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Three-Layer Precision, Three-Layer Recall, and Three-Layer F1 Score

This section needs fixing, it seems there are some problems with the math command. See here for a working version.

Three-layer precision ($P_{3}$), three-layer recall ($R_{3}$), and three-layer $F_{1}$ score ($F_{3}$) are defined as follows:

$F_{3}(\Pi ,\Xi )={\frac {2P_{3}(\Pi ,\Xi )R_{3}(\Pi ,\Xi )}{P_{3}(\Pi ,\Xi )+R_{3}(\Pi ,\Xi )}},$

where

Failed to parse (syntax error): {\displaystyle P_3(\Pi, \Xi) &=& \frac{1}{n_\mathcal{Q}} \sum_{j = 1}^{n_\mathcal{Q}} \max \{ F_2(\mathcal{P}_i, \mathcal{Q}_j) \mid i = 1,\ldots, n_\mathcal{P} \},\\[.2cm] R_3(\Pi, \Xi) &=& \frac{1}{n_\mathcal{P}} \sum_{i = 1}^{n_\mathcal{P}} \max \{ F_2(\mathcal{P}_i, \mathcal{Q}_j) \mid j = 1,\ldots, n_\mathcal{Q} \},\\[.2cm] F_2(\mathcal{P}, \mathcal{Q}) &=& \frac{2 P_2(\mathcal{P}, \mathcal{Q}) R_2(\mathcal{P}, \mathcal{Q})} {P_2(\mathcal{P}, \mathcal{Q}) + R_2(\mathcal{P}, \mathcal{Q})},\\[.2cm] P_2(\mathcal{P}, \mathcal{Q}) &=& \frac{1}{m_Q} \sum_{l = 1}^{m_Q} \max \{ F_1(P_k, Q_l) \mid k = 1,\ldots, m_P \},\\[.2cm] R_2(\mathcal{P}, \mathcal{Q}) &=& \frac{1}{m_P} \sum_{k = 1}^{n_P} \max \{ F_1(P_k, Q_l) \mid l = 1,\ldots, m_Q \},\\[.2cm] F_1(P, Q) &=& \frac{2 P_1(P, Q) R_1(P, Q)}{P_1(P, Q) + R_1(P, Q)},\\[.2cm] P_1(P, Q) &=& |P \cap Q|/|Q|,\\[.2cm] R_1(P, Q) &=& |P \cap Q|/|P|. }