Problem adapted from Applied Predictive Modeling by Kuhn, and Johnson

Infrared $($IR$)$ spectroscopy technology can be used to determine the chemical makeup of a

substance. The theory of IR spectroscopy holds that unique molecular structures absorb IR

frequencies differently. In practice a spectrometer fires a series of IR frequencies into a sample

material, and the device measures the absorbance of the sample at each individual frequency.

This series of measurements creates a spectrum profile which can then be used to determine the

chemical make$$up of the sample material.

A Tecator Infratec Food and Feed Analyzer instrument was used to analyze $215$ samples of meat

across $100$ frequencies. A sample of these frequency profiles are displayed in Figure $1.$ In addition

an IR profile, analytical chemistry was used to determine the percent content of water, fat, and

protein for each sample. If we can establish a predictive relationship between IR spectrum and

fat content, then food scientists could predict a sample$$s fat content instead of using analytical

chemistry to determine the content. This would provide costs savings, since analytical chemistry

is a more expensive, time consuming process.

Load the data using following commands. See $?$tecator for details.

library$($caret$)$

## Loading required package: lattice

##

## Attaching package: 'caret'

## The following object is masked from 'package:purrr':

##

## lift

data$($tecator$)$

You should now have access to two matrices:

$$ absorp: absorbance data for $215$ samples. Each row is one observation, and each column is one predictor.

$$ endpoints: outcome variables. Measured are the percentages of water, fat and protein in the samples

$($columns $1,2$ and $3,$ resectively$).$

We want to predict fat content based on the absorbance data.

$($a$)$ In the absorbance data, the predictors are the measurements at the individual frequencies. Because the

frequencies lie in a systematic order $(8501,050$ nm$,$ each column corresponds to one such frequency$),$

the predictors have a high degree of correlation. Hence, the data lie in a smaller dimension than the

total number of predictors $(100).$ Plot the predictors as functions of frequencies $($i$.$e$,$ plot each row of

the data as a line$)$ in one plot. Explain if you see any patterns.

$($b$)$ Do you think the data is indeed $100-$dimensional, or could we reduce the dimension to a lower number?

Use PCA on the absorbance data and explore the results, addressing each of the points below.

How many PCs would you like to keep?

Examine the loadings of the PCs you would like to keep. Is there a nice interpretation of the

loadings in terms of how they contribute to the corresponding PCs$,$ or what patterns they are

capturing?

Do the PCs you want to keep correlate with the response?

Do not print pages after pages of output. Instead, present summary of your exploration using appropriate

tables$/$figures$.$

$($c$)$ Investigate the following models using your choice of data splitting method $($holdout$,$ CV etc$)$ and

compare their predictive performance.

Principal components regression with number of components chosen by $10-$fold CV$.$

Partial least squares regression with number of components chosen by $10-$fold CV$.$

Note: as before, parameter tuning happens in $1$ nner loop, and you choice of data splitting method is outer

loop.

$($d$)$ Investigate the loadings of the first PLS component. Can you interpret the loadings, and the corre$-$

sponding PLS component is a nice way?

$($e$)$ Create a plot showing absolute value of correlation between the response of the PLS components.

Explain any patterns you observe.

$($b$)$ Do you think the data is indeed $100-$dimensional, or could we reduce the dimension to a lower number?

Use PCA on the absorbance data and explore the results, addressing each of the points below.

How many PCs would you like to keep?

Examine the loadings of the PCs you would like to keep. Is there a nice interpretation of the

loadings in terms of how they contribute to the corresponding PCs$,$ or what patterns they are

capturing?

Do the PCs you want to keep correlate with the response?

Do not print pages after pages of output. Instead, present summary of your exploration using appropriate

tables$/$figures$.$

$($c$)$ Investigate the following models using your choice of data splitting method $($holdout$,$ CV etc$)$ and

compare their predictive performance.

Principal components regression with number of components chosen by $10-$fold CV$.$

Partial least squares regression with number of components chosen by $10-$fold CV$.$

Note: as before, parameter tuning happens in inner loop, and you choice of data splitting method is outer

loop.

$($d$)$ Investigate the loadings of the first PLS component. Can you interpret the loadings, and the corre$-$

sponding PLS component is a nice way?

$($e$)$ Create a plot showing absolute value of correlation between the response of the PLS components.

Explain any patterns you observe.

SHOW ALL R$-$CODE!!!