| --- |
| license: cc-by-4.0 |
| task_categories: |
| - other |
| language: |
| - en |
| tags: |
| - spatial-statistics |
| - DBSCAN |
| - point-process |
| - cluster-detection |
| - Monte-Carlo |
| pretty_name: DBSCAN Cluster Density Ratios on CSR Point Fields |
| size_categories: |
| - 1B<n<10B |
| --- |
| |
| # DBSCAN Cluster Density Ratios — CSR Simulation Data |
|
|
| Monte-Carlo simulation of DBSCAN clustering applied to complete-spatial-randomness (CSR) |
| point fields. Used to characterize the null distribution of detected cluster density ratios |
| and calibrate anomaly detection p-values (look-elsewhere correction). |
|
|
| ## Simulation parameters |
|
|
| | Parameter | Value | |
| |---|---| |
| | N (points per field) | 10,000 | |
| | Domain | Disk, radius R=100 | |
| | Domain area S₀ | π·100² ≈ 31,415.93 | |
| | Background intensity λ₀ | N/S₀ = 1/π ≈ 0.31831 | |
| | DBSCAN min_samples | 10 | |
| | Post-filter min_cluster_size | 10 | |
| | Post-filter min_area | 0.5 | |
| | eps sweep | 0.80 → 2.86 (step 0.01, 207 values) | |
| | Iterations per eps | ~12,000 – 1,000,000 | |
|
|
| **Total compute**: ~60 h on a single workstation (16-process parallel). |
|
|
| ## Schema |
|
|
| Each parquet file corresponds to one `eps` value (filename: `data/eps_X.XX.parquet`). |
|
|
| | Column | Type | Description | |
| |---|---|---| |
| | `S_prime` | float64 | Convex-hull area of detected cluster. **-1.0 = no cluster found** (placeholder row) | |
| | `N_prime` | int64 | Point count of detected cluster. **-1 = no cluster found** | |
| | `iteration` | int64 | Field index (1-based). Multiple clusters per iteration are all recorded. | |
|
|
| **Always filter** `S_prime != -1` before analysis. Placeholder rows mark iterations with no |
| valid DBSCAN cluster (common at low eps where clustering is rare). |
|
|
| ## Key derived quantities |
|
|
| ```python |
| lambda0 = 10000 / (np.pi * 100**2) # ≈ 0.31831 |
| R = (df.N_prime / df.S_prime) / lambda0 # density ratio (main statistic) |
| R_tilde = R * eps**2 # eps-collapsed statistic (eps-invariant master) |
| ``` |
|
|
| The density ratio `R` follows a heavy-tailed distribution (tail index α≈7). After |
| rescaling to `R̃ = R·eps²`, the distribution collapses to a single master inverse-gamma |
| (shape≈20.5) across all eps — the central empirical finding of this dataset. |
|
|
| ## Usage |
|
|
| ```python |
| import pandas as pd |
| import numpy as np |
| |
| # Load one eps slice |
| eps = 1.20 |
| df = pd.read_parquet(f"hf://datasets/Winternewt/cluster-distribution-simdata/data/eps_{eps:.2f}.parquet") |
| df = df[df.S_prime != -1] # drop placeholder rows |
| |
| lambda0 = 10000 / (np.pi * 100**2) |
| df['R'] = (df.N_prime / df.S_prime) / lambda0 |
| df['R_tilde'] = df['R'] * eps**2 |
| print(df.R_tilde.describe()) |
| ``` |
|
|
| ## Repository |
|
|
| Source code and analysis: https://github.com/winternewt/cluster_distribution |
| Analytic findings: see `docs/analytic_findings.md` in the source repo. |
|
|
