Graphics Reference
In-Depth Information
Visualization
at the Object Recovery Stage
5.3
Anyfunctional data setconsists ofacollection ofcontinuous functional objects,such
as a set of continuous curves describing the temperature changes over the course of
a year, or the price increases in an online auction. Despite their continuous nature,
limitations in human perception and measurement capabilities allow us to observe
these curves only at discrete time points. Moreover, the presence of human and mea-
surement error results in discrete observations that are noisy realizations of the un-
derlying continuous curve. hus, the first step in every functional data analysis is to
recover the underlying continuous functional object from the observed data. his is
typically done with the help of smoothing techniques.
A variety of different smoothers exist. One very flexible and computationally ef-
ficient choice is the penalized smoothing spline (Ruppert et al.,
). Let τ
,...,τ
L
be a set of knots. hen, a polynomial spline of order p is given by
L
l
=
+
,
β
t
β
p
t
p
f
(
t
)=
β
+
β
t
+
+ċċċ+
+
β
pl
(
t
−
τ
l
)
(
.
)
where u
+
uI
[
u
]
denotes the positive part of the function u. Define the roughness
=
penalty
∫
D
m
f
dt ,
PEN
m
(
t
)=
(
t
)
(
.
)
where D
m
f , m
,
,
,...,denotesthemth derivative of the function f .hepenal-
ized smoothing spline f minimizes the penalized squared error
=
∫
dt
PENSS
λ,m
=
y
(
t
)−
f
(
t
)
+
λ PEN
m
(
t
)
(
.
)
where y
denotes the observed data at time t and the smoothing parameter λ con-
trols the trade-off between the data fit and the smoothness of the function f .Using
m
t
(
)
in (
.
) leads to the commonly encountered cubic smoothing spline. Other
possible smoothers include the use of B-splines or radial basis functions (Ruppert
et al.,
).
We want to emphasize that we use a common set of smoothing parameters across
all functional objects. For instance, for the penalized smoothing splines, we pick
a common set of knots τ
,...,τ
L
, a common spline order p, and a common penal-
izing term λ, and apply this common set of smoothing parameters to all functional
objects,
=
n. he rationale behind using a common set is that it allows us to
make comparisons among the individual functional objects. Conversely, if one were
to use, say, a large value of λ for object i but a small value for object i
′
,thenitisnot
quite clear whether an observed difference between i and i
′
is attributable to a dif-
ference in the underlying population or instead to the difference in the smoothing
parameters.
i
