Multiallelic Walsh Transforms in Genotype Analysis

• Multiallelic Walsh transforms generalize the binary Walsh-Hadamard method by extending orthogonal decompositions to genotype spaces with multiple alleles per locus.
• The framework computes transform coefficients efficiently, allowing precise quantification of additive effects and multiway epistatic interactions in fitness landscapes.
• Its rich algebraic structure links combinatorial circuits and polyhedral geometry to offer actionable insights into the statistical and biological analysis of genotype-phenotype maps.

A multiallelic Walsh transform generalizes the classical Walsh-Hadamard transform from biallelic (two-level) systems to genotype spaces in which each locus may take an arbitrary number of allelic states. This framework supports the rigorous decomposition of multiallelic genotype-phenotype maps (e.g., fitness landscapes) into orthogonal additive and higher-order interaction components, enabling the quantitative measurement of additive effects, dominance deviations, and multiway epistasis in systems with several alleles per locus. The algebraic structure admits efficient computation, precise statistical interpretation, and connections to combinatorial and polyhedral methods such as circuits and triangulations (Crona et al., 2024, Greene, 2023).

1. Genotype Space and Basis Construction

Given $k$ loci, with locus $i$ admitting $m_i$ alleles labeled by $0,1,\ldots, m_i-1$, the genotype space is $G = A_1 \times \cdots \times A_k$, where each $A_i = \{0,1,\ldots, m_i-1\}$, and $N = |G| = \prod_{i=1}^k m_i$.

For each locus $i$, an orthogonal basis of real-valued contrast functions $\varphi_{i,\alpha_i}: A_i \to \mathbb{R}$ is chosen, indexed by $\alpha_i=0,\ldots, m_i-1$. These satisfy:

• $i$0
• $i$1
• $i$2 for $i$3

The multiallelic Walsh basis on $i$4 is then constructed by taking products across loci:

$i$5

Orthonormality holds:

$i$6

These basis functions are key to all subsequent transforms and decompositions (Crona et al., 2024).

2. Transform Formulas and Inversion

Given any function $i$7 (traditionally, a fitness landscape), one defines its multiallelic Walsh coefficients:

• Forward transform:

$i$8

• Inverse transform:

$i$9

Here, both sums are over all $m_i$0 possible multi-indices (over all possible allele combinations).

The orthonormality of the basis ensures that the transform and its inverse are exact, and—crucially—support a decomposition of $m_i$1 into interpretable statistical components (Crona et al., 2024, Greene, 2023).

A different but equivalent explicit transform construction is given via the kernels $m_i$2 and $m_i$3 as follows. For each locus $m_i$4, let $m_i$5 if $m_i$6 and $m_i$7, $m_i$8 otherwise; $m_i$9 if $0,1,\ldots, m_i-1$0 or $0,1,\ldots, m_i-1$1, $0,1,\ldots, m_i-1$2 if $0,1,\ldots, m_i-1$3, $0,1,\ldots, m_i-1$4 otherwise. The transforms $0,1,\ldots, m_i-1$5 and $0,1,\ldots, m_i-1$6 act as tensor products of local matrices, and are mutual (scaled) inverses:

$0,1,\ldots, m_i-1$7

where $0,1,\ldots, m_i-1$8 and $0,1,\ldots, m_i-1$9. In the biallelic case ($G = A_1 \times \cdots \times A_k$0), these reduce to the standard Walsh-Hadamard transform ($G = A_1 \times \cdots \times A_k$1 kernel) (Greene, 2023).

3. Interaction and Variance Decomposition

Walsh coefficients can be systematically grouped by the number of nonzero entries in $G = A_1 \times \cdots \times A_k$2 (“interaction order”):

• Order 0: overall mean ($G = A_1 \times \cdots \times A_k$3)
• Order 1: single-locus effects (additive + dominance)
• Order 2: pairwise interactions (epistasis)
• Up to order $G = A_1 \times \cdots \times A_k$4: $G = A_1 \times \cdots \times A_k$5-way interactions (higher-order epistasis)

The functional decomposition becomes:

$G = A_1 \times \cdots \times A_k$6

(Crona et al., 2024, Greene, 2023)

The variance of $G = A_1 \times \cdots \times A_k$7 over $G = A_1 \times \cdots \times A_k$8 in the Walsh basis admits the decomposition:

$G = A_1 \times \cdots \times A_k$9

Additive, pairwise, $A_i = \{0,1,\ldots, m_i-1\}$0, $A_i = \{0,1,\ldots, m_i-1\}$1-way interaction variances are obtained by summing $A_i = \{0,1,\ldots, m_i-1\}$2s of appropriate orders.

