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README.md

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 ---
 license: mit
+language:
+  - en
+tags:
+  - fractal
+  - fractal-dimension
+  - box-counting
+  - hurst-exponent
+  - self-affine
+  - time-series
+  - signal-processing
+  - benchmark
+  - ground-truth
+  - synthetic
+  - roughness
+  - estimator-validation
+pretty_name: Fractal Dimension Benchmark — synthetic signals with exactly known dimension
+size_categories:
+  - n<1K
+configs:
+  - config_name: default
+    data_files: data/signals.parquet
+  - config_name: index
+    data_files: data/signals_index.csv
 ---
+
+# Fractal Dimension Benchmark
+
+**Ground truth for fractal dimension estimators.** 76 synthetic time series whose box-counting
+dimension is known exactly — in closed form, per file, not in expectation.
+
+If you are building or testing a **fractal dimension estimator**, a **Hurst exponent
+estimator**, a **roughness measure**, a box-counting implementation, a variogram or
+detrended-fluctuation method, this is a test set where you already know every answer.
+
+Dimensions span **1.05 to 1.95** in steps of 0.05, at 19,684 samples each. Deterministic and
+reproducible bit for bit.
+
+Companion tool and derivation: [10.5281/zenodo.22052384](https://doi.org/10.5281/zenodo.22052384)
+
+---
+
+## Why fBm is not enough
+
+The usual test signal for a dimension estimator is **fractional Brownian motion**. Its box
+dimension is `2 − H`, which is correct **on average**. But any single realization you hand
+someone deviates from that by an unknown amount, so when your estimator reads 1.47 on a signal
+labelled 1.50, you cannot tell whether your estimator is off, or the realization is.
+
+You are checking a ruler against a stick that is roughly a metre, usually.
+
+These signals are deterministic. Given the file, the true value is a closed-form expression.
+Given the seed, the file regenerates bit for bit on any machine.
+
+---
+
+## The construction
+
+Each signal is a *closed-leg recursion*: replace a segment with three legs whose signed heights
+sum to one, then replace each leg with a scaled copy of the whole figure, repeatedly. With
+equal durations the box dimension is exactly
+
+```
+D = 1 + log₃( |d₁| + |d₂| + |d₃| )
+```
+
+so to hit a target `D` you solve `Σ|dᵢ| = 3^(D−1)` subject to `Σdᵢ = 1`. The balanced solution
+
+```
+b = (3^(D−1) − 1) / 2 ,   s = 1 + b ,   legs = ( s/2 , −b , s/2 )
+```
+
+keeps every leg under 1 in magnitude across the whole range, which is the condition for the
+limit curve to exist.
+
+This is Barnsley's affine fractal interpolation function (1986). The dimension formula is
+classical. Nothing here is claimed as new mathematics — what is offered is the test set.
+
+---
+
+## Two kinds of signal, and the difference matters
+
+**`shuffled_*` — 57 signals, 3 seeds at each dimension.**
+The leg *order* is shuffled at every subdivision. Because addition does not care about order,
+`Σ|dᵢ|` is untouched and the box dimension is **still exactly** the stated value. What changes
+is that the curve is no longer exactly self-similar, only statistically so — which averages
+away multifractality and makes these the right target for a plain single-number estimator.
+
+**`fixed_*` — 19 signals, one at each dimension.**
+The same legs in fixed order. Identical exact box dimension, but genuinely **multifractal**:
+roughness varies from point to point, so an increment-based estimator will legitimately
+disagree with the box dimension. These test whether your estimator *notices* that one number is
+not enough.
+
+An estimator that scores well on `fixed_*` without flagging the disagreement is not measuring
+what it claims to measure.
+
+---
+
+## Contents
+
+```
+data/signals.parquet       all 76 signals, 1,495,984 rows, long format:
