  Multivariate data are typically represented by a rectangular matrix (table)
in which the rows are the objects (cases) and the columns are the variables
(measurements). When there are many variables one often reduces the dimension
by principal component analysis (PCA), which in its basic form is not robust to
outliers. Much research has focused on handling rowwise outliers, i.e. rows
that deviate from the majority of the rows in the data (for instance, they
might belong to a different population). In recent years also cellwise outliers
are receiving attention. These are suspicious cells (entries) that can occur
anywhere in the table. Even a relatively small proportion of outlying cells can
contaminate over half the rows, which causes rowwise robust methods to break
down. In this paper a new PCA method is constructed which combines the
strengths of two existing robust methods in order to be robust against both
cellwise and rowwise outliers. At the same time, the algorithm can cope with
missing values. As of yet it is the only PCA method that can deal with all
three problems simultaneously. Its name MacroPCA stands for PCA allowing for
Missingness And Cellwise & Rowwise Outliers. Several simulations and real data
sets illustrate its robustness. New residual maps are introduced, which help to
determine which variables are responsible for the outlying behavior. The method
is well-suited for online process control.
