On this page we show some examples of using Moore Rejection Sampling with selected targets

The function of Levy

The function of Levy is $\Re^2 \rightarrow \Re$ with

$f(x,y) = \sum_{i=1}^5 i \cos((i - 1)x + i)\sum_{j=1}^5j\cos((j + 1)y + j)$

Function of Levy on a domain [-2,2]x[-2,2]

We convert the above function into a target shape by exponentiating its negative that is scaled by a temperature parameter $T$ :

$l_{T}(x) = \exp \left( - \frac{1}{T}\sum_{i=1}^5 i \cos((i - 1)x_1 + i) \sum_{j=1}^5j\cos((j + 1)x_2 + j) \right)$

Levy Target Shape $l_{40}(x_1,x_2): [-10,10]\times [-10,10] \to \Re$ is plotted below.

Levy Target Shape on a domain [-10,10]x[-10,10]

Rosenbrock's Multi-dimensional function

Rosenbrock's $ D $ -dimensional function $\Re^2 \rightarrow \Re$ with

$f(x) = \exp \left( - \sum_{i=2}^D \left( 100(x_i - x_{i-1}^2)^2 + ( 1 - x_{i-1})^2 \right) \right)$

Rosenbrock's D=2 dimensional function on a domain [-2,2]x[-1,3]