# dot-PACE: Calculate univariate functional PCA In MFPCA: Multivariate Functional Principal Component Analysis for Data Observed on Different Dimensional Domains

## Description

This function is a slightly adapted version of the `fpca.sc` function in the refund package for calculating univariate functional principal components based on a smoothed covariance function. The smoothing basis functions are penalized splines.

## Usage

 ``` 1 2 3 4 5 6 7 8 9 10``` ```.PACE( X, Y, Y.pred = NULL, nbasis = 10, pve = 0.99, npc = NULL, makePD = FALSE, cov.weight.type = "none" ) ```

## Arguments

 `X` A vector of xValues. `Y` A matrix of observed functions (by row). `Y.pred` A matrix of functions (by row) to be approximated using the functional principal components. Defaults to `NULL`, i.e. the prediction is made for the functions in `Y`. `nbasis` An integer, giving the number of B-spline basis to use. Defaults to `10`. `pve` A value between 0 and 1, giving the percentage of variance explained in the data by the functional principal components. This value is used to choose the number of principal components. Defaults to `0.99` `npc` The number of principal components to be estimated. Defaults to `NULL`. If given, this overrides `pve`. `makePD` Logical, should positive definiteness be enforced for the covariance estimate? Defaults to `FALSE`. `cov.weight.type` The type of weighting used for the smooth covariance estimate. Defaults to `"none"`, i.e. no weighting. Alternatively, `"counts"` (corresponds to `fpca.sc` in refund) weights the pointwise estimates of the covariance function by the number of observation points.

## Value

 `fit` The approximation of `Y.pred` (if `NULL`, the approximation of `Y`) based on the functional principal components. `scores` A matrix containing the estimated scores (observations by row). `mu` The estimated mean function. `efunctions` A matrix containing the estimated eigenfunctions (by row). `evalues` The estimated eigenvalues. `npc` The number of principal comopnents that were calculated. `sigma2` The estimated variance of the measurement error. `estVar` The estimated smooth variance function of the data.

## References

Di, C., Crainiceanu, C., Caffo, B., and Punjabi, N. (2009). Multilevel functional principal component analysis. Annals of Applied Statistics, 3, 458–488. Yao, F., Mueller, H.-G., and Wang, J.-L. (2005). Functional data analysis for sparse longitudinal data. Journal of the American Statistical Association, 100, 577–590.

`PACE`