Cross-validation for PCA
Internal cross-validation can be used for estimating the level of structure in a data set and to optimise the choice of number of principal components.
Q2(object, originalData = completeObs(object), fold = 5, nruncv = 1,
type = c("krzanowski", "impute"), verbose = interactive(),
variables = 1:nVar(object), ...)object |
A |
originalData |
The matrix (or ExpressionSet) that used to obtain the pcaRes object. |
fold |
The number of groups to divide the data in. |
nruncv |
The number of times to repeat the whole cross-validation |
type |
krzanowski or imputation type cross-validation |
verbose |
|
variables |
indices of the variables to use during cross-validation calculation. Other variables are kept as they are and do not contribute to the total sum-of-squares. |
... |
Further arguments passed to the |
This method calculates Q^2 for a PCA model. This is the cross-validated version of R^2 and can be interpreted as the ratio of variance that can be predicted independently by the PCA model. Poor (low) Q^2 indicates that the PCA model only describes noise and that the model is unrelated to the true data structure. The definition of Q^2 is:
Q^2=1 - sum_i^k sum_j^n (x - \hat{x})^2 / ∑_i^k ∑_j^n(x^2)
for the matrix
x which has n rows and k columns. For a given
number of PC's x is estimated as \hat{x}=TP' (T are scores
and P are loadings). Although this defines the leave-one-out
cross-validation this is not what is performed if fold is less
than the number of rows and/or columns. In 'impute' type CV,
diagonal rows of elements in the matrix are deleted and the
re-estimated. In 'krzanowski' type CV, rows are sequentially left
out to build fold PCA models which give the loadings. Then,
columns are sequentially left out to build fold models for
scores. By combining scores and loadings from different models, we
can estimate completely left out values. The two types may seem
similar but can give very different results, krzanowski typically
yields more stable and reliable result for estimating data
structure whereas impute is better for evaluating missing value
imputation performance. Note that since Krzanowski CV operates on
a reduced matrix, it is not possible estimate Q2 for all
components and the result vector may therefore be shorter than
nPcs(object).
A matrix or vector with Q^2 estimates.
Henning Redestig, Ondrej Mikula
Krzanowski, WJ. Cross-validation in principal component analysis. Biometrics. 1987(43):3,575-584
data(iris) x <- iris[,1:4] pcIr <- pca(x, nPcs=3) q2 <- Q2(pcIr, x) barplot(q2, main="Krzanowski CV", xlab="Number of PCs", ylab=expression(Q^2)) ## q2 for a single variable Q2(pcIr, x, variables=2) pcIr <- pca(x, nPcs=3, method="nipals") q2 <- Q2(pcIr, x, type="impute") barplot(q2, main="Imputation CV", xlab="Number of PCs", ylab=expression(Q^2))
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