NDCGk

NDCGk, or normalized discounted cumulative gain at rank k, is a widely used metric for evaluating ranked lists when items have graded relevance. It measures how well a ranking matches the true relevance ordering, up to a cutoff position k, and is normalized to a maximum of 1 for an ideal ranking.

The core component is DCGk, defined as the sum of gains discounted by position: DCGk = sum_{i=1}^k (2^{rel_i} - 1) / log2(i + 1). Here rel_i is the relevance grade assigned to the item at position i. The term (2^{rel_i} - 1) provides higher rewards for more relevant items, while the log2(i + 1) factor reduces the contribution of items appearing later in the list.

IDCGk is the ideal DCG at rank k, obtained by arranging the items in nonincreasing order of their relevance and computing DCG on this best possible ranking. NDCGk is then the ratio NDCGk = DCGk / IDCGk, yielding a value between 0 and 1, where 1 indicates a perfect ranking relative to the given judgments.

Practical considerations include the choice of k, as it determines how much of the ranking is evaluated, and the relevance scale, since the gains depend on the grading scheme. The common convention uses log base 2 for discounting and geometric gains via 2^{rel} - 1, though variants exist. Handling incomplete or noisy relevance judgments is also important in real datasets.

NDCGk is widely used in information retrieval, search engine evaluation, and recommender systems to assess ranking quality, especially when graded relevance is available. It provides a robust, interpretable measure that emphasizes early, highly relevant results.