Problem Statement Consider a set of scalars {x1,x2,...,xN} drawn from U(0,1) 012 and a corresponding set {y1,y2,...,yN} where:
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yi = sin(2πxi)+ ϵi
and ϵi is drawn from N(0,σnoise). Given the following functional form:
yˆi = ∑wjϕj (xi | µj,σj)+ b M
j=1
with:
(1)
(2)
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024(3)
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find estimates ˆb, {µˆj}, {σˆj}, and {wˆj} that minimize the loss function:
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029(4)
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031 for all (xi,yi) pairs. Estimates for the parameters must be found using stochastic
032 gradient descent. A framework that supports automatic differentiation must be
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034 used. Set N = 50,σnoise = 0.1. Select M as appropriate. Produce two plots. First,
035 show the data-points, a noiseless sinewave, and the manifold produced by the
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regression model. Second, show each of the M basis functions. Plots must be of
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038 suitable visual quality.
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056 −4 −2 0 2 4 −4 −2 0 2 4 x x
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058 Figure 1: Example plots for models with equally spaced sigmoid and gaussian basis functions.
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