How do you incorporate prior knowledge or domain expertise into robust PCA and factor analysis?

2 How to regularize PCA and FA?

One way to make PCA and FA more robust is to add some regularization to the optimization problem. Regularization is a technique that penalizes overly complex or extreme solutions and encourages simpler or more sparse ones. For example, you can add a penalty term to the objective function of PCA or FA that penalizes the magnitude of the principal components or the factor loadings. This can help you avoid overfitting, reduce noise, and enhance interpretability. Another example is to use sparse PCA or sparse FA, which impose sparsity constraints on the principal components or the factor loadings. This can help you identify the most relevant variables or factors and eliminate the irrelevant ones.

3 How to constrain PCA and FA?

Another way to make PCA and FA more robust is to add some constraints to the solution space. Constraints are conditions that limit the possible values or relationships of the parameters or variables. For example, you can add some non-negativity constraints to the principal components or the factor loadings, which can make them more interpretable and meaningful in some contexts. Another example is to add some orthogonality constraints to the factor loadings, which can make them more independent and easier to compare. Constraints can also incorporate some prior knowledge or domain expertise into the dimensionality reduction, such as known relationships, patterns, or structures in the data.

4 How to use Bayesian inference for PCA and FA?

A third way to make PCA and FA more robust is to use Bayesian inference instead of frequentist inference. Bayesian inference is a framework that combines prior information and data evidence to update beliefs and uncertainties about the parameters or variables. For example, you can use Bayesian PCA or Bayesian FA, which assign prior distributions to the principal components or the factor loadings, and update them with the data likelihood using Bayes' rule. This can help you incorporate some prior knowledge or domain expertise into the dimensionality reduction, such as expected values, ranges, or shapes of the parameters or variables. Bayesian inference can also handle outliers, noise, or missing values better than frequentist inference, by using robust likelihood functions or imputation methods.

5 What are the benefits and challenges of robust PCA and FA?

Robust PCA and FA can offer several advantages compared to standard PCA and FA, such as improved accuracy and reliability of the dimensionality reduction, enhanced interpretability of the low-dimensional representation, increased flexibility and adaptability, and better integration of prior knowledge or domain expertise into the analysis. Nevertheless, robust PCA and FA also come with some challenges, such as increased computational complexity and cost of optimization or inference, difficulty in choosing the appropriate regularization, constraint, or prior parameters, potential loss of information or variability due to regularization, constraint, or prior bias, and a trade-off between robustness and simplicity of the dimensionality reduction.