calculating the lag time.

:: Dynamics a | the value to delay |

-> Dynamics Int | the delay as a multiplication of the corresponding number and the integration time step |

-> Dynamics a | the initial value |

-> Simulation (Dynamics a) | the delayed value |

Return the delayed value by the specified initial value and a positive number of integration time steps used for calculating the lag time. It allows creating a loop back.

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

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

:: Dynamics Double | the time start |

-> Dynamics Double | the interval width |

-> Dynamics Double | the time period |

-> Dynamics 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.

:: Dynamics Double | the slope parameter |

-> Dynamics Double | the time start |

-> Dynamics Double | the end time |

-> Dynamics Double |

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

# Financial Functions

:: Dynamics Double | the stream |

-> Dynamics Double | the discount rate |

-> Dynamics Double | the initial value |

-> Dynamics Double | factor |

-> Simulation (Dynamics 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

:: Dynamics Double | the stream |

-> Dynamics Double | the discount rate |

-> Dynamics Double | the initial value |

-> Dynamics Double | factor |

-> Simulation (Dynamics 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