House ownership – data rescaling

For each person, we are given their age, yearly income, and whether or not they own a house:

Age	Annual income in USD	House ownership status
23	50,000	Non-owner
37	34,000	Non-owner
48	40,000	Owner
52	30,000	Non-owner
28	95,000	Owner
25	78,000	Non-owner
35	130,000	Owner
32	105,000	Owner
20	100,000	Non-owner
40	60,000	Owner
50	80,000	Peter

House ownership and annual income

The aim is to predict whether Peter, aged 50, with an income of $80,000 per year, owns a house and could be a potential customer for our insurance company.

Analysis

In this case, we could try to apply the 1-NN algorithm. However, we should be careful about how we measure the distances between the data points, since the income range is much wider than the age range. Income levels of USD 115 k and USD 116 k are USD 1,000 apart. The two data points for these incomes would be very far apart. However, relative to each other, the difference between these data points isn't actually that big. Because we consider both measures (age and yearly income) to be about as important as each other, we would scale both from 0 to 1 according to the following formula:

In our particular case, this reduces to the following:

After scaling, we get the following data:

Age	Scaled age	Annual income in USD	Scaled annual income	House ownership status
23	0.09375	50,000	0.2	Non-owner
37	0.53125	34,000	0.04	Non-owner
48	0.875	40,000	0.1	Owner
52	1	30,000	0	Non-owner
28	0.25	95,000	0.65	Owner
25	0.15625	78,000	0.48	Non-owner
35	0.46875	130,000	1	Owner
32	0.375	105,000	0.75	Owner
20	0	100,000	0.7	Non-owner
40	0.625	60,000	0.3	Owner
50	0.9375	80,000	0.5	?

Now, if we apply the 1-NN algorithm with the Euclidean metric, we will find out that Peter more than likely owns a house. Note that, without rescaling, the algorithm would yield a different result. Refer to Exercise 1.5 for more information.