For a set of exponential rates lam, computes the inclusion-exclusion
expansion of \(\prod_{i} (1 - e^{-\lambda_i t})\).
Value
A list with elements:
- sign
Integer vector of signs (\(+1\) or \(-1\)).
- rate_sum
Numeric vector of summed rates for each term.
Details
Returns sign and rate-sum vectors such that: $$\prod_i (1 - e^{-\lambda_i t}) = \sum_k \text{sign}_k \cdot e^{-\text{rate\_sum}_k \cdot t}$$
The number of terms is \(2^m\) where \(m\) is length(lam).
Examples
ie <- ie_expand(c(1, 2))
# Product: (1 - exp(-t)) * (1 - exp(-2t))
# = 1 - exp(-t) - exp(-2t) + exp(-3t)
ie$sign # c(1, -1, -1, 1)
#> [1] 1 -1 -1 1
ie$rate_sum # c(0, 1, 2, 3)
#> [1] 0 1 2 3