Statistically, coefficients in subspaces $A_i = \{0,1,\ldots, m_i-1\}$3 correspond to the additive model, while nonzero coefficients in $A_i = \{0,1,\ldots, m_i-1\}$4 and higher strictly indicate epistatic interactions (Greene, 2023).

4. Circuits and Polyhedral Geometry

The genotype points (encoded as standard basis vectors in $A_i = \{0,1,\ldots, m_i-1\}$5) form the vertices of the polytope $A_i = \{0,1,\ldots, m_i-1\}$6.

A circuit is a minimally affinely dependent subset of vertices, corresponding to linear relations that vanish under additivity. For instance, a two-locus circuit (parallelogram circuit) for alleles $A_i = \{0,1,\ldots, m_i-1\}$7 at locus $A_i = \{0,1,\ldots, m_i-1\}$8 and $A_i = \{0,1,\ldots, m_i-1\}$9 at locus $N = |G| = \prod_{i=1}^k m_i$0 is:

$N = |G| = \prod_{i=1}^k m_i$1

Deviation from zero signals non-additive epistasis confined to the $N = |G| = \prod_{i=1}^k m_i$2 block defined by the chosen alleles. Circuits thus detect local, unaveraged sources of epistasis (Crona et al., 2024).

Triangulations of the genotype polytope relate to the geometry of the fitness landscape. Each triangulation corresponds to a partition induced by lifting each vertex $N = |G| = \prod_{i=1}^k m_i$3 to height $N = |G| = \prod_{i=1}^k m_i$4 and taking the lower convex hull. The sign pattern of circuit relations determines accessible evolutionary pathways, fitness peaks, and ridges (Crona et al., 2024).

5. Statistical and Biological Significance

The multiallelic Walsh transform enables rigorous decomposition of genotype-phenotype maps in biological systems with multiple alleles per locus. This supports the quantitative analysis of additive, dominance, and epistatic genetic variance components, foundational in quantitative genetics and evolutionary biology. Testing for nonzero higher-order Walsh coefficients provides direct statistical tests for evidence of epistasis (Crona et al., 2024, Greene, 2023).

Beyond biology, the orthogonal decomposition is equivalent to high-order interaction decomposition in multi-category factorial data. Applications thus extend to categorical analysis of variance (ANOVA), contingency tables, and signal processing with $N = |G| = \prod_{i=1}^k m_i$5-ary categorical inputs (Greene, 2023).

6. Efficient Computation and Generalizations

The tensor-product structure of the multiallelic Walsh transform allows efficient computation. For loci with allele counts $N = |G| = \prod_{i=1}^k m_i$6, applying the transform to $N = |G| = \prod_{i=1}^k m_i$7 requires $N = |G| = \prod_{i=1}^k m_i$8 arithmetic operations, where $N = |G| = \prod_{i=1}^k m_i$9, and $i$0. Each transform reduces to local matrix-vector application along each coordinate axis, akin to the classic fast Walsh-Hadamard transform in the binary case (Greene, 2023).

Mathematically, the construction generalizes to arbitrary orthogonal systems on product spaces and relates to generalized Walsh bases in analysis and signal processing via representation theory of Cuntz algebras (Dutkay et al., 2018).

7. Examples

For two loci each with three alleles ($i$1), the basis functions are:

• $i$2
• $i$3
• $i$4

The forward Walsh coefficient for $i$5 is:

$i$6

Pairwise circuit relations enumerate all $i$7 and certain $i$8-point (triangle) dependencies among genotypes, each serving as a local test for deviation from additivity—the molecular signature of epistasis in such systems (Crona et al., 2024, Greene, 2023).

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References:

(Crona et al., 2024) Walsh coefficients and circuits for several alleles (Greene, 2023) Multiallelic Walsh transforms (Dutkay et al., 2018) On generalized Walsh bases