Action: TD_EXPONENTIALLY_MODIFIED_GAUSSIAN

Module	ves
Description	Usage
Target distribution given by a sum of exponentially modified Gaussian distributions (static).

Details and examples

Target distribution given by a sum of exponentially modified Gaussian distributions (static).

Employ a target distribution that is given by a sum where each term is a product of one-dimensional exponentially modified Gaussian distributions,

$$p(\mathbf{s}) = \sum_{i} \, w_{i} \prod_{k}^{d} \frac{\lambda_{k,i}}{2} \, \exp\left[ \frac{\lambda_{k,i}}{2} (2 \mu_{k,i} + \lambda_{k,i} \sigma_{k,i}^2 -2 s_{k}) \right] \, \mathrm{erfc}\left[ \frac{\mu_{k,i} + \lambda_{k,i} \sigma_{k,i}^2 - s_{k})}{\sqrt{2} \sigma_{k,i}} \right]$$

where $(\mu_{1,i},\mu_{2,i},\ldots,\mu_{d,i})$ are the centers of the Gaussian component, $(\sigma_{1,i},\sigma_{2,i},\ldots,\sigma_{d,i})$ are the standard deviations of the Gaussian component, $(\lambda_{1,i},\lambda_{2,i},\ldots,\lambda_{d,i})$ are the rate parameters of the exponential component, and $\mathrm{erfc}(x)=1-\mathrm{erf}(x)$ is the complementary error function. The weights $w_{i}$ are normalized to 1, $\sum_{i}w_{i}=1$.

The centers $(\mu_{1,i},\mu_{2,i},\ldots,\mu_{d,i})$ are given using the numbered CENTER keywords, the standard deviations $(\sigma_{1,i},\sigma_{2,i},\ldots,\sigma_{d,i})$ using the the numbered SIGMA keywords, and the rate parameters $(\lambda_{1,i},\lambda_{2,i},\ldots,\lambda_{d,i})$ using the numbered LAMBDA keywords. The weights are given using the WEIGHTS keywords, if no weights are given are all terms weighted equally.

Examples

An exponentially modified Gaussian distribution in one-dimension

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

td1: TD_EXPONENTIALLY_MODIFIED_GAUSSIANTarget distribution given by a sum of exponentially modified Gaussian distributions (static). More details CENTER1The center of each exponentially modified Gaussian distributions=-10.0 SIGMA1The sigma parameters for each exponentially modified Gaussian distributions=1.0 LAMBDA1The lambda parameters for each exponentially modified Gaussian distributions=0.25
The TD_EXPONENTIALLY_MODIFIED_GAUSSIAN action with label td1 calculates something

A sum of two one-dimensional exponentially modified Gaussian distributions

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

TD_EXPONENTIALLY_MODIFIED_GAUSSIANTarget distribution given by a sum of exponentially modified Gaussian distributions (static). More details ...
 CENTER1The center of each exponentially modified Gaussian distributions=-10.0 SIGMA1The sigma parameters for each exponentially modified Gaussian distributions=1.0 LAMBDA1The lambda parameters for each exponentially modified Gaussian distributions=0.5
 CENTER2The center of each exponentially modified Gaussian distributions=+10.0 SIGMA2The sigma parameters for each exponentially modified Gaussian distributions=1.0 LAMBDA2The lambda parameters for each exponentially modified Gaussian distributions=1.0
 WEIGHTSThe weights of the distributions=2.0,1.0
 LABELa label for the action so that its output can be referenced in the input to other actions=td1
... TD_EXPONENTIALLY_MODIFIED_GAUSSIAN
The TD_EXPONENTIALLY_MODIFIED_GAUSSIAN action with label td1 calculates something

A sum of two two-dimensional exponentially modified Gaussian distributions

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

TD_EXPONENTIALLY_MODIFIED_GAUSSIANTarget distribution given by a sum of exponentially modified Gaussian distributions (static). More details ...
 CENTER1The center of each exponentially modified Gaussian distributions=-5.0,+5.0 SIGMA1The sigma parameters for each exponentially modified Gaussian distributions=1.0,1.0 LAMBDA1The lambda parameters for each exponentially modified Gaussian distributions=0.5,0.5
 CENTER2The center of each exponentially modified Gaussian distributions=+5.0,+5.0 SIGMA2The sigma parameters for each exponentially modified Gaussian distributions=1.0,1.0 LAMBDA2The lambda parameters for each exponentially modified Gaussian distributions=1.0,1.0
 WEIGHTSThe weights of the distributions=1.0,1.0
 LABELa label for the action so that its output can be referenced in the input to other actions=td1
... TD_EXPONENTIALLY_MODIFIED_GAUSSIAN
The TD_EXPONENTIALLY_MODIFIED_GAUSSIAN 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 exponentially modified Gaussian distributions
SIGMA	optional	not used	The sigma parameters for each exponentially modified Gaussian distributions
LAMBDA	optional	not used	The lambda parameters for each exponentially modified Gaussian distributions
WEIGHTS	optional	not used	The weights of the distributions
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