.
Generate 100 random points from the following model:
X
∼ Unif(0, π) and
Y
=
g(X) +
_
with independent
_|x
∼
N(0, g(x)2/64), where
g(x) = 1 + sin{x2}/x2. Smooth your data with a constant-span (symmetric nearest neighbor) running-mean
smoother. Select a span of 2k
+ 1 for 1 ≤
k
≤ 11 chosen by cross-validation. Does a running-median smooth with the same span seem very different?
.
Use the data to investigate kernel smoothers as described below:
a.
Smooth the data using a normal kernel smoother. Select the optimal standard deviation of the kernel using cross-validation.
b.
Define the symmetric triangle distribution as
f
(x;
μ, h) =
⎧⎪⎨
⎪⎩
0 if |x
−
μ|
> h,
(x
−
μ
+
h)/a2 if
μ
−
h
≤
x
(μ
+
h
−
x)/a2 if
μ
≤
x
≤
μ
+
h.
The standard deviation of this distribution is
a/
√
6. Smooth the data using a symmetric
triangle kernel smoother. Use cross-validation to search the same set of
standard deviations used in the first case, and select the optimum.
,
c.where
z
= (x
−
μ)/h
and
c
is a constant. Plot this density function. The standard deviation of this density is about 0.90h. Smooth the data using a kernel smoother with this kernel. Use cross-validation to search the same set of standard deviations used previously, and select the optimum.
d.
Compare the smooths produced using the three kernels. Compare their CVRSS values at the optimal spans. Compare the optimal spans themselves. For kernel smoothers, what can be said about the relative importance of the kernel and the span?