# Is there a way to quantitatively calculate how similar one curve is to another curve?

Or at the very least say curve A is more or less similar to curve C than curve B is.

My friend needs help with some analysis for his research project, and he asked me for help and I told him that you guys could offer some help. Specifically, he has a reference curve and a bunch of experimental curves and would like to know a way to see which one is the most similar to the reference curve.

Thanks in advance for any help.

Note by Michael Tong
4 years, 10 months ago

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Interesting problem. One could start by sampling several x-values, and the y-value at each of those points for each of those curves. Then you could calculate the mean difference between the y-values of the experimental curves with the reference curve. This could also be done in the s direction too. Or, one could sample several different points on the experimental curves and calculate their shortest distances to the reference curve. The average of these values would be a measure of similarity. You may want to use set distances along the actual curve to sample the points, and not along the x axis, saying oh we'll sample x = 0, 1, 2, 3, ... because there could be differing amount of variation in the curve between set x values. Also using this sampling method, one could calculate the slope of the tangent line at each of those points, and average them, comparing them to the reference curve. This one makes the most sense, since averaging the derivatives at certain points would get rid of the confounding variables of shifts. However, stretches and shrinks could affect this greatly.

All of these could be taken from a calculus approach, but if the curves have no defined formula, it makes just as much sense to calculate these values statistically.

- 4 years, 10 months ago

Yes, there are many depending on your need. Similarity of curve can mean different meanings depending upon the criteria. For example $$y = f(x)$$ and $$y = f(x + 2)$$ are very similar in contrast that they are just shifted by 2 units along x axis but can be very different if your consider their position along y axis in reference to the a particular x coordinate.

Anyway, calculating the area between two curves might be a good idea in most of the cases to determine their similarity.

Hope this helps.

- 4 years, 10 months ago