Datasets:

Trackertracker2
/

closed-leg-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

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.

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

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:

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:

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
The Closed-Leg Atlas — a browser for self-affine and self-similar curves. 10.5281/zenodo.22040679

@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

Prepared with AI assistance for numerical verification and code. Every value in the answer key is a closed-form expression, independently recomputed.

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Fractal Dimension Benchmark