@QMuse @andreasmaaan There has been a lot of debate previously on the interpretation of the SM metric. It is a complicated topic that is headache-inducing and quite counter-intuitive. I would urge you to read
this post if you are interested in understanding how SM is defined and how it works. The current version of
Loudspeaker Explorer now has SM computation as a feature and you can see a detailed computation with charts for every step (under "Olive Preference Score", "SM"). See
here for an example computation.
The simplest definition of SM I was able to come up with is:
SM describes how much of the curve deviation from a flat, horizontal line can be explained by the overall slope (as opposed to just jagginess). Which is hard enough to wrap your head around as it is. And even then, that description glosses over a few details.
And to further clarify: Assuming that all data points are equally-spaced from the PIR slope (and all else is equal), a speaker with a steeper PIR slope will always receive a better rating for SM_PIR. Correct?
And furthermore, there is no effective limit to this, i.e. a hypothetical speaker with a PIR slope approaching ∞ dB/decade would (all else equal) achieve a superior rating for SM_PIR than a speaker with a shallower PIR slope (once again assuming all data points are equally-spaced from each speaker's respective PIR slope). Also correct?
Yes, this is correct on both points. However, do keep in mind that the
total Olive score might not improve with larger slope, because NBD_PIR is part of the score too, and that variable
worsens as slope increases. So in the end, there is a delicate (and non-linear, I suspect) balancing act going on between SM_PIR and NBD_PIR. For this reason, it is very difficult to draw conclusions from SM_PIR or NBD_PIR in isolation.
Given that correlation coefficient biases SM in favour of steeper slope, why do we speculate that Olive's model uses it (as opposed to a formula that is independent of it)?
It's not speculation. The Olive paper literally
defines SM as the squared correlation coefficient.
It seems also the case that correlation coefficient would be biased in favour of a steeply upward-sloping SM_PIR slope - is this correct?
Yep, and actually that's a very good point which I didn't realize until now. That's a great illustration of why the model might perform poorly with speakers that do not behave similarly to the speakers used in the original study!
I don't think so. -30deg slope and +30deg slope should result in the same corr. coefficient, just the sign would be opposite.
Sadly, SM is defined as r², i.e. the correlation coefficient
squared. It will favor any kind of tilt in any direction.
Ok, so do we think that Olive deliberately built in SM_PIR such that it would advantage speakers with the steepest possible downward-sloping PIR?
I doubt that. Looking at the wording of the paper, it's way more likely Olive defining SM as r² was an accident, because it makes very little sense to use r² to represent the "smoothness" of a curve. It's quite possible Olive got confused about what r² meant, which is understandable given that
@MZKM,
@bobbooo, myself and others were just as confused too (
@daverosenthal is probably the only one who
got it right from the very beginning). That doesn't make the model invalid though, just weirder and harder to reason about.