I have a collection of data points, $(x_0, y_0)...(x_{n-1}, y_{n-1})$ of the function $y = y(x)$ where the values of x are in ascending order. I'm working out an algorithm for smoothing splines, following de Boor (A Practical Guide to Splines, 2001), but I'm confused about one aspect of his explanation.
On page 207, de Boor writes the weighted least squares regression term as $\sum_{i=0}^{n-1}(\frac{y_i - f(x_i)}{\delta y_i})^2$, where $\delta y_i$ is "an estimate of the standard deviation in $y_i$". (Here, $f(x_i)$ is the smoothing function to be constructed.)
If all I have is my collection of data points, how can $(\delta y_i)^2$ be calculated (or estimated)?
I am tempted to use something like $(\delta y_i)^2 = |y_i - \bar{y}|^2$ as $y_i$'s "contribution" toward the overall sample variance, but should $\bar{y}$ be the mean of all y, or something more "local" to $y_i$? I suspect that this isn't the right approach.
Any help would be appreciated in sorting out how to calculate or estimate $\delta y_i$.