Theoretical Results

Theoretical Results#

Our accompanying paper shows that virtually every spatially-variable-gene (SVG) detection method reduces to a single quadratic-form statistic, the Q-statistic. This includes graph-based methods like Moran’s I, parametric models, and non-parametric dependence tests. The Q-statistic is

\[Q_n = \mathbf{z}^\top \mathbf{K} \mathbf{z},\]

where \(\mathbf{z}\) is the standardised feature vector and \(\mathbf{K}\) is a kernel matrix that encodes spatial structure. Under the null hypothesis of spatial independence, \(Q_n\) follows a weighted \(\chi^2\) distribution whose weights are the eigenvalues of \(\mathbf{K}\). Moment-matching approximations turn that distribution into a fast p-value, giving the Q-test. The kernel choice critically affects the consistency and the power of the resulting test.

This page summarises the key theoretical results.

Theorem 1: Q-tests detect mean shifts only#

All spatial Q-tests detect mean-shift patterns (\(\mathbb{E}[\mathbf{x} \mid S = \mathbf{s}] \neq \mathbb{E}[\mathbf{x}]\)).

This follows directly from using a linear kernel \(l(x_i, x_j) = x_i x_j\) in the quadratic form, which reduces the conditional \(X \mid S = s_i\) to its mean. To probe higher moments (variance, distributional changes), swap in a non-linear kernel. For example, apply a Gaussian or polynomial kernel to \(\mathbf{z}^2\) rather than \(\mathbf{z}\).

In spatial transcriptomics the distributional information is typically absent. We observe only one realisation \((x_i, s_i)\) per location, which blurs the line between mean independence and statistical independence. Treating the signal as a deterministic element of a Hilbert space \(f \in L^2(\mathcal{S})\) and applying spectrum theory of kernel operators yields the consistency condition below.

Theorem 2: Consistency requires positive definiteness#

A spatial Q-test is universally consistent (power approaches 1 as \(n \to \infty\)) for every non-constant deterministic pattern if and only if \(\mathbf{K}\) is strictly positive definite.

Under \(H_0\), \(Q_n \approx \sum_i \lambda_i \chi^2_1\). When some \(\lambda_i < 0\) (indefinite kernel), signals aligned with the negative eigenspace cancel signals aligned with the positive eigenspace. We call this spectral cancellation, and it costs the test power on composite patterns.

The implication is that you should pick a kernel with a non-negative spectrum.

Kernel	Spectrum	Consistency
Gaussian	Strictly positive	Guaranteed.
Matérn	Strictly positive	Guaranteed.
Moran’s I	Indefinite	Spectral cancellation.
Graph Laplacian	Non-negative	Guaranteed (high-frequency Moran).
CAR (inverse Laplacian)	Strictly positive	Guaranteed (low-frequency Moran).

CAR is a scalable correction to Moran’s I#

The Conditional Autoregressive (CAR) kernel is strictly positive definite:

\[\mathbf{K} = (\mathbf{I} - \rho \tilde{\mathbf{W}})^{-1},\]

where \(\tilde{\mathbf{W}}\) is the row-normalised adjacency matrix and \(0 < \rho < 1\) is the autoregressive parameter (default \(0.9\)). The matrix \(\mathbf{I} - \rho \tilde{\mathbf{W}}\) is the CAR precision matrix. It is sparse with \(\mathcal{O}(nk)\) non-zeros even though \(\mathbf{K}\) itself is dense, which is what makes CAR scalable on large graphs.

Key properties:

Strictly positive definite for any \(0 < \rho < 1\).
Theoretically consistent (Theorem 2).
Scales via sparse-precision LU solves, with no \(\mathcal{O}(n^2)\) materialisation.
Polynomial spectral decay that emphasises smooth, large-scale patterns while keeping a heavy tail for mid- and high-frequency components.

Use CAR as the default for graph-flavoured spatial-pattern detection.

Null-distribution approximations#

Under \(H_0\), \(Q_n\) follows a weighted \(\chi^2\) mixture

\[Q_n \;\sim\; \sum_{i=1}^{m} \lambda_i \chi^2_1, \qquad m = n - 1,\]

where \(\lambda_i\) are the eigenvalues of the double-centred kernel \(\tilde{\mathbf{K}} = \mathbf{H}\mathbf{K}\mathbf{H}\). quadsv ships three moment-matching fits, selected through the method argument of compute_null_params():

"clt": a normal fit to \((c_1, c_2)\). Valid for any \(\mathbf{K}\), including indefinite kernels. This is the only sensible choice for Moran’s I.
"welch": a scaled central \(\chi^2\) fit to \((c_1, c_2)\). PSD kernels only. This is the default when it applies.
"liu": a shifted non-central \(\chi^2\) fit to \((c_1, c_2, c_3, c_4)\). PSD kernels only. Tightest tail.

Here \(c_p = \operatorname{tr}(\tilde{\mathbf{K}}^p)\) is the \(p\)-th spectral power sum. quadsv also applies a finite-\(n\) Dirichlet(1/2) correction to \(\operatorname{Var}[Q_n]\). See Scalable Computation for the formula, the cumulant-evaluation paths (FFT / NUFFT analytic, Matrix Frobenius, Hutchinson probes), and the full complexity table.

R-test: bivariate spatial co-expression#

The R-statistic extends the Q-test to two features at a time:

\[R_{xy} = \mathbf{x}^\top \mathbf{K} \mathbf{y},\]

where \(\mathbf{x}\) and \(\mathbf{y}\) are standardised. Under \(H_0\), \(R_{xy} \sim \mathcal{N}\bigl(0, \operatorname{tr}(\mathbf{K}^2)\bigr)\), which gives a fast Normal p-value.

A typical workflow:

Identify SVGs via the univariate Q-test.
Test pairwise R-statistics among the top SVGs.
Control FDR across comparisons.

Drop-in replacement for Moran’s I#

Method	Test consistency	Use case
Moran’s I	Spectral cancellation.	Classical autocorrelation; backwards compatibility only.
Graph Laplacian	Guaranteed.	High-frequency, local variation.
CAR	Guaranteed.	Low-frequency, smooth patterns.

In practice:

On a graph, use the CAR kernel for consistent, high-power detection across functional patterns.
On 2-D physical space, use the FFT- and NUFFT-accelerated forms of any PSD kernel. Matérn is a common starting point.