Estimation and prediction using generalized Wendland covariance functions under fixed domain asymptotics

Bevilacqua, Moreno; Faouzi, Tarik; Furrer, Reinhard; Porcu, Emilio

doi:10.1214/17-aos1652

Cited by 81 publications

(104 citation statements)

References 40 publications

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“…In each case, we estimated the parameters of the covariance function using ML and used them to compute the SCCC given in equation (6). Three set of parameters were considered, one set each for σ 1 = σ 2 = 1, ν = 4 and k = 1: The ML estimates of the parameters of the Wendland-Gneiting covariance function had low bias and standard errors, and agreed with previously published results (e.g., Bevilaqua et al, 2019). Using these estimates, we computed the SCCC in each case for 0 < ||h|| < 2.…”

Section: Monte Carlo Simulationssupporting

confidence: 70%

A spatial concordance correlation coefficient with an application to image analysis

et al. 2020

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In this work we define a spatial concordance coefficient for second-order stationary processes. This problem has been widely addressed in a non-spatial context, but here we consider a coefficient that for a fixed spatial lag allows one to compare two spatial sequences along a 45 • line. The proposed coefficient was explored for the bivariate Matérn and Wendland covariance functions. The asymptotic normality of a sample version of the spatial concordance coefficient for an increasing domain sampling framework was established for the Wendland covariance function. To work with large digital images, we developed a local approach for estimating the concordance that uses local spatial models on non-overlapping windows. Monte Carlo simulations were used to gain additional insights into the asymptotic properties for finite sample sizes. As an illustrative example, we applied this methodology to two similar images of a deciduous forest canopy. The images were recorded with different cameras but similar fields-of-view and within minutes of each other. Our analysis showed that the local approach helped to explain a percentage of the non-spatial concordance and to provided additional information about its decay as a function of the spatial lag.As a consequence of the asymptotic normality of the sample CCC, an approximate hypothesis testing problem of the form H 0 : ρ c = ρ 0 versus H 1 : ρ c = ρ 0 for a fixed ρ 0 can be constructed. Alternatively, an approximate confidence interval of the form ρ c ± z α/2 σ 2 Z can be used, where z α/2 is the upper quantile of order α/2 of the standard normal distribution. Applications and extensions of Lin's coefficient can be found in Lin et al. (2012), among others.

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Section: Monte Carlo Simulationssupporting

confidence: 70%

A spatial concordance correlation coefficient with an application to image analysis

et al. 2020

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“…In this section, we let d = 1, 2 or 3 and extend the results for the Matérn covariance functions of Section 4 to the isotropic Wendland family of covariance functions on [0, 1] d (Gneiting, 2002;Bevilacqua et al, 2019). Here k θ = k θ,s,µ , with θ = (σ 2 , ρ), is given by…”

Section: Microergodic Parameter Estimation For the Isotropic Wendlandmentioning

confidence: 97%

“…The Wendland covariance function is given by k θ,s,µ (u, v) = k θ,s,µ (u − v). The parameter s drives the smoothness of the Wendland covariance function, similarly as for the Matérn covariance function (Bevilacqua et al, 2019). The parameters σ 2 > 0 and ρ > 0 are interpreted similarly as for the Matérn covariance functions and are to be estimated.…”

Section: Microergodic Parameter Estimation For the Isotropic Wendlandmentioning

confidence: 99%

Maximum likelihood estimation for Gaussian processes under inequality constraints

Bachoc¹,

Lagnoux²,

López-Lopera³

2019

Electron. J. Statist.

