Compute overlap rate for (phoneme alignment) intervals

Usage

compute_overlap_rate(x1, x2, y1, y2)

Arguments

x1, x2: start and end times for the first interval
y1, y2: start and end times for the second interval

Value

the overlap rate

Details

Paulo and Oliveira (2004) provide an "overlap rate" statistic for computing the amount of overlap between two (time) intervals. To my knowledge, nobody has described the Overlap Rate in this way, but it is the Jaccard index applied to time intervals.

Let $X=[x_\text{min}, x_\text{max}]$ and $Y=[y_\text{min}, y_\text{max}]$ be the sets of times spanned by the intervals $x$ and $y$. Then, $X \cap Y$ is the intersection or the times covered by both intervals, and $X \cup Y$ is the union or the times covered by either interval. The size of a set $A$ is denoted $|A|$. Then the overlap rate is the Jaccard index or the proportion of elements that the two sets have in common:

$$\text{overlap rate} = \frac{|X \cap Y|}{|X \cup Y|}$$

References

Paulo, S., & Oliveira, L. C. (2004). Automatic Phonetic Alignment and Its Confidence Measures. In J. L. Vicedo, P. Martínez-Barco, R. Muńoz, & M. Saiz Noeda (Eds.), Advances in Natural Language Processing (pp. 36–44). Springer. https://doi.org/10.1007/978-3-540-30228-5_4

Examples

compute_overlap_rate(
  c(0.0, 0.0, 0.0, 0.0),
  c(1.0, 1.0, 1.0,  NA),
  c(0.5, 2.0, 1.0, 1.0),
  c(2.0, 3.0, 2.0, 2.0)
)
#> [1] 0.25 0.00 0.00   NA