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Unformatted text preview: 1, . . . , m, with xi , yi ∈ R. We will assume that xi are
sorted, i.e., x1 < x2 < · · · < xm . Let a0 < a1 < a2 < · · · < aK be a set of ﬁxed knot points, with
a0 ≤ x1 and aK ≥ xm . Explain how to ﬁnd the convex piecewise linear function f , deﬁned over
[a0 , aK ], with knot points ai , that minimizes the least-squares ﬁtting criterion
i=1 (f (xi ) − yi )2 . You must explain what the variables are and how they parametrize f , and how you ensure convexity
of f .
Hints. One method to solve this problem is based on the Lagrange basis, f0 , . . . , fK , which are the
piecewise linear functions that satisfy
fj (ai ) = δij , i, j = 0, . . . , K. Another method is based on deﬁning f (x) = αi x + βi , for x ∈ (ai−1 , ai ]. You then have to add
conditions on the parameters αi and βi to ensure that f is continuous and convex.
43 Apply your method to the data in the ﬁle pwl_fit_data.m, which contains data with xj ∈ [0, 1].
Find the best aﬃne ﬁt (which corresponds to a = (0, 1)), and the best piecewise-linear convex
function ﬁt for...
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- Fall '13
- The Aeneid