Biomedical Engineering Reference
In-Depth Information
distances are estimated by monotonic regression. In nonmetric MDS it is assumed
that
d
ij
≈
f(δ
ij
)
, therefore
f(δ
ij
)
are often referred as the disparities [
157
] in con-
trast to the original dissimilarities
δ
ij
, on one hand, and the distances
d
ij
of the
configuration space on the other hand. In this context, the disparity is a measure of
how well the distance
d
ij
matches the dissimilarity
δ
ij
. There is no rigorous sta-
tistical method to evaluate the quality and the reliability of the results obtained by
an MDS analysis. However, there are two methods often used for that purpose: the
Shepard plot and the stress. The Shepard plot is a scatter plot of the dissimilari-
ties and disparities against the distances, usually overlaid with a line having unitary
slope. The plot provides a qualitative evaluation of the goodness of fit. On the other
hand, the stress value gives a quantitative evaluation. Additionally, the stress plotted
as a function of dimensionality can be used to estimate the adequate
p
i
-dimension.
When the curve ceases to decrease significantly, the resulting “elbow” may corre-
spond to a substantial improvement in fit.
In order to obtain the MDS mapping, the respiratory impedances from the same
patient groups as presented in the previous chapter have been calculated using rela-
tion (
3.8
), for the 4 to 48 Hz frequency interval in increments of 2 Hz. As a result,
we have the impedance for each patient consisting of a complex vector of real and
imaginary parts, with 23 frequency points. Consequently, the distance between the
real parts of the impedance Re, respectively the imaginary parts of the impedance
Im, between various patients, can be calculated with some distance relations. We
propose two such relations, as follows:
M
1
,M
2
(
Re
1
−
Im
2
)
2
k
Re
2
)
2
D
1
=
+
(
Im
1
−
(8.15)
k
=
1
and
M
1
,M
2
k
=
Re
2
)
2
Im
2
)
2
[
(
Re
1
−
+
(
Im
1
−
]
k
1
D
2
=
(8.16)
M
1
,M
2
k
[
(
Re
1
+
Re
2
)
2
+
(
Im
1
+
Im
2
)
2
]
k
=
1
where both sums run over all distances between the patients in each group (
M
1,
respectively,
M
2 are the total number of patients in each group used for calculating
distance measurement). The
(
M
2
)
symmetric MDS ma-
trix is constructed with the values resulting from the calculus of the corresponding
distances, where
ℵ
M
1
+ℵ
M
2
)
×
(
ℵ
M
1
+ℵ
is the cardinal of the data set. The matrix can be visualized as
a three-dimensional plot, which takes an
n
i
×
ℵ
n
i
distance matrix
D
, and returns an
n
i
×
p
i
configuration matrix
Y
[
101
]. Rows of
Y
are the coordinates of
n
i
points in
p
-dimensional space for some
p
i
<n
i
. When
D
is a Euclidean distance matrix, the
distances between those points are given by
D
. The variable
p
i
is the dimension of
the smallest space in which the
n
i
points whose inter-point distances are given by
D
can be embedded. One can specify
D
as either a full dissimilarity matrix form of
D
,
or in upper triangle vector form (such as, e.g. the output by PDIST in Matlab). A full
dissimilarity matrix must be real and symmetric, and have zeros along the diagonal
and positive elements everywhere else. A dissimilarity matrix in upper triangle form
