Characteristic function-based tests for spatial randomness

We introduce a new type of test for complete spatial randomness that applies to mapped point patterns in a rectangle or a cube of any dimension This is the first test of its kind to be based on characteristic functions and utilizes a weighted L2-distance between the empirical and uniform characteristic functions The test shows surprising connections to Ripley’s K-function and Zimmerman’s w2 statistic It is also simple to calculate and does not require adjusting for edge effects

Test Development

The core of the new test involves a weighted L2-like distance, scaled by n, defined as Δ = ∫ R D (1/n) Σ j=1 n exp(i t ⊺ x j) − φˆX(t) w(t) d t The weight function w(t) is usually chosen as the squared modulus of the CF, φ0 (t)2, or the Gaussian or double exponential (Laplace) density (Meintanis, 2016>

Weight Function Choices

The paper explores several choices for the weight function w(t), including:

1. Bessel-like weight function: This produces a simple rectangular φr The expectation of the CF test statistic under the null hypothesis is E0 (Δr) = 1 − E0(φr (x1 − x2))

2. Normalized squared modulus of the null distribution CF φ0 (t): This is a common choice for weight function in the CF-based test

3. Centered Gaussian weight function: This is the only weight function that is both isotropic and separable, making it valuable in high dimensions

4. Cauchy weight function: The corresponding statistic in [0, 1] D is Δr = (1/n) Σ j,k=1 exp (-1/r Σ d=1 x j d - x k d! − 2 Σ j=1 Ö D d=1 r n squared − e(-x j d / r) − e(-(1−x j d) / r) o + n/n 2r

Asymptotic Distribution and Efficiency

The paper develops efficient algorithms for the asymptotic and small-sample null distributions of the Cauchy-weighted statistic in Section 4 The Cauchy-weighted CF test is shown to have a much lower computational cost than the isotropicedge-corrected L-test, so it is chosen as the focus of investigation The Bonferroni-corrected omnibus CF test is demonstrated to be more well rounded than the individual CF tests

Simulation and Application

Simulations assess the accuracy of both proposed null distributions for the test statistic, showing that the two-tailed test using either method has a type I error rate very close to the theoretical value 0.05 The power of the CF test under different alternatives is compared against tests like Zimmerman’s w2-test and Clark-Evans test In application, the CF test outperforms other tests in most simulation settings of heterogeneity, aggregation and regularity, especially when the sample size is large The analysis applies the CF test to four example point patterns used in Zimmerman (1993) to demonstrate CSR, aggregation, regularity and heterogeneity

Theoretical Foundations

The asymptotic distribution of Δ converges in distribution to Í∞ j=1 λj Z2 j, where (Z j) is a sequence of independent standard normal variables and (λ j) are the eigenvalues of the linear operator T The paper derives the eigenvalues and asymptotic distribution using an efficient algorithm based on a special matrix form under certain conditions The moments of any order may be obtained in closed form in this case, which is not true at least for the variance when other weights like Gaussian or double exponential density functions are used The mth cumulant of test statistic Δ satisfies κm = (n − 1)/(2n) m-1/(2m) D/rD + o(rD) (m ≥ 2)

Conclusion and Future Directions

The exact relationship between the choice of the weight function and the power of the CF tests for different alternative hypotheses is still unclear The empirical CF has a unified formula even for non-rectangular study regions, such as disks, polygons, or disconnected shapes The test can also be applied to other types of null hypotheses for spatial point processes The three-dimensional CF statistic may be used to test for space-time CSR, but in practice testing for space-time interaction is usually of more interest than testing for CSR

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The paper develops efficient algorithms for the asymptotic and small-sample null distributions of the Cauchy-weighted statistic in Section 4 The Cauchy-weighted CF test is shown to have a much lower computational cost than the isotropicedge-corrected L-test, so it is chosen as the focus of investigation

Simulations assess the accuracy of both proposed null distributions for the test statistic, showing that the two-tailed test using either method has a type I error rate very close to the theoretical value 0.05 The power of the CF test under different alternatives is compared against tests like Zimmerman’s w2-test and Clark-Evans test In application, the CF test outperforms other tests in most simulation settings of heterogeneity, aggregation and regularity, especially when the sample size is large

The asymptotic distribution of Δ converges in distribution to Í∞ j=1 λj Z2 j, where (Z j) is a sequence of independent standard normal variables and (λ j) are the eigenvalues of the linear operator T The moments are given in Section 4.

1. Improved spatial pattern analysis for complete spatial randomness by implementing a new test based on weighted L2-distance between empirical and uniform characteristic functions. This allows AI systems to determine whether a mapped point pattern follows complete spatial randomness, which is crucial for validating models of spatial uniformity, heterogeneity, and regularity in various dimensions.

2. Enhanced detection of spatial interactions at different scales by utilizing the scale parameter r in the Cauchy weighted characteristic function test statistic. AI systems can now precisely identify whether aggregation or regularity occurs at specific distance scales by analyzing the behavior of 1 - Delta r against r, as shown in Figure 4 of the paper.

3. Development of efficient algorithms for finding the asymptotic null distribution of the Cauchy-weighted CF test statistic under a Cauchy weight function. This enables AI systems to perform fast and accurate hypothesis testing without relying on computationally expensive Monte Carlo simulations for large sample sizes, reducing computational overhead significantly compared to older tests like the isotropicedge-corrected L-test.

4. Creation of an omnibus Bonferroni-corrected CF test that combines multiple parameter values of r. This allows AI systems to create a more powerful test than the L-test or Clark-Evans test in most simulation settings, offering better performance against heterogeneity, aggregation, and regularity when sample sizes are large.

5. Ability to extend spatial tests to non-rectangular or sparsely sampled point patterns using the empirical characteristic function. This means AI can analyze complex spatial data structures beyond simple rectangles, as the empirical CF has a simple unified formula regardless of what the sampling region looks like—a disk, a polygon, or even a disconnected (or sparsely sampled) region.