Spatio-temporal variograms and covariance models

Variograms and covariance functions are the fundamental tools for modeling dependent data observed over time, space, or space-time. This paper aims at constructing nonseparable spatio-temporal variograms and covariance models. Special attention is paid to an intrinsically stationary spatio-temporal random field whose covariance function is of Schoenberg-Lévy type. The correlation structure is studied for its increment process and for its partial derivative with respect to the time lag, as well as for the superposition over time of a stationary spatio-temporal random field. As another approach, we investigate the permissibility of the linear combination of certain separable spatio-temporal covariance functions to be a valid covariance, and obtain a subclass of stationary spatio-temporal models isotropic in space.

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Globally covering a-priori regional gravity covariance models

Advances in Geosciences ◽

10.5194/adgeo-1-143-2003 ◽

2003 ◽

Vol 1 ◽

pp. 143-147 ◽

Cited By ~ 8

Author(s):

D. Arabelos ◽

C. C. Tscherning

Keyword(s):

Gravity Anomaly ◽

Covariance Function ◽

Signal To Noise Ratio ◽

A Priori ◽

Scale Factor ◽

Covariance Functions ◽

Covariance Models ◽

Free Air Gravity ◽

Regional Gravity ◽

Degree Variances

Abstract. Gravity anomaly data generated using Wenzel’s GPM98A model complete to degree 1800, from which OSU91A has been subtracted, have been used to estimate covariance functions for a set of globally covering equal-area blocks of size 22.5° × 22.5° at Equator, having a 2.5° overlap. For each block an analytic covariance function model was determined. The models are based on 4 parameters: the depth to the Bjerhammar sphere (determines correlation), the free-air gravity anomaly variance, a scale factor of the OSU91A error degree-variances and a maximal summation index, N, of the error degree-variances. The depth of Bjerhammar-sphere varies from -134km to nearly zero, N varies from 360 to 40, the scale factor from 0.03 to 38.0 and the gravity variance from 1081 to 24(10µms-2)2. The parameters are interpreted in terms of the quality of the data used to construct OSU91A and GPM98A and general conditions such as the occurrence of mountain chains. The variation of the parameters show that it is necessary to use regional covariance models in order to obtain a realistic signal to noise ratio in global applications.Key words. GOCE mission, Covariance function, Spacewise approach`

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COVARIANCE FUNCTION MODELLING IN LOCAL GEODETIC APPLICATIONS USING THE SIMPLEX METHOD

Boletim de Ciências Geodésicas ◽

10.1590/s1982-21702016000200019 ◽

2016 ◽

Vol 22 (2) ◽

pp. 342-357

Author(s):

Carlo Iapige De Gaetani ◽

Noemi Emanuela Cazzaniga ◽

Riccardo Barzaghi ◽

Mirko Reguzzoni ◽

Barbara Betti

Keyword(s):

Simplex Method ◽

Covariance Function ◽

Real Data ◽

Data Sets ◽

Covariance Functions ◽

The Earth ◽

Function Modelling ◽

Covariance Models ◽

Definition Of ◽

Function Models

Collocation has been widely applied in geodesy for estimating the gravity field of the Earth both locally and globally. Particularly, this is the standard geodetic method used to combine all the available data to get an integrated estimate of any functional of the anomalous potential T. The key point of the method is the definition of proper covariance functions of the data. Covariance function models have been proposed by many authors together with the related software. In this paper a new method for finding suitable covariance models has been devised. The covariance fitting problem is reduced to an optimization problem in Linear Programming and solved by using the Simplex Method. The procedure has been implemented in a FORTRAN95 software and has been tested on simulated and real data sets. These first tests proved that the proposed method is a reliable tool for estimating proper covariance function models to be used in the collocation procedure

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Estimating and modeling spatio-temporal complex-valued covariance functions

10.5194/egusphere-egu21-16382 ◽

2021 ◽

Author(s):

Sabrina Maggio ◽

Donato Posa ◽

Sandra De Iaco ◽

Claudia Cappello

Keyword(s):

Complex Data ◽

Temporal Dependence ◽

Covariance Functions ◽

Covariance Models ◽

Unit Of Measurement ◽

Oceanographic Data ◽

Spatio Temporal ◽

Complex Valued ◽

Vectorial Data

Oceanographic data belong to the wide class of vectorial data, for which the decomposition in modulus and direction is meaningful, and the vectorial components are characterized by homogeneous quantities, with the same unit of measurement. Another feature of oceanographic data is that they exhibit spatio-temporal dependence. In Geostatistics, such data can be properly modelled by recalling the theory of complex-valued random fields. However, in the literature, only techniques for modeling and predicting the spatial evolution of these phenomena were proposed; while the temporal dependence was analyzed separately from the spatial one, or just time-varying complex covariance models were used. Thus, the novelty of this paper regards some advances of the complex formalism for analyzing complex data in space-time and new classes of spatio-temporal complex covariance models. A case study on spatio-temporal complex estimating and modeling with oceanographic data is provided and a comparison between two classes of complex covariance models is also proposed.

