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plDirac

Definition 16   Let $\Omega=\{X_1,X_2,...,X_n\}$ and $\omega^\ast\in \Omega$. A plDirac on $\Omega$ given $\omega^\ast$ is a kernel where the compute function is defined as follows:

$$Compute(\omega) = \left\{ \begin{array}{ll} 1 & \mbox{if } \omega=\omega^\ast \\ 0 & \mbox{otherwise} \end{array}\right.$$ (3.15)

Example 10   Given $\Omega =\{X,Y\}$, $X=[1,2,\ldots,10]$ and $Y=[-10,10]$ construct a plDirac on $\Omega$ given $\omega ^\ast =(6,7.78)$.

図 3.10: A dirac kernel on $\Omega =\{X,Y\}$, $X=[1,2,\ldots,10]$, $Y=[-10,10]$ and $\omega ^\ast =(6,7.78)$

The plDirac is constructed as follow:

  Vxy[X] = 6;
  Vxy[Y] = 7.78;

  plDirac Pxy(X^Y,Vxy);

The output of the plDirac kernel example shows as follow:

X = {x} with x in [0,1,...,10]
Y = {y} with y in [-10:10)
P(x y) =  if (x y) = (6 7.78) then 1.0 else 0.0

Generating 5 random values
draw # 0 = { x=6 y=7.78 }
draw # 1 = { x=6 y=7.78 }
draw # 2 = { x=6 y=7.78 }
draw # 3 = { x=6 y=7.78 }
draw # 4 = { x=6 y=7.78 }

Generating 5 best values
best # 0 = { x=6 y=7.78 }
best # 1 = { x=6 y=7.78 }
best # 2 = { x=6 y=7.78 }
best # 3 = { x=6 y=7.78 }
best # 4 = { x=6 y=7.78 }

Examples of compute
compute({ x=6 y=7.78 } )= 1
compute({ x=7 y=6.67 } )= 0
compute({ x=3 y=7.9 } )= 0
compute({ x=4 y=-9.7 } )= 0
compute({ x=10 y=-7.9 } )= 0