Measuring "consistent" growth versus "spiked" growth

Question

I have a time-series database of users and their growth on our platform over time. I am trying to determine from this database users who are showing a consistent growth over time (and especially exponential growth). However, I am struggling to determine between users who have grown consistently over a period of time, against those who have grown very quickly over 1-2 days and then stopped growing.

(excuse the crudely drawn diagrams)

Effectively I want to find people with this type of growth (and give them a high score):

And give people a lower score when their growth looks like this:

Adding to this, its more important to see the increase in growth relatively, rather than absolutely.

I would be interested to know what sort of approach would allow me to score these correctly.

Answer 1

You can look at the variance of the growth derivative (or its discrete approximation, the difference between neighboring points).

For those with a consistent growth, the derivative will likely be some positive number, more or less constant over time, i.e. with low variance.

For spiked growth, the derivative will be zero when no growth is taking place, and quite large at "spikes".

Edit:

Assume your users' growth can be represented by the images:

The log(1+daily differences) is then:

If you plot their histogram, you'll see that the consistent users have a more-or-less uniform log-growth, while the for the spiky ones it is mostly close to zero, but occasionally quite large:

You can quantify it by kurtosis:

e1071::kurtosis(dy1)
[1] -1.050228

e1071::kurtosis(dy2)
[1] 4.022098