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Bayesian spatio-temporal models for mapping TB mortality risk and its relationship with social inequities in a region from Brazilian Legal Amazon

Transactions of the Royal Society of Tropical Medicine and Hygiene ◽

10.1093/trstmh/traa008 ◽

2020 ◽

Vol 114 (5) ◽

pp. 323-331 ◽

Cited By ~ 1

Author(s):

Josilene D Alves ◽

Francisco Chiaravalloti-Neto ◽

Luiz H Arroyo ◽

Marcos A M Arcoverde ◽

Danielle T Santos ◽

...

Keyword(s):

Mortality Risk ◽

Ecological Study ◽

Cultural Variations ◽

Mortality Risks ◽

Political Crises ◽

Social Inequities ◽

Temporal Models ◽

Risk Areas ◽

Spatio Temporal ◽

Over Time

Abstract Background Reducing TB mortality is a great challenge in Brazil due to its territorial extension, cultural variations and economic and political crises, which impact the health system. This study aimed to estimate in space and time the risk of TB mortality and test its relationship with social inequities. Methods This was an ecological study that included deaths from TB between 2006 and 2016 in Cuiabá, Brazilian Legal Amazon. Bayesian models based on the integrated nested Laplace approximation approach were used to estimate spatio-temporal RRs. RRs for TB mortality were obtained according to the covariables representative of social inequities. Results The risk of TB mortality was stable between 2006 and 2016 and high-risk areas were identified throughout the municipality studied. Regarding social inequities, income was an important factor associated with TB mortality risk, as an increase of 1 SD in income resulted in a 35.4% (RR 0.646; CI 95% 0.476 to 0.837) decrease in risk. Conclusions The results provided evidence of areas with higher TB mortality risks that have persisted over time and are related to social inequities. Advancing social policies and protections in these areas will contribute to achieving the WHO's End TB strategy.

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On the Family of Covariance Functions Based on ARMA Models

Engineering Proceedings ◽

10.3390/engproc2021005037 ◽

2021 ◽

Vol 5 (1) ◽

pp. 37

Author(s):

Till Schubert ◽

Jan Martin Brockmann ◽

Johannes Korte ◽

Wolf-Dieter Schuh

Keyword(s):

Spatial Data ◽

Covariance Function ◽

Moving Average ◽

Stochastic Methods ◽

Autoregressive Moving Average ◽

Shape Parameters ◽

Covariance Functions ◽

Covariance Modeling ◽

Model Complex ◽

Covariance Models

In time series analyses, covariance modeling is an essential part of stochastic methods such as prediction or filtering. For practical use, general families of covariance functions with large flexibilities are necessary to model complex correlations structures such as negative correlations. Thus, families of covariance functions should be as versatile as possible by including a high variety of basis functions. Another drawback of some common covariance models is that they can be parameterized in a way such that they do not allow all parameters to vary. In this work, we elaborate on the affiliation of several established covariance functions such as exponential, Matérn-type, and damped oscillating functions to the general class of covariance functions defined by autoregressive moving average (ARMA) processes. Furthermore, we present advanced limit cases that also belong to this class and enable a higher variability of the shape parameters and, consequently, the representable covariance functions. For prediction tasks in applications with spatial data, the covariance function must be positive semi-definite in the respective domain. We provide conditions for the shape parameters that need to be fulfilled for positive semi-definiteness of the covariance function in higher input dimensions.

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Updatable Spatio-Temporal Probabilistic Corrosion Modeling for Offshore Structures

Volume 11B: Honoring Symposium for Professor Carlos Guedes Soares on Marine Technology and Ocean Engineering ◽

10.1115/omae2018-78684 ◽

2018 ◽

Author(s):

Karoline M. Neumann ◽

Ole Tom Vårdal ◽

Sören Ehlers

Keyword(s):

