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Orthogonality and Dimensionality in Airline Cluster Analysis using PCA and Kernel PCA

This paper replicates and extends a study on US airline profit cycles by demonstrating that while the original six-cluster taxonomy is geometrically robust across linear and nonlinear PCA spaces, the dataset structurally supports only three clusters, a finding obscured in the original analysis by collinearity in the raw 7-dimensional space.

Original authors: Andreas Schlapbach

Published 2026-06-09

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Andreas Schlapbach

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ 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 trying to sort a long line of 26 people (representing 26 years of airline history from 1995 to 2020) into different groups based on how much money they made. A previous study tried to do this using seven different measurements (like fuel prices, wages, and ticket sales) and ended up with six distinct groups.

This paper is a "reality check." The author, Andreas Schlapbach, took that same data and asked three simple questions:

Are those seven measurements actually telling us six different stories, or are they just repeating the same story over and over?
Is six the right number of groups, or is there a simpler answer?
Did the math used to find these groups get tricked by the way the data is structured?

Here is what the paper found, explained simply:

1. The "Echo Chamber" Effect (Collinearity)

The author discovered that the seven measurements used in the original study were like seven people in a room all shouting the same thing at slightly different volumes. In technical terms, they are highly collinear.

The Analogy: Imagine trying to measure the size of a growing tree. You measure its height, the length of its branches, the width of its trunk, and the number of leaves. All of these grow together. If you use all of them to sort the tree into "small," "medium," and "large," you aren't getting six different pieces of information; you are just measuring the tree's growth six different ways.
The Result: The author found that the data actually only has about 2.5 independent dimensions of real information, not seven. The original study was essentially measuring the same "growth trend" over and over.

2. The "Stretchy Rubber Band" (Why the Groups Stayed the Same)

Even though the data was "noisy" with these repeating measurements, the original six groups remained exactly the same when the author re-ran the math using a cleaner, simpler version of the data.

The Analogy: Imagine the years of airline history are beads on a long, stretchy rubber band. The original study used a rubber band that was stretched out very unevenly (because of the repeating measurements). The author stretched it even more evenly.
The Surprise: Even though the rubber band changed shape, the order of the beads didn't change. The beads representing the "low profit years" were still at one end, and the "high profit years" were at the other. Because the groups were arranged in a straight line of time (chronological order), stretching the line didn't mix them up. The "six groups" were just six slices of that same time-sausage.

3. The "Three vs. Six" Debate (The Real Number of Groups)

The most important finding is about how many groups actually exist. The original study claimed there were six distinct eras. However, when the author used a "quality check" tool (called the Silhouette criterion) to see how well the groups fit together, the data screamed that there are really only three natural groups.

The Three Real Groups:
1. The Struggle Era (1995–2005): Low profits, crises, and low ticket prices.
2. The Recovery Era (2006–2014): Rebuilding, high fuel costs, but profits coming back.
3. The Golden Era (2015–2019): Record-breaking profits.
The Outlier: The year 2020 (COVID) was so weird it didn't fit into any group; it was a "singularity" or a glitch in the matrix.

The author argues that the original study's "six groups" were a bit like cutting a loaf of bread into six slices when the loaf was only big enough for three distinct chunks. The six slices are still bread, but the data naturally prefers three chunks. The "noise" from the repeating measurements (collinearity) hid the fact that three was the better answer.

4. The "Curvature Check" (Is the Data Bending?)

The author wanted to make sure the data wasn't secretly curved or twisted in a way that simple math couldn't see. They used a fancy technique called Kernel PCA (which looks for hidden curves) with six different mathematical "lenses."

The Result: All six lenses agreed: The data is flat. There are no hidden curves or secret shapes. The airline profit cycles move in a straight, predictable line. This confirms that the simple math used in the original study was actually fine; it just needed to be interpreted with the right number of groups.

The Bottom Line

The paper concludes that the original study's main idea—that airline profits go through different cycles and return to normal after a crisis—is correct and robust. The math didn't break, and the groups didn't disappear.

However, the study suggests a correction:

Don't use all seven measurements (they are redundant).
Don't force six groups (the data naturally wants three).
Do recognize that 2020 was a unique outlier that broke the pattern.

In short: The airline profit story is a straight line with three main chapters and one weird footnote, not a complex maze with six hidden rooms.

Technical Summary: Orthogonality and Dimensionality in Airline Cluster Analysis using PCA and Kernel PCA

Problem Statement

This study replicates and critically evaluates the clustering methodology presented in Renold et al. [6], which characterizes US airline profit cycles from 1995 to 2020. The original study utilized k-means clustering on seven highly collinear variables, combined with Principal Component Analysis (PCA) and system dynamic modelling, to identify a six-cluster taxonomy. Two specific methodological gaps in the original work prompted this replication:

Collinearity Distortion: The seven input variables are driven by secular growth trends, creating high multicollinearity. It remains unexamined whether this collinearity distorts cluster assignments when moving from the raw 7-dimensional space to a geometrically cleaner principal component (PC) score space.
Cluster Justification: The choice of $k=6$ clusters lacks quantitative justification. The study questions whether the data structurally supports six clusters or if a different number is more appropriate.

The analysis aims to answer three core questions:

Does collinearity materially distort cluster assignments between raw and PC spaces?
What number of clusters does the data structurally support?
Does including the fourth principal component (capturing the COVID-19 yield shock) improve cluster quality?

