Inverse, exponentially spaced representation of a count field?

Question

var minCount = 10; // Used as 0, anything below 10 uses will be ignored

User hasMany Attributes

Attribute (id, value, count, worth)
// Count - number of users who associate with this attribute
// Worth - float 0.1 (or a better non-zero number?) to 1

An attribute with count of minCount has worth 1. The attribute(s) with the highest count has the lowest possible worth. The lesser used attributes are worth more - lower count -> higher worth.

The difference in worth should be exponential:

• Commonly used attributes (high count) - smaller gap between worth
• Uncommon attributes (low count) - larger gap between worth

I'll be using php for now, but psuedo code is equally acceptable - I'd rather just understand how to do it.

Answer 1

So what's the problem?

Use the following simple formula:

currentWorth = exp(minCount / currentCount - 1)

where: exp - exponent of e; currentCount - current attribute's count; minCount - least attribute's count.

Brief explanation:

First consider formula without exponent application:

currentWorth = minCount / currentCount

For attribute having minimal count (currentCount = minCount):

currentWorth = minCount / minCount = 1

For attributes having any other count (obviously greater than minCount):

(currentWorth = minCount / currentCount) < 1

Now let's apply exponential law:

currentWorth = exp(minCount / currentCount - 1)

For attribute having minimal count (currentCount = minCount):

currentWorth = exp(minCount / minCount - 1) = exp(1 - 1) = exp(0) = 1

For attributes having any other count (currentCount > minCount):

currentWorth = exp(minCount / currentCount - 1)

Assuming:

t = 1 - minCount / currentCount > 0

We'll have:

(currentWorth = exp(-t) = 1 / exp(t)) < 1