Ranking algorithm with missing values and bias

Question

The problem is :

A set of 5 independent users where asked to rate 50 products given to them. All 50 products would have been used by the users in some point of time. Some users have more bias towards certain products. One user did not truly complete the survey and gave random values. It is not necessary for the users to rate all the products. Now given a 4 sample dataset , rank the products based on ratings

datset :
product #user1 #user2 #user3 #user4 #user5
 0        29    -        10   90     12 
 1         -    -         -    -      7
 2         -    -        95    6      1
 3         -    -         -    -      2
 4         -    -         -    -     50
 5         -    35       21    13     -
 6         -     -        -     -     5
 7         4     -        -    30     -
 8        11     -        -     -    14  
 .
 .
 .

How to come out with a ranking for the products.

This is a remodeled problem very close to the original problem.

Solution: I tried to clean the data and fill missing values using PCA and apply NMF but i'm not sure about the solution .

Any help will be deeply appreciated

Answer 1

If you don't care about the absolute scores and are mostly interested in a consistent relative ranking, you can view your problem as an instance of the rank aggregation problem : given a list of (partial or total) rankings, derive a consensus ranking that minimizes the total disagreement with the input rankings. There are several possible ways to formalize disagreement, and to postulate reasonable conditions that should hold. One example of such a condition is the Condorcet criterion: If an item defeats every other item in simple pairwise majority voting, then it should rank first.

This excellent paper contains a good motivation and literature review of consensus ranking approaches. Kemeny optimal aggregation minimizes Kendall-Tau distance, ie, the total count of pairwise disagreements between lists. While this optimal aggregation is NP-hard, the authors propose reasonable heuristic approaches.

Answer 2

In this case, two imputation methods can be used:

• As everyone would try at first, fill with the most likely value ie average mean.
• Predict based on other attributes which is called imputation by regression.

Actually, I think the second method seems better for this dataset where users mostly rank more than one product.

Also, if you have another datasets depending on users, you may use it too for prediction of the missing values in this dataset.