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Local depth-based classification of directional data

This paper proposes and evaluates a novel classification method for directional data that utilizes a local depth-based approach within a DD-plot framework, demonstrating its effectiveness through extensive simulations and real-world applications.

Original authors: Giuseppe Gismondi, Rebecca Rivieccio, Giuseppe Pandolfo

Published 2026-02-24

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Giuseppe Gismondi, Rebecca Rivieccio, Giuseppe Pandolfo

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

The Big Picture: Navigating a World of Directions

Imagine you are trying to sort a massive pile of arrows. Some arrows point North, some South, some point toward a mountain, and others toward a river. In statistics, this is called directional data. Unlike regular numbers (like height or weight) that sit on a straight line, these "arrows" live on the surface of a sphere (like the Earth).

The problem? It's hard to sort them. If you just look at the "average" direction, you might get confused. For example, if you have a group of arrows pointing North and another group pointing South, the "average" points straight up (out of the Earth), which tells you nothing about the actual groups.

The Old Way: The "Global" Map

For a long time, statisticians used a method called Global Depth. Think of this like looking at a map of the entire world from space. You try to find the "center" of the whole country.

How it works: It asks, "How close is this arrow to the center of everything?"
The Flaw: This works great if everyone is living in one big city. But what if the arrows are actually living in three different, far-away villages? The "global center" might end up in the middle of the ocean, far away from any village. If you try to sort arrows based on how close they are to that ocean center, you'll get the villages mixed up.

The New Idea: The "Local" Neighborhood Watch

The authors of this paper (Giuseppe Gismondi, Rebecca Rivieccio, and Giuseppe Pandolfo) say: "Stop looking at the whole world. Let's look at the neighborhood."

They propose a new method called Local Cosine Distance Depth (LCDD).

The Analogy: The Coffee Shop vs. The Whole City
Imagine you are trying to figure out if a new person belongs to the "Artists" group or the "Engineers" group.

Global Approach: You ask, "How close is this person to the average citizen of the entire city?" If the city has a mix of artists and engineers, the average person is a boring mix of both. This doesn't help you sort the new person.
Local Approach: You ask, "Who are the 5 people standing closest to this new person in the coffee shop?"
- If the 5 closest people are all wearing berets and holding paintbrushes, you classify the new person as an Artist.
- If the 5 closest people are all wearing hard hats and holding blueprints, you classify them as an Engineer.

This is what LCDD does. Instead of measuring distance to the "center of the universe," it measures how central a point is within its own small, local neighborhood.

How They Did It (The "Mirror" Trick)

To make this math work on a sphere (where directions are tricky), the authors used a clever trick involving mirrors.

The Reflection: Imagine you are standing in a room full of people. To understand your "local neighborhood," the authors imagine a mirror placed right in front of you. They reflect everyone else in the room across that mirror.
The Calculation: They then count how "central" you are to this new, mirrored crowd.
The Result: If you are in the middle of a cluster, your mirrored reflection makes you look very central. If you are on the edge of a cluster, the reflection makes you look far away. This helps the computer spot the "villages" (clusters) even if they are far apart or shaped weirdly.

The Test Drive: Simulations and Real Life

The team put their new method to the test in two ways:

1. The Simulation Lab (The Video Game)
They created fake worlds with millions of arrows.

Scenario A: Arrows were scattered in messy, multi-colored blobs.
Scenario B: Arrows were in weird, donut-shaped rings.
The Result: The old "Global" method got confused and mixed the colors. The new "Local" method saw the distinct blobs and rings perfectly, sorting them with much higher accuracy.

2. Real-World Examples

The Grocery Store: They looked at data from a wholesale distributor (restaurants vs. retail stores). The local method figured out the spending habits of these two groups much better than the old method.
The Spam Filter: They looked at emails to see if they were "Spam" or "Not Spam." Emails are like high-dimensional arrows (lots of words). The local method found the subtle patterns that distinguish spam from real mail, reducing errors by about 8%.

The Takeaway

Why does this matter?
In a complex world, things aren't always simple and uniform. Sometimes data comes in clusters, sometimes in rings, and sometimes in weird shapes.

Global Depth is like a general who looks at the whole battlefield from a helicopter. It's good for big, simple battles.
Local Depth is like a scout on the ground. It looks at the immediate surroundings to make the best decision.

By using this "scout" approach, the authors created a smarter way to sort directional data (like wind directions, magnetic fields, or even word frequencies in emails), making our computers better at understanding the complex, multi-faceted world around us.

1. Problem Statement

Context: Directional data (unit vectors on a hypersphere $S^{q-1}$ ) arises in fields like geology, meteorology, and text mining (via compositional data transformations). Unlike Euclidean data, directional data lacks a natural ordering and a unique reference direction.
The Challenge:

Global Depth Limitations: Traditional statistical depth functions (e.g., Angular Tukey Depth, Global Cosine Distance Depth) provide a "center-outward" ordering based on the entire distribution. While effective for unimodal, convex distributions, they fail to capture local structures in multimodal or non-convex data (common in mixture models or clustering).
Classification Failure: In complex directional distributions, global depth may assign similar centrality values to points in different local modes, leading to poor separation in classification tasks.
Goal: The authors aim to develop a local depth function for directional data that can adapt to the scale of locality, thereby improving the performance of the Depth-vs-Depth (DD) classifier for complex, multimodal directional datasets.

