Multidimensional scaling and nonlinear biplots - Understanding Biplots - page 220

In the case of a two-dimensional biplot the intersection L ∩ N ( µ ) follows by setting
all elements of ( y , y m + 1 ) except the first two equal to zero. Denote these two nonzero
coordinates as z = ( z 1 , z 2 ) .Then
d µ d n + 1 (µ) 1

λ 1 y 1 λ 2 y 2 z =− n 1 d µ d n + 1 (µ)

1

1

. (5.16)

t ∗ .
Multiplying the coefficient vector a and t ∗ with any constant will result in an equally
valid equation for the intersection space L ∩ N (µ) . Therefore we reparameterize the
equation as l (µ) z = t such that l 1 (µ) + l 2 (µ) = 1.

Thus the intersection space becomes a line with equation of the form a z =

5.4.2.1 Normal projection

With normal projection, we find a trajectory that is normal to all intersection spaces
L ∩ N . If we denote the k th two-dimensional biplot trajectory at the marker µ by

b 1 (µ)
b 2 (µ)

,

β k (µ) =

then the direction of the tangent at β k (µ) is defined by

d µ b 1 (µ)
d
d

b 2 (µ)

µ

and the gradient of the tangent is given by

d
d

b 2 (µ)

µ

b 1 (µ) .

d
d µ

The tangent must be orthogonal to the direction of the intersection space L ∩ N ( µ)

with gradient − l 1 (µ)/ l 2 (µ) , therefore

d
d

b 1 (µ)

l 1 (µ)

µ

−

l 2 (µ) =−

b 2 (µ) .

d
d µ

Furthermore, since β k (µ) lies on the intersection space, it can be expressed as
l (µ) β k (µ) = t (µ) ,

which after differentiation becomes
d µ l 1 (µ) b 1 (µ) + d µ b 1 (µ) l 1 (µ) + d µ l 2 (µ) b 2 (µ) + d µ b 2 (µ) l 2 (µ) =

d
d

t (µ).

µ

After some algebraic manipulation it follows that

b 1 (µ)
l 2 (µ)

d
d µ

t (µ)
l 2 (µ)

d
d µ

= l 1 (µ) . (5.17)