A similarity-based approach to prediction

摘要

Assume we are asked to predict a real-valued variable y_t based on certain characteristics x_t = (x], x~d_t), and on a database consisting of (x_i~1..., x_i~d, y_i) for i = 1,..., n. Analogical reasoning suggests to combine past observations of x andy with the current values of x to generate an assessment of y by similarity-weighted averaging. Specifically, the predicted value of y, y_t~s, is the weighted average of all previously observed values y_i, where the weight of y_i, for every i = 1,..., n,is the similarity between the vector x_t~1, x_t~d, associated withyt, and the previously observed vector, x_i~1,..., x_i~d. The "empirical similarity" approach suggests estimation of the similarity function from past data. We discuss this approach as astatistical method of prediction, study its relationship to the statistical literature, and extend it to the estimation of probabilities and of density functions.