Orthogonal Projections to Latent Structures

This function fits an Orthogonal Projections to Latent Structures (OPLS) model to the provided x (predictor) and y (response) data.

Usage

opls(
  x,
  y,
  ncomp = NULL,
  center = TRUE,
  scale = FALSE,
  crossval = 7,
  permutation = 0,
  ncomp.ortho = deprecated()
)

Arguments

x: A numeric matrix or data frame of the predictor variables.
y: A numeric vector, or a matrix or data frame with one column, of the response variable.
ncomp: The number of orthogonal components removed. If NULL (default), it is determined automatically by cross-validation.
center: A logical value indicating whether to mean-center x and y. Default is TRUE.
scale: A logical value indicating whether to scale x and y to unit variance. Default is FALSE.
crossval: An integer giving the number of cross-validation groups (default 7), between 2 and the number of observations. With crossval = 0, the model is not cross-validated (\(Q^2\) is NA), which requires a fixed ncomp and no permutation.
permutation: An integer giving the number of permutations for the permutation test. Default is 0 (no test).
ncomp.ortho: Use ncomp.

Value

An object of class specproc_opls (a list), which predict() applies to new data, with the following components:

x_scores: The predictive scores \(\textbf{t}\) (one column, p1).
x_loadings: The predictive loadings \(\textbf{p}\).
x_weights: The predictive weights \(\textbf{w}\).
orthoScores: The orthogonal scores \(\textbf{T}_o\) (columns o1, o2, ...).
orthoLoadings: The orthogonal loadings \(\textbf{P}_o\).
orthoWeights: The orthogonal weights \(\textbf{W}_o\).
y_weights: The y-weight \(c\) of the predictive component.
y_scores: The y-scores \(\textbf{u} = \textbf{y}/c\).
correction: The OPLS-filtered x, \(\textbf{X} - \textbf{T}_o\textbf{P}_o^T\) (preprocessed).
fitted: The fitted response, in the units of y.
coefficients: The regression coefficients of the preprocessed filtered x, \(\textbf{w}c\).
vip, ortho_vip: The predictive and orthogonal variable importance in projection (Galindo-Prieto et al., 2014), one value per variable.
components: A tibble with one row per component: R2X, R2Y and Q2 (the increase due to the component), their cumulative values, and significance ("R1" significant, "NS" \(Q^2\) increase below 0.01, "N4" \(R^2Y\) increase below 0.01).
summary: A one-row data frame with R2X(cum), R2Y(cum), Q2(cum), RMSEE (root mean squared error of estimation), the numbers of predictive (pre) and orthogonal (ort) components and, with a permutation test, pR2Y and pQ2.
permutation: With a permutation test, a tibble of R2Y(cum), Q2(cum) and sim (the correlation between the permuted and the original response) of the model (first row) and of each permuted model.
center, scale: The column centers and scales applied to x.
y_center, y_scale: The center and scale applied to y.

Details

OPLS is a supervised modeling technique used to find the multidimensional direction in the x-space that explains the maximum multidimensional variance in the y-space. It separates the systematic variation in x into two parts: one that is linearly related to y (predictive components) and one that is statistically uncorrelated to the response variable y (orthogonal components).

The model has one predictive component and ncomp orthogonal components, fitted with the NIPALS algorithm of Trygg and Wold (2002). For each orthogonal component:

The weight \(\textbf{w} = \textbf{X}^T\textbf{y}/\|\textbf{X}^T\textbf{y}\|\), the score \(\textbf{t} = \textbf{Xw}\) and the loading \(\textbf{p} = \textbf{X}^T\textbf{t}/(\textbf{t}^T\textbf{t})\) are computed.
The orthogonal weight is the part of \(\textbf{p}\) orthogonal to \(\textbf{w}\), \(\textbf{w}_o = \textbf{p} - (\textbf{w}^T\textbf{p})\textbf{w}\), normalized, with score \(\textbf{t}_o = \textbf{Xw}_o\) and loading \(\textbf{p}_o = \textbf{X}^T\textbf{t}_o/(\textbf{t}_o^T\textbf{t}_o)\).
\(\textbf{X}\) is deflated by \(\textbf{t}_o\textbf{p}_o^T\).

