Optimal Experimental Design¶

Optimal designs generate the most information-rich set of experiments for a user-specified statistical model. Unlike space-filling or classical RSM designs, optimal designs let you say "I expect these specific effects to matter — give me the minimum runs that estimate them as precisely as possible."

Core Concept¶

An optimal design maximizes (or minimizes) a mathematical criterion on the information matrix $\mathbf{X}'\mathbf{X}$, where $\mathbf{X}$ is the design matrix. The three supported criteria are:

Criterion	Optimizes	Best for
D-optimal	Maximize $	\mathbf{X}'\mathbf{X}
A-optimal	Minimize $\text{tr}[(\mathbf{X}'\mathbf{X})^{-1}]$	Average parameter variance
I-optimal	Minimize integrated prediction variance	Prediction accuracy over the space

D-optimal is the recommended default for most applications — it maximizes the information in the design for estimating all model parameters simultaneously.

Specifying the Model¶

You specify which effects to include in the model. ALchemist builds the design matrix accordingly. The intercept is always included automatically.

Model Type Shortcuts¶

The quickest way to specify a model — use a shortcut name:

Shortcut	Includes
`"linear"`	Intercept + all main effects
`"interaction"`	Intercept + main effects + all two-factor interactions
`"quadratic"`	Intercept + main effects + interactions + quadratic terms (continuous only)

Custom Effects¶

For full control, pass an explicit list of effects using variable names:

Main effect:        "Temperature"
Two-factor interaction: "Temperature*Pressure"
Quadratic term:     "Temperature**2"

Example — a selective model:

effects = [
    "Temperature",           # main effect
    "Pressure",              # main effect
    "Catalyst",              # main effect (categorical)
    "Temperature*Pressure",  # interaction
    "Temperature**2",        # quadratic (continuous only)
]

Rules: - Use * for interactions: "Temperature*Pressure" - Use **2 for quadratic: "Temperature**2" - Quadratic terms are only valid for Real and Integer variables (not Categorical or Discrete with exactly 2 values) - Categorical variables can have main effects and interactions, but not quadratic terms

Tip: Use the Preview button in the web UI to see how many model columns your specification produces, and the recommended run count, before generating the design.

Run Count¶

You can specify run count in two ways:

Mode	Parameter	When to use
Absolute	`n_points`	Fixed budget of experiments
Multiplier	`p_multiplier`	Proportional to model complexity

The p_multiplier approach is recommended: p_multiplier=2.0 gives 2× as many runs as model columns (p). This ensures the design is estimable with some degrees of freedom for lack-of-fit testing.

Rule of thumb: Use at least p_multiplier=1.5 for reliable estimation; 2.0 is a solid default; 3.0 for high-quality estimation with outlier robustness.

Exchange Algorithms¶

Optimal designs are found by iteratively exchanging candidate points to improve the criterion. Five algorithms are available:

Algorithm	Speed	Quality	When to use
`sequential` (Dykstra)	Fastest	Lower	Quick prototyping, large spaces
`simple_exchange`	Fast	Moderate	General use
`fedorov`	Moderate	High	Recommended default
`modified_fedorov`	Moderate	High	Similar to Fedorov
`detmax` (Mitchell)	Slowest	Highest	Final design, critical applications

Fedorov is the default — it performs full pairwise candidate exchanges and reliably finds high-quality designs. Use DetMax when you want the best possible design and runtime is not a concern.

Web UI Walkthrough¶

Define your variable space.
In the Initial Design panel, switch to the Optimal Design tab.
Choose Quick (model type shortcut) or Custom (effect checkboxes + text input).
Click Preview to see the model terms and recommended run count.
Optionally adjust n_points to the recommended value, or set a p_multiplier.
Choose criterion (D/A/I) and algorithm.
Click Generate.
Review design quality metrics (D-efficiency, model terms).
Click Stage Suggestions to queue the design.

AI-Assisted Effect Selection: The Optimal Design panel includes an AI suggestion tool. See LLM-Assisted Effect Suggestion for details.

Python API¶

Preview before generating¶

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"])

# Get model info (dry run — no design generated yet)
info = session.get_optimal_design_info(model_type="quadratic")
print(f"Model has {info['p_columns']} terms: {info['model_terms']}")
print(f"Recommended runs: {info['n_points_recommended']}")

Generate design¶

# Using a model type shortcut
points, info = session.generate_optimal_design(
    model_type="quadratic",
    p_multiplier=2.0,
    criterion="D",
    algorithm="fedorov",
    random_seed=42,
)

# Using a custom effects list
points, info = session.generate_optimal_design(
    effects=["Temperature", "Pressure", "Temperature*Pressure", "Temperature**2"],
    n_points=20,
    criterion="D",
    algorithm="detmax",
)

# Design quality metrics
print(f"D-efficiency: {info['D_eff']:.1f}%")
print(f"A-efficiency: {info['A_eff']:.1f}%")
print(f"Model terms: {info['model_terms']}")

REST API¶

Preview model terms¶

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

{
  "model_type": "quadratic"
}

Response:

{
  "p_columns": 9,
  "model_terms": ["Intercept", "Temperature", "Pressure", "Catalyst[Pt]", "Catalyst[Pd]", "Temperature*Pressure", "Temperature**2", "Pressure**2", "..."],
  "n_points_minimum": 9,
  "n_points_recommended": 18
}

Generate design¶

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

{
  "model_type": "quadratic",
  "p_multiplier": 2.0,
  "criterion": "D",
  "algorithm": "fedorov",
  "random_seed": 42
}

Custom effects:

POST /api/v1/sessions/{session_id}/optimal-design
{
  "effects": ["Temperature", "Pressure", "Temperature*Pressure"],
  "n_points": 15,
  "criterion": "D",
  "algorithm": "fedorov"
}

D-efficiency¶

The design response includes a D-efficiency percentage:

\[D\text{-eff} = \frac{100}{p} \cdot |\mathbf{X}'\mathbf{X}|^{1/p}\]

A D-efficiency of 100% is a theoretical ideal (rarely achieved in practice). Values above 90% are generally excellent; above 80% is acceptable.

Limitations¶

Categorical variables (experimental)

Optimal design with categorical variables uses dummy coding (k−1 indicator columns per variable). The statistical properties are well-understood but designs may be harder to interpret than classical factorial methods. For categorical screening with well-established properties, consider GSD or Full Factorial instead.

Discrete variables with exactly 2 allowed values cannot include quadratic terms.
Very large candidate sets (many variables × many levels) may be slow. Reduce n_levels (default 5) to speed up.

For the conceptual background on optimality criteria and design matrices, see Background: Design of Experiments.