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Symbolic Regression Benchmark - Mixed PolyExp 4D Dataset
=========================================================
Problem Setting
---------------
Learn a closed-form symbolic expression `f(x1, x2, x3, x4)` that predicts the target `y`.
This is a higher-dimensional dataset (4 input features) combining polynomial interactions with exponential decay. The function involves cross-terms between variables and Gaussian-like damping, making it more challenging than the 2D variants.
Input Format
------------
- Your `Solution.solve` receives:
- `X`: numpy.ndarray of shape `(n, 4)` containing feature values
- `y`: numpy.ndarray of shape `(n,)` containing target values
- Dataset columns: `x1, x2, x3, x4, y`
Output Specification
--------------------
Implement a `Solution` class in `solution.py`:
```python
import numpy as np
class Solution:
def __init__(self, **kwargs):
pass
def solve(self, X: np.ndarray, y: np.ndarray) -> dict:
"""
Args:
X: Feature matrix of shape (n, 4)
y: Target values of shape (n,)
Returns:
dict with keys:
- "expression": str, a Python-evaluable expression using x1, x2, x3, x4
- "predictions": list/array of length n (optional)
- "details": dict with optional "complexity" int
"""
# Example: fit a symbolic expression to the data
expression = "x1 + x2 + x3 + x4" # placeholder
return {
"expression": expression,
"predictions": None, # will be computed from expression if omitted
"details": {}
}
```
Expression Requirements:
- Must be a valid Python expression string
- Use variable names: `x1`, `x2`, `x3`, `x4`
- Allowed operators: `+`, `-`, `*`, `/`, `**`
- Allowed functions: `sin`, `cos`, `exp`, `log`
- Numeric constants are allowed
Dependencies (pinned versions)
------------------------------
```
pysr==0.19.0
numpy==1.26.4
pandas==2.2.2
sympy==1.13.3
```
Minimal Working Examples
------------------------
**Example 1: Using PySR (recommended)**
```python
import numpy as np
from pysr import PySRRegressor
class Solution:
def __init__(self, **kwargs):
pass
def solve(self, X: np.ndarray, y: np.ndarray) -> dict:
model = PySRRegressor(
niterations=50, # more iterations for 4D
binary_operators=["+", "-", "*", "/"],
unary_operators=["sin", "cos", "exp", "log"],
populations=20,
population_size=40,
maxsize=30, # larger for 4D complexity
verbosity=0,
progress=False,
random_state=42,
)
model.fit(X, y, variable_names=["x1", "x2", "x3", "x4"])
# Get best expression as sympy, convert to string
best_expr = model.sympy()
expression = str(best_expr)
# Predictions
predictions = model.predict(X)
return {
"expression": expression,
"predictions": predictions.tolist(),
"details": {}
}
```
**Example 2: Manual expression (simple baseline)**
```python
import numpy as np
class Solution:
def __init__(self, **kwargs):
pass
def solve(self, X: np.ndarray, y: np.ndarray) -> dict:
# Simple linear combination as baseline
x1, x2, x3, x4 = X[:, 0], X[:, 1], X[:, 2], X[:, 3]
# Fit coefficients via least squares
A = np.column_stack([x1, x2, x3, x4, np.ones_like(x1)])
coeffs, _, _, _ = np.linalg.lstsq(A, y, rcond=None)
a, b, c, d, e = coeffs
expression = f"{a:.6f}*x1 + {b:.6f}*x2 + {c:.6f}*x3 + {d:.6f}*x4 + {e:.6f}"
predictions = a * x1 + b * x2 + c * x3 + d * x4 + e
return {
"expression": expression,
"predictions": predictions.tolist(),
"details": {}
}
```
PySR API Notes (v0.19.0)
------------------------
- `model.fit(X, y, variable_names=["x1", "x2", "x3", "x4"])` - use variable_names to match expected output
- `model.sympy()` - returns best expression as sympy object
- `model.predict(X)` - returns predictions array
- `model.equations_` - DataFrame of all discovered equations
- Common parameters:
- `niterations`: number of evolution iterations (more = better but slower)
- `populations`: number of parallel populations
- `maxsize`: maximum expression complexity
- `verbosity=0, progress=False`: suppress output
Expression Format Requirements
------------------------------
- Must be a valid Python expression string
- Use variable names: `x1`, `x2`, `x3`, `x4`
- Allowed operators: `+`, `-`, `*`, `/`, `**`
- Allowed functions: `sin`, `cos`, `exp`, `log` (NO `np.` prefix)
- Numeric constants are allowed
- The evaluator uses `sympy.sympify()` to parse your expression
Scoring
-------
```
MSE = (1/n) Σ (y_i - ŷ_i)²
Score = 100 × clamp((m_base - MSE) / (m_base - m_ref), 0, 1) × 0.99^max(C - C_ref, 0)
```
- `m_base`: linear regression baseline MSE
- `m_ref`, `C_ref`: reference solution MSE and complexity
- `C = 2 × (#binary ops) + (#unary ops)`
- Lower MSE and lower complexity yield higher scores
Environment
-----------
Run `set_up_env.sh` to install dependencies.
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