# ICA via JADE Algorithm Computes ICA decomposition using Cardoso and Souloumiac's (1993, 1996) Joint Approximate Diagonalization of Eigenmatrices (JADE) approach. ```r icajade(X, nc, center = TRUE, maxit = 100, tol = 1e-6, Rmat = diag(nc)) ``` ## Arguments - `X`: Data matrix with `n` rows (samples) and `p` columns (variables). - `nc`: Number of components to extract. - `center`: If `TRUE`, columns of `X` are mean-centered before ICA decomposition. - `maxit`: Maximum number of algorithm iterations to allow. - `tol`: Convergence tolerance. - `Rmat`: Initial estimate of the `nc`-by-`nc` orthogonal rotation matrix. ## Returns - **S**: Matrix of source signal estimates (`S = Y %*% R`). - **M**: Estimated mixing matrix. - **W**: Estimated unmixing matrix (`W = crossprod(R, Q)`). - **Y**: Whitened data matrix. - **Q**: Whitening matrix. - **R**: Orthogonal rotation matrix. - **vafs**: Variance-accounted-for by each component. - **iter**: Number of algorithm iterations. - **converged**: Logical indicating if algorithm converged. ## References Cardoso, J.F., & Souloumiac, A. (1993). Blind beamforming for non-Gaussian signals. **IEE Proceedings-F, 140**(6), 362-370. tools:::Rd_expr_doi("https://doi.org/10.1049/ip-f-2.1993.0054") Cardoso, J.F., & Souloumiac, A. (1996). Jacobi angles for simultaneous diagonalization. **SIAM Journal on Matrix Analysis and Applications, 17**(1), 161-164. tools:::Rd_expr_doi("10.1137/S0895479893259546") Helwig, N.E. & Hong, S. (2013). A critique of Tensor Probabilistic Independent Component Analysis: Implications and recommendations for multi-subject fMRI data analysis. **Journal of Neuroscience Methods, 213**(2), 263-273. tools:::Rd_expr_doi("https://doi.org/10.1016/j.jneumeth.2012.12.009") ## Author(s) Nathaniel E. Helwig ## Details ICA Model The ICA model can be written as `X = tcrossprod(S, M) + E`, where `S` contains the source signals, `M` is the mixing matrix, and `E` contains the noise signals. Columns of `X` are assumed to have zero mean. The goal is to find the unmixing matrix `W` such that columns of `S = tcrossprod(X, W)` are independent as possible. Whitening Without loss of generality, we can write `M = P %*% R` where `P` is a tall matrix and `R` is an orthogonal rotation matrix. Letting `Q` denote the pseudoinverse of `P`, we can whiten the data using `Y = tcrossprod(X, Q)`. The goal is to find the orthongal rotation matrix `R` such that the source signal estimates `S = Y %*% R` are as independent as possible. Note that `W = crossprod(R, Q)`. JADE The JADE approach finds the orthogonal rotation matrix `R` that (approximately) diagonalizes the cumulant array of the source signals. See Cardoso and Souloumiac (1993,1996) and Helwig and Hong (2013) for specifics of the JADE algorithm. ## See Also `icafast` for FastICA `icaimax` for ICA via Infomax ## Examples ```r ########## EXAMPLE 1 ########## # generate noiseless data (p == r) set.seed(123) nobs <- 1000 Amat <- cbind(icasamp("a", "rnd", nobs), icasamp("b", "rnd", nobs)) Bmat <- matrix(2 * runif(4), nrow = 2, ncol = 2) Xmat <- tcrossprod(Amat, Bmat) # ICA via JADE with 2 components imod <- icajade(Xmat, nc = 2) acy(Bmat, imod$M) cor(Amat, imod$S) ########## EXAMPLE 2 ########## # generate noiseless data (p != r) set.seed(123) nobs <- 1000 Amat <- cbind(icasamp("a", "rnd", nobs), icasamp("b", "rnd", nobs)) Bmat <- matrix(2 * runif(200), nrow = 100, ncol = 2) Xmat <- tcrossprod(Amat, Bmat) # ICA via JADE with 2 components imod <- icajade(Xmat, nc = 2) cor(Amat, imod$S) ########## EXAMPLE 3 ########## # generate noisy data (p != r) set.seed(123) nobs <- 1000 Amat <- cbind(icasamp("a", "rnd", nobs), icasamp("b", "rnd", nobs)) Bmat <- matrix(2 * runif(200), 100, 2) Emat <- matrix(rnorm(10^5), nrow = 1000, ncol = 100) Xmat <- tcrossprod(Amat,Bmat) + Emat # ICA via JADE with 2 components imod <- icajade(Xmat, nc = 2) cor(Amat, imod$S) ```

icajade function