How to summarize a time series independent variable for regression?

Question

One of the things I've struggled with is the issue of effectively summarizing some sort of time series metric (i.e. independent variable) for use in a regression analysis. The most common solution that I've encountered (which may not be the best) is to compute a meaningful average. Although I can get a single number I still have the nagging feeling that it doesn't really capture what I need.

Here's an example graph:

Let's assume that there are 3 data points which when plotted over time we can see different types of growth. What I wish to capture/model is that the "flattens" represents the possibility that item sale (or whatever it represents) is eventually going to take a downturn or just stop selling. The "growth" curve is something that won't immediately stop but for the time being (e.g. 1 month) it's a safe bet to assume item popularity.

Here are my questions:

1. How can/should we capture the "shapes" of such time series based independent variables for regression? Does it even matter or is the average the best we can do?
2. I thought of computing "slopes" (start, end) but as the graph shows (approximately) that there such a computation could lead to similar values for both flattens and growth curves. Is this even a thing?
3. What could/should I use?

Note: These shapes are just examples and the actual time series could be rather fluctuating. What I think I'm interested in is a way to capture a notion of a general trend in some way (if at all). I understand trends could be captured via moving averages, but I don't know how to capture "up/down-ness" or summarize it in a meaningful way.

I'm hoping this doesn't become a two-step problem where we first train a model on "learning" these curves and what they represent and then feeding the output into the regression. That would be too complicated for now but I don't even know if that's a valid solution or should I just stick with averages.

Answer 1

One possible approach that I've been investigating is the use of the second derivative (double differencing) with averaging and/or summing the values (more on that in a bit).

Here's an example in Python:

import numpy as np
d = np.array([0,7.5,9,9,9,8,8,8,8,8,8,8]) #flattens curve
avg = np.average(np.diff(np.diff(d)))     # = -0.75
sum = np.sum(np.diff(np.diff(d)))         # = -7.5

The average provides better "trend" of the second derivative IMHO. The summation seems better at "locating" the approximate local maxima/minima "point".

Here's how to interpret the values:

avg = 0 -> flat trend
avg < 0 -> decreasing trend
avg > 0 -> increasing trend

We could make this a little more tolerant by assuming flatness is between $\pm 1$ if necessary. For all different time series patterns (more than about a 100 in my data set) this seems to provide a decent solution to address the need that is at the right level of simplicity.

For now, this is what I will be going with unless someone provides a better answer. I may be inclined to include both the "overall average" of the entire time series independent variable plus the average of the second derivative. I will surely do a correlation test to ascertain if I should only keep one of them, but I have a feeling that both could be valuable (though I'm not sure if this commonly practiced).