I've got 2 curves. One was produced by hand, the other was produced by an algorithm which aims to produce a very similar curve. When testing my algorithm, I'd like to be able to quantify the "similarity" between curves so that when making changes I can judge if I've improved the algorithm or not. For example, if $\{h_i\}$ is the hand crafted data and $\{a_i\}$ and $\{b_i\}$ are curves A and B from 2 versions of my model, I'd like to be able to compare $\{a_i\}$ to $\{h_i\}$ and $\{b_i\}$ to $\{h_i\}$ and then justify picking model version A over version B.
This makes me think of performing residual analysis similar to regression analysis where the hand-crafted curve is $y_{true}$ and the algorithm curve is $y_{fitted}$. In a way, I've already fitted a model via my algorithm and now I just want to analyse the fit. However, I've never done this for data that was not actually fitted via a standard model (e.g. regression) and I am getting some unusual values that I haven't seen before such as a very negative $R^2$. Are there any methods to compare 2 curves that are not related by a model?
Another idea that might or might not work is to regress the model curve against the hand-crafted curve and then analyse the regression. Would this make sense?