5. Tests with synthetic landscapes¶

This module generates synthetic landscapes that allows testing and direct interpretation of the statistical methods used

5.2. Random landscapes¶

Studying random landscapes provides a principled way to characterize how measurement noise and batch effects alone shape the statistical properties of empirical landscapes. By construction, random maps correspond to the limiting case in which no biological signal is present, i.e. the underlying deterministic phenotype satisfies $\bar{F}(\mathbf{x}) = 0$. As such, they define a natural null baseline against which structured biological interactions can be detected and interpreted.

In this framework, each replicate $r$ of the observed landscape is modeled as

\begin{equation} \tilde{F}_r(\mathbf{x}) = a_r + b_r\,\xi_r(\mathbf{x}), \end{equation}

where $a_r$ and $b_r$ represent additive and multiplicative batch effects, respectively, and $\xi_r(\mathbf{x})$ denotes the $r$-th realization of a random field $\xi(\mathbf{x})$ defined over configuration space.

The random variable $\xi(\mathbf{x})$ is characterized by its first and second moments, with mean $\mathbb{E}[\xi(\mathbf{x})] = \mu(\mathbf{x})$, variance $Var[\xi(\mathbf{x})] = \sigma_\xi^2(\mathbf{x})$, and, in general, a nontrivial covariance structure $\mathbb{E}[\xi(\mathbf{x})\,\xi(\mathbf{x}^\prime)]$ across configurations. Different assumptions on these moments define different ensembles of random landscapes, allowing us to probe how unstructured variability, heteroscedasticity, or correlations alone impact downstream quantities such as Walsh–Hadamard spectra and inferred epistatic interactions.

The synlands module in epistasia provides a collection of generators implementing these different random landscape ensembles, which can be used both as synthetic test cases and as explicit null models for empirical analyses.

5.2.1. Gaussian i.i.d. random landscape (no batch effects)¶

The simplest null ensemble is the Gaussian i.i.d. random landscape, in which the noise term $\xi(\mathbf{x})$ is drawn independently for each configuration $\mathbf{x}$ and each replicate from a normal distribution with zero mean and fixed variance,

\begin{equation} \xi(\mathbf{x}) \sim \mathcal{N}(0,\sigma^2) \end{equation}

In this model, all configurations are statistically equivalent: fluctuations are uncorrelated across states and have identical variance. As a result, any apparent structure observed in the inferred landscape—such as nontrivial Walsh–Hadamard spectra or apparent high-order interactions—arises purely from combinatorial effects and finite-sample fluctuations, rather than from genuine biological coupling.

In the following example, we use gaussian_random_landscape() to generate a simple Gaussian i.i.d. landscape and visualize its basic structure. This case will serve as a reference baseline for comparison with more structured random ensembles and with empirical data.

Now we can visualize how the epistatic coefficients are distributed across different interaction orders. For this, we use the function epistasis_distributions_by_order from the epistasis module, which computes all epistatic coefficients for all orders in all backgrounds simultaneously.

For each order $k$, the number of coefficients is given by the combinatorial factor (corresponding to the number of possible combinations of $k$ interacting loci) multiplied by the total number of posible backgrounds \begin{equation} \text{comb}(N, k)\times2^{N-k}. \end{equation} In the plots below, we show the histograms of the $\varepsilon$ --values for each order. Some distributions, especially at very low or very high orders, may display irregular or “pathological” shapes — this happens because there are too few coefficients to build a smooth histogram. When the number of coefficients is sufficiently large, the distributions appear approximately Gaussian, as expected for random (iid) landscapes.

Observe that the variance of the distributions, reflected in the spread along the horizontal axis, increases systematically with interaction order. This behavior is a direct consequence of the way local epistatic coefficients are constructed: each coefficient $\varepsilon$ is obtained by combining $2^{k}$ phenotypic measurements corresponding to the vertices of a $k$-dimensional hypercube. As a result, the variance of $\varepsilon$ grows with order, even in the absence of any underlying biological structure.

To make this scaling more explicit, in the next example we compute and visualize the standard deviation of the epistatic coefficient distributions as a function of interaction order. This provides a quantitative illustration of how local epistasis becomes increasingly noisy at higher orders, purely due to combinatorial amplification of measurement noise.

As discussed above, Walsh–Hadamard (WH) coefficients provide a natural representation of background-averaged epistatic interactions. Rather than repeating the formal derivation, here we focus on their interpretation in the context of random landscapes, which serve as a reference null model in the absence of biological structure.

For a given interaction order $k$, the relevant quantity is the second moment of the interaction strengths across all subsets of size $k$, $\langle \mathcal{E}^2 \rangle_k$, often referred to as the epistasis amplitude. This quantity enters the order-wise functional variance spectrum as $V(k) = 4^{-k}\binom{N}{k}\,\langle \mathcal{E}^2 \rangle_k$.

In random landscapes with unstructured fluctuations across configurations, $\langle \mathcal{E}^2 \rangle_k$ can be predicted analytically for simple ensembles and is observed to grow systematically with interaction order. Importantly, this growth does not reflect genuine high-order biological interactions, but instead arises from the combinatorial accumulation of noise across the $2^k$ vertices of a $k$-dimensional hypercube.

In practice, this behavior is captured numerically by the variance decomposition routine, which extracts an order-dependent interaction factor $U(k)$ directly from the WH coefficients. In random landscapes, $U(k)$ follows the theoretical scaling predicted by the null model, providing a baseline against which deviations observed in empirical landscapes can be meaningfully interpreted.

