textreuse (version 0.1.2)

lsh_probability: Probability that a candidate pair will be detected with LSH

Description

Functions to help choose the correct parameters for the lsh and minhash_generator functions. Use lsh_threshold to determine the minimum Jaccard similarity for two documents for them to likely be considered a match. Use lsh_probability to determine the probability that a pair of documents with a known Jaccard similarity will be detected.

Usage

lsh_probability(h, b, s)
lsh_threshold(h, b)

Details

Locality sensitive hashing returns a list of possible matches for similar documents. How likely is it that a pair of documents will be detected as a possible match? If h is the number of minhash signatures, b is the number of bands in the LSH function (implying then that the number of rows r = h / b), and s is the actual Jaccard similarity of the two documents, then the probability p that the two documents will be marked as a candidate pair is given by this equation.

$$p = 1 - (1 - s^{r})^{b}$$

According to http://infolab.stanford.edu/~ullman/mmds/book.pdf{MMDS}, that equation approxmiates an S-curve. This implies that there is a threshold (t) for s approximated by this equation.

$$t = \frac{1}{b}^{\frac{1}{r}}$$

References

Jure Leskovec, Anand Rajaraman, and Jeff Ullman, http://www.mmds.org/#book{Mining of Massive Datasets} (Cambridge University Press, 2011), ch. 3.

Examples

# Threshold for default values
lsh_threshold(h = 200, b = 40)

# Probability for varying values of s
lsh_probability(h = 200, b = 40, s = .25)
lsh_probability(h = 200, b = 40, s = .50)
lsh_probability(h = 200, b = 40, s = .75)