This is not remotely a question about MATLAB. I'll answer it only because I have some time, and because I do understand interpolation reasonably well. :)
Which interpolant is better? Both have their advantages. So asking which is better is like asking if an apple is better than an orange.
A linear interpolant has the advantage that it will be shape preserving in one respect, i.e., that it will never exceed the range of your data. And since spectra have the property that they tend to be peaky, and then rapidly dive down to zero, a classic spline can often be problematic here, since a spline will probably oscillate above and below zero. That is meaningless behavior when used to interpolate an always positive spectral curve. As such a linear interpolant can be viewed as a better method there.
At the same time, since spectra will be sampled at a finite set of points that are often too coarse to approximate the peaks of your curve well, a linear interpolant will be poor there since a linear interpolant is just a straight line, connect the dots interpolant.
If you read the last two paragraphs, I am tell you that a spline is better near the peaks of your spectra, while the linear interpolant is better near the baseline. You can't win here. Or, said differently, they both suck.
Statistically, there is nothing you can say, since statistics tells you nothing about an interpolatatory curve fit. So I have no idea what you are asking there. Well, with one caveat. I once put together an analysis showing that in the presence of significant noise on a curve, a linear interpolant can be a lower variance predictor than a cubic spline. But that applies only to a curve where the noise really is pretty significant. Then the spline will have oscillations in it, because the spline will also be interpolating the noise. Again, this requires that the noise is significant, and spectra are typically sufficiently carefully measured that the noise is not that large of a component. So a spline will usually win in this respect on spectra.
As an example of what I am saying, consider the following vaguely spectral curve:
F = @(x) x.*(sin(5*x.^2) + 1).^4;
fplot(F,[0,2.5],'g--')
hold on
xint = linspace(0,2.5,40);
plot(xint,F(xint),'ro')
hold off
The above curve in blue will be recognized as ground truth. Just pretend it is a spectrum for something. I choce it because it has peaks of varying widths, and because it then dives down to zero, where it will be flat, but not go below zero. I probably should have chosen a curve where some of the peaks come near each other, to partially overlap. So sue me. This is good enough.
But now plot the linear interpolant on that set of points. (Note that I can just do connect the dots as plot does for me.)
plot(xint,F(xint),'ro',xint,F(xint),'b-')
Do you accept that the curve, where the point straddle a peak, will approximate the spectra poorly? At the same time, a linear interpolant will NEVER pass below zero. What does a spline do here?
xf = linspace(0,2.5,1000);
spl = spline(xint,F(xint));
plot(xint,F(xint),'ro',xf,fnval(spl,xf),'b-')
hold on
fplot(F,[0,2.5],'g--')
hold off
So the green dashed line is ground truth here. The spline is smoother, but it still fails terribly. It oscillates in places where it should never do so, going below zero, which is meaningless. It does a little better at hitting the peaks, but it still performs poorly where the peaks are sharp and poorly approximated by only a few scattered points.
You can do a little better if you use pchip as the interpolant, because it will never pass below zero. So the baseline oscillations go away. But pchip will now perform more poorly at the peaks, much like the linear interpolant there.
pspl = pchip(xint,F(xint));
plot(xint,F(xint),'ro',xf,fnval(pspl,xf),'b-')
hold on
fplot(F,[0,2.5],'g--')
hold off
You can't really win for trying here. Sorry. Is there a decent compromise? There are tricks using log transformations I've tried in the past. But they tend to be difficult to get to work perfectly. I recall that long ago, in a galaxy far, far away, I wrote an interpolant specifically designed to work well on spectra, that had the good behaviors of a cubic spline near the peaks, but the good behaviors of pchip near the valleys. That code lives in another universe though - the universe of APL. That also means it was written close to 40 years ago.
