ZigZag Swing Pattern Classifier

Complete taxonomy for classifying ZigZag swing patterns by structure and market regime. Every confirmed ZigZag sequence maps to exactly one variant — no gaps, no overlaps.

Self-Evolving Skill: This skill improves through use. If instructions are wrong, parameters drifted, or a workaround was needed — fix this file immediately, don't defer. Only update for real, reproducible issues.

When to Use

• Classifying a confirmed L₀→H₁→L₂ (two-pivot) or L₀→H₁→L₂→H₃ (three-pivot) swing
• Looking up market regime implications of a specific pattern
• Answering "how many distinct patterns exist?" and proving exhaustiveness
• Implementing pattern labeling in code (Rust

  qta

  crate, Python pipelines)
• Understanding the epsilon tolerance band that defines "equal"
• Applying Freedman-Diaconis binning for sub-classification depth

Notation (Single Source of Truth)

Price Level Comparisons

Part 1: Two-Pivot Patterns (UP-DOWN)

Pattern: L₀ → H₁ → L₂ (one up-leg, one down-leg)

Base Classification (3 Classes)

Every UP-DOWN triplet falls into exactly one:

Granular Classification: 9 FD-Binned Variants

HL and LL each decompose into 4 sub-classes using Freedman-Diaconis binning on normalized coordinates:

• HL bins use z = (L₂ − L₀) / (H₁ − L₀), ranging 0 to 1
• LL bins use o = (L₀ − L₂) / ATR₁₄, ranging 0 to ∞

Trading Implications

Optional Sub-Classification Flags

Attach to any variant for richer context:

Example labels:

HL-FD2+C

LL-FD4+S+C

EL+X

Part 2: Three-Pivot Patterns (UP-DOWN-UP)

Pattern: L₀ → H₁ → L₂ → H₃ (up-leg, down-leg, up-leg)

9 Exhaustive Variants

Two independent dimensions × 3 values each = 3×3 = 9 mutually exclusive, collectively exhaustive variants.

Dimension 1 — L₂ vs L₀: {HL, EL, LL} Dimension 2 — H₃ vs H₁: {HH, EH, LH}

Exhaustiveness Proof

All variants satisfy these mandatory constraints:

• L₀ < H₁ (first uptrend exists)
• L₂ < H₁ (retracement doesn't exceed peak)
• H₃ > L₂ (second uptrend exists)

L₂ has exactly 3 relationships to L₀ (higher, equal, lower). H₃ has exactly 3 relationships to H₁ (higher, equal, lower). These dimensions are independent — no constraint eliminates any combination. Therefore 3 × 3 = 9 variants, all feasible, none missing.

Natural Groupings

• Bullish (4): HL+HH, EL+HH, HL+EH, LL+HH
• Neutral (3): HL+LH, EL+EH, LL+EH
• Bearish (2): EL+LH, LL+LH

Regime Assignments

Part 3: 27-Way Extension

Adding H₃ vs L₀ as a third independent dimension:

• L₂ vs L₀: {HL, EL, LL}
• H₃ vs H₁: {HH, EH, LH}
• H₃ vs L₀: {Above, Equal, Below}

This yields 3×3×3 = 27 sub-variants. Some are mathematically impossible due to constraints (e.g., HL+LH+H₃<L₀ requires H₃ < L₀ < L₂ < H₃, a contradiction).

Analysts often simplify this third dimension to a binary: "reclaims L₀" vs "fails to reclaim L₀".

The Epsilon Tolerance Band

The tolerance band ε determines what "equal" means. It adapts to volatility and microstructure.

Core Formula

ε = min(ε_max, max(ε_min, √[(a·S)² + (b·ATR₁₄)²]))

Where:

• S = rolling median spread (bid-ask)
• ATR₁₄ = 14-bar Average True Range
• a = 2.0 (spread scaling)
• b = 0.05 (M5-M30) or 0.07 (H1-D1)

Bounds (EURUSD defaults)

ε_min = max(3 ticks, 1×S) = max(0.00003, S)
ε_max = min(5 pips, 0.20 × swing) = min(0.00050, 0.20 × W)

Classification Using ε

ε_r = ε / W  (relative tolerance)

EL if |z| ≤ ε_r
HL if z > ε_r
LL if z < −ε_r

Practical Fallbacks

If bid-ask spread unavailable:

• ATR-only: ε = 0.05 × ATR₁₄ (M5-M30), 0.07 × ATR₁₄ (H1-D1)
• Fixed band: ε = 3 pips (intraday), 5 pips (swing), 10 pips (daily)

For complete worked examples with sensitivity analysis, read

references/epsilon-tolerance-detail.md

Freedman-Diaconis Binning

The FD rule computes statistically optimal bin edges from data:

h = 2 × IQR(X) × n^(-1/3)    (bin width)
K = clip(⌈(max−min) / h⌉, 3, 6)  (number of bins)
edges = linspace(min, max, K+1)

Procedure for HL Patterns

1. Collect 2-3 years of UP-DOWN triplets
2. Filter to HL (z > ε_r)
3. Winsorize z at 0.5%-99.5%
4. Compute FD bin edges on z ∈ (ε_r, 1)
5. Label: HL-FD1 (shallowest) through HL-FDK (deepest)

Procedure for LL Patterns

1. Filter to LL (z < −ε_r)
2. Compute o = (L₀ − L₂) / ATR₁₄
3. Winsorize o at 0.5%-99.5%
4. Compute FD bin edges on o ∈ (ε_r, ∞)
5. Label: LL-FD1 (micro) through LL-FDK (extreme)

Recompute bin edges monthly or quarterly to track regime drift. Fall back to quantile binning if sample < 400.

Implementation Reference

The

qta

crates/qta/

• ZigZagConfig::new(reversal_depth, epsilon_multiplier, bar_threshold_dbps)

• ZigZagState::process_bar(&bar) → ZigZagOutput

  with

  completed_segment

  containing

  base_class

  (EL/HL/LL) and

  z

  score
• The crate computes τ and ε dynamically per pivot from

  bar_threshold_dbps

The classification framework in this skill extends the crate's output with FD-binning, three-pivot analysis, and market regime labeling.

Deep Reference Files

For ASCII visualizations, worked examples, and implementation pseudocode, read these reference files as needed:

references/notation-definitions.md

references/two-pivot-variants.md

references/three-pivot-variants.md

references/epsilon-tolerance-detail.md

references/binning-methodology.md

references/data-pipeline.md

references/eurusd-validation-scenarios.md

Post-Execution Reflection

After this skill completes, check before closing:

1. Did the command succeed? — If not, fix the instruction or error table that caused the failure.
2. Did parameters or output change? — If the underlying tool's interface drifted, update Usage examples and Parameters table to match.
3. Was a workaround needed? — If you had to improvise (different flags, extra steps), update this SKILL.md so the next invocation doesn't need the same workaround.

Only update if the issue is real and reproducible — not speculative.