Action: TD_CHI

Module	ves
Description	Usage
Chi distribution (static).

Details and examples

Chi distribution (static).

Employ a target distribution given by a chi distribution that is defined as

$$p(s) = \frac {2^{1-\frac{k}{2}}} {\sigma \, \Gamma\left(\frac{k}{2}\right) } \, \left(\frac{s-a}{\sigma}\right)^{k-1} \, \exp\left(- \frac{1}{2} \left(\frac{s-a}{\sigma}\right)^2\right),$$

where $a$ is the minimum of the distribution that is defined on the interval $[a,\infty)$, the parameter $k$ (given as a positive integer larger than 1) determines how far the peak of the distribution is from the minimum (known as the "degrees of freedom"), and the parameter $\sigma>0$ determines the broadness of the distribution.

The minimum $a$ is given using the MINIMUM keyword, the parameter $k$ is given using the KAPPA keyword, and the parameter $\sigma$ is given using the SIGMA keyword.

This target distribution action is only defined for one dimension, for multiple dimensions it should be used in combination with the TD_PRODUCT_DISTRIBUTION action.

Examples

Chi distribution with $a=10.0$, $\sigma=2.0$, and $k=2$

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

td: TD_CHIChi distribution (static). More details  MINIMUMThe minimum of the chi distribution=10.0  SIGMAThe sigma parameter of the chi distribution given as a positive number=2.0  KAPPAThe k parameter of the chi distribution given as positive integer larger than 1=2
The TD_CHI action with label td calculates something

The Chi distribution is only defined for one dimension so for multiple dimensions we have to use it in combination with the TD_PRODUCT_DISTRIBUTION action as shown in the following example where we have a uniform distribution for argument 1 and a Chi distribution for argument 1

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

td_uni: TD_UNIFORMUniform target distribution (static). More details

The TD_UNIFORM action with label td_uni calculates somethingtd_chi: TD_CHIChi distribution (static). More details  MINIMUMThe minimum of the chi distribution=-10.0  SIGMAThe sigma parameter of the chi distribution given as a positive number=2.0  KAPPAThe k parameter of the chi distribution given as positive integer larger than 1=2

The TD_CHI action with label td_chi calculates somethingtd_pd: TD_PRODUCT_DISTRIBUTIONTarget distribution given by a separable product of one-dimensional distributions (static or dynamic). More details DISTRIBUTIONSLabels of the one-dimensional target distribution actions for each argument to be used in the product distribution=td_uni,td_chi
The TD_PRODUCT_DISTRIBUTION action with label td_pd 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
MINIMUM	compulsory	none	The minimum of the chi distribution
SIGMA	compulsory	none	The sigma parameter of the chi distribution given as a positive number
KAPPA	compulsory	none	The k parameter of the chi distribution given as positive integer larger than 1
WELLTEMPERED_FACTORThis keyword do not have examples	optional	not used	Broaden the target distribution such that it is taken as [p(s)]^(1/gamma) where gamma is the well tempered factor given here
SHIFT_TO_ZEROThis keyword do not have examples	optional	false	Shift the minimum value of the target distribution to zero
NORMALIZEThis keyword do not have examples	optional	false	Renormalized the target distribution over the intervals on which it is defined to make sure that it is properly normalized to 1