Final answer:
In kNN, distance is calculated using metrics such as Euclidean, Manhattan, Canberra, or Bray-Curtis to determine the closeness between data points. Euclidean distance is the most common, measuring the straight-line distance in a multidimensional space.
Step-by-step explanation:
In the k-nearest neighbors (kNN) algorithm, distance is calculated to determine the closeness or similarity between data points. The choice of distance metric could be Euclidean, Manhattan, Canberra, or Bray-Curtis, among others. Euclidean distance is the most common metric used, measuring the straight-line distance between points in a multidimensional space. Manhattan distance calculates the sum of the absolute differences of their coordinates; it's like moving across a grid-based path rather than a straight line. Canberra and Bray-Curtis distances are other alternatives, the former being a weighted version of the Manhattan distance and the latter used particularly for ecological data.
To find the distance between neighbors, one must choose a distance metric that aligns with the nature of the data and then compute the distance between each data point and its potential neighbors to determine the closest ones. For example, in a two-dimensional space, the Euclidean distance between two points (x1, y1) and (x2, y2) is calculated as the square root of the sum of the squares of the differences between corresponding coordinates: sqrt((x1-x2)^2 + (y1-y2)^2).