next up previous contents index
Next: Functional random variables Up: PDF discretisation Previous: Sampling - Fixed intervals   Contents   Index

Approximation by boxes - Variable intervals

For many PDFs, variable interval sizes allow a better approximation using less values compared to fixed ones. This is especially useful if the function consists of large, almost constant parts and sharp peaks on the other hand. Variable interval sizes allow the specification of almost constant areas by one large interval and the use of multiple, fine grained intervals for sharp peaks, which need to be described in more detail.

We have a set of intervals $ I$ so that for each two intervals $ J_1, J_2 \in I$, $ J_1 \neq J_2$ the disjunction is the empty set $ J_1 \cap J_2 = \emptyset$ and the union of all intervals forms a new interval from zero to $ x \in \mathbb{R}^+$, $ \cup_{J \in I} = [0, x[$. Intuitively, this means that the intervals do not overlap and that there are no gaps between the intervals.

To ensure both properties mentioned above, the intervals are specified by their right hand value only. Thus, we have a set $ I_X$ whose values define the right hand sides of all intervalls. Suppose we can define an order on the set such that $ x_1 < x_2 < \ldots < x_{n-1} < x_n$. Then the $ i$th interval is $ [x_{i-1}, x_i[$ for $ i > 1$ and $ [0, x_1[$ for $ i = 1$. This allows us to specify $ n$ intervals by $ n$ values only and to ensure that the intervals neither do overlap nor have gaps inbetween. Now the probability $ p_i$ for the $ i$th interval is given by

$\displaystyle p_i = \int_{x_{i-1}}^b f(x) dx, \qquad lim_{b \rightarrow x_i}
$

for $ i > 1$ and

$\displaystyle p_i = \int_{0}^b f(x) dx, \qquad lim_{b \rightarrow x_1}
$

for $ i = 1$.


next up previous contents index
Next: Functional random variables Up: PDF discretisation Previous: Sampling - Fixed intervals   Contents   Index
Snowball 2007-03-16