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Cellwise Robust Discriminant Analysis

This paper proposes Cellwise Robust Discriminant Analysis (cellQDA and cellLDA), a new method that utilizes cellwise and casewise robust estimators to handle outlying cells and missing values during both training and prediction, thereby preserving information that would otherwise be lost by discarding entire outlier cases.

Original authors: Fabio Centofanti, Can Hakan Dagidir, Mia Hubert, Peter J. Rousseeuw

Published 2026-05-29

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CC0 1.0

Original authors: Fabio Centofanti, Can Hakan Dagidir, Mia Hubert, Peter J. Rousseeuw

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a teacher trying to sort students into different study groups based on their test scores. In a perfect world, every student's score is a true reflection of their knowledge. But in the real world, data is messy. Sometimes a student gets a bad grade because they were sick (a "case" outlier), and sometimes they just made a silly typo on one specific question (a "cell" outlier).

This paper introduces a new, smarter way to sort these students, called Cellwise Robust Discriminant Analysis (or cellLDA and cellQDA). Here is how it works, broken down into simple concepts:

1. The Old Way vs. The New Problem

The Old Way (Classical Analysis):
Imagine a teacher who calculates the "average student" for each group. If one student has a terrible score on one math problem, the old teacher might throw that whole student's entire report card in the trash. They treat the student as a total failure because of one bad number. This is called handling "case outliers."

The New Problem (Cell Outliers):
But what if that student actually knows the material, but just made a typo on one question? Throwing away their whole report card wastes all the good information they have. This is a "cell outlier"—a single bad entry in a sea of good data. The old methods often fail here because they don't know how to ignore just the typo without ignoring the whole student.

2. The Solution: A "Spot the Typo" System

The authors propose a two-step system that acts like a very careful detective:

Step A: Training the Detective (The Classroom)
First, the system looks at the "clean" data from each group to learn what a "normal" student looks like.

It uses a special tool (called cellMCD) that scans the data.
If it sees a weird number (like a 100-year-old child or a negative weight), it flags that specific number as suspicious but keeps the rest of the student's data.
It builds a "map" of what normal looks like for each group, ignoring the typos.

Step B: The Exam (The Test Data)
Now, new students arrive for classification. This is where the magic happens.

The Flagging: When a new student arrives with a weird number (e.g., a very high salt content in a dessert), the system doesn't panic. It asks: "Is this number a typo?"
The Penalty: The system has a rule: "If you have to ignore too many numbers to make this student fit a group, they probably don't belong there."
The Decision: It calculates a score for every group. If a student fits Group A perfectly except for one weird number, the system says, "Okay, we'll ignore that one weird number, and they belong in Group A." But if the student is weird in every way, the system says, "This student doesn't fit any group," and puts them in a "Unknown" bin.

3. The "Cellwise Contamination" Model

The paper invents a new mathematical story (a model) to explain why this works.

Imagine every data point is a mix of Truth (a normal bell-curve distribution) and Noise (a wild, unpredictable distribution).
The system tries to figure out: "Is this specific number part of the Truth, or is it part of the Noise?"
If it's Noise, it gets flagged and ignored for the final decision. If it's Truth, it counts.

4. Why This is Better (The Results)

The authors tested this on computer simulations and real data about Swiss sweets (cookies, ice cream, cakes, puddings).

The Simulation: When they added "typos" (outliers) to the test data, the old methods got confused and sorted students into the wrong groups. The new method (cellQDA) kept sorting them correctly because it knew how to ignore the typos.
The Real Data: When classifying sweets, the new method was much more accurate (83% vs. 57% for the old method).
Missing Data: The system also handles missing information (blank cells) naturally. If a student didn't take a test, the system just ignores that question and decides based on the rest, rather than rejecting the whole student.

5. Visualizing the Results

The paper includes cool pictures called Classmaps and Cellmaps:

Classmaps: These show where students fall. If a student is far away from their group, they are flagged.
Cellmaps: These are like heatmaps. If a specific sweet has a "suspicious" amount of salt, that specific square turns red. This helps humans see exactly which ingredient caused the confusion.

Summary

In short, this paper teaches computers how to be gracious. Instead of rejecting a whole person (or data point) because of one mistake, the new method identifies the mistake, ignores it, and makes a fair decision based on the rest of the information. It works for both linear and quadratic sorting methods and handles missing data without breaking a sweat.

Technical Summary: Cellwise Robust Discriminant Analysis

Problem Statement

Classical Discriminant Analysis (DA), including Linear (LDA) and Quadratic (QDA) variants, relies on the arithmetic mean and empirical covariance matrix to estimate class parameters. These estimators are highly sensitive to outliers. While traditional robust statistics have focused on casewise outliers (entire data points lying far from the class center), modern data analysis increasingly encounters cellwise outliers. These are unexpected entries within a data matrix where only a few cells of a case are corrupted, while the remaining cells contain valuable information.

Removing an entire case due to a few outlying cells results in significant information loss. Furthermore, casewise robust methods often fail in the presence of cellwise contamination because even a small percentage of outlying cells can pollute a majority of cases. Existing literature has addressed cellwise robust estimation for training data (e.g., Aerts and Wilms, 2017), but a critical gap remains: robust out-of-sample prediction. Current methods struggle to classify new test data that may themselves contain outlying cells or missing values (NA's), often leading to misclassification.

