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Constructing Large Orthogonal Minimally Aliased Response Surface Designs Through Enumeration and Combination of Weighing Designs

This paper introduces an algorithmic framework that constructs large Orthogonal Minimally Aliased Response Surface (OMARS) designs by enumerating and combining weighing designs, thereby enabling the scalable generation of high-quality experimental layouts for complex, high-dimensional studies.

Original authors: Jade Lejeune Herman, Peter Goos

Published 2026-08-06

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Jade Lejeune Herman, Peter Goos

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a detective trying to solve a mystery, but instead of a single clue, you have a mountain of suspects, each with a hidden motive. In the world of science and industry, these "suspects" are variables—like temperature, pressure, or chemical mixtures—that might be influencing a result. The goal is to figure out which ones actually matter and how they work together. This is the art of "Design of Experiments." Think of it as a master plan for testing things. If you just guess and check, you might miss the real culprit or waste years of time and money. But if you plan your tests carefully, you can learn the maximum amount of information with the minimum number of tries.

For a long time, scientists had great plans for small mysteries with just a few suspects. But now, thanks to robots and super-fast machines, we can test hundreds of things at once. This is like moving from solving a crime in a small village to investigating a massive city with thousands of people. The old plans don't work for the city; they get too messy and confusing. We need a new kind of map that stays organized even when the number of suspects explodes. This paper is all about building that new map, ensuring that even in a chaotic city of thousands of variables, we can still tell exactly who did what, without getting tricked by false clues.

The authors of this paper, Jade Lejeune Herman and Peter Goos, are tackling a specific type of map called an "Orthogonal Minimally Aliased Response Surface" (OMARS) design. That's a mouthful, so let's break it down with a simple analogy. Imagine you are baking a cake and you want to know how sugar, flour, and eggs affect the taste.

Orthogonal means your tests are perfectly balanced. If you change the sugar, you don't accidentally change the flour at the same time. It's like having a scale where you can weigh one ingredient without the others tipping the balance.
Minimally Aliased means you avoid "aliasing," which is like a case of mistaken identity. In a bad experiment, you might think the sugar made the cake sweet, but actually, it was the honey you added at the same time. Aliasing is when two suspects look so similar you can't tell them apart. "Minimally aliased" means you've set up the tests so that the suspects are as distinct as possible.
Response Surface means you aren't just looking for a straight line (more sugar = sweeter); you are looking for curves. Maybe too much sugar makes it cloying, or maybe a little bit of salt makes the sugar taste better. You want to see the whole shape of the flavor.

The problem is that making these perfect maps for huge experiments is incredibly hard. It's like trying to arrange millions of puzzle pieces so that no two pieces ever look the same, and every single piece fits perfectly with every other piece. Previous methods were like trying to solve this puzzle by hand; they worked for small puzzles but crashed when the puzzle got too big.

This paper introduces a clever new way to build these massive maps. The authors realized that instead of trying to build the giant puzzle all at once, they could build smaller, perfect "weighing designs" first. Imagine a weighing design as a small, perfect set of scales. These scales have a special rule: every column (or ingredient) has a fixed number of empty spots (zeros) and a fixed number of weights (ones or minus ones). The magic is that these scales are perfectly balanced.

The authors developed a computer program that acts like a super-organized librarian. This librarian goes through every possible way to arrange these small scales, checking them one by one to find the unique, perfect ones. They didn't just stop at the small ones, though. They found a way to stack these small, perfect scales on top of each other to create giant ones. It's like taking a stack of small, perfect Lego towers and snapping them together to build a skyscraper without any wobbling.

