1. Introduction

2. CHAID Algorithm and Chi-Square Detection

2.2. Introduction of Chi-Square Detection

2.2.1. The Concept and Significance of Chi-Square Detection

Chi-square detection is the deviation between the theoretical value and the actual value of the statistical sample. The degree of deviation determines the size of the chi-square value. If the degree of deviation is smaller, the chi-square value will be smaller. On the contrary, the larger the chi-square value is, if the actual value and the theoretical value are calculated, the chi-square value is equal to 0.

2.2.2. The Basic Idea of Chi-Square Detection

Chi-square test is a commonly used hypothesis test based on the chi-square distribution.

Firstly, assume that the hypothesis: the expected frequency is not different from the observed frequency’ holds. Under this premise, the chi-squared values of the theoretical and actual values are calculated. The probability that hypothesis holds under the current statistical sample can be determined based on the chi-squared distribution and degrees of freedom. The following figure shows some chi-square value probability tables:

Table 1. A probability table of partial chi-square distributions.

Table 1. A probability table of partial chi-square distributions.

$\mathbf{P}({\mathbf{x}}^2\ge \mathbf{k}$)	k	$\mathbf{P}({\mathbf{x}}^2\ge \mathbf{k}$)	k
0.50	0.455	0.05	3.841
0.40	0.708	0.025	5.024
0.25	1.323	0.010	6.635
0.15	2.072	0.005	7.879
0.10	2.706	0.001	10.828

If the P-value is small, then the probability of the hypothesis ${\mathrm{H}}_0$ holding is small, the hypothesis ${\mathrm{H}}_0$ should be rejected, indicating a significant difference between the theoretical value and the actual value; if the P-value is large, the hypothesis ${\mathrm{H}}_0$ cannot be rejected and there is a difference between the theoretical value and the actual situation represented by the actual value.

2.2.3. Formula for Chi-Square Detection

$$\begin{gathered} x^2={\displaystyle \sum _{}^{}\frac{{(A-E)}^2}{E}}={\displaystyle \sum _{i=1}^k\frac{{(A_i-E_i)}^2}{E_i}}={\displaystyle \sum _{i=1}^k\frac{{(A_i-n{p}_i)}^2}{n{p}_i}} \\ (i=1,2,3,\dots ,k) \end{gathered}$$

(1)

In this formula,where ${\mathrm{x}}^2$is the chi-square value obtained by the actual and theoretical values, k is the number of cells in the two-dimensional table, ${\mathrm{A}}_{\mathrm{i}}$ is the actual value of i, ${\mathrm{E}}_{\mathrm{i}}$ is the expected value of i, n is the total number of samples,${\mathrm{p}}_{\mathrm{i}}$ is the expected frequency of i, and ${\mathrm{E}}_{\mathrm{i}}$ = (n: the total number of samples) * (${\mathrm{p}}_{\mathrm{i}}$: the expected probability of i).

2.2.4. Steps for Chi-Square Detection

Firstly, assuming that ${\mathrm{H}}_0$holds, determine the degree of freedom (degree of freedom = (row-1) * (column-1), where the row, the column is the number of rows and columns in the two-dimensional table). Then the theoretical frequency number is obtained with the maximum likelihood estimation, At last, substitute it into the formula to solve.

Table 2. Chi-square detection sample data.

Table 2. Chi-square detection sample data.

	Male	Female
Make up	15 (55)	95 (55)	110
Do not make up	85 (45)	5 (45)	90
	100	100	200

Assume that in the above example “whether or not to make up has no relationship with gender”.

Maximum likelihood estimation yields the expected value ${\mathrm{E}}_{\mathrm{i}}$: ${\mathrm{E}}_1$=100 * 110 / 200=55 (100 is the number of men actually surveyed, 110 / 200 is the number of people in all the surveys, from which the likelihood estimate of men made up).

$${\mathrm{E}}_2=100\ast 110/200=55$$

$${\mathrm{E}}_3=100\ast 90/200=45$$

$${\mathrm{E}}_4=100\ast 90/200=45$$

There is a significant difference between the value obtained in maximum likelihood estimation (inside of parentheses) and the actual value (outside of parentheses).

Generation formula:

$${\mathrm{x}}^2=\sum _{\mathrm{i}=1}^{\mathrm{k}}\frac{{\left({\mathrm{A}}_{\mathrm{i}}-\mathrm{n}{\mathrm{p}}_{\mathrm{i}}\right)}^2}{\mathrm{n}{\mathrm{p}}_{\mathrm{i}}}=\frac{{\left(95-55\right)}^2}{55}+\frac{{\left(15-55\right)}^2}{55}+\frac{{\left(85-45\right)}^2}{45}+\frac{{\left(5-45\right)}^2}{45}=129.3>10.828$$

(2)

Because the desired result, ${\mathrm{x}}^2$> 10.828, means that the null hypothesis of 0.001 might hold, the 99.9% probability is that no makeup is significantly associated with sex;

2.3. The Field Selection Procedure for CHAID Algorithm

After understanding the process of chi-square detection, let’s take a look at the field selection process of CHAID algorithm.

CHAID field is selected using a chi-square statistic. Chi-square distribution is actually whether the two category fields are distributed or not. And the numerical field will help you automatically discrete, into a category field, to analyze whether it is related to the target field. Larger indicates that the more significant the relationship, otherwise not obvious.

The row of the above table is the income level, and the column is whether to buy a computer. The calculation of chi-square statistics should first draw the frequency from the data, and then find their expectations. Then the squared difference between the two is found and summed.

Table 3. Income and actual data.

Table 3. Income and actual data.

income	Yes	No	Total
high	2	2	4
medium	4	2	6
low	3	1	4
Total	9	5	14

Table 4. Income and buying a computer expectation data.

Table 4. Income and buying a computer expectation data.

income	Yes	No	Total
high	2.571429	1.428571	4
medium	3.857143	2.142857	6
low	2.571429	1.428571	4
Total	9	5	14

Table 5. The squared difference between the expected data and the actual data.

Table 5. The squared difference between the expected data and the actual data.

income	Yes	No
high	0.126984	0.228571
medium	0.005291	0.009524
low	0.071429	0.128571

The first table is a table of the actual data, derived from the actual data statistics. The second table is a table of seeking expectations, which is equivalent to thinking that the two are independent, so you can multiply them directly by the probability, and then multiply the total.

The third table is seeking the variance. After summing, ${\mathrm{x}}^2$ is equal to 0.57, and the corresponding probability is 75%, indicating that the two are relatively weak. The full chi-square probabilities show as follows.

Table 6. All chi-square probabilities.

Table 6. All chi-square probabilities.

age	student	credit_rating	income
$x^2$=3.54667 P = 0.16977	$x^2$=2.80000 P = 0.09426	$x^2$=0.93333 P = 0.33400	$x^2$=0.57037 P = 0.75188

After comparison, we can see that the age feature value is the largest in ${\mathrm{x}}^2$=3.54667 value, indicating that the age is the most closely related to whether to buy a computer, so we chose age variable as the variable of the decision tree to produce the leaf node of the next level.

3. Comparison of Decision Tree Algorithm and Accuracy Analysis of CHAID Algorithm

Figure 3. ID3 decision tree.

Figure 3. ID3 decision tree.

4. Discussion, Conclusion and Future Work

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