# Efficient False Positive Control Algorithms in Big Data Mining
## 2 Related Concepts and Techniques
### False Positive Control-Related Methods  
False positive control methods for multiple hypothesis testing can be broadly classified into two categories: FWER control methods and FDR control methods. FWER control methods are more stringent than FDR control methods, and FDR control methods will achieve better efficacy than FWER control methods. Therefore, for multiplex problems that require strict control of the number of false positives, the FWER control method is required. For a multiple testing problem in an exploratory study, the FDR control method is preferred. After further problem analysis from the perspective of hypothesis testing, this paper will use two types of labels \(W_{1}\), \(W_{0}\) to denote the \"range\" of parameters, since \"hypothesis\" is a kind of virtual determination of the range to which the real parameters belong. Since \"hypothesis\" is a virtual determination of the range of the real parameters, then the null hypothesis \(H_{0}\) can be regarded as the range of the real parameters belonging to the label \(W_{1}\), and the alternative hypothesis \(H_{1}\) can be regarded as the range of the real parameters belonging to the label \(W_{0}\). In this paper, we will choose the transaction dataset as the real parameter, and obviously, the null hypothesis \(H_{0}\) becomes that transaction \(T_{i}\) belongs to label \(W_{1}\). Let \(S_{i}\) be the set of items contained in a transaction \(T_{i}\), then if a transaction \(T_{i}\) contains the set of items \(S_{i}\) and the label of that transaction is \(W_{1}\), then we can define a rule L: \(S_{i}\) \(\Rightarrow\)\(W_{1}\), which obviously also becomes a false positive control problem for multiple hypothesis testing in association rule mining. This section briefly describes three methods for correcting multiple hypothesis testing in association rule mining: the direct adjustment method, the permutation-based approach, and the holdout evaluation method. 1. Direct adjustment method: The direct adjustment method is the direct control of false positives using the implementation algorithm of FWER or FDR. A common direct adjustment method for FWER is the Bonferroni correction [27; 28], which calculates the hypothesized \(p\)-value and considers it significant if the \(p\)-value is not greater than \(\alpha/n\). A common direct adjustment method for FDR is the BH procedure [19], where the \(p\)-values are sorted in ascending order \(p_{1},\ldots,p_{n}\), if \(p_{i}\leq i\alpha/n\), \(i=n,\ldots,1\) holds, it is considered that \(H_{1},\ldots,H_{i}\) is statistically significant. 2. Permutation-based approach: The permutation-based approach [29] is to randomly disrupt the class labels and then recombine them with the transactions and recalculate the \(p\)-values [30; 31]. Since the individual hypothesis tests are dependent on each other,the random disturbance is used to break the association between the transactions and the class labels. The distribution of the recalculated \(p\)-values is, therefore, an approximation of the null distribution, which allows a more precise determination of the truncation threshold (corrected significance threshold) of the \(p\)-values. To keep the FWER under the \(\alpha\) level, an operation is performed that randomly generates a set of n labels is performed as a way to break the association between transactions and class labels. A truncated \(p\)-value (significance threshold) is eventually found such that the probability of having at least one false positive error is no greater than \(\alpha\). To find the truncated \(p\)-value, the smallest \(p\)-value obtained after the calculation in each permutation is ranked from lowest to highest, and the \(\alpha\)th value among them is used as the truncation threshold. To control the FDR at the \(\alpha\) level, n-label permutations are randomly generated and adjusted for each \(p\)-value by the following method.