ZigZag Swing Pattern Classifier Skill

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.

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 |

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

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 |

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

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

Procedure for LL Patterns

Filter to LL (z < −ε_r)
Compute o = (L₀ − L₂) / ATR₁₄
Winsorize o at 0.5%-99.5%
Compute FD bin edges on o ∈ (ε_r, ∞)
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 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.

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 |

Agent Skills: ZigZag Swing Pattern Classifier

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