Stat 992: Lecture 35
Smoothing periodic functional data
April 23, 2004
1. Solution to Problem 34. Diffusion smoothing on
a unit circle. Motivation would be when we have
periodic functional data g(θ) = g(θ+L). This
periodic condition can be transformed to
g¡L
2π
2π
Lθ¢=g³L
2π¡2π
Lθ+ 2π¢´.
So any period Lfunction can be transformed to a
periodic function with period 2π. So without loss
of generality, we consider periodic data of the form
g(θ) = g(θ+ 2π)which is viewed as data defined
on a unit circle.
Laplacian on S1is trivially ∆ = ∂2
θ. This can
be seen from the line element dl =dθ so the
Laplacian on a circle must be the Laplacian in 1D
Euclidean space. We need to solve ∂tf=∂2
θf
with initial condition f(θ, 0) = g(θ). The so-
lution to an isotropic diffusion equation on mani-
fold is given as a heat kernel convolution by find-
ing the eigenfunctions of Laplacian first. We solve
∂2
θH+λH = 0. Trivially two independent so-
lutions are H= cos √λθ, sin √λθ with λ > 0.
To grantee the periodicity of the eigenfunctions,
√λ=j∈Z+. So we have sequence of eigenfunc-
tions Hj= sinj θ, ˜
Hj= cosj θ.j= 0,1,2,···.
Note that R2π
0sin2jθ dθ =R2π
0cos2jθ dθ =π
from symmetry. So we normalize the eigenfunc-
tions Hj=1
πsin jθ, ˜
Hj=1
πcos jθ. This gives an
orthonormal system of basis functions on S1.
From Lecture 21, heat kernel expansion,
Kt(θ1, θ2) = 1
π2
∞
X
j=0
e−j2t[sin(jθ1) sin(jθ2)
+ cos(jθ1) cos(jθ2)]
=1
π2
∞
X
j=0
e−j2tcos[j(θ1−θ2)].
This makes intuitive sense since the kernel must
be isotropic. Then the solution to the diffusion
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
−6
−4
−2
0
2
4
6
Figure 1: Left: White noise N(0,1) and Kσ∗N(0,1).
Right: 1000 zero mean unit variance Gaussian field. We
are seeing the boundary effect of kernel smoothing.
smoothing is given by
f(θ, t) = Z2π
0
Kt(θ, φ)g(φ)dφ
=1
π2
∞
X
j=0
e−j2tZ2π
0
g(φ) cos[j(θ−φ)] dφ.
Obviously it is much easier to discretize the diffu-
sion equation than implementing the above analyti-
cal solution exactly.
2. Solution to Problem 36 (Tulaya Limpiti). Simulate
zero mean isotropic Gaussian field in [0,1] and find
the corrected P-value numerically. First we gener-
ate 5000 Gaussian white noise N(0,1) apply iter-
ated Gaussian kernel smoothing.
weight= K(1,[-1 0 1])
>>weight =
0.2119 0.5761 0.2119
GRF=zeros(5000,99);
for k=1:5000
w=normrnd(0,1,101,1);
tem=w;
for j=1:20
for i=2:100
tem(i) = dot(w((i-1):(i+1)),weight);
end;
w=tem;
end;
