rchemo (version 0.1-3)

aicplsr: AIC and Cp for Univariate PLSR Models

Description

Computation of the AIC and Mallows's $Cp$ criteria for univariate PLSR models (Lesnoff et al. 2021). This function may receive modifications in the future (work in progress).

Usage

aicplsr(
    X, y, nlv, algo = NULL,
    meth = c("cg", "div", "cov"),
    correct = TRUE, B = 50, 
    print = FALSE, ...)

Value

crit

dataframe with $n$, and the etimated criteria ($df$, $ct$, $ssr$, $aic$, $cp1,cp2$) for 0 to $nlv$ latent variables in the model.

delta

dataframe with the differences between the estimated values of $aic$, $cp1$ and $cp2$, and those of the model with the lowest estimated values of $aic$, $cp1$ and $cp2$, for models with 0 to $nlv$ latent variables

opt

vector with the optimal number of latent variables in the model (i.e. minimizing aic, cp1 and cp2 values)

Arguments

X

A $nxp$ matrix or data frame of training observations.

y

A vector of length $n$ of training responses.

nlv

The maximal number of latent variables (LVs) to consider in the model.

algo

a PLS algorithm. Default to NULL (plskern is used).

meth

Method used for estimating $df$. Possible values are "cg" (dfplsr_cg), "cov" (dfplsr_cov)or "div" (dfplsr_div).

correct

Logical. If TRUE (default), the AICc corection is applied to the criteria.

B

For meth = "div": the number of observations in the data receiving perturbation (maximum is $n$; see dfplsr_cov). For meth = "cov": the number of bootstrap replications (see dfplsr_cov).

print

Logical. If TRUE, fitting information are printed.

...

Optionnal arguments to pass in algo.

Details

For a model with $a$ latent variables (LVs), function aicplsr calculates $AIC$ and $Cp$ by:

$AIC(a)=n\ast log(SSR(a))+2\ast (df(a)+1)$

$Cp(a)=SSR(a)/n+2\ast df(a)\ast s2/n$

where $SSR$ is the sum of squared residuals for the current evaluated model, $df(a)$ the estimated PLSR model complexity (i.e. nb. model's degrees of freedom), $s2$ an estimate of the irreductible error variance (computed from a low biased model) and $n$ the number of training observations.

By default (argument correct), the small sample size correction (so-called AICc) is applied to AIC and Cp for deucing the bias.

The functions returns two estimates of Cp ($cp1$ and $cp2$), each corresponding to a different estimate of $s2$.

The model complexity $df$ can be computed from three methods (argument meth).

References

Burnham, K.P., Anderson, D.R., 2002. Model selection and multimodel inference: a practical informationtheoretic approach, 2nd ed. Springer, New York, NY, USA.

Burnham, K.P., Anderson, D.R., 2004. Multimodel Inference: Understanding AIC and BIC in Model Selection. Sociological Methods & Research 33, 261-304. https://doi.org/10.1177/0049124104268644

Efron, B., 2004. The Estimation of Prediction Error. Journal of the American Statistical Association 99, 619-632. https://doi.org/10.1198/016214504000000692

Eubank, R.L., 1999. Nonparametric Regression and Spline Smoothing, 2nd ed, Statistics: Textbooks and Monographs. Marcel Dekker, Inc., New York, USA.

Hastie, T., Tibshirani, R.J., 1990. Generalized Additive Models, Monographs on statistics and applied probablity. Chapman and Hall/CRC, New York, USA.

Hastie, T., Tibshirani, R., Friedman, J., 2009. The elements of statistical learning: data mining, inference, and prediction, 2nd ed. Springer, NewYork.

Hastie, T., Tibshirani, R., Wainwright, M., 2015. Statistical Learning with Sparsity: The Lasso and Generalizations. CRC Press

Hurvich, C.M., Tsai, C.-L., 1989. Regression and Time Series Model Selection in Small Samples. Biometrika 76, 297. https://doi.org/10.2307/2336663

Lesnoff, M., Roger, J.M., Rutledge, D.N., Submitted. Monte Carlo methods for estimating Mallows's Cp and AIC criteria for PLSR models. Illustration on agronomic spectroscopic NIR data. Journal of Chemometrics.

Mallows, C.L., 1973. Some Comments on Cp. Technometrics 15, 661-675. https://doi.org/10.1080/00401706.1973.10489103

Ye, J., 1998. On Measuring and Correcting the Effects of Data Mining and Model Selection. Journal of the American Statistical Association 93, 120-131. https://doi.org/10.1080/01621459.1998.10474094

Zuccaro, C., 1992. Mallows'Cp Statistic and Model Selection in Multiple Linear Regression. International Journal of Market Research. 34, 1-10. https://doi.org/10.1177/147078539203400204

Examples

data(cassav)

Xtrain <- cassav$Xtrain
ytrain <- cassav$ytrain

nlv <- 25
res <- aicplsr(Xtrain, ytrain, nlv = nlv)
names(res)
headm(res$crit)

z <- res$crit
oldpar <- par(mfrow = c(1, 1))
par(mfrow = c(1, 4))
plot(z$df[-1])
plot(z$aic[-1], type = "b", main = "AIC")
plot(z$cp1[-1], type = "b", main = "Cp1")
plot(z$cp2[-1], type = "b", main = "Cp2")
par(oldpar)