Original vectors and interp1 vectors result in a different correlation coefficients?

I am curious as to why I can be receiving a different correlation coefficient (R) value between the original random vectors and their interpolation. When I investigate the interpolated values, both vectors have different numbers than the original yet plot the same. In calculating the R values of both original and both interpolated vectors, the results are not similar. Any ideas of why this is?

[Jan Orwat]

Could you describe between what kind of data you are calculating this correlation? Is it like vector x and y and then x_interpolated and y_interpolated? You wrote about random vectors, then those are interpolated. It seems it may increasing the strength of relationship between the data thus changing correlation, even if interpolated data looks similar. In other words interpolation decreases randomness.

[Aundrea Dolan]

Jan- thank you for your timely response. A little background on the data: I used a program run through Arc to quantify the amount of shoreline change over time. It supplies me with, among others, calculations of shoreline change over time (in meters) and the linear regression rate of change to provide insight into if the change is erosive or accretionary. Some of the data has one or two gaps that I have assigned NaNs to- which is why I have interpolated. The pre and post interpolated vectors have differing numbers and, thus, different correlation coefficients. Where most of my confusion stems from the fact that I have one script that does this for many shoreline change envelopes (SCE) and linear regression rates (LRR), so each file gets interpolated. When I looked into the original SCE and LRR for the data I'm working on currently, I found there to be no NaNs so, in theory, the pre and post interpolated R values (and values contained in the vectors) should be the same. Is this right to assume?

I hope I've explained my issue clearly. Again, thank you for your feedback!

Aundrea

Answer 1

[John D'Errico]

There is NO expectation that a new set of interpolated points will have the same correlation coefficient as that for the base set. For example, suppose we have a set of points that follow a nonlinear relationship.

x = (0:1:10)';
y = exp(x);

We can compute the correlation coefficient.

corrcoef([x,y])
ans =
                       1         0.691404156500157
       0.691404156500157                         1

If I then interpolate the points, to get a NEW set of values, i.e.,

xi = linspace(0,10,50)':
yi = interp1(x,y,xi,'spline');

corrcoef([xi,yi])
ans =
                       1         0.687541374878435
       0.687541374878435                         1

As you see, the correlation coefficient is not the same for the new set as for the old one. That is as expected.

It would not matter had I used spline, cubic, or linear interpolation in interp1. The correlation coefficient will generally be close to the original set, but it need not be the same at all.

This is a nonlinear relationship. There is NO expectation that the correlation be the same for interpolated points as for the original. The correlation coefficient is NOT something that is maintained by interpolation. In fact, if I get creative in how I choose my data and the new set of points, I can trivially come up with an example where the correlation coefficient changes sign.

x = (0:1:10)';
y = sin(x);

corrcoef([x,y])
ans =
                       1        -0.116741765087288
      -0.116741765087288                         1

So a moderately small negative correlation.

xi = (0:.1:1)';
yi = interp1(x,y,xi,'spline');

corrcoef([xi,yi])
ans =
                       1         0.994300310123026
       0.994300310123026                         1

So a correlation coefficient that is near 1, on an interpolated set from the original data, when the original set had a negative correlation.

In all cases I have shown, the interpolated values will plot neatly on top of the curve from the original set.