PCA-based analysis of wave–wind state space
Principal Component Analysis (PCA) is used to characterize the dominant structure of the observational feature space defined by peak frequency, mean square slope (MSS), directional spread, and spectral tail properties. After standardizing variables, PCA provides an orthogonal set of modes that capture the primary axes of variability across all buoy observations, enabling interpretation of the data as a low-dimensional state space rather than a set of discrete regimes.
This is represented by : where n is number of observations and p is features each observation is reduced to z-score values :
Figure 1: PCA Mode Loading Coefficients
PC Descriptions
PC1: Anti correlation between break frequency and mss. As peak frequency increases, mss decreases. Simultaneously, the spectrum becomes broader and the high frequency tail becomes flatter. Summarized by: inverse relationship between MSS and peak frequency, modulated by spread and tail structure. Spectral redistribution/ maturity axis.
PC2: Inverse relationship between tail exponent and directional spreading. As the spectral tail gets steeper, the directional distribution gets narrower. As the directional distribution gets wider, the spectral tail is flatter. Spectral steepness vs directional spreading
PC3: High side of axis corresponds to steep spectral tail, low frequency peak, large directional spreading and high mss. Likely the axis that corresponds to energetic mixed/bimodal seas.
PC4: Unlike PC1 this axis does not separate frequency peak and mss, but highlights their covariance. High mss and high frequency peaks appear together while the directional spread decreases. This is likely reflecting strong local forcing (high winds)
The manifold is dominantly organized by energy redistribution and directional organization (PC1 & PC2). End members can be defined by the extremes of PC1 and PC2.
The leading modes reveal physically interpretable combinations of variables: the first principal component (PC1) captures a redistribution between spectral scale and surface slope (i.e., a contrast between peak frequency and MSS, modulated by spectral breadth), while the second component (PC2) describes variation in directional organization and spectral structure (e.g., narrow, steep spectra versus broad, mixed conditions). Together, these modes define a curved, low-dimensional manifold along which wave–wind states evolve.
Rather than partitioning the data into discrete clusters, PCA is used here to
- Identify dominant axes of variability governing wave–wind coupling,
- Map observations into a continuous state space, and
- Analyze transitions between physically distinct states by examining how spectral properties and directional structure vary along and across principal components.
This framework enables interpretation of tropical cyclone wave fields as a continuum of states structured by underlying physical processes, with persistent “endmember” conditions emerging as extremes along the manifold rather than as isolated clusters.
PC Interpretation
Reframe to : how does the system move through a structured state space? Within a physically motivated feature space, the observations occupy a structured continuum rather than density-separated clusters.
Figure 2: PC1 x PC2 Colored by Physical Variables
The figure shows the distribution of observations in the reduced PC1–PC2 space, with each panel colored by one of the original physical variables. A key result is that all four variables, mean square slope (MSS), peak frequency, directional spread, and tail exponent, exhibit smooth, coherent gradients across the same two-dimensional space. Rather than forming isolated clusters or discontinuous groupings, the data occupy a continuous, densely populated manifold. This indicates that the variability captured by the original four-dimensional feature space is not independent, but instead strongly coupled and effectively constrained to a low-dimensional structure.
The leading principal components provide a physically interpretable coordinate system for this manifold. The first mode (PC1) primarily represents a redistribution between spectral peak and surface roughness, with peak frequency increasing and MSS decreasing along the axis, accompanied by changes in spectral breadth. The second mode (PC2) captures variation in directional organization and spectral structure, separating more narrowly directed, steep-tailed spectra from broader, more mixed conditions.
The fact that each variable varies smoothly and systematically across these axes demonstrates that wave–wind states evolve continuously, rather than transitioning between discrete regimes. This structure explains the lack of robust clustering in the data and supports a framework in which tropical cyclone wave fields are described as a continuum of states governed by a small number of coupled physical processes.
Framework
Wave–wind states evolve primarily through changes in spectral scale and surface slope, while directional structure and spectral tail properties introduce secondary variability that modulates this primary pathway.
Figure 3: PC1 x PC2 Colored By physical Variables with Density Distributions
The marginal distributions further clarify the role of each principal component. MSS and peak frequency vary almost exclusively along PC1, exhibiting near-monotonic behavior and minimal dependence on PC2. This indicates that PC1 represents a dominant, quasi-one-dimensional control governing the redistribution between the location of peak spectral energy and surface slope.
In contrast, directional spread and spectral tail exponent vary across both PC1 and PC2, demonstrating that these variables contribute to secondary structure within the manifold. Together, these results show that the feature space is not only low-dimensional but hierarchically organized, with a primary axis of variability modulated by additional degrees of freedom related to directional and spectral structure.
Although the PCA axes are defined within a selected feature space, the smooth organization of observations across PC1–PC2 is not imposed by the method. The analysis shows that these independently derived wave-state metrics covary coherently, with MSS and peak frequency forming a dominant transition axis and directional spread/tail exponent providing secondary modulation.
Global State Space & Trajectories
Projecting individual storm observations into the global PC1–PC2 space reveals that each storm traces a coherent, continuous trajectory through the same low-dimensional manifold.
These trajectories exhibit smooth temporal evolution, without discontinuous transitions or jumps between regions of the state space, indicating that the manifold represents physically accessible wave–wind states rather than a statistical artifact.
The paths are nonlinear and often curved, reflecting coupled variation between peak frequency, surface slope, and directional structure. Importantly, multiple storms occupy and traverse similar regions of the manifold, suggesting the existence of a shared state space governing wave–wind coupling across different events.
Figure 4: Individual Storm Trajectories Through PC1 x PC2 Space Figure 5: Idalia Trajectory Through PC1 x PC2 Space
Projecting storm-relative variables onto the PCA state space reveals that the manifold is strongly organized by physical forcing.
The primary axis (PC1) exhibits clear gradients with both wind speed and radius from the storm center, indicating that the dominant variability in spectral scale and surface slope is directly tied to wind forcing and wave development stage. In contrast, azimuthal binning shows that different storm-relative sectors occupy distinct regions and follow different trajectories through the manifold, reflecting asymmetries in wind–wave interaction and fetch. These results demonstrate that the low-dimensional PCA space is not merely a statistical reduction, but a physically meaningful coordinate system describing how wave–wind states evolve under varying storm conditions.
The system can be described as dynamics on a low-dimensional, wind-organized manifold.