The predictive component is then computed from the filtered \(\textbf{X}\). The filtered data are the same as those of projected_osc() with the same ncomp, and of o2pls() with nx = ncomp.

Preprocessing. x and y are centered (center) and scaled to unit variance (scale) with their means and standard deviations, as in the other orthogonalization methods. For Pareto scaling, common in metabolomics, apply pareto_scale() to x first (or step_pareto_scale() in a recipe): it gives the same filter, scores and \(R^2\) and \(Q^2\) values.

Cross-validation. The observations are split into crossval interleaved groups (observation i is in group (i - 1) %% crossval + 1). Each group is predicted by a model fitted to the others, with the same number of orthogonal components, giving the predictive residual sum of squares (PRESS) and \(Q^2 = 1 - \text{PRESS}/\text{SS}_y\), where \(\text{SS}_y\) is the total sum of squares of the preprocessed y. The data are preprocessed once, with the centers and scales of all the observations.

Number of orthogonal components. If ncomp = NULL, components are added (up to min(10, n, p) - 1) while each is significant: a component is significant if it increases \(R^2Y\) by at least 0.01 and \(Q^2\) by at least 0.01. If the predictive component alone is not significant, no model is built. If the first orthogonal component is not significant, the model has no orthogonal component (a one-component PLS model), with a warning.

Permutation test. The response is permuted permutation times and the model refitted with the same number of components. pR2Y and pQ2 are the proportions of permuted models whose \(R^2Y\) and \(Q^2\) are at least those of the model, \((1 + \#\{\text{perm} \geq \text{model}\})/\text{permutation}\). Use set.seed() for reproducible p-values. The test is skipped by default (permutation = 0), as it refits the model as many times.

The results reproduce those of ropls::opls() with predI = 1 and the same preprocessing (scaleC), crossvalI, orthoI and permI (apart from rounding): specProc no longer depends on ropls. Unlike ropls, variables with zero variance are kept (their centered values are zero), and when ncomp = NULL a model without orthogonal components is returned instead of no model when only the predictive component is significant.

References

Trygg, J., and Wold, S., (2002). Orthogonal projections to latent structures (O-PLS). Journal of Chemometrics, 16(3):119-128.
Galindo-Prieto, B., Eriksson, L., and Trygg, J., (2014). Variable influence on projection (VIP) for orthogonal projections to latent structures (OPLS). Journal of Chemometrics, 28(8):623-632.
Thévenot, E.A., Roux, A., Xu, Y., Ezan, E., and Junot, C., (2015). Analysis of the human adult urinary metabolome variations with age, body mass index, and gender by implementing a comprehensive workflow for univariate and OPLS statistical analyses. Journal of Proteome Research, 14(8):3322-3335.

Author

Christian L. Goueguel

Examples

data(forageLIBS)
wl <- suppressWarnings(as.numeric(names(forageLIBS)))
spectra <- forageLIBS[which(wl > 760 & wl < 780)]  # the K I resonance lines
cal <- 1:300
fit <- opls(spectra[cal, ], forageLIBS$K[cal], permutation = 5)
fit
#> Orthogonal projections to latent structures (OPLS)
#> 
#> Variables:               245
#> Observations:            300
#> Scaling:                 centered
#> Predictive components:   1
#> Orthogonal components:   6
#> 
#>       R2X(cum) R2Y(cum) Q2(cum) RMSEE pR2Y pQ2
#> Total    0.995     0.66   0.616 0.311  0.2 0.2
#> 
#> Use predict(<model>, newdata, type = ) to filter new data or predict the response.
head(predict(fit, spectra[-cal, ], type = "response"))
#> [1] 1.947951 2.248777 2.443311 1.985011 2.025831 1.736219

# Pareto scaling of the spectra
opls(pareto_scale(spectra[cal, ]), forageLIBS$K[cal], ncomp = 2)
#> Orthogonal projections to latent structures (OPLS)
#> 
#> Variables:               245
#> Observations:            300
#> Scaling:                 centered
#> Predictive components:   1
#> Orthogonal components:   2
#> 
#>       R2X(cum) R2Y(cum) Q2(cum) RMSEE
#> Total    0.896    0.596   0.569 0.337
#> 
#> Use predict(<model>, newdata, type = ) to filter new data or predict the response.