Comparison of model selection criteria for Partial Least Squares Regression.

This function computes the test error over several runs for different model selection strategies.

benchmark.pls(
  X,
  y,
  m = ncol(X),
  R = 20,
  ratio = 0.8,
  verbose = TRUE,
  k = 10,
  ratio.samples = 1,
  use.kernel = FALSE,
  criterion = "bic",
  true.coefficients = NULL
)

Arguments

X	matrix of predictor observations.
y	vector of response observations. The length of `y` is the same as the number of rows of `X`.
m	maximal number of Partial Least Squares components. Default is `m=ncol(X)`.
R	number of runs. Default is 20.
ratio	ratio no of training examples/(no of training examples + no of test examples). Default is 0.8
verbose	If `TRUE`, the functions plots the progress of the function. Default is `TRUE`.
k	number of cross-validation splits. Default is 10.
ratio.samples	Ratio of (no of training examples + no of test examples)/`nrow(X)`. Default is 1.
use.kernel	Use kernel representation? Default is `use.kernel=FALSE`.
criterion	Choice of the model selection criterion. One of the three options aic, bic, gmdl. Default is "bic".
true.coefficients	The vector of true regression coefficients (without intercept), if available. Default is `NULL`.

Value

MSE

data frame of size R x 5. It contains the test error for the five different methods for each of the R runs.

data frame of size R x 5. It contains the optimal number of components for the five different methods for each of the R runs.

DoF

data frame of size R x 5. It contains the Degrees of Freedom (corresponding to M) for the five different methods for each of the R runs.

TIME

data frame of size R x 4. It contains the runtime for all methods (apart from the zero model) for each of the R runs.

M.CRASH

data frame of size R x 2. It contains the number of components for which the Krylov representation and the Lanczos representation return negative Degrees of Freedom, hereby indicating numerical problems.

if true.coefficients are available, this is a data frame of size R x 5. It contains the model error for the five different methods for each of the R runs.

SIGMAHAT

data frame of size R x 5. It contains the estimation of the noise level provided by the five different methods for each of the R runs.

Details

The function estimates the optimal number of PLS components based on four different criteria: (1) cross-validation, (2) information criteria with the naive Degrees of Freedom DoF(m)=m+1, (3) information criteri with the Degrees of Freedom computed via a Lanczos represenation of PLS and (4) information criteri with the Degrees of Freedom computed via a Krylov represenation of PLS. Note that the latter two options only differ with respect to the estimation of the model error.

In addition, the function computes the test error of the "zero model", i.e. mean(y) on the training data is used for prediction.

If true.coefficients are available, the function also computes the model error for the different methods, i.e. the sum of squared differences between the true and the estimated regression coefficients.

References

Kraemer, N., Sugiyama M. (2011). "The Degrees of Freedom of Partial Least Squares Regression". Journal of the American Statistical Association 106 (494) https://www.tandfonline.com/doi/abs/10.1198/jasa.2011.tm10107

Author

Nicole Kraemer

Examples


# generate artificial data
n<-50 # number of examples
p<-5 # number of variables
X<-matrix(rnorm(n*p),ncol=p)
true.coefficients<-runif(p,1,3)
y<-X%*%true.coefficients + rnorm(n,0,5)
my.benchmark<-benchmark.pls(X,y,R=10,true.coefficients=true.coefficients)
#> iteration no  1  
#> iteration no  2  
#> iteration no  3  
#> iteration no  4  
#> iteration no  5  
#> iteration no  6  
#> iteration no  7  
#> iteration no  8  
#> iteration no  9  
#> iteration no  10