BEE – Bifurcated Extremal Entropy

BEE extracts local structure around extrema in time series.
For every detected peak and trough it computes:

• PE – Permutation Entropy (local randomness)
• H_RS, H_DFA, H_VT, H_SP – Hurst exponents via Range/Standard deviation, Detrended Fluctuation Analysis, Variance–Time, and Spectral methods

Per-method modifiers are supported:

• .z(w) – apply rolling Z-score to the smoothed series before that Hurst method (default window w = 100 if omitted)
• .perst() – apply a persistence transform to the Hurst output - absolute value of (2H − 1)

All metrics are bifurcated: evaluated separately on upward (peak) and downward (trough) regimes and then interpolated to continuous traces.

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Install

pip install beevol         
# or from source
pip install .

Requires Python ≥ 3.8.

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Pipeline (Conceptual Overview)

Let $x_t$ denote the raw series:

1. Savitzky–Golay smoothing (mandatory) The raw series is smoothed with a Savitzky–Golay filter (window sg_window, polynomial order polyorder). This smoothed series is the basis for all subsequent computations.

2. Extrema detection Peaks and troughs are detected on the smoothed series using a prominence threshold. Two index sets result: peaks (upward extrema) and troughs (downward extrema).

3. Permutation Entropy (PE) At each extremum, PE is computed using an embedding dimension (m) and a delay (delay). PE quantifies local ordinal complexity.

4. Per-method preprocessing for Hurst bases Each Hurst base method (RS, DFA, VT, SP) may specify .z(w). If present, a rolling Z-score (window w) is applied to the smoothed series before computing that method. If omitted, the smoothed series is used directly.

5. Hurst computation at each extremum For each Hurst base method, a trailing window (hurst_window_h_*) ending at the extremum is selected and the base method is applied to that segment.

6. Optional persistence transform If .perst() is attached, the resulting Hurst value is transformed to |2H − 1|, emphasizing deviation from pure randomness.

7. Bifurcation and interpolation Values at peaks populate the “max” stream; values at troughs populate the “min” stream. Each stream is linearly interpolated over time, producing continuous traces. Their difference (max − min) indicates directional divergence.

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Rationale for Bifurcation

Time series often exhibit asymmetric behavior in rising versus falling phases (e.g., explosive growth vs. corrective decay). Partitioning metrics by direction allows:

• Quantification of asymmetry in complexity and memory.
• Identification of regime shifts via divergence between up and down traces.
• Targeted transformations (persistence, Z-scoring) applied only where specified.

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Quick Start (CLI)

Show defaults

python -m beevol.cli --show-defaults

Run on a CSV with two columns

python -m beevol.cli --data_source data.csv --target-columns colA colB

Specify Hurst methods with modifiers

python -m beevol.cli \
  --run-hurst H_RS.perst() H_DFA.perst().z() H_VT H_SP.z(50)

• H_RS.perst() applies persistence to RS output.
• H_DFA.perst().z() applies a Z-score (default window 100) to the smoothed series before DFA, then persistence to the DFA result.
• H_SP.z(50) applies spectral Hurst on a Z-scored (window 50) smoothed series.

Suppress plots and printing

python -m beevol.cli --output dfs

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Python API

from beevol.swarm import sting

cfg = dict(
    data_source="TEST",                 # or "path/to.csv"
    target_columns=["value"],           # if CSV
    run_hurst=["H_RS", "H_DFA.z(200).perst()", "H_SP"],
    run_PE=True,
    output=[],                          # e.g. ["plots", "dfs"]
    return_figs=False,
    verbose=False
)

full, pe_df, he_df = sting(**cfg)

• full: wide DataFrame with interpolated traces (e.g., col__max_PE, col__min_PE, col__max_H_RS, …).
• pe_df, he_df: long tables with one row per extremum and a target column.

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Hurst Spec Grammar

BASE[.perst()][.z()][.z(W)]
BASE ∈ {H_RS, H_DFA, H_VT, H_SP}

• .z() without an argument uses hurst_z_window (default 100).
• Modifiers can appear in any order; .z() affects the input, .perst() affects the output.

from beevol.hive import parse_hurst_spec
parse_hurst_spec("H_DFA.perst().z(50)")
# returns ("H_DFA", {"perst": True, "z": True, "z_window": 50})

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Configuration Keys (YAML or dict)

Key	Type	Meaning	Default
data_source	str	"TEST" or path to CSV	"TEST"
target_columns	list[str]	CSV columns to process	["value"]
sg_window, polyorder	int	Savitzky–Golay parameters	7, 2
prominence, tolerance	float, int	Extrema detection threshold and alignment tolerance	0.5, 5
entropy_window	int	PE rolling window	20
hurst_window_h_*	int	Window per Hurst base (RS, DFA, VT, SP)	20 each
run_PE	bool	Toggle PE	True
run_hurst	list[str]	Hurst specs with modifiers	see default
hurst_z_window	int	Default Z window when .z() has no argument	100
verbose	bool	Print alignment and informative output	True
output	list[str]	Any of plots, extrema_accuracy, dfs, all	[]
return_figs	bool	Return Matplotlib figures	False

Synthetic generator parameters (N, phi1, phi2, sigma1, sigma2, T, drift, scale, seed) are also configurable.

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Outputs

• full_wide: interpolated/rolled columns per target and direction (e.g., target__max_H_RS, target__min_H_RS).
• pe_long / he_long: per-extremum tables (idx, value, metrics, target).

Optional plots include PE up/down traces and their difference, and analogous plots for each Hurst base.

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Tests and Their Purpose

The bundled test suite (tests/test_all.py) verifies:

Test block	Purpose
Permutation entropy on constants/noise	Confirms PE → 0 for constants and >0 for noise
Hurst estimators on white noise	Ensures H ≈ 0.5 bounds (0.3–0.7) for all bases
Synthetic generator variance	Confirms generated AR process has nontrivial spread
Extrema routines	Validates peak/trough detection and tolerance logic
End-to-end feature computation	Checks PE + Hurst extraction at extrema (new API)
CLI defaults and CSV failure modes	Confirms CLI behavior and error handling
Multi-column processing	Ensures bifurcation and column naming across targets

The persistence transform is bounded in [0, 1] by construction (absolute value of 2H − 1). White-noise checks ensure estimators do not produce spurious persistence.

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Citation

Woltman, N. (2025). Bifurcated Extremal Entropy. GitHub / PyPI: beevol

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License

MIT