# add_es: Adds two eigenspaces using block-wise incremental SVD (with... In idm: Incremental Decomposition Methods

## Description

This function implements two procedures for updating existing decomposition. When `method="esm"` it adds two eigenspaces using the incremental method of Hall, Marshall & Martin (2002). The results correspond to the eigenspace of the mean-centered and concatenated data. When `method = "isvd"` it adds the eigenspace of an incoming data block to an existing eigenspace using the block-wise incremental singular value decomposition (SVD) method described by Zha & Simon (1999), Levy and Lindenbaum (2000), Brand (2002) and Baker (2012). New data blocks are added row-wise. The procedure can optionally keep track of the data mean using the orgn argument, as described in Ross et al. (2008) and Iodice D'Enza & Markos (2015).

## Usage

 `1` ```add_es(eg, eg2, current_rank, ff = 0, method = c("esm", "isvd")) ```

## Arguments

 `eg` A list describing the eigenspace of a data matrix, with components `u` Left eigenvectors `v` Right eigenvectors `m` Number of cases `d` Eigenvalues `orgn` Data mean `method` refers to the procedure being implemented: `"esm"` refers to the eigenspace merge (Hall et al., 2002); `"isvd"` refers to the incremental SVD method, with or without keeping track of the data mean. `eg2` (*)A list describing the eigenspace of a data matrix, with components `u` Left eigenvectors `v` Right eigenvectors `m` Number of cases `d` Eigenvalues `orgn` Data mean `current_rank` Rank of approximation; if empty, the full rank is used `ff` (**)Number between 0 and 1 indicating the forgetting factor used to down-weight the contribution of earlier data blocks to the current solution. When ff = 0 (default) no forgetting occurs

(*) for `method = "esm"` only; (**) for `method = "isvd"` only.

## Value

A list describing the SVD of a data matrix, with components

 `u` Left singular vectors `d` Singular values `v` Right singular vectors `m` Number of cases `orgn` Data mean; returned only if `orgn` is given as input

## References

Zha, H., & Simon, H. D. (1999). On updating problems in latent semantic indexing. SIAM Journal on Scientific Computing, 21(2), 782-791.

Levy, A., & Lindenbaum, M. (2000). Sequential Karhunen-Loeve basis extraction and its application to images. IEEE Transactions on Image Processing, 9(8), 1371-1374.

Brand, M. (2002). Incremental singular value decomposition of uncertain data with missing values. In Computer Vision-ECCV 2002 (pp. 707-720). Springer Berlin Heidelberg.

Ross, D. A., Lim, J., Lin, R. S., & Yang, M. H. (2008). Incremental learning for robust visual tracking. International Journal of Computer Vision, 77(1-3), 125-141.

Baker, C. G., Gallivan, K. A., & Van Dooren, P. (2012). Low-rank incremental methods for computing dominant singular subspaces. Linear Algebra and its Applications, 436(8), 2866-2888.

Iodice D' Enza, A., & Markos, A. (2015). Low-dimensional tracking of association structures in categorical data, Statistics and Computing, 25(5), 1009-1022.

`do_es`, `i_pca`, `i_mca`, `update.i_pca`, `update.i_mca`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28``` ```## Example 1 - eigenspace merge (Hall et al., 2002) #Iris species data("iris", package = "datasets") X = iris[,-5] #obtain two eigenspaces eg = do_es(X[1:50, ]) eg2 = do_es(X[c(51:150), ]) #add the two eigenspaces keeping track of the data mean eg12 = add_es(method = "esm", eg, eg2) #equivalent to the SVD of the mean-centered data (svd(scale(X, center = TRUE,scale = FALSE))) ## Example 2 - block-wise incremental SVD with mean update, full rank (Ross et al., 2008) data("iris", package = "datasets") # obtain the eigenspace of the first 50 Iris species X = iris[,-5] eg = do_es(X[1:50, ]) #update the eigenspace of the remaining species to eg_new = add_es(method = "isvd", eg, data.matrix(X[c(51:150), ])) #equivalent to the SVD of the mean-centered data (svd(scale(X, center = TRUE, scale = FALSE))) ##Example 3 - incremental SVD with mean update, 2d approximation (Ross et al., 2008) data("iris", package = "datasets") # obtain the eigenspace of the first 50 Iris species X = iris[,-5] eg = do_es(X[1:50, ]) #update the eigenspace of the remaining species to eg = add_es(method = "isvd", eg, data.matrix(X[c(51:150), ]),current_rank = 2) #similar to PCA on the covariance matrix of X (SVD of the mean-centered data) ```