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# Response Surface Methodology (RSM)
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Designs optimized for fitting response surface models and process optimization. These designs efficiently estimate quadratic effects and interaction terms, making them ideal for modeling non-linear relationships and finding optimal factor settings.
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## Capabilities
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### Box-Behnken Design
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Three-level designs that efficiently estimate quadratic response surfaces using fewer experimental runs than full factorial designs.
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```python { .api }
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def bbdesign(n, center=None):
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    """
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    Create a Box-Behnken design
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    Parameters:
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    - n: int, number of factors in the design (minimum 3)
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    - center: int, optional, number of center points to include (default 1)
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    Returns:
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    - mat: 2d-array, design matrix with levels -1, 0, +1
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    """
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```
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**Key Features:**
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- Uses 3 levels for each factor: -1 (low), 0 (center), +1 (high)
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- Each factor appears at its extreme levels in combination with center levels of other factors
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- Efficient for quadratic model fitting with fewer runs than central composite designs
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- Suitable for 3 or more factors
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**Usage Example:**
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```python
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import pyDOE3
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# 3-factor Box-Behnken design
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design = pyDOE3.bbdesign(3)
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print(f"Design shape: {design.shape}")  # (13, 3) - 12 edge points + 1 center point
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# 4-factor design with 3 center points
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design = pyDOE3.bbdesign(4, center=3)
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print(f"Design shape: {design.shape}")  # (27, 4) - 24 edge points + 3 center points
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```
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### Central Composite Design
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Comprehensive response surface designs combining factorial points, star points, and center points for quadratic model estimation.
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```python { .api }
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def ccdesign(n, center=(4, 4), alpha="orthogonal", face="circumscribed"):
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    """
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    Create a Central Composite Design (CCD)
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    Parameters:
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    - n: int, number of factors in the design
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    - center: tuple of 2 ints, number of center points in (factorial, star) blocks
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    - alpha: str, design property - "orthogonal" or "rotatable"
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    - face: str, star point placement - "circumscribed", "inscribed", or "faced"
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    Returns:
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    - mat: 2d-array, design matrix with factorial, star, and center points
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    """
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```
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**Alpha Options:**
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- **"orthogonal"** (default): Minimizes correlation between regression coefficients
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- **"rotatable"**: Provides constant prediction variance at constant distance from center
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**Face Options:**
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- **"circumscribed"** (CCC): Star points extend beyond factorial space (5 levels)
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- **"inscribed"** (CCI): Star points within factorial limits, scaled design (5 levels)  
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- **"faced"** (CCF): Star points at face centers of factorial cube (3 levels)
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**Usage Example:**
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```python
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import pyDOE3
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# Standard 3-factor CCD with orthogonal design
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design = pyDOE3.ccdesign(3)
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print(f"Design shape: {design.shape}")  # (20, 3)
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# Rotatable design with face-centered star points
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design = pyDOE3.ccdesign(3, alpha="rotatable", face="faced")
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# Custom center points: 2 in factorial block, 6 in star block
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design = pyDOE3.ccdesign(4, center=(2, 6))
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```
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### Doehlert Designs
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Uniform space-filling designs for response surface methodology with efficient coverage of spherical experimental regions.
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#### Doehlert Shell Design
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Builds designs by adding concentric shells of points around the center for uniform space coverage.
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```python { .api }
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def doehlert_shell_design(num_factors, num_center_points=1):
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    """
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    Generate a Doehlert shell design matrix
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    Parameters:
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    - num_factors: int, number of factors (minimum 1)
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    - num_center_points: int, number of center points (default 1)
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    Returns:
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    - design: 2d-array, design matrix with N = k² + k + C points
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                       where k = factors, C = center points
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    """
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```
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#### Doehlert Simplex Design
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Creates simplex-based Doehlert designs for efficient response surface exploration.
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```python { .api }
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def doehlert_simplex_design(num_factors):
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    """
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    Generate a Doehlert simplex design matrix
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    Parameters:
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    - num_factors: int, number of factors in the design
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    Returns:
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    - design: 2d-array, design matrix with k² + k + 1 points
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                       where k = number of factors
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    """
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```
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**Key Features of Doehlert Designs:**
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- Uniform distribution in spherical experimental domain
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- Fewer runs than central composite designs
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- Good for sequential experimentation and optimization
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- Not orthogonal or rotatable, but acceptable variance properties
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**Usage Example:**
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```python
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import pyDOE3
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# 3-factor Doehlert shell design with 2 center points
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shell_design = pyDOE3.doehlert_shell_design(3, num_center_points=2)
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print(f"Shell design shape: {shell_design.shape}")  # (13, 3)
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# 4-factor Doehlert simplex design  
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simplex_design = pyDOE3.doehlert_simplex_design(4)
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print(f"Simplex design shape: {simplex_design.shape}")  # (21, 4)
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```
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## Design Selection Guidelines
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### When to Use Each RSM Design:
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**Box-Behnken Designs:**
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- **Best for:** 3-5 factors, spherical or cuboidal experimental regions
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- **Advantages:** Fewer runs than CCD, no extreme combinations, good for constrained regions
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- **Use when:** Cannot run extreme factor combinations, want 3-level design
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**Central Composite Designs:**
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- **Best for:** 2-6 factors, comprehensive quadratic modeling
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- **Advantages:** Most flexible RSM design, well-studied properties, many variants
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- **Use when:** Need maximum model flexibility, can accommodate 5 levels
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**Doehlert Designs:**  
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- **Best for:** Sequential optimization, spherical experimental regions
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- **Advantages:** Uniform space-filling, fewer runs, good for augmentation
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- **Use when:** Space-filling properties important, sequential experimentation
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### RSM Design Comparison:
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| Design Type | Factors | Levels | Runs (3 factors) | Best For |
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|-------------|---------|--------|------------------|----------|
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| Box-Behnken | 3+ | 3 | 13-15 | Constrained regions |
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| Central Composite | 2+ | 3-5 | 15-20 | General RSM |
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| Doehlert Shell | 1+ | Continuous | 10-13 | Space-filling |
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| Doehlert Simplex | 1+ | Continuous | 13 | Sequential optimization |
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### Model Fitting Capabilities:
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All RSM designs support fitting second-order polynomial models:
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```
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y = β₀ + Σβᵢxᵢ + Σβᵢᵢxᵢ² + ΣΣβᵢⱼxᵢxⱼ + ε
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```
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Where:
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- β₀: intercept
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- βᵢ: linear effects  
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- βᵢᵢ: quadratic effects
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- βᵢⱼ: interaction effects
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## Types
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```python { .api }
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import numpy as np
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from typing import Union, Tuple
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# Type aliases for RSM designs
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DesignMatrix = np.ndarray
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CenterPoints = Union[int, Tuple[int, int]]
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AlphaType = str  # "orthogonal" or "rotatable"
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FaceType = str   # "circumscribed", "inscribed", or "faced"
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```