The “not-a-knot” condition on these functions requires continuity of the third derivative at the second and next-to-last points. For 10 points each:
[10m] Name these piecewise polynomial functions. Control points are used to adjust the shape of their “B” type.
ANSWER: splines [accept cubic splines or B-splines]
[10e] Splines are preferred over summing together polynomial basis functions for this task of approximating a function from a set of discrete data points.
ANSWER: interpolation [or interpolating a function or interpolate a function]
[10h] Unlike regression splines or natural splines, these splines do not require selecting knots, since they treat every unique value in the input as a knot. These splines control for overfitting by using a sum of squared errors loss function that is penalized by the integral of the squared second derivative of the spline.
ANSWER: smoothing splines [reject “smooth splines”]
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