[SPARK-26351][mllib]Update doc and minor correction in the mllib evaluation metrics

docs/mllib-evaluation-metrics.md

@@ -413,13 +413,13 @@ A ranking system usually deals with a set of $M$ users
 
 $$U = \left\{u_0, u_1, ..., u_{M-1}\right\}$$
 
-Each user ($u_i$) having a set of $N$ ground truth relevant documents
+Each user ($u_i$) having a set of $N_i$ ground truth relevant documents
 
-$$D_i = \left\{d_0, d_1, ..., d_{N-1}\right\}$$
+$$D_i = \left\{d_0, d_1, ..., d_{N_i-1}\right\}$$
 
-And a list of $Q$ recommended documents, in order of decreasing relevance
+And a list of $Q_i$ recommended documents, in order of decreasing relevance
 
-$$R_i = \left[r_0, r_1, ..., r_{Q-1}\right]$$
+$$R_i = \left[r_0, r_1, ..., r_{Q_i-1}\right]$$
 
 The goal of the ranking system is to produce the most relevant set of documents for each user. The relevance of the
 sets and the effectiveness of the algorithms can be measured using the metrics listed below.
@@ -439,21 +439,21 @@ $$rel_D(r) = \begin{cases}1 & \text{if $r \in D$}, \\ 0 & \text{otherwise}.\end{
         Precision at k
       </td>
       <td>
-        $p(k)=\frac{1}{M} \sum_{i=0}^{M-1} {\frac{1}{k} \sum_{j=0}^{\text{min}(\left|D\right|, k) - 1} rel_{D_i}(R_i(j))}$
+        $p(k)=\frac{1}{M} \sum_{i=0}^{M-1} {\frac{1}{k} \sum_{j=0}^{\text{min}(Q_i, k) - 1} rel_{D_i}(R_i(j))}$
       </td>
       <td>
-        <a href="https://en.wikipedia.org/wiki/Information_retrieval#Precision_at_K">Precision at k</a> is a measure of
+        <a href="https://en.wikipedia.org/wiki/Evaluation_measures_(information_retrieval)#Precision_at_K">Precision at k</a> is a measure of
          how many of the first k recommended documents are in the set of true relevant documents averaged across all
          users. In this metric, the order of the recommendations is not taken into account.
       </td>
     </tr>
     <tr>
       <td>Mean Average Precision</td>
       <td>
-        $MAP=\frac{1}{M} \sum_{i=0}^{M-1} {\frac{1}{\left|D_i\right|} \sum_{j=0}^{Q-1} \frac{rel_{D_i}(R_i(j))}{j + 1}}$
+        $MAP=\frac{1}{M} \sum_{i=0}^{M-1} {\frac{1}{N_i} \sum_{j=0}^{Q_i-1} \frac{rel_{D_i}(R_i(j))}{j + 1}}$
       </td>
       <td>
-        <a href="https://en.wikipedia.org/wiki/Information_retrieval#Mean_average_precision">MAP</a> is a measure of how
+        <a href="https://en.wikipedia.org/wiki/Evaluation_measures_(information_retrieval)#Mean_average_precision">MAP</a> is a measure of how
          many of the recommended documents are in the set of true relevant documents, where the
         order of the recommendations is taken into account (i.e. penalty for highly relevant documents is higher).
       </td>
@@ -462,10 +462,10 @@ $$rel_D(r) = \begin{cases}1 & \text{if $r \in D$}, \\ 0 & \text{otherwise}.\end{
       <td>Normalized Discounted Cumulative Gain</td>
       <td>
         $NDCG(k)=\frac{1}{M} \sum_{i=0}^{M-1} {\frac{1}{IDCG(D_i, k)}\sum_{j=0}^{n-1}
-          \frac{rel_{D_i}(R_i(j))}{\text{ln}(j+2)}} \\
+          \frac{rel_{D_i}(R_i(j))}{\text{log}(j+2)}} \\
         \text{Where} \\
-        \hspace{5 mm} n = \text{min}\left(\text{max}\left(|R_i|,|D_i|\right),k\right) \\
-        \hspace{5 mm} IDCG(D, k) = \sum_{j=0}^{\text{min}(\left|D\right|, k) - 1} \frac{1}{\text{ln}(j+2)}$
+        \hspace{5 mm} n = \text{min}\left(\text{max}\left(Q_i, N_i\right),k\right) \\
+        \hspace{5 mm} IDCG(D, k) = \sum_{j=0}^{\text{min}(\left|D\right|, k) - 1} \frac{1}{\text{log}(j+2)}$
       </td>
       <td>
         <a href="https://en.wikipedia.org/wiki/Discounted_cumulative_gain#Normalized_DCG">NDCG at k</a> is a

mllib/src/main/scala/org/apache/spark/mllib/evaluation/RankingMetrics.scala

@@ -138,6 +138,8 @@ class RankingMetrics[T: ClassTag](predictionAndLabels: RDD[(Array[T], Array[T])]
         var dcg = 0.0
         var i = 0
         while (i < n) {
+          // Base of the log doesn't matter for calculating NDCG,
+          // if the relevance value is binary.
           val gain = 1.0 / math.log(i + 2)
           if (i < pred.length && labSet.contains(pred(i))) {
             dcg += gain