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If the nonlinear relationship had been monotonic rank correlation (Spearman's rho) would be appropriate. In your example there is a clear small region where the curve changes from monotoncally increasing to montonically decreasing like a parabola would do at the point where the first derivative equals $0$.

I think if you have some modeling knowledge (beyond the empiricial information) where that change point occurs (say at $x=a$) then you can characterize the correlation as positive and use Spearman's rho on the set of $(x,y)$ pairs where $x < a$ to provide an estimate of that correlation and use another estimate of Spearman's correlation for $x>a$ where the correlation is negative. These two estimates then characterize the correlation structure between $x$ and $y$ and unlike a correlation estimate that would be near $0$ when estimated using all the data these estimates will both be large and opposite in sign.

Some might argue that just the empirical information (i.e. the observed $(x,y)$ pairs is enough to justify this.

If the nonlinear relationship had been monotonic, rank correlation (Spearman's rho) would be appropriate. In your example there is a clear small region where the curve changes from monotonically increasing to monotonically decreasing like a parabola would do at the point where the first derivative equals $0$.

I think if you have some modeling knowledge (beyond the empirical information) where that change point occurs (say at $x=a$) then you can characterize the correlation as positive and use Spearman's rho on the set of $(x,y)$ pairs where $x < a$ to provide an estimate of that correlation and use another estimate of Spearman's correlation for $x>a$ where the correlation is negative. These two estimates then characterize the correlation structure between $x$ and $y$ and unlike a correlation estimate that would be near $0$ when estimated using all the data these estimates will both be large and opposite in sign.

Some might argue that just the empirical information (i.e. the observed $(x,y)$ pairs) is enough to justify this.

If the nonlinear relationship had been monotonic rank correlation (Spearman's rho) would be appropriate. In your example there is a clear small region where the curve changes from monotoncally increasing to montonically decreasing like a parabola would do at the point where the first derivative equals 0.

I think if you have some modeling knowledge (beyond the empiricial information) where that change point occurs (say at x=a) then you can characterize the correlation as positive and use Spearman's rho on the set of (x,y) pairs where x < a to provide an estimate of that correlation and use another estimate of Spearman's correlation for x>a where the correlation is negative. These two estimates then characterize the correlation structure between x and y and unlike a correlation estimate that would be near 0 when estimated using all the data these estimates will both be large and opposite in sign.

Some might argue that just the empirical information (i.e. the observed (x,y) pairs is enough to justify this.

If the nonlinear relationship had been monotonic rank correlation (Spearman's rho) would be appropriate. In your example there is a clear small region where the curve changes from monotoncally increasing to montonically decreasing like a parabola would do at the point where the first derivative equals $0$.

I think if you have some modeling knowledge (beyond the empiricial information) where that change point occurs (say at $x=a$) then you can characterize the correlation as positive and use Spearman's rho on the set of $(x,y)$ pairs where $x < a$ to provide an estimate of that correlation and use another estimate of Spearman's correlation for $x>a$ where the correlation is negative. These two estimates then characterize the correlation structure between $x$ and $y$ and unlike a correlation estimate that would be near $0$ when estimated using all the data these estimates will both be large and opposite in sign.

Some might argue that just the empirical information (i.e. the observed $(x,y)$ pairs is enough to justify this.

If the nonlinear relationship had been monotonic rank correlation (Spearman's rho) would be appropriate. In your example there is a clear small region where the curve changes from monotoncally increasing to montonically decreasing like a parabola would do at the point where the first derivative equals 0.

I think if you have some modeling knowledge (beyond the empiricial information) where that change point occurs (say at x=a) then you can characterize the correlation as positive and use Spearman's rho on the set of (x,y) pairs where x<a to provide an estimate of that correlation and use another estimate of Spearman's correlation for x>a where the correlation is negative. These two estimates then characterize the correlation structure between x and y and unlike a correlation estimate that would be near 0 when estimated using all the data these estimates will both be large and opposite in sign.

Some might argue that just the empirical information (i.e. the observed (x,y) pairs is enough to justify this.

If the nonlinear relationship had been monotonic rank correlation (Spearman's rho) would be appropriate. In your example there is a clear small region where the curve changes from monotoncally increasing to montonically decreasing like a parabola would do at the point where the first derivative equals 0.

I think if you have some modeling knowledge (beyond the empiricial information) where that change point occurs (say at x=a) then you can characterize the correlation as positive and use Spearman's rho on the set of (x,y) pairs where x < a to provide an estimate of that correlation and use another estimate of Spearman's correlation for x>a where the correlation is negative. These two estimates then characterize the correlation structure between x and y and unlike a correlation estimate that would be near 0 when estimated using all the data these estimates will both be large and opposite in sign.

Some might argue that just the empirical information (i.e. the observed (x,y) pairs is enough to justify this.