Compute robust linear interpolation with edge cases
Overview
This question evaluates understanding of linear interpolation, representation as convex combinations, handling of degenerate input cases, bounding interpolation error under bounded measurement noise, and piecewise-linear interpolation for identifying interpolation versus extrapolation, relevant to data scientist roles in the Statistics & Math domain. It is commonly asked to test numerical reasoning, robustness to edge cases and noise, and the ability to connect interpolation theory with practical numerical methods, thus probing both conceptual understanding and practical application.
You are given two distinct points (x0, y0) and (x1, y1) with x0 != x1. a) Derive the linear interpolation function f(x) on [min(x0, x1), max(x0, x1)] and compute f(19) when (x0, y0) = (10, 100) and (x1, y1) = (25, 175). Provide both the exact rational form and a decimal rounded to 3 decimals. b) Prove that f(x) can be written as a convex combination of y0 and y1 via t = (x − x0)/(x1 − x0). c) Specify a consistent rule for the degenerate case x0 = x1 with y0 != y1 (e.g., error, average, or limiting value) and justify your choice. d) If the observed y-values have independent bounded errors |ε| ≤ 0.5, give a bound on the interpolation error at any x. e) Using piecewise-linear interpolation on the sorted points (0,0), (3,9), (8,4), (10,10), compute f(6) and f(9), and state whether each evaluation is interpolation or extrapolation.