The
orthogonal projection [P.sub.M,I] : [L.sup.2](I) [right arrow] [V.sub.M](I) is defined by
We now give the formulation of
orthogonal projection [P.sub.C] where C is a simply closed convex sets as follows, and in the case that C is not a simply closed convex sets, for instance, C is a halfspace, we can found more the formulation in [33].
At the same time, SIMPCA estimation results are better than SOPIM estimation results, mainly because Column weighting for SIMPCA is introduced, while SOPIM uses
orthogonal projection only.
Suppose there exists a reducing subspace [mathematical expression not reproducible] denote the
orthogonal projection from [([K.sup.2.sub.u]).sup.[perpendicular to]] onto M.
Therefore, an
orthogonal projection matrix [O.sub.p] can be defined to identify any spectra in the corresponding data set Y of the sample profiles that are related to those in X.
(ii) Every element x [member of] X admits at most one
orthogonal projection [P.sup.[perpendicular to].sub.H] (x) onto H.
It follows that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the
orthogonal projection. Since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], we conclude that [P.sub.T]([x.sub.k]) [right arrow] [[pi].sub.N'] [omicron] [DX.sub.T]([sigma])[|.sub.N] as k [right arrow] [infinity].
If we restrict f to be in [L.sup.2](0, 1), then [Q.sub.n]f is exactly the
orthogonal projection of [L.sup.2](0,1) onto [[DELTA].sub.n] [subset] [L.sup.2](0,1) under the [L.sup.2]- norm [[parallel]f[parallel].sup.2] = [square root of ([[integral].sup.1.sub.0] [[absolute value of (f)].sup.2]dm)] of [L.sup.2](0,1).
Consider the
orthogonal projection G' of G onto [H.sub.xy].
For the right-hand GUI widget, I create a cube (made square through the
orthogonal projection).
For accuracy analysis,
orthogonal projection and nonorthogonal projection were considered.