Gaussian kernel

The output of a hidden unit, in case of a Gaussian kernel function is defined as ... 

Fig. 44.20. (a) The trained logical OR. The contour lines in the xl, x2-plane around the centres denote the 20, 40, 60 and 80 percent confidence limits of the Gaussian kernels, (b) The trained logical OR with increased width factors. 

FIGURE 4.28 Visualization of kernel regression with two Gaussian kernels. The point sizes reflect the influence on the regression model. The point sizes in the left plot are for the solid kernel function, those in the right plot are for the dashed kernel function. 

Figure 10.1 A Gaussian kernel is shown in (a), and an exponential kernel is shown in (b).

Figure 10.3 The input images are shown in the first row, the result of a convolution with a Gaussian kernel is shown in the second row, and the result of a convolution with an exponential kernel is shown in the last row.

Figure 10.18 A comparison between the computed and the actual illuminant along a horizontal line of the image (marked by a white line in Figure 10.17). Local space average color is computed using an exponential kernel, a Gaussian kernel, or a resistive grid.

Figure 11.1 Atomic resolution structure of yeast tRNAPhe (PDB accession code 1TRA), rendered as black sticks and reconstructed density (red transparent surface). The reconstructed density was generated from the filtered consensus bead model by smoothing with a Gaussian kernel. Figure adapted from Lipfert ct al. (2007b).

The Analyze software uses the Kernel PLS method [114] with two key parameters, the number of latent variables and sigma. In this study these values were fixed at 5 and 10, respectively. K-PLS uses kernels and can therefore be seen as a nonlinear extension of the PLS method. The commonly used radial basis function kernel or Gaussian kernel was applied, where the kernel is expressed as [142]... 

In the model building step, data were first mean-centered. The PCA model utilizes 5 TVs to capture 97% of the variability in the system. To monitor the process, scores plot between the 1st and 2nd TVs, and the SPE vs plots are considered. KDE with a bivariate Gaussian kernel with a smoothing parameter h = 45, is utilized to define the NOR, the SPEumu, and the 95% upper control limit is chosen for these measures, to... 

Kernels that have constant support over simple geometric regions such as parallelepipeds and ellipsoids can be used for simple bounded problems, but Gaussian kernels are more useful for localized distributions over extended spaces. These distributions... 

The problem is that the appropriate kernel can be heard to find and that a single Gaussian kernel is often not adequate. 

A single Gaussian kernel is appropriate when pB is unimodal at Thi as in Figure 1. In this case, we could substitute... 

The kernel defined by Eqs. (2.5) will be inefficient when pB is multimodal. In this situation we must dissect the conformation space into separate macrostate regions a, b, c,. .. and find Gaussian kernels that match pB in each region. This can be accomplished using window functions and characteristic packets, concepts that have previously been introduced in the context of global optimization [14,15], potential-energy landscape analysis... 

Macromolecules can pose a special problem—even after coarse-grained dissection the macrostate probability distributions at high temperatures are seldom well modeled by Gaussian kernels, even in torsion-angle space. For example, we examined arbitrarily selected macrostates of Met-enkephalin and found that ( . > a useful goodness-of-fit measure, was typically... 

Vanderbilt and Louie (Ref. 43) discuss the use of Gaussian kernels determined by Eq. (2.5) in the closely related situation of Metropolis Monte Carlo-simulated annealing. 

To generate artificial odor maps, a population of 4,000 pseudo-sensors generated from the IR spectrum is projected chemotopically onto a 10x10 SOM lattice (100 nodes). The odor images are then low-pass filtered using a 5x5 Gaussian kernel. 

The Gaussian kernel is used in potential function classifiers, also known as radial basis function networks. A sigmoid kernel implements a multilayer perceptron (cf. Section 8.2) with a single hidden layer. 

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