Methodology

The authors propose cellLDA and cellQDA, a unified framework for cellwise robust discriminant analysis that addresses both the training and prediction stages.

1. Robust Training (In-Sample)

The method estimates the location ( $\mu_g$ ) and scatter matrix ( $\Sigma_g$ ) for each class $g$ using the cellMCD estimator (Raymaekers and Rousseeuw, 2024).

Objective: CellMCD minimizes a penalized negative log-likelihood that includes a partial squared Mahalanobis distance calculated only on unflagged cells.
Flagging: It simultaneously estimates a binary mask matrix $W_g$ where $w_{ij}=0$ indicates a flagged (outlying or NA) cell and $w_{ij}=1$ indicates a clean cell.
LDA Adaptation: For Linear DA, a shared scatter matrix is estimated by pooling centered data and applying cellMCD with a fixed center.

2. Out-of-Sample Prediction (Test Data)

The core novelty lies in the prediction stage, which handles test cases containing cellwise outliers and NA's.

Cellflagger: For a new test point $x$ , a likelihood-based "cellflagger" iteratively flags cells. It minimizes a penalized objective function to determine which cells are outlying relative to a specific class $g$ . A cell is flagged if its standardized residual exceeds a robust tolerance interval (based on the $\chi^2$ quantile).
Contamination Model: The authors introduce a novel statistical model extending the cellwise contamination framework (Alqallaf et al., 2009) to multiple classes. The model assumes:
- Clean cells follow a multivariate normal distribution $N(\mu_g, \Sigma_g)$ .
- Outlying cells follow a heavy-tailed Laplace distribution.
- Outlying cells occur with a specific probability $p_{gj}$ .
Robust Discriminant Function: Based on the joint likelihood of the data and the contamination indicators under the above model, a new discriminant function $\delta_g(x, w_g)$ $δ_{g} (x, w_{g})$ is derived. This function includes:
1. The standard log-likelihood of the unflagged cells.
2. A penalty term for the number of flagged cells (based on the Bernoulli probability of contamination).
3. A term accounting for the magnitude of the outlying cells (based on the Laplace density).
Handling Missing Data: The framework naturally handles NA's by fixing the corresponding weights to 0 (treating them as permanently excluded) before applying the cellflagger, ensuring the discriminant function only sums over available clean cells.
Casewise Outlier Detection: If a test case has more than $d/2$ flagged cells or a large Mahalanobis distance on the remaining cells, it is assigned to a new "outlier" class (class 0).

Key Contributions

Unified Framework: The approach unifies parameter estimation, outlier detection, and classification under a single penalized maximum likelihood principle.
Robust Prediction: Unlike previous cellwise robust methods, this approach provides a mechanism for classifying test data that may contain outlying cells, preventing the misclassification of cases with minor data errors.
Handling Missing Values: The method naturally accommodates missing data without requiring imputation prior to classification.
Interpretability: The authors introduce graphical tools (Classmaps and Cellmaps) to visualize classification results, distinguishing between clean cases, correctly classified cases with outlying cells, and casewise outliers.

Results

Simulations

The methods were evaluated on simulated data with dimensions $d=5$ and $d=20$ , under various contamination scenarios (cellwise, casewise, and mixed) and correlation structures.

Clean Test Data: When training data contained cellwise outliers, cellQDA and cellLDA maintained high accuracy, outperforming classical DA (CQDA/CLDA) and casewise robust methods (RQDA/RLDA).
Contaminated Test Data: When test data contained cellwise outliers, only cellQDA and cellLDA maintained robust performance. Classical and casewise robust methods suffered significant accuracy drops because their discriminant rules were not designed to handle outlying cells in the test set.
Mixed Scenarios: The proposed methods consistently outperformed existing robust baselines (GLQDA/RLDA) when both training and test data were contaminated.

Real Data Application

The method was applied to a dataset of 804 Swiss sweets (Cookies, Ice Cream, Cakes, Puddings) with 9 nutritional variables.

Accuracy: cellQDA achieved the highest cross-validated accuracy (0.832), surpassing CQDA (0.569), RQDA (0.805), and GLQDA (0.767).
Interpretation: Graphical displays revealed that cellQDA correctly classified products with unusual nutritional profiles (e.g., specific "Weight Watchers" brands) that were misclassified by other methods or flagged as global outliers. The cellmaps successfully identified specific unusual ingredients (e.g., unexpectedly high/low salt) driving the classification.
Missing Data: The method maintained high accuracy (76%) even when up to 60% of test cells were set to missing, significantly outperforming classical DA applied to complete data.

Significance and Claims

The paper claims that the primary significance of this work is the development of a robust out-of-sample prediction method for discriminant analysis. While robust training is established, the ability to classify new data containing cellwise outliers without discarding the entire observation is a novel contribution.

The authors emphasize that their approach is coherent, as the estimation, flagging, and classification steps all derive from the same penalized likelihood model. This unification allows for natural handling of missing values and provides a statistically grounded way to penalize the number of flagged cells. The simulation results demonstrate that these methods offer superior performance in realistic scenarios where data contamination occurs in both training and testing phases, without sacrificing efficiency on clean data.