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# SilverSight Symbolic Regression — Design Document
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## Principle
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**No external imports.** Use existing SilverSight infrastructure:
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- `EquationShape` + `classifyEquation` — structural classification
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- `spectral_profile` — 8D spectral features
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- `chaos_game` — IFS contraction search
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- `sidon_address` — deterministic addressing
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- `qubo/` — binary optimization
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## Architecture
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```
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Input: (X, y) data points
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↓
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1. Generate candidate expression trees
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2. Classify each with classifyEquation → Hachimoji state
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3. Compute spectral profile for each expression
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4. Apply linear scaling (solve for a, b)
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5. Compute fitness = BIC(scaled_expression, data)
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6. Use chaos game to navigate state space
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7. Use QUBO to select optimal expression
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Output: Best expression with R² score
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```
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## Missing Pieces (to implement)
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### 1. Expression Tree Data Structure
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```python
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@dataclass
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class ExprNode:
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op: str # '+', '-', '*', '/', 'sqrt', 'sin', 'cos', 'log', 'exp', 'x', 'const'
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left: Optional['ExprNode']
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right: Optional['ExprNode']
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value: Optional[float] # for const nodes
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depth: int = 0
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size: int = 0
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```
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### 2. Linear Scaling (Keijzer 2003)
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Given expression g(x) and target y, solve:
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```
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f(x) = a·g(x) + b
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where a, b = argmin Σ(f(x_i) - y_i)²
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```
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Closed-form solution:
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```
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a = cov(g, y) / var(g)
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b = mean(y) - a·mean(g)
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```
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### 3. BIC Fitness
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```
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fitness = n·ln(MSE) + k·ln(n)
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where:
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n = data points
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k = tree complexity (weighted: var=1, const=3)
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MSE = mean squared error after linear scaling
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```
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### 4. Chaos Game Search
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Use existing `chaos_game.py` to navigate expression space:
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- Each expression maps to a spectral profile
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- Spectral profile maps to a Sidon address
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- Chaos game contracts toward good expressions
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### 5. ε-Lexicase Selection
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Select on individual data points, not aggregate fitness:
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- For each data point, keep only expressions within ε of best
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- This preserves behavioral diversity
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### 6. Operator Diversity via Hachimoji States
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- Φ (trivial): simple expressions (x, x², √x)
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- Σ (symmetric): balanced expressions (x·y, x+y)
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- Λ (quantified): expressions with constants
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- Π (complex): deep expressions (sin(1/x), exp(x²))
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- Ω (contradiction): degenerate expressions (constant, NaN)
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Force diversity by requiring expressions in multiple Hachimoji states.
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## Implementation Plan
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### Phase 1: Core Infrastructure
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1. `python/expr_tree.py` — expression tree data structure
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2. `python/linear_scaling.py` — Keijzer linear scaling
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3. `python/bic_fitness.py` — BIC fitness function
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### Phase 2: Search
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4. `python/chaos_game_search.py` — chaos game navigation
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5. `python/lexicase_selection.py` — ε-lexicase selection
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### Phase 3: Integration
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6. `python/symbolic_regression.py` — main entry point
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7. `tests/test_symbolic_regression.py` — verification
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### Phase 4: QUBO Enhancement
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8. Use QUBO to select optimal expression from candidates
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9. Use Hachimoji states to enforce diversity
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## Key Advantage Over GP-ELITE
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GP-ELITE uses genetic programming (random mutation + crossover).
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SilverSight uses **chaos game** (deterministic IFS contraction) + **Hachimoji states** (structural classification).
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This means:
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- Deterministic: same input → same output
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- Structured: expressions are classified by type, not just fitness
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- Efficient: chaos game contracts faster than random search
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## Test Cases
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1. **Kepler's Third Law**: T = a^1.5 from 8 planets
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2. **Simple polynomial**: y = 2x² + 3x + 1
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3. **Trigonometric**: y = sin(x) + 0.5·cos(2x)
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4. **Exponential**: y = exp(-x²)
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5. **BMCTE entropy**: H(p) = f(N, p) from experiment data
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