# The CIPCA-BPNN Failure Prediction Method Based on Interval Data Compression and Dimension Reduction
## 2 Methods
### Cipca  
The research object of interval data principal component analysis is an interval data matrix \(X_{n\times p}\) containing \(n\) samples; each sample is described by \(p\) interval variables:  
\[ \begin{array}{llll}X_{n\times p}&=\left(X_{1},X_{2},\ldots,X_{p}\right)\\ &=\left[\begin{array}{cccc}\left[x_{11},\overline{x_{11}}\right]&\left[x_{12 },\overline{x_{12}}\right]&\cdots&\left[x_{1p},\overline{x_{1p}}\right]\\ \left[x_{21},\overline{x_{21}}\right]&\left[x_{22},\overline{x_{22}}\right]& \cdots&\left[x_{2p},\overline{x_{2p}}\right]\\ \vdots&\vdots&\ddots&\vdots\\ \left[x_{n1},\overline{x_{n1}}\right]&\left[x_{n2},\overline{x_{n2}}\right]& \cdots&\left[x_{np},\overline{x_{np}}\right]\end{array}\right]\end{array}  \tag{3}\]  
Many principal component analysis methods for interval data have been proposed. [PERSON], [PERSON], and [PERSON] (1997) proposed vertices principal component analysis (VPCA) and centers principal component analysis (CPCA) ([PERSON], 2012). However, these two methods have the disadvantage of using only local information. [PERSON], [PERSON], and [PERSON] (2012) proposed the complete-information-based principal component analysis for interval data (CIPCA) ([PERSON], 2012). This method uses all the information of the interval samples. The modeling results always reflect the internal structural characteristics of the data and are not easily affected by the size of the interval samples. Compared with VPCA and CPVA, CIPCA has higher accuracy and stronger robustness. The same as traditional principal component analysis, in CIPCA, the \(k\)-th interval principal component \(P_{k}\) is a linear combination of \(p\) interval variables, i.e., \(P_{k}=u_{1k}X_{1}+u_{2k}X_{2}+\cdots+x_{pk}X_{p}\), where \(u_{k}=(u_{1k},u_{2k},\ldots,u_{pk})\in R^{p}\) subject to \(u_{k}^{\prime}u_{k}=1\), and \(u_{k}^{\prime}u_{l}=1\). The principal component analysis method is based on the \(k\)-th interval  
\(0\) (\(1\leq l,k\leq p,l\
eq k\)). Using variance to describe the information contained in the principal component of the \(k\)-th interval, we have  
\[ \begin{array}{rl}D_{CI}(P_{k})&=\frac{1}{n}\langle P_{k},P_{k}\rangle\\ &=\frac{1}{n}\langle u_{1k}X_{1}+u_{2k}X_{2}+\cdots+u_{pk}X_{p},\ u_{1k}X_{1}+u_{ 2k}X_{2}+\cdots+u_{pk}X_{p}\rangle\\ &=\frac{1}{n}\langle u_{1k},u_{2k},\ldots,u_{pk}\rangle\left(\begin{array}{ ccccc}\langle X_{1},X_{1}\rangle&\langle X_{1},X_{2}\rangle&\cdots&\langle X_{1},X_{p} \rangle\\ \langle X_{2},X_{1}\rangle&\langle X_{2},X_{2}\rangle&\cdots&\langle X_{2},X_{ p}\rangle\\ \vdots&\vdots&\ddots&\vdots\\ \langle X_{p},X_{1}\rangle&\langle X_{p},X_{2}\rangle&\cdots&\langle X_{p},X_{ p}\rangle\end{array}\right)\left(\begin{array}{c}u_{1k}\\ u_{2k}\\ \vdots\\ u_{pk}\end{array}\right)\\ \end{array}  \tag{4}\]  
According to the principal component analysis, the sum of the variances of the first m principal components \(P_{1},P_{2},\ldots,P_{m}\) should reach the maximum, so \(m\) standard orthogonal vectors \(u_{1},u_{2}\ldots,u_{m}\) should be solved to maximize \(\sum_{k=1}^{m}D_{CI}(P_{k})\) and satisfy \(D_{CI}(P_{1})\geq D_{CI}(P_{2})\geq\cdots\geq D_{CI}(P_{m})\) at the same time; it can be expressed as  
\[s.t.\left\{ \begin{array}{c}\max\sum_{k=1}^{m}u_{k}^{\zeta C_{l}}u_{k}\\ u_{k}^{\prime}u_{k}=1\\ u_{1}^{\prime}u_{k}=0\\ u_{1}^{\prime}S^{CI}u_{1}\geq u_{2}^{\prime}S^{CI}u_{2}\geq\cdots\geq u_{m}^{ \prime}S^{CI}u_{m}\\ l=1,2,\ldots,m\ (l\
eq k)\end{array} \right. \tag{5}\]  
The modeling steps of CIPCA are as follows:  
Step 1: Normalize all interval variables to obtain the standardized interval data matrix \(X_{n\times p}^{*}\). The normalization method is as follows:  
\[x_{ij}^{*}=\left[\frac{x_{ij}-E_{CI}(X_{j})}{D_{CI}(X_{j})},\frac{\overline{x_ {ij}}-E_{CI}(X_{j})}{D_{CI}(X_{j})}\right] \tag{6}\]  
Step 2: Calculate the covariance matrix \(S^{CI}\) of \(X_{n\times p}^{*}\).