Equilibrium ranking with a bias towards the target metric
Formulation of the problem:
There is a discrete set of objects. And also a set of signs for them. We assume that the signs are numerical. It is necessary to find the best object or group of objects only based on the available characteristics.
Let's make a small digression. Many can already say at this point that such problems are solved using the collaborative filtering method. And in general they will be right. But there are cases when filtering is not suitable or is not enough. For example, let's imagine ourselves in the role of a car salesman who is thinking about what new brand or model of car he should start selling. Let's say he has 1000 options to choose from. And here it becomes clear that the idea of collaborative filtering does not fit very well into this case. The seller wants to make a choice not based on the preferences of other sellers, but based on certain characteristics that determine the benefit of the object.
The bottom line is that we have n features. What needs to be done with them to achieve what you want? You can sum up the value of all the features for an object and get a final score that reflects the cumulative total of all knowledge about the object. But what's wrong with such a simple approach?
Issues
All features are expressed at different scales and taken from different distributions. We find that some of them will make a greater contribution to the final grade. Standardization will not change the type of distribution and will not solve the problem of different influences of characteristics. Switching to ranks could solve this problem, but do not forget that attribute values for different objects can and most likely will be repeated. Accordingly, the solution to the problem will be to deprive the set of features of the above problems.
Solution
The solution to the problem will be to define a mapping A that transfers each feature to a new space in which all features will be from the same distribution. The distribution itself can be anything, and now we will not go into detail about how to take it, since in part it will depend on the specific case. Let’s just say that in this article we will take a normal distribution with an average value of 50, with a minimum value of 0 and a maximum value of 99. We select the variance in such a way that the distribution density at point 99, multiplied by the number of objects, is equal to one.
We come to the picture when the X axis shows possible ratings for assigning to any object based on the value of a specific attribute. And along the Y axis is the number of objects that can be given a particular rating. For now we are looking at one specific sign. We sort it in descending order. And finally, each object is assigned a rating value according to the following rule:
A score of 99 from the distribution can only be assigned to one object. We put it in the categories with the highest attribute value or zero index (due to sorting). Next, a score of 98 from the distribution can be assigned to two objects (for example). We assign it to categories with indices 1 and 2. And so on for subsequent assessments.
We carry out similar actions for all available signs. And we get the desired result: all the features are distributed equally, and the sum of the values of all of them for a single object will not contradict anything
Let's briefly summarize what we did: Each characteristic led to one predetermined distribution. In this case, for 3 features we get a ball in three-dimensional space, from which we can easily select the required Top objects.
But we haven’t talked about one nuance in this approach until now:
Let’s imagine a situation where we assign a rating for attribute X, for example, 75. We can assign it to 10 objects. And in front of us there are 20 objects that have the same metric value and are suitable for giving a rating of 75. What to do? How to choose 10 objects out of 20 available?
Firstly, you need to understand how many such cases there are, how many there are. The simplest thing that comes to mind is to take the ratio of the unique number of occurring attribute values and the total number of objects.
Secondly, let’s introduce a certain target metric, another feature. And let's move on to a couple of signs: the target metric and the X feature. Thus, if we did not eliminate the uncertainty in the choice of assessment, then at least we should have greatly reduced it.
