Action: TD_GENERALIZED_NORMAL

Module	ves
Description	Usage
Target distribution given by a sum of generalized normal distributions (static).

Details and examples

Target distribution given by a sum of generalized normal distributions (static).

Employ a target distribution that is given by a sum where each term is a product of one-dimensional generalized normal distributions (version 1, also know as an exponential power distribution), defined as

$$p(\mathbf{s}) = \sum_{i} \, w_{i} \prod_{k}^{d} \frac{\beta_{k,i}}{2 \, \alpha_{k,i} \, \Gamma(1/\beta_{k,i})} \exp\left( -\left\vert \frac{s_{k}-\mu_{k,i}}{\alpha_{k,i}} \right\vert^{\beta_{k,i}} \right)$$

where $(\mu_{1,i},\mu_{2,i},\ldots,\mu_{d,i})$ are the centers of the distributions, $(\alpha_{1,i},\alpha_{2,i},\ldots,\alpha_{d,i})$ are the scale parameters of the distributions, $(\beta_{1,i},\beta_{2,i},\ldots,\beta_{d,i})$ are the shape parameters of the distributions, and $\Gamma(x)$ is the gamma function. The weights $w_{i}$ are normalized to 1, $\sum_{i}w_{i}=1$.

Employing $\beta=2$ results in a Gaussian (normal) distributions with mean $\mu$ and variance $\alpha^2/2$, $\beta=1$ gives the Laplace distribution, and the limit $\beta \to \infty$ results in a uniform distribution on the interval $[\mu-\alpha,\mu+\alpha]$.

The centers $(\mu_{1,i},\mu_{2,i},\ldots,\mu_{d,i})$ are given using the numbered CENTER keywords, the scale parameters $(\alpha_{1,i},\alpha_{2,i},\ldots,\alpha_{d,i})$ using the numbered SCALE keywords, and the shape parameters $(\beta_{1,i},\beta_{2,i},\ldots,\beta_{d,i})$ using the numbered SHAPE keywords. The weights are given using the WEIGHTS keywords, if no weights are given are all terms weighted equally.

Examples

A generalized normal distribution in one-dimensional

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

td1: TD_GENERALIZED_NORMALTarget distribution given by a sum of generalized normal distributions (static). More details CENTER1The center of each generalized normal distribution=+20.0  ALPHA1The alpha parameters for each generalized normal distribution=5.0  BETA1The beta parameters for each generalized normal distribution=4.0
The TD_GENERALIZED_NORMAL action with label td1 calculates something

A sum of two one-dimensional generalized normal distributions

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

TD_GENERALIZED_NORMALTarget distribution given by a sum of generalized normal distributions (static). More details ...
 CENTER1The center of each generalized normal distribution=+20.0  ALPHA1The alpha parameters for each generalized normal distribution=5.0  BETA1The beta parameters for each generalized normal distribution=4.0
 CENTER2The center of each generalized normal distribution=-20.0  ALPHA2The alpha parameters for each generalized normal distribution=5.0  BETA2The beta parameters for each generalized normal distribution=3.0
 LABELa label for the action so that its output can be referenced in the input to other actions=td1
... TD_GENERALIZED_NORMAL
The TD_GENERALIZED_NORMAL action with label td1 calculates something

A sum of two two-dimensional generalized normal distributions

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

TD_GENERALIZED_NORMALTarget distribution given by a sum of generalized normal distributions (static). More details ...
 CENTER1The center of each generalized normal distribution=-20.0,-20.0 ALPHA1The alpha parameters for each generalized normal distribution=5.0,3.0 BETA1The beta parameters for each generalized normal distribution=2.0,4.0
 CENTER2The center of each generalized normal distribution=-20.0,+20.0 ALPHA2The alpha parameters for each generalized normal distribution=3.0,5.0 BETA2The beta parameters for each generalized normal distribution=4.0,2.0
 WEIGHTSThe weights of the generalized normal distribution=2.0,1.0
 LABELa label for the action so that its output can be referenced in the input to other actions=td1
... TD_GENERALIZED_NORMAL
The TD_GENERALIZED_NORMAL action with label td1 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
CENTER	optional	not used	The center of each generalized normal distribution
ALPHA	optional	not used	The alpha parameters for each generalized normal distribution
BETA	optional	not used	The beta parameters for each generalized normal distribution
WEIGHTS	optional	not used	The weights of the generalized normal distribution
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