Distance-based partial least squares regression
dbplsr is a variety of partial least squares regression where explanatory information is coded as distances between individuals. These distances can either be computed from observed explanatory variables or directly input as a squared distances matrix.
Since distances can be computed from a mixture of continuous and qualitative explanatory variables or, in fact, from more general quantities, dbplsr is a proper extension of plsr.
Notation convention: in distance-based methods we must distinguish observed explanatory variables which we denote by Z or z, from Euclidean coordinates which we denote by X or x. For explanation on the meaning of both terms see the bibliography references below.
## S3 method for class 'formula' dbplsr(formula,data,...,metric="euclidean", method="ncomp",weights,ncomp) ## S3 method for class 'dist' dbplsr(distance,y,...,weights,ncomp=ncomp,method="ncomp") ## S3 method for class 'D2' dbplsr(D2,y,...,weights,ncomp=ncomp,method="ncomp") ## S3 method for class 'Gram' dbplsr(G,y,...,weights,ncomp=ncomp,method="ncomp")
formula: an object of class formula. A formula of the form y~Z. This argument is a remnant of the plsr function, kept for compatibility.
data: an optional data frame containing the variables in the model (both response and explanatory variables, either the observed ones, Z, or a Euclidean configuration X).
y: (required if no formula is given as the principal argument). Response (dependent variable) must be numeric, matrix or data.frame.
distance: a dist or dissimilarity class object. See functions dist in the package stats and daisy
in the package cluster.
D2: a D2 class object. Squared distances matrix between individuals.
G: a Gram class object. Weighted centered inner products matrix of the squared distances matrix D2. See details in dblm.
metric: metric function to be used when computing distances from observed explanatory variables. One of "euclidean" (default), "manhattan", or "gower".
method: sets the method to be used in deciding how many components needed to fit the best model for new predictions. There are five different methods, "AIC", "BIC", "OCV", "GCV" and "ncomp" (default). OCV and GCV find the number of components that minimizes the Cross-validation coefficient (ocv or gcv). AIC and BIC find the number of components that minimizes the Akaike or Bayesian Information Criterion (see AIC
for more details).
weights: an optional numeric vector of weights to be used in the fitting process. By default all individuals have the same weight.
ncomp: the number of components to include in the model.
...: arguments passed to or from other methods to the low level.
Partial least squares (PLS) is a method for constructing predictive models when the factors (Z) are many and highly collinear. A PLS model will try to find the multidimensional direction in the Z space that explains the maximum multidimensional variance direction in the Y space. dbplsr is particularly suited when the matrix of predictors has more variables than observations. By contrast, standard regression (dblm) will fail in these cases.
The various possible ways for inputting the model explanatory information through distances, or their squares, etc., are the same as in dblm.
The number of components to fit is specified with the argument ncomp.
A list of class dbplsr containing the following components:
residuals: a list containing the residuals (response minus fitted values) for each iteration.
fitted.values: a list containing the fitted values for each iteration.
fk: a list containing the scores for each iteration.
bk: regression coefficients. fitted.values = fk*bk
Pk: orthogonal projector on the one-dimensional linear space by fk.
ncomp: number of components included in the model.
ncomp.opt: optimum number of components according to the selected method.
weights: the specified weights.
method: the using method.
y: the response used to fit the model.
H: the hat matrix projector.
G0: initial weighted centered inner products matrix of the squared distance matrix.
Gk: weighted centered inner products matrix in last iteration.
gvar: total weighted geometric variability.
gvec: the diagonal entries in G0.
gvar.iter: geometric variability for each iteration.
ocv: the ordinary cross-validation estimate of the prediction error.
gcv: the generalized cross-validation estimate of the prediction error.
aic: the Akaike Value Criterium of the model.
bic: the Bayesian Value Criterium of the model.
Boj E, Delicado P, Fortiana J (2010). Distance-based local linear regression for functional predictors. Computational Statistics and Data Analysis 54, 429-437.
Boj E, Grane A, Fortiana J, Claramunt MM (2007). Implementing PLS for distance-based regression: computational issues. Computational Statistics 22, 237-248.
Boj E, Grane A, Fortiana J, Claramunt MM (2007). Selection of predictors in distance-based regression. Communications in Statistics B - Simulation and Computation 36, 87-98.
Cuadras CM, Arenas C, Fortiana J (1996). Some computational aspects of a distance-based model for prediction. Communications in Statistics B - Simulation and Computation 25, 593-609.
Cuadras C, Arenas C (1990). A distance-based regression model for prediction with mixed data. Communications in Statistics A - Theory and Methods 19, 2261-2279.
Cuadras CM (1989). Distance analysis in discrimination and classification using both continuous and categorical variables. In: Y. Dodge (ed.), Statistical Data Analysis and Inference. Amsterdam, The Netherlands: North-Holland Publishing Co., pp. 459-473.
Boj, Eva evaboj@ub.edu, Caballe, Adria adria.caballe@upc.edu, Delicado, Pedro pedro.delicado@upc.edu and Fortiana, Josep fortiana@ub.edu
When the Euclidean distance is used the dbplsr model reduces to the traditional partial least squares (plsr).
summary.dbplsr for summary.
plot.dbplsr for plots.
predict.dbplsr for predictions.
#require(pls) library(pls) data(yarn) ## Default methods: yarn.dbplsr <- dbplsr(density ~ NIR, data = yarn, ncomp=6, method="GCV")
Useful links