# Efficient False Positive Control Algorithms in Big Data Mining
## 2 Related Concepts and Techniques
### Concepts Related to False Positives
#### 2.1.2 Multiple Hypothesis Testing  
Hypothesis testing can solve the single hypothesis testing problem, but in the era of big data, the amount of data involved is huge, and hypothesis testing is no longer sufficient to deal with such a huge amount of data. Therefore, multiple hypothesis testing is used in order to satisfy the problem of dealing with large-scale data [17; 18]. Multiple hypothesis testing is an effective method for calculating large-scale statistical inference problems. It takes all the individual hypothesis tests proposed in the sample as a whole, i.e., a test cluster, and tests each hypothesis in the test cluster simultaneously. For example, n hypotheses \(\{H_{1},H_{2},\ldots,H_{n}\}\) can be proposed in a given sample and each re-evaluationof the hypothesis test commits the first I type of error, and the first II class error; for each heavy hypothesis test, the summary of the results can be obtained as shown in Table 1.  
As shown in Table 1, the results calculated in the n-weight hypothesis test are obtained in four cases, denoted by U, V, T, and S, respectively. The meaning of R in the table is the number of rejections of the null hypothesis \(H_{0}\). The number of correct rejections of the null hypothesis \(H_{0}\) is S, the number of correct acceptances of the null hypothesis \(H_{0}\) is U, the number of committing the I type errors (false positives) is V, and the number of II type errors (false negatives) is T. Similar to the single hypothesis testing, the false positive error of the I type in the process of multiple hypothesis testing can cause incalculable harm to daily applications and subsequent scientific research, so this paper focuses on multiple hypothesis testing in the false positive control problem. In Table 1, the number of false positive errors committed in n-fold hypothesis testing is V. In order to reduce the harm caused by the false positive phenomenon to daily applications and subsequent scientific research, it is necessary to control the false positive phenomenon, i.e., to reduce the number of false positive errors V.  
In multiple hypothesis tests, as in a single hypothesis test controlling for \(p\leq\alpha\), even though \(\alpha\) is a small value, it can lead to an overall significant level that is too high after the multiple hypothesis tests, resulting in a large number of false positives. For example, if the significant level in an n-weight hypothesis test is \(\alpha\), then the number of false positives generated in that n-weight hypothesis test is \(n\alpha\), and if n is very large, \(n\alpha\) will also become very large, which will generate a large number of false positives. Therefore, it is necessary to correct for multiple hypothesis tests to reduce the occurrence of false positives.  
The FWER (family-wise error rate) is the probability of making at least one false positive error in an n-fold hypothesis test. The use of the cluster error rate is the more commonly used control method for multiple hypothesis testing. The commonly used methods for correcting FWER are the Bonferroni correction method [7], the step-down algorithm [9], and the step-up algorithm [10].  
The FDR (false discovery rate) [19] indicates the number of false positives as a proportion of the rejected null hypothesis. The FDR method relaxes the control of false positives compared to the above methods but can significantly improve the power. The commonly used methods for FDR correction are the BH method [19], ABH method [20], TST method [21], etc.