On the Use of the Variogram in Checking for Independence in Spatial Data

Diblasi, Ángela Magdalena; Bowman, Adrian

doi:10.1111/j.0006-341x.2001.00211.x

Cited by 42 publications

(33 citation statements)

References 12 publications

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“…Once the tendency is adjusted, a first analysis of the spatial structure of the residuals has been conducted. The residuals have been represented graphically in the geographical coordinates ( Figure 6) and a contrast test has been performed to check the spatial independence of the observations [30]. The nugget effect (see variogram part of Table 4) can be attributed to measurement errors or spatial sources of variation at smaller distances than the sampling interval (or both).…”

Section: Resultsmentioning

confidence: 99%

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Spatial Statistical Analysis of Urban Noise Data from a WASN Gathered by an IoT System: Application to a Small City

et al. 2016

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EU Directive 49/2002 and Spanish law 37/2006 urge cities to develop strategic noise maps and action plans to evaluate noise exposure and to establish noise abatement procedures in critical areas. However, noise mapping involves costly and cumbersome measurement procedures that can become a real issue in practice. This paper describes a distributed noise monitoring system based on WASN (Wireless Acoustic Sensor Network) and the application of a geo-statistical methodology for statistical spatial-temporal prediction of noise levels in semi-open areas, such as a typical, small Mediterranean city (Algemesí, València, Spain). This methodology is applied to the study of the spatial evolution in time of the noise pollution. To this end, a spatial statistical model is developed by using the noise pollution measurements obtained over a set of points located at some strategic locations. The geo-statistical time model allows for estimating specific noise levels and characterizing the spatial-temporal variation of the noise pollution. The results show that the developed model provides a good approximation of the measurements obtained experimentally.

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Section: Resultsmentioning

confidence: 99%

“…where µ(x) = E[Z(x)] and δ(x) is a Gaussian process intrinsically stationary with zero mean, whose spatial dependence characterization is given by the variogram γ [30]:…”

Section: Spatial Analysismentioning

confidence: 99%

Spatial Statistical Analysis of Urban Noise Data from a WASN Gathered by an IoT System: Application to a Small City

et al. 2016

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“…The corresponding properties of d ij = |Z i − Z j | can then be derived from mean and variance results in Diblasi and Bowman (2001) and a correlation result in Cressie (1985):…”

Section: Standard Errors and Confidence Intervalsmentioning

confidence: 99%

“…In the case of independent spatial data, the distributional properties of the sample variogram can be evaluated and exploited to form the basis of a test for the presence of spatial correlation, as described by Diblasi and Bowman (2001). Similarly, a test of goodness-of-fit assuming known parameters can be constructed, as discussed in Maglione and Diblasi (2004).…”

Section: Introductionmentioning

confidence: 99%

Inference for variograms

Bowman

Crujeiras

2013

Computational Statistics & Data Analysis

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The empirical variogram is a standard tool in the investigation and modelling of spatial covariance. However, its properties can be difficult to identify and exploit in the context of exploring the characteristics of individual datasets. This is particularly true when seeking to move beyond description towards inferential statements about the structure of the spatial covariance which may be present. A robust form of empirical variogram based on a fourth-root transformation is used. This takes advantage of the normal approximation which gives an excellent description of the variation exhibited on this scale. Calculations of mean, variance and covariance of the binned empirical variogram then allow useful computations such as confidence intervals to be added to the underlying estimator. The comparison of variograms for different datasets provides an illustration of this. The suitability of simplifying assumptions such as isotropy and stationarity can then also be investigated through the construction of appropriate test statistics and the distributional calculations required in the associated p-values can be performed through quadratic form methods. Examples of the use of these methods in assessing the form of spatial covariance present in datasets are shown, both through hypothesis tests and in graphical form. A simulation study explores the properties of the tests while pollution data on mosses in Galicia (North-West Spain) are used to provide a real data illustration.

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“…The use of these tests is restricted to regularly sampled data, which is not always the case in spatial analysis. Following the idea of Diblasi and Bowman (2001), which adapted the regression methods to construct a test for independence in spatial data based on a smoothed variogram, Bowman and Crujeiras (2013) proposed some tests for assessing isotropy and stationarity. The hypotheses to test are nonparametric, but they can be rewritten in terms of the variogram function.…”

Section: (D) Non-and Semiparametric Hypothesesmentioning

confidence: 99%

Rejoinder on: An updated review of Goodness-of-Fit tests for regression models

González–Manteiga

Crujeiras

2013

TEST

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First for all, we would like to thank the discussants for reading our paper and for taking time to prepare such interesting and valuable contributions. The feeling, after revising the discussions and going back to the original version of the paper, is that there is still too much to say about Goodness-of-Fit (GoF) tests for regression models and the discussants have given a good proof of this. Although they qualify the review as thorough and nearly exhaustive, it is clear after their comments there are quite a few issues and open problems that were left untreated.We have tried to organize this rejoinder according to the different topics that have been brought up in the discussions, instead of answering each discussion separately. Bearing this in mind, some brief comments will be made on (a) tests based on the error distribution; (b) tests with dependence between error and covariate; (c) bandwidth selection and calibration; (d) non-and semiparametric hypotheses; (e) tests in survival analysis and (f) some further topics.(a) Tests based on the error distribution In Sect. 2.4, concerned with this topic, two main methodological approaches are presented. The first one, collected in the paper by Van Keilegom et al. (2008), is based on the empirical distribution of the residuals, whereas Huskova and Meintanis (2009) considered an alternative route This rejoinder refers to the comments available at

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On the Use of the Variogram in Checking for Independence in Spatial Data

Cited by 42 publications

References 12 publications

Spatial Statistical Analysis of Urban Noise Data from a WASN Gathered by an IoT System: Application to a Small City

Spatial Statistical Analysis of Urban Noise Data from a WASN Gathered by an IoT System: Application to a Small City

Inference for variograms

Rejoinder on: An updated review of Goodness-of-Fit tests for regression models

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