28/2/2012

PCA for NIR Spectra_part 005: "Reconstruction"

We saw how to plot the raw spectra (X), how to calculate the mean spectrum, how to center the sprectra (subtracting the mean spectrum from every spectra of the original matrix X). After that we have developed the PCAs with the NIPALS algorithm, getting two matrices: T (scores) and P (loadings).
We have to decide the number of PCs, looking to the plots, or to the numbers (explained variance).
Depending of the numbers of PCs, these matrices will have more or less columns.
With these two matrices we can reconstruct again the X centered matrix, but we´ll get also a residual matrix “E”.
Xc = T.Pt+E
This post just shows this in R:
> P3pc_nipals<-P_nipals[,1:3]
> tP3pc_nipals<-t(P3pc_nipals)
> Xc3pc_reconst<-T3pc_nipals
> Xc3pc_reconst<-T3pc_nipals%*%tP3pc_nipals
> matplot(wavelengths,t(Xc3pc_reconst),lty=1,
  + pch=1,xlab="data_points",ylab="log(1/R)")
> resid3pc<-Xc- Xc3pc_reconst
> matplot(wavelengths,t(resid3pc),lty=1,
  + pch=1,xlab="data_points",ylab="log(1/R)")


We can see the plots of the X centered matrix reconstructed and the plot representing the residual variance or Error matrix “E”.
If we add the mean spectrum to every spectra of the centered matrix we will get the X matrix reconstructed.

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