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)

Symbol	Definition	Example
L₀	Initial low (first confirmed pivot)	1.0800
H₁	Swing high (reversal peak)	1.0850
L₂	Second low (retracement)	1.0810
H₃	Second high (three-pivot only)	1.0870
W	Swing magnitude: H₁ − L₀	0.0050
z	Normalized retracement: (L₂ − L₀) / (H₁ − L₀)	0.20
o	Volatility-normalized overshoot: (L₀ − L₂) / ATR₁₄	0.35
ε	Tolerance band for "equal" classification	5 pips
τ	ZigZag reversal threshold	6.6 pips

Price Level Comparisons

Code	Meaning	Condition
HL	Higher Low	L₂ > L₀ + ε
EL	Equal Low	|L₂ − L₀| ≤ ε
LL	Lower Low	L₂ < L₀ − ε
HH	Higher High	H₃ > H₁ + ε
EH	Equal High	|H₃ − H₁| ≤ ε
LH	Lower High	H₃ < H₁ − ε

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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:

Class	Condition	Meaning	Frequency
EL	|L₂ − L₀| ≤ ε	Equal Low (retest)	~20–30%
HL	L₂ > L₀ + ε	Higher Low (pullback)	~50–60%
LL	L₂ < L₀ − ε	Lower Low (undercut)	~10–20%

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 ∞

Variant	z or o Range	Market Regime	Probability
EL	|z| ≤ ε_r	Retest — support re-established	20–30%
HL-FD1	0.75 < z < 1.0	Shallow pullback — buyers in control	20–25%
HL-FD2	0.50 < z ≤ 0.75	Moderate pullback — Fib 38–50%	15–20%
HL-FD3	0.25 < z ≤ 0.50	Deep pullback — Fib 61.8%	10–15%
HL-FD4	0 < z ≤ 0.25	Near-complete retrace — barely held	5–10%
LL-FD1	o ≤ q₂₀	Micro undercut — brief false break	8–12%
LL-FD2	q₂₀ < o ≤ q₄₀	Shallow undercut — moderate panic	4–8%
LL-FD3	q₄₀ < o ≤ q₆₀	Deep undercut — structural break	2–5%
LL-FD4	o > q₆₀	Extreme undercut — tail event	<2%

Trading Implications

Variant	Entry Signal	Stop Loss	Target
EL	Long from L₀+ε	L₀−ε	H₁ + ΔH
HL-FD1	Long from L₂	L₂−ε	H₁ + ΔH
HL-FD2	Long at confirmation	L₂−ε	Prior H + ΔH
HL-FD3	Reduced size; wait	L₂−ε	Support + ΔH
HL-FD4	Extreme risk; avoid	L₂−ε	Critical
LL-FD1	Short spike trade	L₂+spike	L₀
LL-FD2	Wait for reversal	Breakout	L₀
LL-FD3	Short continuation	Reversal	New lows
LL-FD4	Crisis mode; hedge	Capitulation	TBD

Optional Sub-Classification Flags

Attach to any variant for richer context:

Flag	Meaning	Signal
+C	Any close < L₀ between H₁→L₂	Stronger bearish commitment
+S	L₂ occurs in single bar (spike)	Sharp reversal; mean reversion
+X	Wicks below L₀ only, no close	Liquidity grab; false break

Example labels: HL-FD2+C, LL-FD4+S+C, EL+X

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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}

#	L₂ vs L₀	H₃ vs H₁	Name	Market Regime
1	HL	HH	Continuation impulse	Bull trend continuation
2	HL	EH	Double-top test	Range, bullish bias
3	HL	LH	Triangle compression	Neutral consolidation
4	EL	HH	Range break up	Bullish transition
5	EL	EH	Rectangle	Balanced range
6	EL	LH	Lower-high at flat base	Range, bearish bias
7	LL	HH	V-reversal / spring	Bullish reversal
8	LL	EH	Undercut then stall	Volatile range
9	LL	LH	Rally failure	Bear trend continuation

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

Regime	Variants
Trend Continuation	HL+HH (bull), LL+LH (bear)
Range Consolidation	HL+LH, EL+EH, LL+EH
Bullish Transition	EL+HH, HL+EH
Bearish Transition	EL+LH
Reversal	LL+HH

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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".

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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.

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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.

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Implementation Reference

The qta Rust crate (crates/qta/) implements the core ZigZag state machine that produces the pivots consumed by this classification framework:

• 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.

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Deep Reference Files

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

File	Contents
references/notation-definitions.md	Single source of truth: all symbols, formulas, abbreviations, pattern classes
references/two-pivot-variants.md	All 9 two-pivot ASCII diagrams, trading rules, flag examples
references/three-pivot-variants.md	All 9 three-pivot ASCII diagrams, 27-way extension, HL+LH granular sub-variants
references/epsilon-tolerance-detail.md	Complete ε formula, worked calculation, sensitivity analysis, pseudocode
references/binning-methodology.md	Freedman-Diaconis algorithm details, FD vs quantile comparison, worked example
references/data-pipeline.md	11-step end-to-end pipeline: raw quotes → OHLC → ATR → pivots → classify → output
references/eurusd-validation-scenarios.md	3 worked market scenarios (Normal, Volatile, Crash) validating ε and τ

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.