The next example illustrates the null behavior of the variance spectrum and of the epistasis amplitude across interaction orders. In particular, it allows us to verify that the systematic increase of interaction strength with order persists even in the absence of any underlying biological interactions, and to quantify the corresponding uncertainty and null fluctuations using bootstrap procedures.

By construction, any apparent high-order structure observed in this landscape must arise solely from combinatorial amplification of measurement noise. The results therefore provide a reference baseline against which empirical landscapes can be meaningfully compared in subsequent sections.

The apparent mismatch between the null bootstrap confidence interval and the uncertainty confidence interval does not indicate a methodological inconsistency, but rather a finite-sampling effect. By construction, the null bootstrap enforces the hypothesis $\bar{F}(\mathbf{x}) = 0$, corresponding to a landscape with no biological signal, and is therefore centered exactly at zero.

In contrast, the uncertainty confidence interval is built around the empirical estimate of $\bar{F}(\mathbf{x})$, obtained from a finite number $R$ of replicates. In random Gaussian landscapes, this empirical mean fluctuates around zero with a standard deviation scaling as $R^{-1/2}$, leading to a small but systematic offset that vanishes in the limit $R \to \infty$.

We then compute the epistasis amplitude across interaction orders using epistasis_amplitude, together with both uncertainty and null bootstraps. In this setting, any apparent order-dependent structure should be interpreted as a null baseline shaped by combinatorial effects and finite-$R$ variability, rather than genuine biological interactions.

The table above summarizes the epistasis amplitude $\langle \mathcal{E}^2\rangle_k$ across interaction orders for a purely random Gaussian landscape with a small number of replicates ($R=5$). As expected from the null model, the epistasis amplitude increases rapidly with interaction order, reflecting the combinatorial amplification of unstructured noise rather than genuine biological interactions.

Crucially, the observed estimates remain statistically consistent with the null prediction across all orders: the null confidence intervals closely overlap the uncertainty intervals (deviations come from finite sampling effects), the signal-to-null ratios remain of order unity, and the order-wise $p$-values do not indicate significant excess variance. This example illustrates how large apparent high-order epistasis can arise in random landscapes when $R$ is small, and highlights the importance of null comparisons for interpreting epistatic structure in empirical data.

5.3. Parametric vs. wild bootstrap consistency test¶

To validate that our wild (residual-based) bootstrap provides reliable uncertainty estimates for the epistasis amplitude, we perform a controlled test on a fully synthetic Gaussian i.i.d. landscape. In this setting the generative model is known exactly: for each configuration $x$ and replicate $r$, we sample $F(x,r) \sim \mathcal{N}(0,\sigma^2)$. This allows us to compare two uncertainty estimators:

• Wild bootstrap: the non-parametric procedure used throughout the package, which resamples replicate-level residuals and propagates noise through the Walsh–Hadamard analysis pipeline.

• Parametric bootstrap: the ground-truth procedure for this synthetic setting, obtained by repeatedly sampling new datasets from the known Gaussian model and recomputing the epistasis amplitude for each sample.

Because the parametric bootstrap must estimate tail quantiles of the amplitude distribution order by order, it requires a sufficiently large number of replicates $R$ (and bootstrap draws) to stabilize the confidence intervals, especially at high interaction orders where variance is strongly amplified. In practice, we observe that the wild bootstrap produces slightly wider (more conservative) confidence intervals than the parametric bootstrap, while remaining consistent with it across orders. This agreement supports that the wild bootstrap correctly captures noise propagation in the Walsh–Hadamard basis.

5.4. Flat epistatic landscapes and recoverability tests¶

To disentangle the role of combinatorics from that of interaction strength, we consider a class of synthetic landscapes with uniform epistatic amplitude across interaction orders, referred to as flat epistatic landscapes. In these maps, all interaction orders $k = 1,\dots,N$ have the same typical interaction strength,

\begin{equation} \langle \mathcal{E}_k^2 \rangle = K^2, \end{equation}

so that no order is intrinsically stronger or weaker than the others at the level of background-averaged epistasis. This condition is enforced in the Walsh–Hadamard representation by sampling coefficients with zero mean and order-dependent variance

\begin{equation} \text{Var}(f_{\mathbf{s}}) = K^2\,4^{-|\mathbf{s}|}. \end{equation}

As a result, the associated background-averaged interactions $\mathcal{E}_{\mathbf{s}} = 2^{|\mathbf{s}|} f_{\mathbf{s}}$ have constant variance across orders, while the order-wise functional variance

\begin{equation} V(k) = 4^{-k}\binom{N}{k}\langle \mathcal{E}_k^2 \rangle \end{equation}

remains purely shaped by combinatorial factors. Flat epistatic landscapes therefore provide a controlled ground-truth system in which epistatic interactions of all orders are present with equal strength. Any structure observed in the variance spectrum arises solely from geometry and entropy, rather than from a hierarchy of interaction strengths, making these landscapes a natural benchmark for inference.

To assess the recoverability of epistatic interactions, we combine this flat signal with synthetic noise, such as i.i.d.\ Gaussian noise, wild (residual-based) noise, or correlated multivariate Gaussian noise. The resulting mixed landscape $F_{\mathrm{mix}}(\mathbf{x}) = F_{\mathrm{flat}}(\mathbf{x}) + F_{\mathrm{noise}}(\mathbf{x})$ mimics realistic measurement conditions while preserving a known ground truth.

By varying the noise amplitude and the number of replicates, we quantify up to which interaction order $k$ the epistasis amplitude $\langle \mathcal{E}_k^2 \rangle$ can be distinguished from the bootstrap null distribution. As noise increases—or the number of replicates decreases—the higher-order signal is progressively masked, and the observed variance spectrum becomes dominated by noise, defining a practical limit of detectability.