Exploring the Viability of Fisher Discriminants in Galaxy Morphology Classification

1. Problem Statement

Accurate galaxy classification is a fundamental challenge in astronomy, particularly distinguishing between morphological types (e.g., spiral vs. elliptical, or the presence of a central bulge). While complex machine learning algorithms (such as Deep Learning, Boosted Decision Trees, and Neural Networks) have demonstrated high performance, they often suffer from:

• High computational costs: Long training and inference times.
• Complexity: Difficulty in interpretability ("black box" nature).
• Resource intensity: High storage and processing requirements.

The authors investigate whether a simpler, linear algorithm—the Fisher Discriminant—can achieve competitive accuracy in galaxy morphology classification, potentially offering a more efficient alternative for specific astronomical tasks.

2. Methodology

Data Sources

• Input Data: Photometric data from the Sloan Digital Sky Survey (SDSS) DR8.
• Labels: Morphological classifications from Galaxy Zoo 2 (GZ2), specifically focusing on the decision tree branch distinguishing galaxies with a "rounded" central bulge versus "no-bulge."
• Dataset Construction:
  • Initial sample: 1,530 galaxies (1,170 rounded, 360 no-bulge).
  • Balancing: Due to class imbalance, the final dataset was balanced to 720 galaxies (360 rounded, 360 no-bulge).
  • Split: Divided into training, validation, and evaluation sets in a 2:2:1 ratio.
• Input Features: 11 SDSS photometric parameters derived from the r-band (chosen for low uncertainty):
  • 4 Apparent magnitudes (PSF, fiber, model, Petrosian).
  • 3 Petrosian radii (effective, 50% light, 90% light).
  • 3 Model fit probabilities (Star, Exponential, de Vaucouleurs).
  • 1 Adaptive shape measure (ellipticity).

Algorithms Compared

The study utilized the ANNz2 software package (built on TMVA/ROOT) to compare four classifiers:

1. Fisher Discriminant: A linear method maximizing between-class variance while minimizing within-class variance.
2. Artificial Neural Networks (ANN): A Multi-Layer Perceptron (MLP) with two hidden layers (8 and 4 nodes), using tanh activation and BFGS backpropagation.
3. Boosted Decision Trees (BDT): A forest of 500 trees using bagging.
4. k-Nearest Neighbours (kNN): Using a Gaussian kernel and Euclidean distance.

Pre-processing Transformations

To optimize performance, five variable transformations were applied to the input features prior to training:

1. No Transformation (X)
2. Normalization (N): Scales features to [0, 1].
3. Decorrelation (D): Removes linear correlations.
4. Principal Component Analysis (PCA): Reduces dimensionality.
5. Uniformisation (U): Transforms variables to a uniform distribution.
6. Gaussianisation (G): Transforms variables to a Gaussian distribution.

Evaluation Metrics

Performance was assessed using a Confusion Matrix to derive:

• Accuracy: Overall correctness.
• Precision: Reliability of positive predictions.
• Recall: Sensitivity to positive instances.
• F1 Score: Harmonic mean of precision and recall.
• Gaussian Kernel Density Estimation (KDE): Used to visualize class separation and overlap.

3. Key Results

Impact of Pre-processing

• Fisher Discriminant: Significantly benefited from Uniformisation and Gaussianisation. Uniformisation yielded the highest accuracy (0.9310).
• ANN: Also improved with Uniformisation and Gaussianisation, but showed the most dramatic improvement in Recall with Normalization (0.9583).
• BDT and kNN: Generally performed worse or remained static with most transformations. Decorrelation and PCA degraded BDT performance.

Algorithm Comparison (Best Performers)

• Accuracy: Fisher Discriminant (with Uniformisation) achieved the highest score (0.9310), outperforming ANN (0.9132), BDT (0.9201), and kNN (0.8681).
  • Improvement over ANN: +1.93%
  • Improvement over BDT: +0.42%
  • Improvement over kNN: +3.08%
• Precision: Fisher Discriminant (with Gaussianisation) achieved the highest precision (0.9697), significantly outperforming others.
• Recall: ANN (with Normalisation) achieved the highest recall (0.9583), indicating it was better at identifying "rounded" galaxies (minimizing false negatives).
• F1 Score: Fisher Discriminant (with Uniformisation) achieved the highest F1 score (0.9290).

Behavioral Analysis

• Class Separation: The Fisher Discriminant showed high density peaks for "no-bulge" galaxies (high precision) but lower density for "rounded" galaxies, leading to lower recall (more false negatives).
• Stability: The Fisher Discriminant produced robust results in a single run (40 seconds). In contrast, ANNs showed high sensitivity to random seed initialization, requiring multiple runs and significant time (5 minutes) to stabilize, with performance varying drastically based on the seed.

4. Key Contributions

1. Viability of Simple Models: Demonstrated that a linear classifier (Fisher Discriminant) can outperform complex non-linear models (ANN, BDT, kNN) in specific galaxy morphology tasks when paired with appropriate pre-processing (Uniformisation).
2. Pre-processing Sensitivity: Highlighted that the effectiveness of variable transformations is algorithm-dependent. While Fisher and ANN benefit from distributional transformations (Uniformisation/Gaussianisation), tree-based and distance-based methods (BDT/kNN) do not necessarily benefit and may degrade.
3. Efficiency vs. Performance Trade-off: Provided empirical evidence that the Fisher Discriminant offers a superior balance of speed (40s vs. 5 mins) and stability without the hyperparameter tuning required by ANNs.

5. Significance and Conclusion

The study concludes that while complex deep learning models are powerful, simpler algorithms like the Fisher Discriminant are highly viable for specific, well-defined astronomical classification tasks, particularly those with smaller sample sizes or where interpretability and speed are critical.

• Practical Application: For tasks requiring rapid processing of large survey data where "no-bulge" identification is the primary goal, the Fisher Discriminant with Uniformisation is the optimal choice.
• Limitations: The Fisher Discriminant struggles with recall (identifying all rounded galaxies) compared to ANNs, suggesting it may be less suitable if the goal is to capture every instance of a specific feature.
• Future Work: The authors suggest extending feature selection, testing on other GZ2 decision tree branches, and applying the method to higher redshift samples to validate generalizability.

In summary, the paper argues that in the era of big data astronomy, simplicity should not be discarded; the Fisher Discriminant remains a robust, fast, and accurate tool for galaxy morphology classification when properly pre-processed.