Noise Class

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Latest SysCAD Version: 19 March 2024 - SysCAD 9.3 Build 139.35102

Related Links: Defining a Class, Example PGM Files

Related Links: Noise Controller


Description

The Noise class can be used to add noise to a value.

  • All random number distributions use a single random number generator.
  • If Init or SetType are not called, it is assumed that type 0 is required (Gaussian). If Init or SetGlobal are not called, it is assumed that a global random generator must be used.
  • If the NoiseType specified is outside the specified range of methods available, then the nearest number will be used. e.g. if a large number is specified, then the last method available (highest method number) will be used.

NOTE: The Controller -- Noise model, is superseding this topic. However, this form is still available for users familiar with the PGM coding style.

Data Members

NOTE: The meaning of the data members Mean and StdDev depend on the Noise class Type! See table below.

Mean: For Gaussiana nd Flat Distributions, this is the value around which the random number must be generated. See table below for its meaning in other distributions. The default is 0.0.

StdDev: The Standard Deviation for Gaussian distribution (68.27% of the numbers generated will be between (Mean - StdDev) and (Mean + StdDev). For the Flat distribution StdDev represents the minimum and maximum distance from the Mean. See table below for its meaning in other distributions. The default is 1.0.

Output: The last value generated by GetVal. This is a constant and cannot be set.

Generator Type Mean parameter StdDev parameter
Gaussian Mean Standard deviation
Flat Mean Maximum deviation
Poisson Mean Not used
Gamma Alpha Beta
Weibull Alpha Beta
Bernoulli Probability, p Not used
Binomial n Probability, p
Geometric Probability, p Not used
Exponential Mean Not used
Extreme Value Alpha Beta
Log Normal m s
Cauchy Alpha Beta
Fisher F m n
Student t n Not used

Member Functions

The examples in the table below are based on the following variables and 'Noise Class' declarations.

Noise a ;declares 'noise' instances
Real d
Call Functionality Return Type Parameters Example

Init (Mean, StdDev, NoiseType, Global, Seed )

This can be used to set all the parameters. The random number generator is initialised and reset.

Mean :Refer to description above
StdDev :Refer to description above
NoiseType :Refer to SetType function below' Global :Refer to SetGlobal function below
Seed :Refer to SetSeed function below

a.Init(5.0,1.0,1,0,3)

Setup (Mean, StdDev, NoiseType, Global)

This function can be used to set all the parameters. The random number generator is initialised and reset.

Mean :Refer to description above
StdDev :Refer to description above
NoiseType :Refer to SetType function below' Global :Refer to SetGlobal function below

a.Setup(4.0,0.9,4,1)

SetType (NoiseType)

This sets the random number generator type.

NoiseType: Is a 'byte' field which specifies the noise generator type required. See below the table for a description of 'NoiseTypes'.

a.SetType(6)

SetGlobal (Global)

This sets the global option. Must be used before the function SetSeed!

Global: Is a 'bit' field which specifies if the random number must be obtained from a shared global random generator or from its own random number generator.

a.SetGlobal(1)

SetSeed (Seed)

This function sets the seed value.

  • This is typically a negative prime integer.
  • If the global option is not set and the same seed is used, the same sequence of random numbers will be generated.
  • If SetGlobal is called after this the seed setting may be discarded.

Seed: Is a data type 'long' field which specifies the seed value for generating the first random number.

a.SetSeed(5)

GetVal ()

This will return the value generated by the random number generator.

  • The method used will depend on the type of distribution selected.
  • The value returned depends on the value of Mean and StdDev.
  • The value returned is equal to the data member Output.

Real

None

d = a.GetVal()

Noise Types

Descriptions of the NoiseType are shown in the table below.

Field Value Generator Type
0 Gaussian generator
1 Gaussian generator
2 Flat generator
3 Flat generator
4 Poisson generator
5 Poisson generator
6 Gamma generator
7 Gamma generator
8 Weibull generator
9 Weibull generator
10 Bernoulli generator
11 Binomial generator
12 Geometric generator
13 Exponential generator
14 Extreme Value generator
15 Log Normal generator
16 Cauchy generator
17 Fisher F generator
18 Student t generator

Notes:

  • The Gaussian random noise generator's output characteristic is the "Bell curve" where 68.27% of the numbers generated are within +/- one StdDev from the Mean.
  • The Flat random noise generator's output characteristic is flat where random numbers are generated between +/- StdDev from the Mean with equal probability.

Probability Density Functions

The random number generator used for all probability functions is from the C++ standard library (http://www.cplusplus.com/reference/random/).

The Probability Density functions used for the Noise generation are shown in the following table. In all cases the following convention is used:

[math]\displaystyle{ \mu }[/math] = Mean Value
[math]\displaystyle{ \sigma }[/math] = Standard Deviation

For all methods, the tag PlantModel.RandomSeedAtStart (set on the Plant Model - FlwSolve page) affects the repeatability of the random values generated. For the same seed value, a repeatable set of random numbers will be generated. For a different seed, a different set of random numbers will be generated. If "*" is used, then a new random set of values will be generated on every run.

