Infrared (IR) spectroscopy technology is 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 makeup of the sample material. A Tecator Infratec Food and Feed Analyzer instrument was used to analyse 215 samples of meat across 100 frequencies. A sample of these frequency profiles is displayed in Fig. 6.20. In addition to an IR profile, analytical chemistry determined 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 with IR instead of using analytical chemistry. This would provide costs savings, since analytical chemistry is a more expensive, time-consuming process:

(a) Start R and use these commands to load the data: > library(caret) > data(tecator) > # use? tecator to see more details The matrix absorp contains the 100 absorbance values for the 215 samples, while matrix endpoints contain the percent of moisture, fat, and protein in columns 1–3, respectively.

(b) In this example the predictors are the measurements at the individual frequencies. Because the frequencies lie in a systematic order (850–1,050nm), the predictors have a high degree of correlation. Hence, the data lie in a smaller dimension than the total number of predictors (215). Use PCA to determine the effective dimension of these data. What is the effective dimension?

(c) Split the data into a training and a test set, pre-process the data, and build each variety of models described in this chapter. For those models with tuning parameters, what are the optimal values of the tuning parameter(s)?