Relying on interpolated data?

Let us look at the issue of ``relying on data not paid for''. First we'll see that interpolation and extrapolation need do no harm. Conventional inversion can be expressed by minimizing the quadratic form

$$Q(\bold x) {=} \sum_k (d_k -\bold a_k \cdot \bold x)^2 \ +\ \lambda\, \bold x' \cdot\bold x$$ (4)

where $\bold x$ is the model vector, $\bold a_k$ is the kth row of a partial derivative matrix, d_k is the kth known data point, $\lambda$ is a scalar, and to avoid clutter I have neglected the residual weighting. The missing data philosophy is to add terms involving the missing data d_m, namely

$$Q(\bold x, d_m) {=} \sum_k (d_k -\bold a_k \cdot \bold x)^2... ...old a_m \cdot \bold x)^2 \ +\ \lambda\, \bold x' \cdot\bold x$$ (5)

Since the missing data d_m are freely adjusted variables, it is evident that the value of Q at the minimum is the same both with and without the missing data. Missing data interpolated or extrapolated in this way cannot perturb the solution $\bold x$.Of course there are many other ways to interpolate and extrapolate, and they are generally less sophisticated than the model-based method I am describing. How can they be of value?