Error estimation in weighted cubic fitting

Question

Given the dataset $(t_i, y_i)$ with absolute errors $\varepsilon_{y_i}$, I want to fit a weighted cubic function and get the errors on interpolated values $\hat{y}(t)$:

$$ \hat{y}(t) = \sum^{3}_{j=0}c_jt^j $$

where $c_j$ are the coefficients determined by the weighted fitting process.

Calculating the residuals and chi squared is straightforward:

$$ r_i = y_i - \hat{y}(t_i) $$ $$ \chi^2 = \sum^n_{i=1}\left( \frac{r_i}{\varepsilon_{y_i}}\right)^2 $$

...but it's unclear to me where to go from here to get errors on $\hat{y}$ values. How do I estimate the error on the interpolation $\hat{y}(t)$?

Answer 1

From the description you want to fit the data but not to interpolate them. Interpolation is about finding a model function that passes over all data points. On the other hand, fitting approximates some function to data.

To problem is a weighted least squares. You need to calculate first the estimators and covariance matrix $ V_c$ of the parameters $c_j$. This can be done in Python with WLS in statsmodels. The error in the fitted model is calculated by propagating the errors of the estimators,

$$ Var(\hat{y}) = \left( \nabla \hat{y} \right)^T \; V_c \; \nabla \hat{y} $$

With $\nabla \hat{y} $ the gradient wrt the $ c_j $'s. The error you are after is the square root of this variance. This as a function of the independent variable t also gives de error band $$\hat{y} \pm \sigma (\hat{y}) $$.