Online Canonical Component Analysis
OnlineCCA.jl binarizes multiple CSV files, summarizes the information of data matrices and, performs some online-CCA functions for extreamly large scale matrices.
Except for SIMPLS, multiple (over three) datasets can be specified against each algorithm (multiple CCA; mCCA).
Contrary to PCA, which is the eigenvalue problem of the covariance matrix w'X'Xw with the constraint condition norm(w) = 1, CCA is solved as the generalized eigenvalue problem of the cross-product matrix w'X'Yv with the constraint condition cor(Xw, Yv) = 1.
When the column vectors of the matrices X and Y are standarized (mean 0, variance 1), the problem is simplified as the eigenvalue problem of the cross-product matrix [ I, w'X'Yv ; v'Y'Xw, I ] with the constraint condition norm(w) = norm(v) = 1 and called Orthogonal CCA. When the column vectors of X and Y is centerized (mean 0), this problem is also known as PLS-SVD. OnlineCCA.jl can perform both approarches by specifying whether the arguments "colvarlist" as local csv file or nothing.
It is known that simultaneous orthogonalization of both w (eigenvectors/loadings/weight) and Xw (latent variables/scores) is difficult. When using the latent variables for prediction problem such as regression/discriminant analysis, the orthogonalization of latent variables is preferred from the viewpoint of the multicolinearity (cf. PLS-R, PLS-DA, NIPALS, SIMPLS, OPLS, and O2PLS). OnlineCCA.jl provides NIPALS to support such approarch.
- Orthogonal eigenvectors-type
- Gradient-based
- GD-CCA
- SGD-CCA
- RSGD-CCA
- Krylov subspace-based
- Orthgonal Iteration (A power method to calculate multiple eigenvectors at once)
- Horst Iteration (Horst-Jacobi algorithm)
- Random projection-based
- oocMCCA (Out-of-core MCCA)
- Gradient-based
- Orthogonal latent variables-type
- NIPALS
- Robbins-Monro : Herbert Robbins, et. al., 1951
- Momentum : Ning Qian, 1999
- Nesterov's Accelerated Gradient Descent（NAG） : Nesterov, 1983
- ADAGRAD : John Duchi, et. al., 2011
# push the key "]" and type the following command.
(v1.0) pkg> add https://github.com/rikenbit/OnlineCCA.jl
(v1.0) pkg> add PlotlyJS
# After that, push Ctrl + C to leave from Pkg REPL mode...
...
...
...
...
...
...
...
...
...