Ranking with multiple weights/ features 
We have entities where every entity has start ($s_i$) and end ($e_i$) times and count $c_i$. 
  An entity is important if its interval ($e_i - s_i$) is large and if its $c_i$ is large.

Here's what I did:
In a time window $t$, we take the set of entities that are active (their interval intersect with our window), $entities_t$
Let's define:
$$S_t = \min_{i\in entities_t} s_i \\ E_t = \max_{i\in entities_t} e_i$$
So at time $t$ we can define the features:
$$i\in entities_t: a_{it} = \frac{e_i - s_i}{E_t - S_t}$$
Similarly, we can define:
$$C_t = \max_{i \in entities_t} c_i$$
and the corresponding features:
$$i\in entities_t: b_{it} = \frac{c_i}{C_t}$$
What I'd like to achieve:
I need to devise a formula which need to resemble "how many" important entities are within the time window.


*

*Are $a$ and $b$ good as features? If so, how to use them together?

*if not, how would you define them?

*I currently don't have quantitive way or some test to asses it, but the $a$ feature is "stronger" than the $b$ feature (In words, the time feature is stronger than the count). 

*Given that, I'd probably need some hyper parameter to weight the two features, right?

 A: Let's begin by examining $a$ and $b$. $a$ is really a proportion of time, which runs from $0$ to $1$. If $a_{it} = 1$, then you have 1 (presumably important) entity in your time window. Likewise if $a_{it} = 0$, then presumably that particular entity is somewhat unimportant. 
Similarly, $b$ is also a proportion running from $0$ to $1$, with a value of $1$ indicating one dominating (presumably important) entity, and $0$ indicating a presumably unimportant entity.
You can also assign a weight to $a$ and $b$ to adjust for the fact that $a$ matters more than $b$. Let's call those values $w_a$ and $w_b$. Let's make $w$ also between 0 and 1, so we can keep the notation nice. 
Now, since $a$ and $b$ have the same support set, we can take a weighted average to get an measure of importance, let's call it $I$. Then
$I_{it} = w_a*a_{it} + w_b*b_{it}$
With $I$ running from $0$ to $1$. For each entity, if $I_{it}$ exceeds some predetermined value, then it is important. 
Each timestamp would be assigned a count of how many important values it has. 
There is one immediate danger that I can see
A time stamp with 1 important event either has one important event and a bunch of miniscule events, or 1 important event and no other events at all. Therefore, it may be worth keeping track of the ratio of important events to all events within a timeframe, so you have an idea which case you're in. 
Hope this helps. 