Probabilistic Models ◽

Offshore Structures ◽

Current State ◽

Time Space ◽

Temporal Models ◽

Corrosion Model ◽

Practical Engineering ◽

Industry Practice ◽

Spatio Temporal ◽

Corrosive Environments

Corrosion models are important to assess how the corrosion influences current and future structural strength. For this purpose it is desirable to describe the uneven corrosion diminution of the irregular surface (i.e. space) and progression (i.e. time) in various corrosive environments. Thickness measurements give an indication of the current state, and should be considered in the corrosion model. The inherit uncertainty in corrosion argues for a probabilistic type of corrosion model. Probabilistic models to describe corrosion in time and space, and that can be updated with observations exist, but are typically too complicated for practical engineering use for in-service corrosion assessment. Simpler models exist, that do not describe all of the mentioned aspects (probabilistic, updatable, describe corrosion in time, space and various environments). Here, a simple model covering these aspects is described in two parts. First bayes updating is used to estimate the parameters of the corrosion distribution for each unique environment. The second part uses this resulting distribution and describes how this distribution develops with time. The model is demonstrated with an example and compared to similar spatio-temporal models. The model is promising for improvement from simplistic uniform description of surface and linear progression used in current industry practice.

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UNIVERSAL KRIGING FOR SPATIO‐TEMPORAL DATA

Mathematical Modelling and Analysis ◽

10.3846/13926292.2003.9637230 ◽

2003 ◽

Vol 8 (4) ◽

pp. 283-290 ◽

Cited By ~ 1

Author(s):

E. Lesauskiene ◽

K. Dučinskas

Keyword(s):

Linear Prediction ◽

Real Data ◽

Universal Kriging ◽

Covariance Functions ◽

Data Prediction ◽

Optimal Linear ◽

Squared Prediction Error ◽

Covariance Models ◽

Spatio Temporal ◽

Mean Function

In this article we have used wide applicable classes of spatio‐temporal nonseparable and separable covariance models. One of the objectives of this paper is to furnish a possibility how to avoid the usage of complicated covariance functions. Assuming regression model for mean function the analytical expressions for the optimal linear prediction (universal kriging) and mean squared prediction error (MSPE) was obtained. Parameterized spatio‐temporal covariance functions were fitted for the real data. Prediction values and MSPE were presented. For visualization of results on graphics are used free available software Gstat.

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Pyramidal Framework: Guidance for the Next Generation of GIS Spatial-Temporal Models

ISPRS International Journal of Geo-Information ◽

10.3390/ijgi10030188 ◽

2021 ◽

Vol 10 (3) ◽

pp. 188

Author(s):

Cyril Carré ◽

Younes Hamdani

Keyword(s):

Conceptual Models ◽

Future Generations ◽

Temporal Data ◽

Temporal Modeling ◽

Temporal Models ◽

Gis Data ◽

Spatio Temporal ◽

New Interpretation ◽

Conventional Literature

Over the last decade, innovative computer technologies and the multiplication of geospatial data acquisition solutions have transformed the geographic information systems (GIS) landscape and opened up new opportunities to close the gap between GIS and the dynamics of geographic phenomena. There is a demand to further develop spatio-temporal conceptual models to comprehensively represent the nature of the evolution of geographic objects. The latter involves a set of considerations like those related to managing changes and object identities, modeling possible causal relations, and integrating multiple interpretations. While conventional literature generally presents these concepts separately and rarely approaches them from a holistic perspective, they are in fact interrelated. Therefore, we believe that the semantics of modeling would be improved by considering these concepts jointly. In this work, we propose to represent these interrelationships in the form of a hierarchical pyramidal framework and to further explore this set of concepts. The objective of this framework is to provide a guideline to orient the design of future generations of GIS data models, enabling them to achieve a better representation of available spatio-temporal data. In addition, this framework aims at providing keys for a new interpretation and classification of spatio-temporal conceptual models. This work can be beneficial for researchers, students, and developers interested in advanced spatio-temporal modeling.

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Basis Expansions for Functional Snippets

Biometrika ◽

10.1093/biomet/asaa088 ◽

2020 ◽

Author(s):

Zhenhua Lin ◽

Jane-Ling Wang ◽

Qixian Zhong

Keyword(s):

Data Analysis ◽

Convergence Rate ◽

Functional Data Analysis ◽

Functional Data ◽

Covariance Function ◽

Covariance Estimation ◽

Simulation Studies ◽

Finite Sample ◽

Covariance Functions ◽

The Mean

Summary Estimation of mean and covariance functions is fundamental for functional data analysis. While this topic has been studied extensively in the literature, a key assumption is that there are enough data in the domain of interest to estimate both the mean and covariance functions. In this paper, we investigate mean and covariance estimation for functional snippets in which observations from a subject are available only in an interval of length strictly (and often much) shorter than the length of the whole interval of interest. For such a sampling plan, no data is available for direct estimation of the off-diagonal region of the covariance function. We tackle this challenge via a basis representation of the covariance function. The proposed estimator enjoys a convergence rate that is adaptive to the smoothness of the underlying covariance function, and has superior finite-sample performance in simulation studies.

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