Here is what they actually found and built:

The Catalog: They completely listed every possible unique "small scale" (weighing design) that has up to 24 tests (rows). They covered cases where each column had either two or three empty spots. For larger designs, they used a smart filtering method to find the best ones without checking every single possibility, which would take too long.
The Big Build: They showed how to take these small, perfect scales and combine them to create massive experiments. They successfully built designs with up to 156 tests (runs) that can handle many factors (columns).
The Proof: They tested these new giant maps to see if they were actually good. They found that the maps they built are excellent at keeping the "suspects" distinct. The "mistaken identities" (aliasing) between the interactions of the variables were kept very low. In fact, for many sizes, their new maps were better than the old standard maps (called Definitive Screening Designs or DSDs) that scientists had been using.

However, the paper is honest about its limits. They didn't solve the problem for every possible size. The "super-librarian" computer program gets too slow if the puzzle gets much bigger than 24 rows. For those huge sizes, they had to rely on their "stacking" trick and a smart guess-and-check method (partial enumeration) rather than checking every single possibility. They also noted that while these maps are great, there is a hard limit on how many factors you can test if you want to see every single possible interaction. It's like a room that can only hold so many people before it gets too crowded to move around. If you try to put too many factors in, you can't estimate the full picture anymore, though you can still find the most important ones using special math tricks.

The authors are very sure about the designs they found within their limits. They didn't just guess; they mathematically proved that their method works and they ran the numbers to show that their new designs are statistically efficient. They didn't claim to have found the "perfect" design for every size in the universe, but they did prove that their method creates a huge new library of high-quality maps that scientists can use right now.

In short, this paper gives scientists a new toolkit. Instead of struggling to build a giant, messy experiment from scratch, they can now grab a pre-made, perfectly balanced block from this new catalog, stack it with others, and instantly have a massive, high-quality experiment ready to go. It turns a chaotic, impossible task into a manageable, systematic one, allowing researchers to explore complex worlds of variables with confidence.

Technical Summary: Constructing Large Orthogonal Minimally Aliased Response Surface Designs

Problem Statement
The paper addresses the challenge of constructing large-scale Orthogonal Minimally Aliased Response Surface (OMARS) designs. While automation and high-throughput experimentation (e.g., in pharmaceuticals and materials science) have enabled studies with many factors, existing design construction methods struggle with scalability. Direct enumeration of OMARS designs becomes computationally intractable as the number of runs increases. Furthermore, previous approaches often rely on pre-existing weighing matrices or integer programming that cannot guarantee the discovery of the optimal designs for larger problem sizes. There is a specific need for frameworks that can generate large experiments while preserving orthogonality among main effects and minimizing aliasing among second-order effects (interactions and quadratic terms).

Methodology
The authors propose an algorithmic framework that constructs large OMARS designs by enumerating and combining weighing designs (rectangular matrices with entries $\{0, \pm 1\}$ ) rather than enumerating OMARS designs directly. The methodology proceeds in three main stages:

Enumeration of Weighing Designs:
- Definition: A weighing design $W(m, z, k)$ is defined as an $m \times z$ matrix with entries in $\{0, \pm 1\}$ , where each column has exactly $k$ non-zero entries, and columns are mutually orthogonal ( $W'W = kI_z$ ).
- Algorithm: A column-by-column enumeration procedure is developed. It starts with a fixed first column and iteratively extends designs by adding one column at a time.
- Two-Stage Process:
  - Stage 1 (Zero-Pattern Enumeration): The algorithm generates all valid binary "zero-pattern" vectors for the next column, ensuring constraints on the maximum number of zeros per row and parity conditions required for orthogonality with the first column.
  - Stage 2 (Sign Assignment & Isomorphism Filtering): For each valid zero-pattern, the algorithm assigns signs ( $\pm 1$ ) to non-zero entries. It rigorously filters out isomorphic designs (those equivalent under row/column permutations or sign flips) using graph isomorphism algorithms on a bipartite graph representation of the matrix.
- Partial Enumeration: For cases where complete enumeration is computationally infeasible, a partial enumeration strategy is employed. Designs are ranked by a quality criterion at each step, and only the top fraction is retained for extension. Validation at $m=16$ showed this approach successfully recovers the top-tier designs.
Construction of OMARS Designs:
- The enumerated weighing designs are transformed into OMARS designs via foldover. An OMARS design $D$ is constructed by concatenating a weighing design $W$ with its negative $-W$ :
  $D = \begin{bmatrix} W \\ -W \end{bmatrix}$
- This construction ensures that main effects are orthogonal to each other and to all second-order terms, while minimizing aliasing among second-order terms.
Concatenation for Larger Sizes:
- To exceed the limits of direct enumeration, the authors propose a concatenation method. Two weighing designs with the same number of columns can be stacked vertically to create a larger weighing design with a weight equal to the sum of the original weights. This recursive property allows the generation of designs with run sizes that are multiples of base enumerated sizes.