Methodology

The analysis utilizes 26 annual observations (1995–2020) from Renold et al. [6], covering operating profit, fuel price, average wages, RPM, load factor, yield per passenger, and other expenses ratio. Data were standardized (zero mean, unit variance) prior to analysis.

Experimental Setup:

Clustering: k-means clustering was performed with 100 random initializations and up to 1000 iterations to ensure convergence.
Spaces Analyzed:
1. 7D-raw: The original 7-dimensional variable space.
2. 3D-pc: The space of the first three principal components (capturing 87.8% of variance).
3. 4D-pc: The space of the first four principal components (capturing 95.5% of variance).
Metrics:
- Cluster Agreement: Measured via the Adjusted Rand Index (ARI).
- Cluster Quality: Assessed using the Silhouette coefficient (higher is better) and the Davies–Bouldin index (lower is better).
- Dimensionality: Effective dimensionality was quantified using the Participation Ratio (PR) derived from the eigenvalue spectrum.
Nonlinearity Check: Kernel PCA (kPCA) was applied under six kernels (Linear, RBF, Polynomial degree 2 & 3, Fisher–Mahalanobis, and Graph Diffusion) to test for hidden manifold curvature.

Key Findings

1. Robustness of Cluster Assignments to Collinearity

The study finds that the six-cluster taxonomy is geometrically robust.

Identity of Assignments: k-means in the 3D-pc space produces bit-for-bit identical cluster assignments to the 7D-raw space for all 26 years (ARI = 1.000).
Mechanism: The lack of distortion is attributed to the temporal alignment of the clusters. The dominant collinear direction (PC1) represents secular US aviation growth. Since the six clusters are contiguous time bands, stretching the temporal axis (the effect of collinearity) preserves the band structure without reordering points across boundaries.
4D-pc Nuance: The 4D-pc space reassigns only the year 2000 (a fuel-spike year) from the 9/11 cluster, separating it based on the yield-per-passenger component (PC4).

2. Structural Support for Three Clusters

Contrary to the original six-cluster framework, the data structurally supports three clusters.

Silhouette Analysis: In the geometrically clean 3D-pc space, the Silhouette score peaks decisively at $k=3$ (0.523), whereas $k=6$ scores 0.448 (14% worse).
Collinearity Suppression: In the 7D-raw space, the Silhouette curve is flat between $k=6$ and $k=7$ , failing to identify $k=3$ as the natural choice. The inflated growth axis suppresses the between-cluster contrast necessary to reveal the optimal partition.
Effective Dimensionality: The Participation Ratio (PR) is calculated at approximately 2.47, indicating that the seven inputs effectively represent only ~2.5 independent measurements due to severe collinearity.
Interpretation: The three natural regimes are identified as:
1. Low-profit era (1995–2005)
2. Recovery era (2006–2014)
3. Peak-profit era (2015–2019)
  (With 2020 as a structural singularity). The six-cluster solution is described as a descriptively coherent subdivision imposed on the data rather than a structure recommended by it.

3. Superiority of the 4D-pc Space

The 4D-pc space is identified as the quantitatively superior clustering space.

It achieves the best Davies–Bouldin index at $k=6$ (0.557 vs. 0.647 in 7D-raw), a 13.9% improvement.
Including PC4 (yield-per-passenger) captures the unique variance structure of the COVID-19 year (2020), making the space internally consistent with the observation that four components are required to model the data fully.

4. Linearity of the Manifold

Kernel PCA analysis confirms that the underlying data manifold is intrinsically linear with no hidden curvature.

Cross-Kernel Agreement: All six kernels (spanning distance-based, information-geometric, and graph-based families) preserve the six-cluster assignment in 2D projections (ARI = 1.000 between linear and non-linear projections).
1D Diagnostic: A 1D diagnostic using the first kernel component reveals a specific behavior: the linear kernel conflates the COVID year (C3) with the peak-profit cluster (C0). However, all five non-baseline kernels shift C3 to overlap with the post-financial-crisis cluster (C5).
Conclusion: The agreement across fundamentally different kernel families confirms the absence of hidden non-linear structure.

Significance and Claims

The paper claims to validate the geometric robustness of Renold et al.'s airline profitability clustering while identifying critical gaps in methodological justification.

Validation of Stability: The headline finding that the six-cluster structure persists when moving from raw 7D space to 3D-PC space is confirmed. The study concludes that collinearity does not distort cluster assignments because the collinear direction (secular growth) is orthogonal to the partitioning criterion (temporal regimes).
Methodological Correction: The study asserts that while the six-cluster solution is empirically defensible for policy analysis requiring fine temporal resolution, it is statistically under-specified. The data structurally supports three clusters, a fact obscured by collinearity in the raw space.
Linearity Confirmation: The cross-kernel validation strengthens the conclusion that the airline profitability manifold is intrinsically linear, closing robustness critiques regarding hidden curvature.
Recommendation: The paper recommends performing k-means on PCA score vectors (specifically the 4D-pc space) rather than raw variables to avoid collinearity-induced signal suppression and to better capture the unique variance of the COVID-19 period.

The study maintains a modest tone, acknowledging that the six-cluster taxonomy remains useful for descriptive purposes and that the original claim of post-pandemic reversion to pre-COVID patterns is robust within the 1995–2020 observation window.