In previous posts we mentioned the difference between distance (dissimilarity) and similarity. Both can be used to describe proximity; i.e., how alike objects are. A distance is a metric function, while similarity is a relative judgment of proximity. A generalization of distance is the Minkowski Distance. The Euclidean Distance cannot be greater than the Manhattan or City Block Distance –named in this way because taxicab drivers in Manhattan can only go from one point to another by driving around rectangular city blocks.
We could extend on this topic and provide math arguments, explaining the Minkowski Distance or what is a metric space wherein a distance ‘live’. We could also add insult to injury and explain why many SEOs don’t have a clue when mistaking distance and similarity, or of the non-sense of talking about ‘similarity distance’ and ‘semantic distance’ in a hyperdimensional space (say ‘Hi’ to SEOs that sell LSI Snake Oil), blah, blah, blah.
Instead, this time I want to provide a real case scenario (taken from Chapter 2 of my upcoming ebook, Keyword Clustering Analysis with Excel) which would help readers and students understand the importance of properly defining distance.
In 2005, the New York Times reported that it was brought to the Court of Appeals’ attention a case wherein a man named James Robbins was accused of selling drugs within 1,000 feet of a school. He was arrested in March 2002 on the corner of Eighth Avenue and 40th Street in Manhattan and charged with selling drugs to an undercover police officer. The nearest school, Holy Cross, is on 43rd Street between Eighth and Ninth Avenues (http://www.nytimes.com/2005/11/23/nyregion/23drugs.html?_r=1&oref=slogin).
Defendant lawyers argued the distance should be measured as a pedestrian would walk city blocks; i.e using the Manhattan or City Block Distance. They claimed the school was more than 1,000 feet away by walking from the site of the arrest.
Law enforcement officials calculated the Euclidean Distance, measuring the distance up Eighth Avenue (764 feet) as one side of a right triangle, and the distance to the church along 43rd Street (490 feet) as another, to find that the length was about 908 feet.
The Court of Appeals upheld his conviction and determined that the Legislature’s intention effectively extended the boundaries of school grounds outward in order to encompass all public areas within a 1000-foot radius of the school (http://www.courts.state.ny.us/ctapps/decisions/nov05/162opn05.pdf). Read reactions to the ruling at http://volokh.com/posts/1132938765.shtml. It is a hilarious discussion.