Action: TD_UNIFORM

Module	ves
Description	Usage
Uniform target distribution (static).

Details and examples

Uniform target distribution (static).

Using this keyword you can define a uniform target distribution which is a product of one-dimensional distributions $p_{k}(s_{k})$ that are uniform over a given interval $[a_{k},b_{k}]$

$$p_{k}(s_{k}) = \left \{\begin{array}{ll} \frac{1}{(b_{k}-a_{k})} & \mathrm{if} \ a_{k} \leq s_{k} \leq b_{k} \\ &\\ 0 & \mathrm{otherwise} \end{array}\right .$$

The overall distribution is then given as

$$p(\mathbf{s}) = \prod^{d}_{k} p_{k}(s_{k}) = \left\{\begin{array}{ll} \prod^{d}_{k} \frac{1}{(b_{k}-a_{k})} & \mathrm{if} \ a_{k} \leq s_{k} \leq b_{k} \ \mathrm{for\ all}\ k \\ \\ 0 & \mathrm{otherwise} \end{array}\right.$$

The distribution is thus uniform inside a rectangular for two arguments and a cube for a three arguments.

The limits of the intervals $a_{k}$ and $b_{k}$ are given with the MINIMA and MAXIMA keywords, respectively. If one or both of these keywords are missing the code should automatically detect the limits.

It is also possible to use one-dimensional distributions that go smoothly to zero at the boundaries. This is done by employing a function with Gaussian switching functions at the boundaries $a_{k}$ and $b_{k}$

$$f_{k}(s_{k}) = \begin{cases} \exp\left(-\frac{(s_{k}-a_{k})^2}{2 \sigma^2_{a,k}}\right) & \mathrm{if}\, s_{k} < a_{k} \\ \\ 1 & \mathrm{if}\, a_{k} \leq s_{k} \leq b_{k} \\ \\ \exp\left(-\frac{(s_{k}-b_{k})^2}{2 \sigma^2_{b,k}}\right) & \mathrm{if}\, s_{k} > b_{k} \end{cases}$$

where the standard deviation parameters $\sigma_{a,k}$ and $\sigma_{b,k}$ determine how quickly the switching functions goes to zero. The overall distribution is then normalized

$$p(\mathbf{s}) = \prod^{d}_{k} p_{k}(s_{k}) = \prod^{d}_{k} \frac{f(s_{k})}{\int d s_{k} \, f(s_{k})}$$

To use this option you need to provide the standard deviation parameters $\sigma_{a,k}$ and $\sigma_{b,k}$ by using the SIGMA_MINIMA and SIGMA_MAXIMA keywords, respectively. Giving a value of 0.0 means that the boundary is sharp, which is the default behavior.

Examples

If one or both of the MINIMA or MAXIMA keywords are missing the code should automatically detect the limits not given. Therefore, if we consider a target distribution that is defined over an interval from 0.0 to 10.0 for the first argument and from 0.2 to 1.0 for the second argument are the following example

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td: TD_UNIFORMUniform target distribution (static). More details
The TD_UNIFORM action with label td calculates something

is equivalent to this one

Click on the labels of the actions for more information on what each action computes

TD_UNIFORMUniform target distribution (static). More details ...
 MINIMAThe minimum of the intervals where the target distribution is taken as uniform=0.0,0.2
 MAXIMAThe maximum of the intervals where the target distribution is taken as uniform=10.0,1.0
 LABELa label for the action so that its output can be referenced in the input to other actions=td
 ... TD_UNIFORM
The TD_UNIFORM action with label td calculates something

and this one

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td: TD_UNIFORMUniform target distribution (static). More details  MAXIMAThe maximum of the intervals where the target distribution is taken as uniform=10.0,1.0
The TD_UNIFORM action with label td calculates something

and also this one

Click on the labels of the actions for more information on what each action computes

td: TD_UNIFORMUniform target distribution (static). More details MINIMAThe minimum of the intervals where the target distribution is taken as uniform=0.0,0,2
The TD_UNIFORM action with label td calculates something

We can also define a target distribution that goes smoothly to zero at the boundaries of the uniform distribution. In the following we consider an interval of 0 to 10 for the target distribution. The following input would result in a target distribution that would be uniform from 2 to 7 and then smoothly go to zero from 2 to 0 and from 7 to 10.

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TD_UNIFORMUniform target distribution (static). More details ...
 MINIMAThe minimum of the intervals where the target distribution is taken as uniform=2.0
 MAXIMAThe maximum of the intervals where the target distribution is taken as uniform=+7.0
 SIGMA_MINIMAThe standard deviation parameters of the Gaussian switching functions for the minima of the intervals=0.5
 SIGMA_MAXIMAThe standard deviation parameters of the Gaussian switching functions for the maximum of the intervals=1.0
 LABELa label for the action so that its output can be referenced in the input to other actions=td
... TD_UNIFORM
The TD_UNIFORM action with label td calculates something

It is also possible to employ a smooth switching function for just one of the boundaries as shown here where the target distribution would be uniform from 0 to 7 and then smoothly go to zero from 7 to 10.

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TD_UNIFORMUniform target distribution (static). More details ...
 MAXIMAThe maximum of the intervals where the target distribution is taken as uniform=+7.0
 SIGMA_MAXIMAThe standard deviation parameters of the Gaussian switching functions for the maximum of the intervals=1.0
 LABELa label for the action so that its output can be referenced in the input to other actions=td
... TD_UNIFORM
The TD_UNIFORM action with label td calculates something

Furthermore, it is possible to employ a sharp boundary by using

Click on the labels of the actions for more information on what each action computes

TD_UNIFORMUniform target distribution (static). More details ...
 MAXIMAThe maximum of the intervals where the target distribution is taken as uniform=+7.0
 SIGMA_MAXIMAThe standard deviation parameters of the Gaussian switching functions for the maximum of the intervals=0.0
 LABELa label for the action so that its output can be referenced in the input to other actions=td
... TD_UNIFORM
The TD_UNIFORM action with label td calculates something

or

Click on the labels of the actions for more information on what each action computes

td: TD_UNIFORMUniform target distribution (static). More details MAXIMAThe maximum of the intervals where the target distribution is taken as uniform=+7.0
The TD_UNIFORM action with label td calculates something

Full list of keywords

The following table describes the keywords and options that can be used with this action

Keyword	Type	Default	Description
MINIMA	optional	not used	The minimum of the intervals where the target distribution is taken as uniform
MAXIMA	optional	not used	The maximum of the intervals where the target distribution is taken as uniform
SIGMA_MINIMA	optional	not used	The standard deviation parameters of the Gaussian switching functions for the minima of the intervals
SIGMA_MAXIMA	optional	not used	The standard deviation parameters of the Gaussian switching functions for the maximum of the intervals