Confidence interval for the percentage of variance retained by the first k components {choosepc}

Confidence interval for the percentage of variance retained by the first \kappa components

Description

Confidence interval for the percentage of variance retained by the first \kappa components.

Usage

eigci(x, k, alpha = 0.05, B = 1000, graph = TRUE)

Arguments

x

A numerical matrix with more rows than columns.

k

The number of principal components to use.

alpha

This is the significance level. Based on this, an (1-\alpha)\% confidence interval will be computed.

B

The number of bootstrap samples to generate.

graph

Should the plot of the bootstrap replicates appear? Default value is TRUE.

Details

The algorithm is taken by Mardia Kent and Bibby (1979, pg. 233–234). The percentage retained by the fist \kappa principal components denoted by \hat{\psi} is equal to

\hat{\psi}=\frac{ \sum_{i=1}^{\kappa}\hat{\lambda}_i }{\sum_{j=1}^p\hat{\lambda}_j },

where \hat{\psi} is asymptotically normal with mean \psi and variance

\tau^2 = \frac{2}{\left(n-1\right)\left(tr\pmb{\Sigma} \right)^2}\left[ \left(1-\psi\right)^2\left(\lambda_1^2+...+\lambda_k^2\right)+ \psi^2\left(\lambda_{\kappa+1}^2+...\lambda_p^2\right) \right],

where a=\left( \lambda_1^2+...+\lambda_k^2\right)/\left( \lambda_1^2+...+\lambda_p^2\right) and \text{tr}\pmb{\Sigma}^2=\lambda_1^2+...+\lambda_p^2.

The bootstrap version provides an estimate of the bias, defined as \hat{\psi}_{boot}-\hat{\psi} and confidence intervals calculated via the percentile method and via the standard (or normal) method Efron and Tibshirani (1993). The funciton gives the option to perform bootstrap.

Value

A list including:

res

If B=1 (no bootstrap) a vector with the esimated percentage of variance due to the first k components, \hat{\psi} and its associated asymptotic (1-\alpha)\% confidence interval. If B>1 (bootstrap) a vector with: the esimated percentage of variance due to the first k components, \hat{\psi}, its bootstrap estimate and its bootstrap estimated bias.

ci

This appears if B>1 (bootstrap). The standard bootstrap and the empirical bootstrap (1-\alpha)\% confidence interval for \psi.

Futher, if B>1 and "graph" was set equal to TRUE, a histogram with the bootstrap \hat{\psi} values, the observed \hat{\psi} value and its bootstrap estimate.

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris [e-mail].

References

Mardia K.V., Kent, J.T. and Bibby, J.M. (1979). Multivariate Analysis. London: Academic Press.

Efron B. and Tibshirani R. J. (1993). An introduction to the bootstrap. Chapman & Hall/CRC.

See Also

pc.choose

Examples

x <- as.matrix(iris[, 1:4])
eigci(x, k = 2, B = 1)