Bayesian PCA missing value estimation

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Description

Implements a Bayesian PCA missing value estimator. The script is a port of the Matlab version provided by Shigeyuki OBA. See also http://ishiilab.jp/member/oba/tools/BPCAFill.html. BPCA combines an EM approach for PCA with a Bayesian model. In standard PCA data far from the training set but close to the principal subspace may have the same reconstruction error. BPCA defines a likelihood function such that the likelihood for data far from the training set is much lower, even if they are close to the principal subspace.

Usage

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bpca(Matrix, nPcs = 2, maxSteps = 100, verbose = interactive(),
  threshold = 1e-04, ...)

Arguments

Matrix

matrix – Pre-processed matrix (centered, scaled) with variables in columns and observations in rows. The data may contain missing values, denoted as NA.

nPcs

numeric – Number of components used for re-estimation. Choosing few components may decrease the estimation precision.

maxSteps

numeric – Maximum number of estimation steps.

verbose

boolean – BPCA prints the number of steps and the increase in precision if set to TRUE. Default is interactive().

threshold

convergence threshold

...

Reserved for future use. Currently no further parameters are used

Details

Scores and loadings obtained with Bayesian PCA slightly differ from those obtained with conventional PCA. This is because BPCA was developed especially for missing value estimation. The algorithm does not force orthogonality between factor loadings, as a result factor loadings are not necessarily orthogonal. However, the BPCA authors found that including an orthogonality criterion made the predictions worse.

The authors also state that the difference between real and predicted Eigenvalues becomes larger when the number of observation is smaller, because it reflects the lack of information to accurately determine true factor loadings from the limited and noisy data. As a result, weights of factors to predict missing values are not the same as with conventional PCA, but the missing value estimation is improved.

BPCA works iteratively, the complexity is growing with O(n^3) because several matrix inversions are required. The size of the matrices to invert depends on the number of components used for re-estimation.

Finding the optimal number of components for estimation is not a trivial task; the best choice depends on the internal structure of the data. A method called kEstimate is provided to estimate the optimal number of components via cross validation. In general few components are sufficient for reasonable estimation accuracy. See also the package documentation for further discussion about on what data PCA-based missing value estimation makes sense.

It is not recommended to use this function directely but rather to use the pca() wrapper function.

There is a difference with respect the interpretation of rows (observations) and columns (variables) compared to matlab implementation. For estimation of missing values for microarray data, the suggestion in the original bpca is to intepret genes as observations and the samples as variables. In pcaMethods however, genes are interpreted as variables and samples as observations which arguably also is the more natural interpretation. For bpca behavior like in the matlab implementation, simply transpose your input matrix.

Details about the probabilistic model underlying BPCA are found in Oba et. al 2003. The algorithm uses an expectation maximation approach together with a Bayesian model to approximate the principal axes (eigenvectors of the covariance matrix in PCA). The estimation is done iteratively, the algorithm terminates if either the maximum number of iterations was reached or if the estimated increase in precision falls below 1e^-4.

Complexity: The relatively high complexity of the method is a result of several matrix inversions required in each step. Considering the case that the maximum number of iteration steps is needed, the approximate complexity is given by the term

maxSteps * row_miss * O(n^3)

Where row_miss is the number of rows containing missing values and O(n^3) is the complexity for inverting a matrix of size components. Components is the number of components used for re-estimation.

Value

Standard PCA result object used by all PCA-based methods of this package. Contains scores, loadings, data mean and more. See pcaRes for details.

Note

Requires MASS.

Author(s)

Wolfram Stacklies

References

Shigeyuki Oba, Masa-aki Sato, Ichiro Takemasa, Morito Monden, Ken-ichi Matsubara and Shin Ishii. A Bayesian missing value estimation method for gene expression profile data. Bioinformatics, 19(16):2088-2096, Nov 2003.

See Also

ppca, svdImpute, prcomp, nipalsPca, pca, pcaRes. kEstimate.

Examples

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## Load a sample metabolite dataset with 5\% missig values (metaboliteData)e
data(metaboliteData)
## Perform Bayesian PCA with 2 components
pc <- pca(t(metaboliteData), method="bpca", nPcs=2)
## Get the estimated principal axes (loadings)
loadings <- loadings(pc)
## Get the estimated scores
scores <- scores(pc)
## Get the estimated complete observations
cObs <- completeObs(pc)
## Now make a scores and loadings plot
slplot(pc)

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