Tutorial: Paper 2 – Resolution Invariance¶

Source paper: Zheng, Low & Wang (2026), Regime Labels Are Not Resolution-Invariant: Evidence Across Five Asset Classes.

This tutorial demonstrates mrv.invariance.res_invariance_validator(), which tests whether a regime model produces consistent labels across sampling frequencies (5m, 15m, 1h, 1d). The core question is: does a model trained on daily bars agree with the same model trained on hourly bars?

Theory background¶

Paper 2 defines resolution invariance as label stability under a change of temporal aggregation. The key empirical finding (Paper 2 Table 2) is that cross-frequency ARI drops significantly below the 0.65 threshold for most asset-frequency pairs, with intraday pairs showing higher agreement than daily-vs-intraday pairs.

ResolutionSpec encodes the Paper 2 panel: four frequencies (5m, 15m, 1h, 1d) and the two intraday-only frequencies for the within-intraday excess statistic.

Step 1: build per-frequency label dicts¶

The validator expects a nested dict: {asset: {freq: label_array}}.

import numpy as np
import pandas as pd

rng = np.random.default_rng(42)
K = 3
N_DAILY = 250

def make_label_seq(n, seed):
    rng_l = np.random.default_rng(seed)
    labels = np.zeros(n, dtype=int)
    state = 0
    trans = {0: [0.88, 0.08, 0.04],
             1: [0.10, 0.78, 0.12],
             2: [0.06, 0.14, 0.80]}
    for i in range(1, n):
        state = rng_l.choice(K, p=trans[state])
        labels[i] = state
    return labels

# 1d bars (250 observations)
labels_1d = make_label_seq(N_DAILY, seed=0)

# 1h bars (approx 8x more observations for equity session)
labels_1h = make_label_seq(N_DAILY * 8, seed=1)

# 15m bars
labels_15m = make_label_seq(N_DAILY * 26, seed=2)

# 5m bars
labels_5m = make_label_seq(N_DAILY * 78, seed=3)

resolution_set = {
    "SPY": {
        "5m":  labels_5m,
        "15m": labels_15m,
        "1h":  labels_1h,
        "1d":  labels_1d,
    }
}

Step 2: run the validator¶

from mrv.invariance import res_invariance_validator, ResolutionSpec

result = res_invariance_validator(
    model_fn=None,
    resolution_set=resolution_set,
    spec=ResolutionSpec(),    # default Paper 2 four-frequency panel
)

result.summary()
print("ARI matrix (SPY):", result.ari_matrix["SPY"])
print("AMI matrix (SPY):", result.ami_matrix["SPY"])

Within-intraday excess¶

Paper 2 shows that within-intraday ARI (e.g., 5m vs 15m) is consistently higher than daily-vs-intraday ARI (1h or 15m vs 1d). The within_intraday_excess field reports the difference:

print("Within-intraday excess (SPY):",
      result.within_intraday_excess.get("SPY"))

A positive value means the model is more consistent among intraday frequencies than it is when compared to the daily bar – the typical finding from Paper 2.

Permutation p-values¶

The validator computes permutation p-values for each frequency pair to guard against inflated ARI from small sample sizes:

if result.perm_pvalue:
    for pair, pv in result.perm_pvalue.get("SPY", {}).items():
        print(f"  {pair}: p={pv:.4f}")

Using PAPER2_FREQS¶

mrv.invariance.PAPER2_FREQS and mrv.invariance.PAPER2_INTRADAY_FREQS are the canonical frequency lists from Paper 2 and are used to construct ResolutionSpec:

from mrv.invariance import PAPER2_FREQS, PAPER2_INTRADAY_FREQS

print("Paper 2 frequencies:", PAPER2_FREQS)
print("Intraday subset:    ", PAPER2_INTRADAY_FREQS)

Interpreting results¶

Field	Interpretation
`ari_matrix[asset]`	DataFrame of cross-frequency ARI values. >= 0.65 is the Paper 2 threshold.
`ami_matrix[asset]`	AMI robustness table (Paper 2 Table S1).
`within_intraday_excess`	Intraday mean ARI minus overall mean ARI (positive = within-intraday more stable).
`perm_pvalue`	Permutation p-values per pair per asset.