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 .. math:: Q_n = \mathbf{z}^\top \mathbf{K} \mathbf{z}, where :math:`\mathbf{z}` is the standardised feature vector and :math:`\mathbf{K}` is a kernel matrix that encodes spatial structure. Under the null hypothesis of spatial independence, :math:`Q_n` follows a weighted :math:`\chi^2` distribution whose weights are the eigenvalues of :math:`\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 (:math:`\mathbb{E}[\mathbf{x} \mid S = \mathbf{s}] \neq \mathbb{E}[\mathbf{x}]`). This follows directly from using a linear kernel :math:`l(x_i, x_j) = x_i x_j` in the quadratic form, which reduces the conditional :math:`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 :math:`\mathbf{z}^2` rather than :math:`\mathbf{z}`. In spatial transcriptomics the distributional information is typically absent. We observe only one realisation :math:`(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 :math:`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 :math:`n \to \infty`) for every non-constant deterministic pattern *if and only if* :math:`\mathbf{K}` is strictly positive definite. Under :math:`H_0`, :math:`Q_n \approx \sum_i \lambda_i \chi^2_1`. When some :math:`\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. .. list-table:: :header-rows: 1 :widths: 24 28 48 * - 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: .. math:: \mathbf{K} = (\mathbf{I} - \rho \tilde{\mathbf{W}})^{-1}, where :math:`\tilde{\mathbf{W}}` is the row-normalised adjacency matrix and :math:`0 < \rho < 1` is the autoregressive parameter (default :math:`0.9`). The matrix :math:`\mathbf{I} - \rho \tilde{\mathbf{W}}` is the CAR *precision* matrix. It is sparse with :math:`\mathcal{O}(nk)` non-zeros even though :math:`\mathbf{K}` itself is dense, which is what makes CAR scalable on large graphs. Key properties: - Strictly positive definite for any :math:`0 < \rho < 1`. - Theoretically consistent (Theorem 2). - Scales via sparse-precision LU solves, with no :math:`\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 :math:`H_0`, :math:`Q_n` follows a weighted :math:`\chi^2` mixture .. math:: Q_n \;\sim\; \sum_{i=1}^{m} \lambda_i \chi^2_1, \qquad m = n - 1, where :math:`\lambda_i` are the eigenvalues of the double-centred kernel :math:`\tilde{\mathbf{K}} = \mathbf{H}\mathbf{K}\mathbf{H}`. ``quadsv`` ships three moment-matching fits, selected through the ``method`` argument of :func:`~quadsv.compute_null_params`: - ``"clt"``: a normal fit to :math:`(c_1, c_2)`. Valid for any :math:`\mathbf{K}`, including indefinite kernels. This is the only sensible choice for Moran's I. - ``"welch"``: a scaled central :math:`\chi^2` fit to :math:`(c_1, c_2)`. PSD kernels only. This is the default when it applies. - ``"liu"``: a shifted non-central :math:`\chi^2` fit to :math:`(c_1, c_2, c_3, c_4)`. PSD kernels only. Tightest tail. Here :math:`c_p = \operatorname{tr}(\tilde{\mathbf{K}}^p)` is the :math:`p`-th spectral power sum. ``quadsv`` also applies a finite-:math:`n` Dirichlet(1/2) correction to :math:`\operatorname{Var}[Q_n]`. See :doc:`/guides/scaling` 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: .. math:: R_{xy} = \mathbf{x}^\top \mathbf{K} \mathbf{y}, where :math:`\mathbf{x}` and :math:`\mathbf{y}` are standardised. Under :math:`H_0`, :math:`R_{xy} \sim \mathcal{N}\bigl(0, \operatorname{tr}(\mathbf{K}^2)\bigr)`, which gives a fast Normal p-value. A typical workflow: 1. Identify SVGs via the univariate Q-test. 2. Test pairwise R-statistics among the top SVGs. 3. Control FDR across comparisons. Drop-in replacement for Moran's I --------------------------------- .. list-table:: :header-rows: 1 :widths: 22 38 40 * - 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. See also -------- - :doc:`/guides/quickstart` for practical recipes. - :doc:`/guides/kernels` for kernel selection and design. - :doc:`/guides/scaling` for null-distribution and operator complexity. - :doc:`/autoapi/quadsv/statistics/index` for the statistical-test API.