Classical & Screening Designs¶

In addition to space-filling methods (LHS, Sobol, etc.), ALchemist supports a full suite of classical Response Surface Methodology (RSM) designs and screening designs. These structured methods follow decades of established statistical practice and are often preferred when variable effects are expected to follow polynomial trends or when you need to screen many factors with minimal runs.

When to Use Classical Designs¶

Situation	Recommended design
Need to estimate main effects + interactions/curvature	CCD or Box-Behnken
Maximum coverage of all factor combinations	Full Factorial
Many factors (≥5), need to screen efficiently	Plackett-Burman or Fractional Factorial
Mixed categorical + continuous factors, efficient screening	GSD
Want maximum statistical freedom (custom model)	Optimal Design
Exploratory, no prior model structure assumed	Space-filling

Method Overview¶

Full Factorial¶

Evaluates every combination of factor levels. Provides complete information about all main effects and interactions.

Run count: n_levels^k where k = number of continuous variables
Use when: You have few factors (2–4) and need complete information
Supports: Continuous and integer variables; categorical variables use their full category set
Key parameter: n_levels — number of evenly-spaced levels per factor (default 2; max 5)

Example (2 variables, 3 levels): 3² = 9 runs at all combinations of (low, mid, high) × (low, mid, high)

Fractional Factorial¶

A subset of a full factorial selected using generator strings. Reduces run count at the cost of some higher-order interaction estimates, which are "aliased" (confounded) with main effects or lower-order interactions.

Run count: 2^(k-p) where p determines the fraction (automatically selected)
Use when: 3–7 continuous factors, want to screen main effects efficiently
Default generators: Built-in Resolution III+ generators for 3–7 factors
Custom generators: Specify your own via the generators parameter (e.g., "a b ab")
Requires: Continuous and integer variables only; no categorical

Resolution describes which effects are confounded: - Resolution III: Main effects aliased with two-factor interactions - Resolution IV: Main effects clear of two-factor interactions - Resolution V+: Two-factor interactions estimable

Central Composite Design (CCD)¶

The most common Response Surface Methodology design. Combines a 2-level factorial core with axial (star) points and center replicates to estimate curvature (quadratic terms).

Run count: 2^k + 2k + n_center where k = number of continuous variables
Use when: 2–5 continuous factors, anticipate curvature, need to fit a full quadratic model
Requires: Continuous and integer variables only; no categorical

Key parameters:

Parameter	Options	Description
`ccd_alpha`	`orthogonal` (default), `rotatable`	Positions axial points for orthogonality or equal prediction variance
`ccd_face`	`circumscribed` (default), `inscribed`, `faced`	Where axial points sit relative to factorial points
`n_center`	integer (default 1)	Center point replicates (helps estimate pure error)

Circumscribed (CCC): Axial points extend outside the factorial cube — gives better prediction but requires a wider variable range
Inscribed (CCI): Keeps all points within the original bounds — useful when extremes are infeasible
Faced (CCF): Axial points at ±1 on each axis — cube-shaped region, no extension

Box-Behnken¶

A spherical design that avoids corner points. Runs are placed at edge midpoints and center, forming a roughly spherical design region.

Run count: Typically ~3k² - 3k + 3 (varies by k)
Use when: 3–5 continuous factors, extremes (corner combinations) are physically infeasible or dangerous
Requires: Exactly 3+ continuous variables; no categorical or integer variables
Does not require extreme corners, making it safer for processes where simultaneous extreme conditions are problematic

Plackett-Burman¶

An ultra-efficient 2-level screening design with run counts that are multiples of 4 (e.g., 12, 20, 24 runs). Designed to estimate main effects with maximum efficiency given the run count.

Run count: Next multiple of 4 ≥ k+1 (e.g., 5 factors → 8 runs)
Use when: Many continuous factors (5–20), want to quickly screen which matter
Requires: Continuous and integer variables only; no categorical
Does not estimate interactions — complex aliasing structure

Generalized Subset Design (GSD)¶

A fractional design for mixed or multi-level factors, including categorical. A GSD with reduction factor r evaluates approximately 1/r of the full factorial.

Run count: total_combinations / gsd_reduction (approximately)
Use when: Mixed categorical + continuous factors, full factorial is too expensive
Supports: Categorical, continuous, and integer variables
Key parameter: gsd_reduction — reduction factor (2 = ~half of full factorial; default 2)

Note: GSD is the recommended classical screening method when your variable space includes categorical variables.

Variable Type Compatibility¶

Method	Real	Integer	Categorical	Discrete
Full Factorial	✅	✅	✅	✅
Fractional Factorial	✅	✅	❌	❌
CCD	✅	✅	❌	❌
Box-Behnken	✅	❌	❌	❌
Plackett-Burman	✅	✅	❌	❌
GSD	✅	✅	✅	✅

If your design includes incompatible variable types, ALchemist will raise an informative error. Use GSD or Full Factorial for mixed-type spaces.

Web UI Walkthrough¶

Define your variable space and load the Initial Design panel.
Select a method from the Design Method dropdown (e.g., CCD).
Adjust method-specific parameters that appear below the dropdown.
Click Generate — the run count is determined automatically from the design structure.
Review the design table, then click Stage Suggestions to queue them for execution.

Unlike space-filling methods, classical designs do not require you to specify a number of points — the run count is determined by the design structure.

Python API¶

from alchemist_core import OptimizationSession

session = OptimizationSession()
session.add_variable("Temperature", "real", min=200, max=350)
session.add_variable("Pressure", "real", min=1, max=10)
session.add_variable("Catalyst", "categorical", categories=["Ni", "Pt", "Pd"])

# Central Composite Design (continuous variables only)
ccd_points = session.generate_initial_design(
    method="ccd",
    ccd_alpha="orthogonal",
    ccd_face="circumscribed",
    n_center=1,
)

# Generalized Subset Design (supports categorical)
gsd_points = session.generate_initial_design(
    method="gsd",
    gsd_reduction=2,
)

# Fractional factorial with default generators
ff_points = session.generate_initial_design(
    method="fractional_factorial",
)

REST API¶

POST /api/v1/sessions/{session_id}/initial-design
Content-Type: application/json

{
  "method": "ccd",
  "ccd_alpha": "orthogonal",
  "ccd_face": "circumscribed",
  "n_center": 1
}

POST /api/v1/sessions/{session_id}/initial-design
{
  "method": "gsd",
  "gsd_reduction": 2
}

Comparison: Space-filling vs. Classical¶

Property	Space-filling (LHS, Sobol)	Classical (CCD, Box-Behnken)
Run count	User-specified	Fixed by design structure
Variable types	All	Restricted (most exclude categoricals)
Statistical model	Agnostic	Polynomial (RSM)
Best for	Exploratory BO	Confirmatory RSM, curvature estimation
Requires prior knowledge	No	Somewhat (know which model to fit)

For designs where you specify exactly which model terms to estimate, see Optimal Experimental Design.