+                           signal_id, kind, target_dimension, true_dimension,
+                           seed, t, value
+data/signals_index.csv     76 rows, one per signal, with the leg coefficients
+```
+
+Dimensions run from **1.05 to 1.95** in steps of 0.05, at 19,684 samples per signal. The `t`
+column is the sample index; samples are evenly spaced by construction, so there is no clock
+time. Sort by `t` within a `signal_id` before measuring.
+
+---
+
+## Loading it
+
+```python
+from datasets import load_dataset
+import numpy as np, pandas as pd
+
+ds = load_dataset("Trackertracker2/closed-leg-benchmark", split="train")
+df = ds.to_pandas()
+
+# one signal
+g = df[df.signal_id == "shuffled_D1.50_s1"]
+y = g.value.to_numpy()
+print(g.true_dimension.iloc[0])       # 1.500000000000
+```
+
+Scoring your estimator across the whole shuffled set:
+
+```python
+errs = []
+for sid, g in df[df.kind == "shuffled"].groupby("signal_id"):
+    y = g.sort_values("t").value.to_numpy().astype(float)
+    errs.append(your_estimator(y) - g.true_dimension.iloc[0])
+
+errs = np.abs(errs)
+print(f"mean |error| {errs.mean():.4f}   worst {errs.max():.4f}")
+```
+
+Straight from parquet, without the `datasets` library:
+
+```python
+df = pd.read_parquet("https://huggingface.co/datasets/Trackertracker2/"
+                     "closed-leg-benchmark/resolve/main/data/signals.parquet")
+```
+
+---
+
+## A reference score
+
+The estimator in the companion deposit — three variogram-of-order-*p* estimators at
+*p* = ½, 1, 2, taking the median — scores:
+
+| set | mean abs. error | worst |
+|---|---|---|
+| `shuffled_*` | **0.021** | 0.093 |
+| `fixed_*` | 0.088 | 0.164 |
+
+Its accuracy is **not uniform**. Errors are smallest around *D* = 1.6 and grow toward the top of
+the range, reaching about 0.09 near *D* = 1.95.
+
+That degradation is a property of the estimator, not of the benchmark, and surfacing it is the
+point. An estimator that does well at 1.5 and poorly at 1.9 is worth knowing about before you
+trust it on data whose answer you do not have.
+
+---
+
+## Caveats, stated plainly
+
+- These are **self-affine graphs**, not general rough signals. An estimator tuned to them may
+  still behave differently on physical data. This benchmark tests correctness against known
+  ground truth; it does not certify performance in the field.
+- **19,684 points** is a moderate record length. Estimators needing longer series will be
+  penalised. That is fair, but it should be said.
+- Large errors on `fixed_*` are **expected**, not failures — those signals are multifractal by
+  construction. Read them alongside whatever multifractality diagnostic your tool provides.
+- The signals occupy a **narrow structural class**. Good scores here are necessary for trusting
+  an estimator, not sufficient.
+
+---
+
+## Citation
+
+The full tool, specification and derivation:
+
+- Roughness — a validated measure of texture for ordered signals.
+  [10.5281/zenodo.22052384](https://doi.org/10.5281/zenodo.22052384)
+- The Closed-Leg Atlas — a browser for self-affine and self-similar curves.
+  [10.5281/zenodo.22040679](https://doi.org/10.5281/zenodo.22040679)
+
+```bibtex
+@software{koch_roughness_2026,
+  author  = {Koch, W. A.},
+  title   = {Roughness: a validated measure of texture for ordered signals},
+  year    = {2026},
+  doi     = {10.5281/zenodo.22052384},
+  url     = {https://doi.org/10.5281/zenodo.22052384}
+}
+```
+
+W. A. Koch · [ORCID 0009-0001-1341-7871](https://orcid.org/0009-0001-1341-7871)
+
+Prepared with AI assistance for numerical verification and code. Every value in the answer key
+is a closed-form expression, independently recomputed.

data/signals.parquet

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data/signals_index.csv

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