Key Contributions

Extended Enumeration Framework: The paper generalizes the concept of weighing matrices to rectangular weighing designs and provides a complete enumeration of non-isomorphic weighing designs for up to 24 runs (covering various factor counts and weights corresponding to 2 or 3 zeros per column).
Scalable Construction: By combining enumeration with a foldover technique and a concatenation strategy, the authors generate catalogs of OMARS designs for run sizes up to 156, significantly expanding the range of feasible experiments beyond previous catalogs (which were limited to ~7 factors).
Quality Assessment: The paper introduces a rigorous evaluation framework using Average Absolute Correlation (AAC), Root-Mean-Square (RMS) correlation, and their geometric mean (GM) to assess aliasing in interaction-interaction (II), interaction-quadratic (IQ), and quadratic-quadratic (QQ) blocks.
Comparative Analysis: The study compares the generated large OMARS designs against Definitive Screening Designs (DSDs) and evaluates their D-, G-, and A-efficiencies for second-order models.

Results

Enumeration Limits: Complete enumeration was achieved for designs with up to 24 rows. The number of non-isomorphic designs initially increases with the number of columns, peaks near half the number of rows, and then decreases. Computational limits (runtime and memory) were identified, motivating the use of partial enumeration and concatenation for larger sizes.
Design Efficiency: The constructed large OMARS designs demonstrate that while D-efficiency remains moderate (starting near 50% and declining to ~30% as factors increase), G- and A-efficiencies for the full second-order model decline rapidly, approaching zero as the number of factors increases. This reflects the inherent trade-off where more factors imply more parameters to estimate with the same information, leading to increased aliasing. However, for submodels (assuming effect sparsity), efficiencies are much higher.
Aliasing Characteristics: While main effects remain orthogonal, second-order effects exhibit residual aliasing. The analysis reveals that high correlations are often concentrated in a small number of factor pairs rather than being uniformly distributed. This allows practitioners to assign critical factors to "safe" columns to minimize specific aliasing.
Comparison with DSDs: For 96-run designs, the OMARS designs generally outperform DSDs in D-, A-, and G-efficiencies and show lower average II and IQ correlations, though DSDs showed lower QQ correlations. However, the paper notes that DSDs estimate quadratic effects less precisely due to fewer middle-level runs.
Estimability Constraints: The foldover structure limits the maximum number of factors ( $c_{max}$ ) that can be included while allowing the estimation of a full second-order model (e.g., $c_{max}=9$ for 96 runs). The authors note that for screening purposes where effect sparsity is assumed, this limitation is mitigated by using advanced model selection techniques (e.g., Lasso, mixed-integer optimization) rather than requiring full model estimability.

Significance
The paper provides a practical methodology for generating high-quality, large-scale OMARS designs suitable for complex industrial and scientific experiments. By moving from direct OMARS enumeration to weighing design enumeration and combination, the authors overcome previous scalability barriers. The resulting catalogs enable researchers to combine screening and response surface estimation in a single step for problem sizes previously unattainable, supporting high-throughput experimentation in fields like pharmaceuticals and materials science. The work acknowledges that while complete enumeration is limited to ~24 runs, the proposed partial enumeration and concatenation strategies effectively extend the utility of these designs to much larger scales.

Technical Summary: Constructing Large Orthogonal Minimally Aliased Response Surface Designs

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