o2m: Perform O2-PLS with two-way orthogonal corrections
In selbouhaddani/O2PLS: O2PLS: Two-Way Orthogonal Partial Least Squares

Description Usage Arguments Details Value See Also Examples

NOTE THAT THIS FUNCTION DOES NOT CENTER NOR SCALES THE MATRICES! Any normalization you will have to do yourself. It is best practice to at least center the variables though.

1 2	o2m(X, Y, n, nx, ny, stripped = FALSE, p_thresh = 3000, q_thresh = p_thresh, tol = 1e-10, max_iterations = 100)

`X`	Numeric matrix. Vectors will be coerced to matrix with `as.matrix` (if this is possible)
`Y`	Numeric matrix. Vectors will be coerced to matrix with `as.matrix` (if this is possible)
`n`	Integer. Number of joint PLS components. Must be positive!
`nx`	Integer. Number of orthogonal components in X. Negative values are interpreted as 0
`ny`	Integer. Number of orthogonal components in Y. Negative values are interpreted as 0
`stripped`	Logical. Use the stripped version of o2m (usually when cross-validating)?
`p_thresh`	Integer. If `X` has more than `p_thresh` columns, a power method optimization is used, see `o2m2`
`q_thresh`	Integer. If `Y` has more than `q_thresh` columns, a power method optimization is used, see `o2m2`
`tol`	double. Threshold for power method iteration
`max_iterations`	Integer, Maximum number of iterations for power method

If both nx and ny are zero, o2m is equivalent to PLS2 with orthonormal loadings. This is a ‘slower’ (in terms of memory) implementation of O2PLS, and is using svd, use stripped=T for a stripped version with less output. If either ncol(X) > p_thresh or ncol(Y) > q_thresh, an alternative method is used (NIPALS) which does not store the entire covariance matrix. The squared error between iterands in the NIPALS approach can be adjusted with tol. The maximum number of iterations in the NIPALS approach is tuned by max_iterations.

A list containing

`Tt`	Joint X scores
`W.`	Joint X loadings
`U`	Joint Y scores
`C.`	Joint Y loadings
`E`	Residuals in X
`Ff`	Residuals in Y
`T_Yosc`	Orthogonal X scores
`P_Yosc.`	Orthogonal X loadings
`W_Yosc`	Orthogonal X weights
`U_Xosc`	Orthogonal Y scores
`P_Xosc.`	Orthogonal Y loadings
`C_Xosc`	Orthogonal Y weights
`B_U`	Regression coefficient in `Tt` ~ `U`
`B_T.`	Regression coefficient in `U` ~ `Tt`
`H_TU`	Residuals in `Tt` in `Tt` ~ `U`
`H_UT`	Residuals in `U` in `U` ~ `Tt`
`X_hat`	Prediction of X with Y
`Y_hat`	Prediction of Y with X
`R2X`	Variation (measured with `ssq`) of the modeled part in X (defined by joint + orthogonal variation) as proportion of variation in X
`R2Y`	Variation (measured with `ssq`) of the modeled part in Y (defined by joint + orthogonal variation) as proportion of variation in Y
`R2Xcorr`	Variation (measured with `ssq`) of the joint part in X as proportion of variation in X
`R2Ycorr`	Variation (measured with `ssq`) of the joint part in Y as proportion of variation in Y
`R2X_YO`	Variation (measured with `ssq`) of the orthogonal part in X as proportion of variation in X
`R2Y_XO`	Variation (measured with `ssq`) of the orthogonal part in Y as proportion of variation in Y
`R2Xhat`	Variation (measured with `ssq`) of the predicted X as proportion of variation in X
`R2Yhat`	Variation (measured with `ssq`) of the predicted Y as proportion of variation in Y

ssq, summary.o2m, plot.o2m, crossval_o2m

test.data <- scale(matrix(rnorm(100)))
hist(replicate(1000,
         o2m(test.data,scale(matrix(rnorm(100))),1,0,0)$B_T.
     ),main='No joint variation',xlab='B_T',xlim=c(0,0.6));
hist(replicate(1000,
         o2m(test.data,scale(test.data+rnorm(100))/2,1,0,0)$B_T.
    ),main='B_T = 0.5 \n 25% joint variation',xlab='B_T',xlim=c(0,0.6));
hist(replicate(1000,
         o2m(test.data,scale(test.data+rnorm(100,0,0.1))/2,1,0,0)$B_T.
    ),main='B_T = 0.5 \n 90% joint variation',xlab='B_T',xlim=c(0,0.6));