Decision Tree Problem, Part 2 |

**Reference**: P. Winston, 1992.

__Factors Affecting Sunburn (continued)__

__Phase 2: From Tree to Rules__

We may now establish rules from the decision tree.

RuleIfthen1 the person's hair color is blonde

the person uses no lotionthe person gets sunburned. 2 the person's hair color is blonde

the person uses lotionnothing happens. 3 the person's hair color is red the person gets sunburned. 4 the person's hair color is brown nothing happens.

__Phase 3: Simplify Rules__

Once a rule set has been generated, simplify it.

[The training data has been multiplied by a factor of 4 to permit the use of the chi-square test below]

Assume a significance level of = 0.05

a) **Eliminate Unnecessary Rule Antecedents**

- Consider the two antecedents in Rule #1: blonde hair and lotion.

Blonde

Actual:

**No Change****Sunburned****Marginal Sum****Blonde**16 16 32 **Not Blonde**20 12 32 **Marginal Sum**36 28 64 Expected:

Sample expected value calculation for contingency cell X

_{11}:**No Change****Sunburned****Blonde**18 14 **Not Blonde**18 14 Sample calculation, where o

and e_{ij}are the observed (actual) and expected values for cell x_{ij}, respectively, and 1_{ij}*i*2, 1*j*2:Sample degrees of freedom calculation:

df = (r - 1)(c - 1) = (2 - 1)(2 - 1) = 1

From the chi-square table, = 3.84

Since , we accept the null hypothesis of independence, H

_{0}.We thus conclude, according to the training examples, that sunburn is independent from blonde hair, and thus we may eliminate this antecedent from Rule #1 and Rule #2.

Lotion

For argument's sake we will also examine the lotion antecedent for independence.

Actual:

**No Change****Sunburned****Marginal Sum****Lotion**12 0 12 **No Lotion**8 12 20 **Marginal Sum**20 12 32 Expected:

**No Change****Sunburned****Lotion**7.5 4.5 **No Lotion**12.5 7.5 = 11.52

df = 1

From the chi-square table, = 3.84

Since , we reject the null hypothesis of independence, H

_{0}, and accept the alternate hypothesis of dependence, H_{a}.Therefore, according to the training examples, sunburn is clearly dependent upon the use of lotion, so we cannot eliminate this antecedent.

b) **Eliminate Unnecessary Rules** (under construction)

The tentative rule set is as follows:

RuleIfthen1 the person uses no lotion the person gets sunburned. 2 the person uses lotion nothing happens. 3 the person's hair color is red the person gets sunburned. 4 the person's hair color is brown nothing happens.