2. Methodology

A. Theoretical Foundation: Local Cosine Distance Depth (LCDD)

The authors propose a local version of the Cosine Distance Depth (CDD).

Global CDD: Defined as $CDD(x, F) = 2 - E_F[d_{cos}(x, W)]$ , where $d_{cos}(x, w) = 1 - \langle x, w \rangle$ . It measures global centrality.
Local Construction: To define a local neighborhood, the authors utilize a symmetric reflector operator $R(x_j, x_i) = 2x_i\langle x_i, x_j \rangle - x_j$ $R (x_{j}, x_{i}) = 2 x_{i} ⟨ x_{i}, x_{j} ⟩ - x_{j}$ .
- For a point $x_i$ , the original sample is augmented with the reflections of all other points through $x_i$ .
- Key Theoretical Result (Proposition 1 & 2): The authors prove that for CDD, the "depth-based neighborhood" (points ordered by depth in the reflected region) is mathematically equivalent to the set of nearest neighbors based on cosine distance.
Definition: The Local Cosine Distance Depth (LCDD) at level $\beta \in (0, 1]$ $β \in (0, 1]$ is defined as the CDD of point $x_i$ $x_{i}$ calculated with respect to the subset of its $\beta(n-1)$ $β (n - 1)$ nearest neighbors (in terms of cosine distance).
- As $\beta \to 1$ , LCDD converges to the global CDD.
- As $\beta \to 0$ , LCDD approaches the maximum possible depth (2).
Properties: LCDD is rotation-invariant, strictly positive, and bounded. Unlike global depth, it is not monotone along rays (allowing for multiple local maxima), which is desirable for multimodal data.

B. Classification Framework: The DD-Classifier

The proposed LCDD is integrated into the DD-plot (Depth vs. Depth plot) framework:

Transformation: For two classes ( $F_1, F_2$ ), each observation is mapped to a 2D point $(LCDD(x, \hat{F}_1), LCDD(x, \hat{F}_2))$ .
Separation: A polynomial separator $s(\cdot)$ is fitted to the DD-plot to minimize the empirical misclassification rate.
Optimization: The locality parameter $\beta$ and the polynomial degree are selected via cross-validation to minimize the classification error.

C. Theoretical Guarantees

Consistency: The paper establishes that the sample LCDD is a uniformly consistent estimator of the population LCDD (Lemma 1).
Bayes Consistency: Theorem 5 proves that under specific regularity conditions (existence of an optimal $\beta^*$ and polynomial separator), the proposed data-driven classifier converges to the Bayes optimal classifier as the sample size increases.

3. Key Contributions

Novel Depth Function: Introduction of the Local Cosine Distance Depth (LCDD), which adapts the global CDD to capture local modes in directional data using a nearest-neighbor approach derived from reflection properties.
Theoretical Equivalence: Proof that for CDD, the complex depth-based neighborhood definition simplifies to standard cosine-distance nearest neighbors, making the method computationally feasible even in high dimensions.
Robust Classification: Development of a LCDD-DD classifier that outperforms global depth-based classifiers in scenarios involving multimodality and non-convex class structures.
Convergence Theory: Rigorous proof of the uniform consistency of the estimator and the Bayes consistency of the resulting classifier.

4. Results

A. Simulation Study

The authors conducted extensive simulations comparing LCDD against global CDD across two scenarios:

Scenario 1 (Von Mises-Fisher Mixtures): Data generated from mixtures of vMF distributions.
- Findings: In unimodal cases, both methods performed similarly. However, in multimodal setups (Setup 2 and 3), LCDD significantly outperformed global CDD, achieving misclassification rates below 1% in low-noise, low-dimensional settings where global methods struggled.
Scenario 2 (Watson Distributions): Data generated from Watson distributions (bipolar or girdle shapes), representing non-convex structures.
- Findings: Global CDD consistently underperformed. LCDD demonstrated superior flexibility, adapting to the bipolar/girdle shapes where global depth failed to distinguish classes effectively.
General Trends: Performance gaps narrowed in high dimensions ( $d=25$ ) and high noise, where both methods converged to similar error rates. The choice of $\beta$ (0.05, 0.1, 0.25) had minimal impact on performance relative to the choice between local and global approaches.

B. Real-Data Applications

Wholesale Customers (Compositional Data):
- Task: Classify customers into "Horeca" vs. "Retail" channels based on spending habits.
- Result: The optimal $\beta$ was 0.05. The local approach improved the misclassification rate by 4.5 percentage points compared to the global approach.
SPAM Database (Text Mining):
- Task: Classify emails as spam vs. non-spam using word frequencies (treated as compositional data on a hypersphere).
- Result: The optimal $\beta$ was 0.01 (indicating very fine-grained local structure). The local approach reduced the misclassification rate by 8 percentage points compared to the global approach.

5. Significance

Handling Complexity: This work addresses a critical gap in directional statistics: the inability of global depth functions to handle multimodal and non-convex data structures.
Computational Efficiency: By proving that LCDD neighborhoods are equivalent to nearest neighbors, the method avoids the high computational cost of geometric depths (like Tukey depth) in high dimensions.
Practical Utility: The results on real-world datasets (marketing and text mining) demonstrate that treating data as directional and applying local depth can yield significant practical improvements in classification accuracy over traditional global methods.
Future Directions: The authors suggest extending this local depth framework to multiclass problems and other types of manifolds.

In summary, the paper successfully bridges the gap between local data structure analysis and directional classification, providing a theoretically sound and empirically superior method for handling complex directional datasets.