Robust fit of linear subspace through multidimensional data
Description
Robust fit of linear subspace through multidimensional data.
Usage
 ## S3 method for class 'matrix'
fitIWPCA(X, constraint=c("diagonal", "baseline", "max"), baselineChannel=NULL, ...,
aShift=rep(0, times = ncol(X)), Xmin=NULL)

Arguments
X 
NxK matrix where N is the number of observations and
K is the number of dimensions (channels).

constraint 
A character string or a numeric value.
If character it specifies which additional contraint to be used
to specify the offset parameters along the fitted line;
If "diagonal" , the offset vector will be a point on the line
that is closest to the diagonal line (1,...,1).
With this constraint, all bias parameters are identifiable.
If "baseline" (requires argument baselineChannel ), the
estimates are such that of the bias and scale parameters of the
baseline channel is 0 and 1, respectively.
With this constraint, all bias parameters are identifiable.
If "max" , the offset vector will the point on the line that is
as "great" as possible, but still such that each of its components is
less than the corresponding minimal signal. This will guarantee that
no negative signals are created in the backward transformation.
If numeric value, the offset vector will the point on the line
such that after applying the backward transformation there are
constraint*N . Note that constraint==0 corresponds
approximately to constraint=="max" .
With the latter two constraints, the bias parameters are only
identifiable modulo the fitted line.

baselineChannel 
Index of channel toward which all other
channels are conform.
This argument is required if constraint=="baseline" .
This argument is optional if constraint=="diagonal" and
then the scale factor of the baseline channel will be one. The
estimate of the bias parameters is not affected in this case.
Defaults to one, if missing.

... 
Additional arguments accepted by iwpca ().
For instance, a N vector of weights for each observation may be
given, otherwise they get the same weight.

aShift, Xmin 
For internal use only.

Details
This method uses reweighted principal component analysis (IWPCA)
to fit a the nodel y_n = a + bx_n + eps_n where y_n,
a, b, and eps_n are vector of the K and x_n
is a scalar.
The algorithm is:
For iteration i:
1) Fit a line L through the data close using weighted PCA
with weights \{w_n\}. Let
r_n = \{r_{n,1},...,r_{n,K}\}
be the K principal components.
2) Update the weights as
w_n < 1 / ∑_{2}^{K} (r_{n,k} + ε_r)
where we have used the residuals of all but the first principal
component.
3) Find the point a on L that is closest to the
line D=(1,1,...,1). Similarily, denote the point on D that is
closest to L by t=a*(1,1,...,1).
Value
Returns a list
that contains estimated parameters and algorithm
details;
a 
A double vector (a[1],...,a[K])with offset
parameter estimates.
It is made identifiable according to argument constraint .

b 
A double vector (b[1],...,b[K])with scale
parameter estimates. It is made identifiable by constraining
b[baselineChannel] == 1 .
These estimates are idependent of argument constraint .

adiag 
If identifiability constraint "diagonal" ,
a double vector (adiag[1],...,adiag[K]), where
adiag[1] = adiag[2] = ... adiag[K], specifying the point
on the diagonal line that is closest to the fitted line,
otherwise the zero vector.

eigen 
A KxK matrix with columns of eigenvectors.

converged 
TRUE if the algorithm converged, otherwise FALSE .

nbrOfIterations 
The number of iterations for the algorithm
to converge, or zero if it did not converge.

t0 
Internal parameter estimates, which contains no more
information than the above listed elements.

t 
Always NULL .

Author(s)
Henrik Bengtsson
See Also
This is an internal method used by the calibrateMultiscan
()
and normalizeAffine
() methods.
Internally the function iwpca
() is used to fit a line
through the data cloud and the function distanceBetweenLines
() to
find the closest point to the diagonal (1,1,...,1).