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plBellShape

Definition 13   Let $\mu\in I\!\!R$ and $\sigma\in I\!\!R^\ast$, a plBellShape on $\Omega=\{X_1\}$, given $\mu$ and $\sigma$ is a kernel where the compute function is defined as follows:

$$compute(\omega) = BellShape(\omega,\mu,\sigma)$$ (3.8)

where

$$BellShape(\omega,\mu,\sigma) = \frac{\exp\left[-\frac{(\omega-\mu)^2}{2\sigma^2}\right]}{\sigma\sqrt{2\pi}}$$ (3.9)

Example 6   Here we define a plBellShape kernel on $\Omega =\{X\}$ with $X=[-30,-29,...,30]$, $\mu =0$ and $\sigma = 8.1$.

図 3.6: A bellShape kernel with $\Omega =\{X\}$, $X=[-30,-29,...,30]$, $\mu =0$ and $\sigma = 8.1$.

The construction of the plBellShape kernel is given by

  plBellShape Px(X,0,8.1);

The output of the plBellShape shows as follows:

X = {x} with x in [-30,-29,...,30]
P(x) = plBellShape(x,0,8.1)

Generating 5 random values
draw # 0 = { x=9 } 
draw # 1 = { x=-3 } 
draw # 2 = { x=5 } 
draw # 3 = { x=5 } 
draw # 4 = { x=-6 } 

Generating 5 best values
best # 0 = { x=0 } 
best # 1 = { x=0 } 
best # 2 = { x=0 } 
best # 3 = { x=0 } 
best # 4 = { x=0 } 

Examples of compute 
compute({ x=0 } )= 0.0492291
compute({ x=-30 } )= 5.2155e-05
compute({ x=29 } )= 8.17183e-05
compute({ x=30 } )= 5.2155e-05

Notice that in contrast with plNormal the compute function is defined for all value on $\omega$ (e.g. on { x=30 }).