Information Technology Reference
In-Depth Information
so that
λ
=
µ/
b
k
b
k
and
µ
b
k
/(
b
k
b
k
)
gives the coordinates of the point on the
b
k
-axis
that is calibrated with a value of
µ
. Normally,
µ
will be set to values 1, 2, 3,
...
,orother
convenient steps for the calibration, to give the values required by the inner products.
Often, the inner product being approximated gives transformed values of some
original variables
a
i
b
k
=
f
(
x
ik
)
and one wants to calibrate in the original units of
measurement. Suppose
represents a value to be calibrated in the original units; then we
must set
µ
=
f
(α)
, where the function will vary with different methods. For example,
in PCA the data are centred, in correspondence analysis (CA) the original counts are
replaced by row and/or column scaled deviations from an independence model, in
metric scaling dissimilarities are defined by a variety of coefficients that are functions
of the original variables, and in nonmetric scaling by monotonic transformations defined
in terms of smooth spline functions or merely by step-functions. Another possibility is
where the calibration steps are kept equal in the transformed units but labelled with
the untransformed values; this is especially common with logarithmic transformations.
Calibrated axes may be constructed for all such methods.
α
2.3.1 Lambda scaling
When plotting points given by the rows of
A
and
B
one set will often be seen to have
much greater dispersion than the other (see, for example, Figure 2.4 where the dispersion
of the sample points overshadows that of the points representing the variables). This can
be remedied as follows. First observe that
AB
=
(λ
A
)(
B
/λ)
,
(2.11)
so that the inner product is unchanged when
A
is scaled by
λ
, provided that
B
is inversely
scaled. This simple fact may be used to choose
λ
in some optimal way to improve the
look of the display. One way of choosing
is to arrange that the average squared distance
of the points in
λ
A
and
B
/λ
is the same. If
A
has
p
rows and
B
has
q
rows and both
are centred, this requires
λ
2
p
=
λ
−
2
B
2
q
.
2
A
λ
(2.12)
giving the required scaling
2
A
q
B
2
4
p
λ
=
.
(2.13)
We term the above method lambda scaling.
Lambda scaling is not the only criterion available; one might prefer to work in terms
of distances rather than squared distances or work in terms of maximum distances. Indeed,
the inner product is invariant for quite general transformations
AB
=
(
AT
)(
B
T
−
1
)
but
such general transformations are liable to induce conflicts such as changing Euclidean
and centroid properties. However, whenever the inner product is maintained everything
written above about the calibration of axes remains valid.
Lambda scaling has only a trivial proportionate effect on distances, but it is important
to be aware that general scaling affects distance severely; this is especially relevant in
PCA, canonical variate analysis (CVA), some forms of CA that approximate Pythagorean
distance, Mahalanobis distance and chi-squared distance.
