>

plGamma

Definition 11   Let $\Omega=\{X_1\}$, $\mu\in I\!\!R$, $\gamma,\beta\in I\!\!R^\ast$. A plGamma given $\Omega$, $\mu,\gamma$ and $\beta$ is a kernel with a compute function defined as follows:

$$compute(\omega) = Gamma(\omega,\gamma,\mu,\beta)$$ (3.3)

where

$$Gamma(\omega,\gamma,\mu,\beta) = \frac{\left(\frac{\omega-\mu}{\beta}\right)^{\gamma-1}\exp(-\frac{\omega-\mu}{\beta})}{\beta\Gamma(\gamma)}$$ (3.4)

with $\omega >= \mu$ and $\Gamma(\cdot)$ being the gamma function. The parameters $\gamma, \mu$ and $\beta$ are called the shape, location and scale parameters respectively.

Example 3   Here we define and test a plGamma kernel on $\Omega=\{[0.0,10]\}$ given $\gamma =5.0$, $\mu =1.0$ and $\beta =0.0$.

図 3.3: A plGamma kernel on $\Omega = \{[0.0,10.0\}$ given $\gamma =5.0$, $\mu =1.0$ and $\beta =0.0$.

The kernel is constructed as follow

plGamma Px(X,2.0,1.0,0.0);

Here is an output example

X = {x} with x in [0..10]
P(x) = plGamma(x,5,1)

Generating 5 random values
draw # 0 = { x=3.27681 } 
draw # 1 = { x=3.33849 } 
draw # 2 = { x=2.26463 } 
draw # 3 = { x=8.31799 } 
draw # 4 = { x=7.01334 } 

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

Examples of compute 
compute({ x=10 } )= 0.192942
compute({ x=9.99 } )= 0.194102
compute({ x=9.99 } )= 0.194102
compute({ x=9 } )= 0.344105
compute({ x=3.5 } )= 1.92581

The corresponding graph is shown by Figure 3.3.