fda.usc (version 0.9.8.1)

fregre.pls: Functional PLS regression with scalar response

Description

Computes functional linear regression between functional explanatory variable $X(t)$ and scalar response $Y$ using partial least squares (PLS). $$Y=\big+\epsilon=\int_{T}{X(t)\beta(t)dt+\epsilon}$$ where $\big< \cdot , \cdot \big>$ denotes the inner product on $L_2$ and $\epsilon$ are random errors with mean zero , finite variance $\sigma^2$ and $E[X(t)\epsilon]=0$. The functional partial least squared (FPLS) are calculated in fdata2pls by alternative formulation of the NIPALS algorithm proposed by Kraemer and Sugiyama (2011).

Usage

fregre.pls(fdataobj, y=NULL, l = NULL,...)

Arguments

fdataobj

fdata class object.

y

Scalar response with length n.

l

Index of components to include in the model.

...

Further arguments passed to or from other methods.

Value

• Return:
• call
  The matched call of fregre.pls function.
• beta.est
  Beta coefficient estimated of class fdata.
• coefficients
  A named vector of coefficients.
• fitted.values
  Estimated scalar response.
• residuals
  y-fitted values.
• H
  Hat matrix.
• df
  The residual degrees of freedom.
• r2
  Coefficient of determination.
• GCV
  GCV criterion.
• sr2
  Residual variance.
• l
  Index of components to include in the model.
• rn
  Amount of shrinkage.
• fdata.comp
  Fitted object in fdata2pls function.
• lm
  Fitted object in lm function
• fdataobj
  Functional explanatory data.
• y
  Scalar response.

Details

Functional partial least squares (FPLS) algorithm maximizes the covariance between $X(t)$ and the scalar response $Y$ via the partial least squares (PLS) components. Let $\left{\tilde{\nu}_k\right}_{k=1}^{\infty}$ the functional PLS components and $X_i(t)=\sum_{k=1}^{\infty}\tilde{\gamma}_{ik}\tilde{\nu}_k$ and $\beta(t)=\sum_{k=1}^{\infty}\tilde{\beta}_k\tilde{\nu}_k$. The functional linear model is estimated by: $$\hat{y}=\big< X,\hat{\beta} \big> \approx \sum_{k=1}^{k_n}\tilde{\gamma}_{k}\tilde{\beta}_k$$

References

Preda C. and Saporta G. PLS regression on a stochastic process. Comput. Statist. Data Anal. 48 (2005): 149{-}158. Kraemer, N., Sugiyama M. (2011). The Degrees of Freedom of Partial Least Squares Regression. Journal of the American Statistical Association. Volume 106, 697-705.

See Also

See Also as: fregre.pls.cv, summary.fregre.fd and predict.fregre.fd. Alternative method: fregre.pc, fregre.basis and fregre.np.

Examples

data(tecator)
x<-tecator$absorp.fdata
y<-tecator$y$Fat
res=fregre.pls(x,y,c(1:8))
summary(res)