Skip to contents

Orthogonal projections

Source: R/ortho_projection.R
ortho_projection.Rd

Performs orthogonal projections of high-dimensional data matrices using principal component analysis (PCA) or partial least squares (PLS).

Usage

ortho_projection(
  Xr, Xu = NULL, Yr = NULL,
  ncomp = ncomp_by_var(0.01),
  method = c("pca", "pca_nipals", "pls", "mpls", "simpls"),
  center = TRUE,
  scale = FALSE,
  tol = 1e-6,
  max_iter = 1000L,
  pc_selection = deprecated(),
  ...
)

# S3 method for class 'ortho_projection'
predict(object, newdata, ...)

# S3 method for class 'ortho_projection'
plot(x, col = "#3B82F6", ...)

# S3 method for class 'ortho_projection'
predict(object, newdata, ...)

Arguments

Xr

A numeric matrix of reference observations (rows) and variables (columns).

Xu

An optional matrix of additional observations to project.

Yr

An optional response matrix. Required for PLS methods ("pls", "mpls", "simpls") and when using ncomp_by_opc().

ncomp

Component selection method. Either:

Default is ncomp_by_var(0.01).

method

A character string specifying the projection method:

  • "pca": PCA via singular value decomposition (default)

  • "pca_nipals": PCA via NIPALS algorithm

  • "pls": PLS via NIPALS algorithm

  • "mpls": Modified PLS via NIPALS (Shenk and Westerhaus, 1991)

  • "simpls": PLS via SIMPLS algorithm (de Jong, 1993)

center

A logical indicating whether to center the data. Default is TRUE. PLS methods always center internally regardless of this setting.

scale

A logical indicating whether to scale the data to unit variance. Default is FALSE.

tol

Convergence tolerance for the NIPALS algorithm. Default is 1e-6. Ignored when method = "simpls".

max_iter

Maximum number of iterations for NIPALS. Default is 1000. Ignored when method = "simpls".

pc_selection

[Deprecated] Use ncomp instead.

...

Additional arguments (currently unused).

object

Object of class "ortho_projection".

newdata

Matrix of new observations to project.

x

An object of class ortho_projection (as returned by ortho_projection).

col

Color for the plot elements. Default is "#3B82F6".

Value

An object of class "ortho_projection" containing:

  • scores: Matrix of projected scores for Xr (and Xu).

  • X_loadings: Matrix of X loadings.

  • Y_loadings: Matrix of Y loadings (PLS only).

  • weights: Matrix of PLS weights (PLS only).

  • projection_mat: Projection matrix for new data (PLS only).

  • variance: List with original and explained variance.

  • scores_sd: Standard deviation of scores.

  • ncomp: Number of components retained.

  • center: Centering vector used.

  • scale: Scaling vector used.

  • method: Projection method used.

  • ncomp_method: The value passed to the ncomp argument.

  • opc_evaluation: opc optimization results (if applicable).

Details

PCA methods

When method = "pca", singular value decomposition factorizes the data matrix \(X\) as:

\[X = UDV^{T}\]

where \(U\) and \(V\) are orthogonal matrices (left and right singular vectors), and \(D\) is a diagonal matrix of singular values. The score matrix is \(UD\) and the loadings are \(V\).

When method = "pca_nipals", the non-linear iterative partial least squares (NIPALS) algorithm is used instead.

PLS methods

Three PLS variants are available:

  • "pls": Standard PLS using the NIPALS algorithm with covariance-based weights.

  • "mpls": Modified PLS using the NIPALS algorithm with correlation-based weights, giving equal influence to all predictors regardless of variance (Shenk and Westerhaus, 1991).

  • "simpls": SIMPLS algorithm (de Jong, 1993), which deflates the cross-product matrix rather than X itself. Computationally faster than NIPALS, especially for wide matrices.

Component selection

When ncomp_by_opc() is used, component selection minimizes RMSD (for continuous Yr) or maximizes kappa (for categorical Yr) between observations and their nearest neighbors. See diss_evaluate.

References

de Jong, S. 1993. SIMPLS: An alternative approach to partial least squares regression. Chemometrics and Intelligent Laboratory Systems 18:251-263.

Martens, H. 1991. Multivariate calibration. John Wiley & Sons.

Ramirez-Lopez, L., Behrens, T., Schmidt, K., Stevens, A., Dematte, J.A.M., Scholten, T. 2013a. The spectrum-based learner: A new local approach for modeling soil vis-NIR spectra of complex data sets. Geoderma 195-196:268-279.

Ramirez-Lopez, L., Behrens, T., Schmidt, K., Viscarra Rossel, R., Dematte, J.A.M., Scholten, T. 2013b. Distance and similarity-search metrics for use with soil vis-NIR spectra. Geoderma 199:43-53.

Shenk, J.S., Westerhaus, M.O. 1991. Populations structuring of near infrared spectra and modified partial least squares regression. Crop Science 31:1548-1555.

See also

ncomp_by_var, ncomp_by_opc, diss_evaluate, mbl

Author

Leonardo Ramirez-Lopez

Examples

# \donttest{
library(prospectr)
data(NIRsoil)

# Preprocess
sg_det <- savitzkyGolay(
  detrend(NIRsoil$spc, wav = as.numeric(colnames(NIRsoil$spc))),
  m = 1, p = 1, w = 7
)

# Split data
train_x <- sg_det[NIRsoil$train == 1 & !is.na(NIRsoil$CEC), ]
train_y <- NIRsoil$CEC[NIRsoil$train == 1 & !is.na(NIRsoil$CEC)]
test_x <- sg_det[NIRsoil$train == 0 & !is.na(NIRsoil$CEC), ]

# PCA with fixed components
proj <- ortho_projection(train_x, ncomp = 5)

plot(proj)


# PCA with variance-based selection
proj <- ortho_projection(train_x, ncomp = ncomp_by_var(0.01))

# PCA with OPC optimization
proj <- ortho_projection(train_x, Xu = test_x, Yr = train_y,
                         ncomp = ncomp_by_opc(40))

#' plot(proj)

# PLS projection (NIPALS)
proj <- ortho_projection(train_x, Xu = test_x, Yr = train_y,
                         method = "pls", ncomp = ncomp_by_opc(40))

# Modified PLS
proj <- ortho_projection(train_x, Yr = train_y,
                         method = "mpls", ncomp = 10)

# SIMPLS (faster for wide matrices)
proj <- ortho_projection(train_x, Yr = train_y,
                         method = "simpls", ncomp = 10)
# }