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Multi-Dimensional Approach (Multi-D)

In the multi-dimensional approach, the joint PDF is parametrized bin by bin. If the correlation is small within each multi-dimensional bin, the correlation between the input variables can be neglected and the product of the marginal distributions is a good approximation of the joint PDF. In the case of a vector of measurements $x_1,x_2, \cdots, x_N$, the marginal distributions depend on the bins $k,\ell, \cdots, n$; where $k, \ell$ and $n$ are the bin index associated with $x_1,x_2$ and $x_N$, respectively. Hence for any value of $\vec{x}$ the PDF is given by the product of Monte Carlo 1-dimensional distribution functions for the corresponding binned grid $(k,\ell, \cdots, n)$:
\begin{displaymath}
P(\vec{x})_{k,\ell, \cdots, n} = \prod_{i=1}^N p(x_i)_{k,\ell, \cdots, n}  .
\end{displaymath} (3)

In practice, one must rely on a finite number of bins so that the PDFs are defined with enough Monte Carlo statistics. Binning effects are an inherent limitation of the multi-D method.



Alain Bellerive 2006-05-19