( Pattern Recognition )

1. A box contains 10,000 ball bearings of similar size: nominally 0.8 cm in diameter. Assume the ball bearing ** masses** are Gaussian distributed with a standard deviation of 0.05 g. Some of the ball bearings are made from a proprietary alloy, which has a density of 4.1 g/cm

^{3}. The others are made from titanium, which has a density of 4.5 g/cm

^{3}. We wish to use these ball bearings in the manufacture of several machines. However, it is important to use the correct type of ball bearing for each application. Some machines can use the proprietary alloy ball bearings, which are relatively inexpensive and not as strong as the titanium. Others must have the titanium ball bearings or they will fail prematurely.

a. Devise a maximum likelihood method to classify each ball bearing as either class 1 (alloy) or class 2 (titanium) based only upon its measured mass. Determine the decision boundary between these classes. Plot the likelihood pdfs and indicate the decision boundary position.

b. Assume it is known that the box contains 9,000 alloy ball bearings and 1,000 titanium ball bearings. Devise a Bayesian method to classify each ball bearing as either class 1 (alloy) or class 2 (titanium) based only upon its measured mass. Determine the decision boundary between these classes. Plot the posterior pdfs and indicate the decision boundary position.

c. Compute the total probability of error (the sum of the individual misclassification errors) for both part (a) and part (b). Show how you derive these values.

d. Let’s characterize the expected loss of using a titanium bearing where an alloy bearing should be used as $0.25. This reflects the case that the titanium bearing is more expensive than the alloy bearing. However, let’s say the expected loss of using an alloy bearing where a titanium bearing should be used as $100. This reflects the cost of premature failure of a machine built with an inadequate ball bearing. With these losses included, repeat part (a) to minimize risk.

e. Repeat part (b) using the losses described in part (d) to minimize risk.

HINT: You’ll need to convert from diameter to mass in order to classify.