Graphics Reference
In-Depth Information
Sliced inverse regression (SIR) was proposed in Li (
)as a way to find compact
representationsthatexploretheintrinsicstructureofhigh-dimensionalobservations.
It has been extended and used in various applications (Chen and Li,
; Li,
;
Chen and Li,
).his technique has been applied with spatial-frequential analysis
in order to segment and diagnose static images, like ultrasound images in Sect.
.
(Chen et al.,
; Chen and Lu,
; Chen et al.,
a,b,
,
). We now
consider the possibility of extending SIR and spatial-frequential analysis to dynamic
images.
When extended to dynamic data, the SIR model is known as dynamic SIR (DSIR)
(Wu and Lu,
). DSIR is combined with spatial-frequential analysis for motion
segmentation. Every pixel in an image is regarded as a realization of a stochastic pro-
cess over space and time. he feature vector for one pixel in one time frame is an-
alyzed through the spatial-frequential analysis of local blocks centered at that pixel.
Assumingthat therelationship between thesefeaturevectorsandclasslabelsremains
similar between successive frames for neighboring pixels, then the intrinsic dimen-
sions of feature vectors in the training images can befoundbyDSIR. heseprojected
feature vectors thus provide prediction rules for forthcoming images in the test set.
Onlyasmallnumberoftraining imagesareneededtodecidetheprojectionoffeature
vectors and prediction rules.
he following model of SIR was introduced in Li (
):
β
′
, β
′
y
=
f
(
x
,
K
x
,є
)
,
(
.
)
where y is a univariate variable,
x
is a random vector with dimension p
, p
K,
the β's are vectors with dimensions p
, є is a random variable that is independent
of
x
,andf is an arbitrary function. he β's are referred to as the effective dimension
reduction (e.d.r.)orprojection directions. Slicedinverse regression(SIR)isamethod
used toestimate the e.d.r. directions based on y and
x
. Under regular conditions, it is
shown that the centered inverse regression curve E
[
x
y
]−
E
[
x
]
occurs in the linear
subspace spanned by β
k
Σ
xx
,whereΣ
xx
denotes the covariance ma-
trix of
x
. Based on these facts, the estimated β's can be obtained by standardizing
x
,
partitioning slices (orgroups) according tothe value of y,calculating the slicemeans
of
x
,andperforming aprincipal component analysis ofthe slice means with weights.
he above model, (
.
), can be extended to dynamic data as follows:
(
k
=
,
, K
)
′
x
′
K
x
y
t
f
β
t
,
, β
t
,є
t
(
.
)
(
)=
(
(
)
(
)
(
))
where y
are response variables and p-dimensional covariates observed
at time t. he projection directions, the β's, are assumed to be invariant over time,
and є
(
t
)
and
x
(
t
)
is the stochastic process of noise. Analogous to the steps in Li (
,
),
thefollowingconditionscanbeassumed,whichprovethesubsequenttheorem(Wu
and Lu,
).
(
t
)
For any b in R
p
, E
b
′
β
′
, β
′
[
x
(
t
)
x
(
t
)
,
K
x
(
t
)]
is linear in
1
Condition
1
β
′
, β
′
b
′
β
′
, β
′
c
β
′
x
(
t
)
,
K
x
(
t
)
for any t.hatis,E
[
x
(
t
)
x
(
t
)
,
K
x
(
t
)] =
c
+
x
(
t
)+
c
K
β
′
+
K
x
(
t
)
for some constants c
, c
,
, c
K
and any t.