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We consider covariance parameter estimation for a Gaussian process under inequality constraints (boundedness, monotonicity or convexity) in fixed-domain asymptotics. We address the estimation of the variance parameter and the estimation of the microergodic parameter of the Matérn and Wendland covariance functions. First, we show that the (unconstrained) maximum likelihood estimator has the same asymptotic distribution, unconditionally and conditionally to the fact that the Gaussian process satisfies the inequality constraints. Then, we study the recently suggested constrained maximum likelihood estimator. We show that it has the same asymptotic distribution as the (unconstrained) maximum likelihood estimator. In addition, we show in simulations that the constrained maximum likelihood estimator is generally more accurate on finite samples. Finally, we provide extensions to prediction and to noisy observations. It is still challenging to obtain results on maximum likelihood estimation of microergodic parameters that would hold for very general classes of covariance functions. Nevertheless, significant contributions have been made for specific types of covariance functions. In particular, when considering the isotropic Matérn family of covariance functions, for input space dimension d = 1, 2, 3, a reparameterized quantity obtained from the variance and correlation length parameters is microergodic (Zhang, 2004). It has been shown in (Kaufman and Shaby, 2013), from previous results in (Du et al., 2009) and(Wang andLoh, 2011), that the maximum likelihood estimator of this microergodic parameter is consistent and asymptotically Gaussian distributed. Anterior results on the exponential covariance function have been also obtained in (Ying, 1991(Ying, , 1993.In this paper, we shall consider the situation where the trajectories of the Gaussian process are known to satisfy either boundedness, monotonicity or convexity constraints. Indeed, Gaussian processes with inequality constraints provide suitable regression models in application fields such as computer networking (monotonicity) (Golchi et al., 2015), social system analysis (monotonicity) (Riihimäki and Vehtari, 2010) and econometrics (monotonicity or positivity) (Cousin et al., 2016). Furthermore, it has been shown that taking the constraints into account may considerably improve the predictions and the predictive intervals for the Gaussian process (Da Veiga and Marrel, 2012;Golchi et al., 2015;Riihimäki and Vehtari, 2010).Recently, a constrained maximum likelihood estimator (cMLE) for the covariance parameters has been suggested in . Contrary, to the (unconstrained) maximum likelihood estimator (MLE) discussed above, the cMLE explicitly takes into account the additional information brought by the inequality constraints. In , it is shown, essentially, that the consistency of the MLE implies the consistency of the cMLE under boundedness, monotonicity or convexity constraints.The aim of this paper is to study the asymptotic conditional distributions of the MLE and the cM...

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“…Low-rank methods project the original process onto representative points or knots (see, e.g., Higdon, 2002;Banerjee, Gelfand, Finley, & Sang, 2008;Cressie & Johannesson, 2008;Stein, 2008); however, these approaches often perform poorly for prediction as they often over-smooth (see Stein, 2014). Alternatively, sparse methods induce zeros either in the covariance matrix using compactly supported covariance functions (see, e.g., Furrer, Genton, & Nychka, 2006;Kaufman, Schervish, & Nychka, 2008;Bevilacqua, Faouzi, Furrer, & Porcu, 2019) or in the precision matrix by assuming conditional independence (Vecchia, 1988;Stein et al, 2004). We ultimately favor approaches that assume conditional independence because predictive performance is generally better (Heaton et al, 2018) and the class of valid covariance models is more expansive.…”

Section: Nngp Modelmentioning

confidence: 99%

Nonseparable covariance models on circles cross time: A study of Mexico City ozone

White

Porcu

2019

Environmetrics

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We propose a continuous spatiotemporal model for Mexico City ozone levels that account for distinct daily seasonality, as well as variation across the city and over the peak ozone season (April and May) of 2017. To account for these patterns, we use covariance models over space, circles, and time. We review relevant existing covariance models and develop new classes of nonseparable covariance models appropriate for seasonal data collected at many locations. We compare the predictive performance of a variety of models that utilize various nonseparable covariance functions. We use the best model to predict hourly ozone levels at unmonitored locations in April and May to infer compliance with Mexican air quality standards and to estimate the respiratory health risk associated with ozone exposure. We find that predicted compliance with air quality standards and estimated respiratory health risk vary greatly over space and time. In some regions, we predict exceedance of national standards for more than a third of the hours in April and May, and on many days, we predict that nearly all of Mexico City exceeds nationally legislated ozone thresholds at least once. In southern Mexico City, we estimate the respiratory risk for ozone to be 55% higher, on average, than the annual average risk and as much at 170% higher on some days.

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Estimation and prediction using generalized Wendland covariance functions under fixed domain asymptotics

Cited by 81 publications

References 40 publications

A spatial concordance correlation coefficient with an application to image analysis

A spatial concordance correlation coefficient with an application to image analysis

Maximum likelihood estimation for Gaussian processes under inequality constraints

Nonseparable covariance models on circles cross time: A study of Mexico City ozone

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