On the Power of Some Nonparametric Isotropy Tests

In this paper, properties of nonparametric significance tests verifying the random field isot‐ ropy hypothesis are discussed. In particular, the subject of the conducted analysis is the probability of rejecting the null hypothesis when it is true. A potential significant difference of empirical rejection probability from the assumed significance level could distort the results of statistical inference. The tests proposed by Guan, Sherman, Calvin (2004) and Lu, Zimmerman (2005) are considered. A simu‐ lation study has been carried out through generating samples from a given theoretical distribution and repeatedly testing the null hypothesis. Isotropic distributions are considered, among others, those based on a multidimensional normal distribution. The main aim of the paper is to compare both con‐ sidered nonparametric significance tests verifying the random field isotropy hypothesis. For this pur‐ pose, the empirical rejection probabilities for both tests have been calculated and compared with the assumed significance level.


Introduction
In spatial statistics, observations of the study variables are treated as realisations of the spatial stochastic process, understood as a collection of random variables indexed by a coordinate vector n T ⊂  . In this paper, weakly stationary spatial processes are considered. These are processes for which the expected value is constant and the covariance function depends only on the shift vector, i.e.
Cov X X C s t E X E X X E X = = . One of the basic tools used to study the variability structure of studied phenomena is the variogram. It is a measure defined in locations shifted by the vector h as An important assumption used in the estimation of variograms is random field isotropy (Sherman, 2010).
The aim of the article is to investigate the properties of nonparametric significance tests verifying the random field isotropy hypothesis. The subject of the analysis will be the empirical probability of rejecting the null hypothesis estimated by percentage of null hypothesis rejections recorded in a simulation. A potential significant difference of empirical rejection probability from the assumed significance level could distort the results of statistical inference.

Isotropy
Isotropy is a property of the stochastic process that occurs when covariance depends only on the distance between locations. In other words, the covariance between realisations of the spatial stochastic process in locations shifted by the vector ( ) , where * C is the covariance function of a one-dimensional random process. However, in reality, anisotropy is common. It is a situation in which the covariance between realisations of the stochastic process in any fixed location and realisations in at least two other locations away by the same distance is different. Anisotropy is a subject of research in many areas, such as computer graphics, chemistry, geology, or physics. In physics, anisotropy in cosmic blackbody radiation is a good example (Smoot, Gorenstein, Muller, 1977).

Simulation
The purpose of the simulation was to estimate the empirical rejection probability.
The simulation was carried out in the following steps: -generation of the realisation of a random field using a given isotropic theoretical distribution on a 16 × 16 square grid; -verification of the null hypothesis about random field isotropy; -repetition of the first two steps 10,000 times; -calculation of the empirical rejection probability.
Three isotropic theoretical distributions were used to generate the realisations of the random field: -256 -dimensional normal distribution with mean 0 and covariance between locations shifted by the vector (x, y) given by the formula -256 -dimensional normal distribution with mean 0 and a unit variance-covariance matrix; for each location, a realisation was independently generated from the uniform distribution over the range (-1, 1).
The spTest package (Weller, 2015) in the R environment was used to verify the hypothesis The two tests used were proposed by: Guan, Sherman and Calvin (2004) and Lu and Zimmerman (2005 H is true, then there exists a full rank matrix A such that 0 AG = (Lu, Zimmerman, 2001), where 0 is the zero matrix. Then the test statistic is given by the formula: where ˆ is an estimator of the variance-covariance matrix and n T is the cardinality of the index set n T .
Σ n According to multivariate Slutsky's theorem (Ferguson, 1996), the test statistic has an asymptotic chi-square distribution with d degrees of freedom, where d is the row rank of A. In the Lu and Zimmerman test, symmetry tests were used to study isotropy. The p-values were calculated by verifying the hypothesis about reflective symmetry and complete symmetry (Hoeting, Weller, 2016). They are defined as follows: Definition 1 A weakly stationary spatial process on a grid is reflection symmetric if Definition 2 A weakly stationary spatial process on a grid is completely symmetric if Both symmetries are field properties weaker than isotropy. Therefore, by rejecting the null hypothesis of symmetry, we have reasons to reject the hypothesis of isotropy. Lu and Zimmerman (2005) used the periodogram as an estimator of the spectral density. They took advantage of the fact that under certain conditions and at certain frequencies and when the null hypothesis of reflection or complete symmetry is true, ratios of periodogram values at different frequencies follow an means Snedecor's F-distribution with parameters 2 and 2. To calculate the p-value, it is preferable to use a Cramér-von Mises goodness-of-fit test (Csörgo, Faraway, 1996) using the appropriate set of periodogram ratios.

Results of experiments
The first of the distributions used is a multidimensional normal distribution with exponential covariance.    Empirical rejection probabilities significantly differ from the assumed significance level. Empirical rejection probability values greater than the significance level mean that the test rejects the null hypothesis more often than the user is willing to accept. Figure 4 shows an example of a multidimensional normal distribution with mean 0 and a unit variance-covariance matrix.   It is worth noting that in the case of the Guan, Sherman and Calvin test, the empirical rejection probabilities are lower than the significance level in each simulated case. This may suggest that it is possible to shift the critical value increasing the test power. Figure 5 shows an example of a realisation created by independently generating (for each location from a 16 × 16 grid) realisations from a uniform distribution over the interval (-1.1).  Tables 5 and 6 respectively.  The difference between empirical rejection probabilities and significance levels is smaller than in the normal distributions case.

Conclusions
In the case of the Guan, Sherman and Calvin test, including correction for the finite sample and the Lu and Zimmerman test (complete symmetry), relative percentage errors were calculated as empirical rejection probability significance level significance level − 100%.
The results are shown in Tables 7 and 8 respectively.  Empirical rejection probabilities significantly differ from the assumed significance level. For the Lu and Zimmerman test and the multivariate normal distribution with exponential covariance, the percentage relative error is up to 164%. A downward trend is visible -the higher the level of significance, the smaller the percentage relative error is. It is advisable to be extremely cautious when using the tests presented in this work.