计算并给出一个信息增益比率的样例。

信息增益比率

因为信息增益中，不同的属性值越多，信息增益也越大，所以通过信息增益比率来做相关尝试。

样例

Outlook	Temperature	Humidity 湿度	Wind 风	Play
Sunny 晴朗	Hot	High	False	No
Sunny 晴朗	Hot	High	True	No
Overcast 低沉	Hot	High	False	Yes
Rainy 雨	Mild 轻微	High	False	Yes
Rainy 雨	Cool 凉	Normal	False	Yes
Rainy 雨	Cool 凉	Normal	True	No
Overcast 低沉	Cool 凉	Normal	True	Yes
Sunny 晴朗	Mild 轻微	High	False	No
Sunny 晴朗	Cool 凉	Normal	False	Yes
Rainy 雨	Mild 轻微	Normal	False	Yes
Sunny 晴朗	Mild 轻微	Normal	False	Yes
Overcast 低沉	Mild 轻微	High	True	Yes
Overcast 低沉	Hot	Normal	False	Yes
Rainy 雨	Mild 轻微	High	True	No

Outlook	Yes	No	Count of each group	Entropy 熵
sunny 晴朗	2	3	5	0.971
overcast 低沉	4	0	4	0.000
rainy 雨	3	2	5	0.971
Results 结果	Values 值
Information 信息	0.694
Overall entropy 总熵	0.940
Information gain 信息增益	0.247
Split information 拆分信息	1.577
Gain ratio 增益比率	0.156

Overall entropy：-5/14 * log2(5/14) - 9/14 * log2(9/14) = 0.940
- No 的数量为 5，Yes 的数量为 9，一共 14 个
Entropy 计算：
- sunny 熵： -2/5 * log2(2/5) - 3/5 * log2(3/5) = 0.971
- overcast 熵： -4/4 * log2(4/4) - 0/4 * log2(0/4) = 0
- sunny 熵： -3/5 * log2(3/5) - 2/5 * log2(2/5) = 0.971
Information: 5/14 * 0.971 + 4/14 * 0 + 5/14 * 0.971 = 0.694
Information gain：0.940 - 0.694 = 0.247 此处有精度问题
Split Information：-5/14 * log2(5/14) - 4/14 * log2(4/14) - 5/14 * log2(5/14) = 1.577
Gain ratio：0.247 / 1.577 = 0.156

overall entropy:

\[Overall = -\sum_{i=1}^{n}P(T_i)log_2P(T_i)\]

Entropy:

\[Entropy(A_j) = -\sum_{i=1}^{n}P(A_j|T_i)log_2P(A_j|T_i)\]

Information:

\[Information(A) = \sum_{j=1}^{m}P(A_j)Entropy(A_j)\]

Information gain (A):

\[InformationGain(A) = Overall - Information(A)\]

Split information:

\[SplitInformation(A) = -\sum_{j=1}^{m}P(A_j)log_2P(A_j)\]

Gain ratio:

\[GainRatio(A) = \frac{InformationGain(A)}{SplitInformation(A)}\]