I'm studying THIS paper which builds an index from a set of observed variables using Principal Component Analysis (PCA).
However, the procedure described in Section 3.1 by formulas (6)-(8) confused me.
In particular, suppose the following model:
$$Y_i=\beta_1 A_i+\beta_2 B_i+\beta_3 C_i+u_i, \ i\in\{1,\dotsc,n\}$$ where explanatory vectors $A,B,C$ are observed, $u$ is an error term and vectors $Y,\beta$ are unknown.
The authors proposed to calculate the correlation matrix: $$R_3=X'X$$ where $X$ is the standardized version of the design matrix $[A,B,C]_{n\times 3}$. With the help of PCA, we can obtain the eigenvalues $\lambda_1\geq\dotsc\geq \lambda_k$ associated to the eigeivectors $\phi_1,\dotsc,\phi_k$. We can then calculate the principal components $P_1,\dotsc,P_k$. For simplicity, suppose $k=3$. The authors proposed an estimator for $Y$ given by $$Y_i=\frac{\sum_{j,k=1}^3 \lambda_j P_{ki}}{\sum_{j=1}^3 \lambda_j },$$ where $P_k=X\lambda_j$.
Question: Is there any error with this formula?
Comments
The notation $P_{ki}$ is more likely to mean the $i$th entry of the $k$th principal component. Also, $\sum_{j,k=1}^3 :=\sum_{j=1}^3\sum_{k=1}^3$ commonly. But if I accept this notation for the summation, an inconsistency arises once $P_k=X\lambda_j$. On the other hand, as $P_k$ is a principal component, I was expecting that $P_k=X\phi_k$.
Did they mean $$Y_i=\frac{\sum_{j=1}^3 \lambda_j(X\phi_{j})_i}{\sum_{j=1}^3 \lambda_j }=\frac{\sum_{j=1}^3 \lambda_j \sum_{k=1}^3 X_{i,k}\phi_{k,j}}{\sum_{j=1}^3 \lambda_j }=\frac{\sum_{j,k=1}^3 \lambda_j X_{i,k}\phi_{k,j}}{\sum_{j=1}^3 \lambda_j }$$ where $\phi_{i,j}$ denotes the $i$th entry of the $j$th eigenvector and $X_{i,j}$ denotes the $(i,j)$th entry of the matrix $X$?
Thanks in advance.