The question here is about a passage in the very frequently (10,000+) cited paper by Storey and Tibshirani "Statistical significance for genomewide studies" (https://doi.org/10.1073/pnas.1530509100).
For their method they need to estimate a parameter named $\pi_0$, which is more or less the overall false discovery rate, for which they compute some $\hat \pi_0(\lambda)$ ($\lambda$ being some bias-variance tradeoff parameter) for various values of $\lambda$ from 0.01, 0.02, 0.03 and so on up to 0.95. Then they supposedly fit some natural cubic spline to that and extrapolate that to $\lambda=1$.
In their own words:
Consider Fig. 3, where we have plotted $\hat π_0(λ)$ versus λ for λ = 0, 0.01, 0.02,..., 0.95. By fitting a natural cubic spline to these data (solid line), we have estimated the overall trend of $\hat π_0(λ)$ as λ increases. We purposely set the degrees of freedom of the natural cubic spline to 3; this means we limit its curvature to be like a quadratic function, which is suitable for our purposes. It can be seen from Fig. 3 that the natural cubic spline fits the points quite well.
Can anybody make sense of this description of the "natural cubic spline fit"? AFAIK the spline should interpolate the data, which it clearly doesn't. It looks more like just one section of a spline, but given natural boundary on the spline, that could only be a straight line IMHO, which it also isn't. I also don't understand the sentence about setting the degrees of freedom and how that "limits the curvature to a quadratic function". The curvature of a cubic spline is related to it's second derivative, which is obviously constant, and not quadratic.
I would shrug that off, if it weren't for those 10000+ citations and the authors being well respected in the field. Maybe I overlooked something? Anyway, the way the method is presented, I can't even implement it as I don't have a clue what's meant here.
Storey, John D., and Robert Tibshirani. "Statistical significance for genomewide studies." Proceedings of the National Academy of Sciences 100.16 (2003): 9440-9445.