Type Probability Density Function P(t) NOTES
Gaussian
  • [math]\displaystyle{ P(t) = \cfrac{1}{\sigma \sqrt{2\pi}}\cdot e^{\dfrac{-(t-\mu)^2}{2\sigma^2}} }[/math]
where [math]\displaystyle{ \sigma \gt 0 }[/math]
Flat
  • [math]\displaystyle{ P(t) = 0 }[/math]
where [math]\displaystyle{ t \lt \mu - m }[/math] or [math]\displaystyle{ t \gt \mu + m }[/math]
  • [math]\displaystyle{ P(t) = \dfrac{1}{2\sigma} }[/math]

where [math]\displaystyle{ \mu - m ≤ t ≤; \mu + m }[/math] and [math]\displaystyle{ m ≥; 0 }[/math] and [math]\displaystyle{ m }[/math] = maximum deviation

Poisson
  • [math]\displaystyle{ P(t) = \cfrac{\mu^t}{t!}\ e^{-\mu} }[/math]
where [math]\displaystyle{ e }[/math] = the base of the natural logarithm system (2.71828...) and [math]\displaystyle{ \mu \gt 0 }[/math]
Gamma
  • [math]\displaystyle{ P(t) = \cfrac{1}{\Gamma(\alpha)\ \beta^{\alpha}}\ t^{\alpha-1}\ e^{-t/\beta} }[/math]

where [math]\displaystyle{ \alpha ≥ 0.5 }[/math] and [math]\displaystyle{ \beta \gt 0 }[/math]

Note: The Chi-squared distribution is a special case of the Gamma distribution
with [math]\displaystyle{ \alpha = n/2 }[/math] and [math]\displaystyle{ \beta = 2 }[/math], where [math]\displaystyle{ n }[/math] = degrees of freedom

Weibull
  • [math]\displaystyle{ P(t) = \alpha^{-\beta}\ \beta\ t^{\beta-1}\ e^{-(t/\alpha)^\beta} }[/math]

where [math]\displaystyle{ t = \alpha (-\ln(U))^{1/\beta} }[/math] and [math]\displaystyle{ \alpha ≥ 0.04 }[/math] and [math]\displaystyle{ \beta ≥ 0.01 }[/math]
[math]\displaystyle{ U }[/math] is a randomly generated number between 0 and 1

Bernoulli
  • [math]\displaystyle{ P(t) = p^{t}\ (1-p)^{1-t} }[/math]

where [math]\displaystyle{ 0 ≤ p ≤ 1 }[/math]

Binomial
  • [math]\displaystyle{ P(t) = \dbinom{n}{t} p^{t}\ (1-p)^{n-t} }[/math]

where [math]\displaystyle{ n }[/math] is a positive integer, [math]\displaystyle{ \dbinom{n}{t} = \cfrac{n!}{t!(n-t)!} }[/math] and [math]\displaystyle{ 0 \lt p ≤ 1 }[/math]

Geometric
  • [math]\displaystyle{ P(t) = p\ (1-p)^{t} }[/math]

where [math]\displaystyle{ 0 \lt p ≤ 1 }[/math]

Exponential
  • [math]\displaystyle{ P(t) = \mu\ e^{-\mu\ t} }[/math]

where [math]\displaystyle{ \mu \gt 0 }[/math]

Extreme Value
  • [math]\displaystyle{ P(t) = \cfrac{1}{\beta}\ z(t)\ e^{-z(t)} }[/math]
where [math]\displaystyle{ z(t) = e^{(\alpha-t)/\beta} }[/math] and [math]\displaystyle{ \beta \gt 0 }[/math]
Log Normal
  • [math]\displaystyle{ P(t) = \cfrac{1}{s\ t\ \sqrt{2\pi;}}\ e^{-(\ln t-m)^2/2s^2} }[/math]
where [math]\displaystyle{ 0 \lt s ≤ 50.70 }[/math]
Cauchy
  • [math]\displaystyle{ P(t) = \cfrac{1}{\pi;\ \beta\ \left [1+\left (\cfrac{t-\alpha}{\beta} \right )^2 \right ]} }[/math]
where [math]\displaystyle{ \beta \gt 0 }[/math]
Fisher F
  • [math]\displaystyle{ P(t) = \cfrac{\Gamma; \left (\cfrac{m+n}{2} \right )}{\Gamma;\left (\cfrac{m}{2}\right )\ \Gamma;\left (\cfrac{n}{2}\right )}\ \frac{\left (\cfrac{m\ t}{n} \right )^{(m/2)}}{t\ \left (1+\cfrac{m\ t}{n} \right )^{(m+n)/2}} }[/math]
where [math]\displaystyle{ 1 ≤ m ≤ 100 }[/math] and [math]\displaystyle{ n ≥ 1 }[/math]
Student t
  • [math]\displaystyle{ P(t) = \cfrac{1}{\sqrt{n\ \pi;}}\ \cfrac{\Gamma;\left (\cfrac{n+1}{2} \right )}{\Gamma;\left (\cfrac{n}{2} \right )}\ \left (1+\cfrac{t^2}{n} \right )^{-(n+1)/2} }[/math]
where [math]\displaystyle{ n ≥ 0.25 }[/math]

References

  1. Press W.H, Teukolsky S.A, Vetterling W.T, Flannery B.P Numerical Recipes in C (2nd Edition) Cambridge University Press 1992.
  2. Sanders D.H Statistics A Fresh Approach McGraw-Hill 1990.
  3. http://www.cplusplus.com/reference/random/ and related links describing each distribution.