Soft Sensing - Advanced Control and Supervision of Mineral Processing Plants - page 182

2

be an orthonormal basis in V m ,so ζ j , ζ k = 0for k = j and ζ k , ζ k = ζ k = 1

for j = k . Then the projection of y into the subspace V m is defined as

m
k = 1 y ,

y = Pr ( y | V m )= ζ k ζ k . (4.90)

= −

It turns out that ε y y is orthogonal to subspace V m , i.e. , ε is orthogonal to

any element in V m , ε ⊥ V m , in particular to each base ζ k (Figure 4.11).

Projection Theorem. Let ℵ be a Hilbert space and let B mon be an orthonormal basis
for a closed subspace V m ⊂ ℵ . Then for y ∈ ℵ the element y for which y − y = ε

is minimum is the projection of y into V m , i.e. [64, 65],

m
k = 1 y ,

y = ζ k ζ k . (4.91)

In other words, projection y is the element in V m which best approximates y .
Moreover,

ε =( y − y )⊥ V m . (4.92)

If basis B m is not orthonormal, then by a Gram-Schmitt procedure [64] it may
be turned into an orthonormal basis and (4.92) still holds, since the Gram-Schmitt
procedure produces just a particular change of coordinates (Figure 4.11). There is a
clear similarity with the case of Euclidean spaces, which is to be expected since the
axioms for inner products and norms are the same.
Let y be a basis - not necessarily orthonormal

- for subspace V m ,andlet g be any element in V m so it may be written as a linear
combination of bases ϕ k (Figure 4.11).
Then, the element in V m which best approximates y (in the sense that

∈ ℵ and let B m = { ϕ 1 , ϕ 2 ,..., ϕ m }

ε =

y − y is minimum) is orthogonal to V m :

y − y , g = 0 ∀ g ∈ V m . (4.93)

In particular, in the case of the basis elements ϕ k ,

y − y , ϕ k = 0 k = 1 ,..., m . (4.94)

=

2. Let y be the projection of y
into the space V 2 spanned by bases ϕ 1 and ϕ 2 ,andlet y a be any other element in V 2 .
Then, by (4.92) εis orthogonal to V 2 and hence to every element in V 2 , in particular
to y

This property is illustrated in Figure 4.12 for m

− y a . By the general Pythagoras Theorem

2 2 2

ε a = ε + y a − y , (4.95)

2 2 , i.e. ,

so ε ≤ ε a ε ≤ ε a ,and ε is minimum with respect to any element

in V 2