Robust fit of linear subspace through multidimensional data

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Description

Robust fit of linear subspace through multidimensional data.

Usage

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## 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 re-weighted 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).

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