Adaptive Gaussians

In addition, the problem of selecting an adequate a is tightly coupled with the definition of the CVs. For small molecules binding to a protein, coordination numbers are often used. These have a sigmoidal functional form that is tailored to change from one to zero when the small molecule is unbinding, but remains constant at values close to zero when the substrate is unbound. In this last case the free dynamics will provide very small fluctuations in CV toward zero, compared to that calculated for the bound state where the value is changing. 

An additional intricacy to be considered is that two variables are often coupled as in the case of the two Ramachandran dihedral angles 4 and T. These angles contain three atoms in common. Therefore, some sort of coupling should be introduced into the scheme to account for their interdependence. 

We recently proposed a more general multivariate Gaussian instead of Eq. [11]  

Here a and runs over the biased CVs and (t is a matrix that, unlike the standard MetaD case, might contain nonzero diagonal elements. Furthermore, in 

In the DA scheme the a matrix is determined with an estimate over the space traveled in CV space  

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The NLME function in S-Plus offers three different estimation algorithms a FOCE algorithm similar to NONMEM, adaptive Gaussian quadrature, and Laplacian approximation. The FOCE algorithm in S-Plus, similar to the one in NONMEM, was developed by Lindstrom and Bates (1990). The algorithm is predicated on normally distributed random effects and normally distributed random errors and makes a first-order Taylor series approximation of the nonlinear mixed effects model around both the current parameter estimates 0 and the random effects t). The adaptive Gaussian quadrature and Laplacian options are similar to the options offered by SAS. 

Branduardi, D., Bussi, G., Parrinello, M. Metadynamics with adaptive Gaussians. JCTC 8, 2247-2254 (2012)... 

Blur the original video frame using region adaptive Gaussian low pass filtering... 

Let gi(jc, y, n) and /(x, y, n) denote the blurred video sequence using region adaptive Gaussian mask and the original video sequence respectively. Now the region adaptive unsharp mask is given by ... 

Fig. 6 Fuzzy weighted, region adaptive Gaussian mask...

The original video frame is blurred by using the fuzzy weighted region adaptive Gaussian mask. 

Douglas A.Reynolds, Thomas F. Quatieri and Robest B. Dunn (2000 January/April/July). Speaker Verification Using Adapted Gaussian Mixture Models. Pages 19-41. 

Moreover, it is worth noting that the chosen width for one region may not be optimal for other regions. This is particularly important with those CVs that present an anisotropic compression of phase space (see below in Section Adaptive Gaussians for a discussion of this issue and a possible solution). 

D. Branduardi, G. Bussi, and M. Parrinello, J. Chem. Theory Comput., 8(7), 2247-2254 (2012). Metadynamics with Adaptive Gaussians. 

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