scholarly journals Spatial Warped Gaussian Processes: Estimation and Efficient Field Reconstruction

Entropy ◽  
2021 ◽  
Vol 23 (10) ◽  
pp. 1323
Author(s):  
Gareth W. Peters ◽  
Ido Nevat ◽  
Sai Ganesh Nagarajan ◽  
Tomoko Matsui
Keyword(s):  
Gaussian Processes ◽  
Random Fields ◽  
Mean Squared Error ◽  
Real Data ◽  
Statistical Properties ◽  
Process Models ◽  
Field Reconstruction ◽  
Minimum Mean Squared Error ◽  
Heavy Tailed ◽  
Non Gaussian

A class of models for non-Gaussian spatial random fields is explored for spatial field reconstruction in environmental and sensor network monitoring. The family of models explored utilises a class of transformation functions known as Tukey g-and-h transformations to create a family of warped spatial Gaussian process models which can support various desirable features such as flexible marginal distributions, which can be skewed, leptokurtic and/or heavy-tailed. The resulting model is widely applicable in a range of spatial field reconstruction applications. To utilise the model in applications in practice, it is important to carefully characterise the statistical properties of the Tukey g-and-h random fields. In this work, we study both the properties of the resulting warped Gaussian processes as well as using the characterising statistical properties of the warped processes to obtain flexible spatial field reconstructions. In this regard we derive five different estimators for various important quantities often considered in spatial field reconstruction problems. These include the multi-point Minimum Mean Squared Error (MMSE) estimators, the multi-point Maximum A-Posteriori (MAP) estimators, an efficient class of multi-point linear estimators based on the Spatial-Best Linear Unbiased (S-BLUE) estimators, and two multi-point threshold exceedance based estimators, namely the Spatial Regional and Level Exceedance estimators. Simulation results and real data examples show the benefits of using the Tukey g-and-h transformation as opposed to standard Gaussian spatial random fields in a real data application for environmental monitoring.

scholarly journals Spatial Warped Gaussian Processes: Estimation and Efficient Field Reconstruction

2021 ◽  
Author(s):  
Gareth William Peters ◽  
Ido Nevat ◽  
Sai Ganesh Nagarajan ◽  
Tomoko Matsui
Keyword(s):  
Gaussian Processes ◽  
Random Fields ◽  
Mean Squared Error ◽  
Real Data ◽  
Statistical Properties ◽  
Process Models ◽  
Multiple Point ◽  
Field Reconstruction ◽  
Minimum Mean Squared Error ◽  
Non Gaussian

A class of models for non-Gaussian spatial random fields is explored for spatial field reconstruction in environmental and sensor network monitoring. The family of models explored utilises a class of transformation functions known as the Tukey g-and-h transformations to create a family of warped spatial Gaussian process models which can support various desirable features such as flexible marginal distributions, which can be skewed, leptokurtic and/or heavy-tailed. The resulting model is widely applicable in a range of spatial field reconstruction applications. To utilise the model in applications in practice, it is important to carefully characterise the statistical properties of the Tukey g-and-h random fields. In this work, we both study the properties of the resulting warped Gaussian processes as well as using the characterising statistical properties of the warped processes to obtain flexible spatial field reconstructions. In this regard, we derive five different estimators for various important quantities often considered in spatial field reconstruction problems. These include the multi-point Minimum Mean Squared Error (MMSE) estimators; the multiple point Maximum A-Posteriori (MAP) estimators; an efficient class of multiple-point linear estimators based on the Spatial-Best Linear Unbiased (S-BLUE) estimators; and two multi-point threshold exceedance based estimators, namely the Spatial Regional and Level Exceedance estimators. Simulation results and real data examples show the benefits of using the Tukey g-and-h transformation as opposed to standard Gaussian spatial random fields in a real data application for environmental monitoring.

scholarly journals MIXED CAUSAL-NONCAUSAL AR PROCESSES AND THE MODELLING OF EXPLOSIVE BUBBLES

2019 ◽  
Vol 35 (6) ◽  
pp. 1234-1270 ◽  
Author(s):  
Sébastien Fries ◽  
Jean-Michel Zakoian
Keyword(s):  
Conditional Distribution ◽  
Stable Distribution ◽  
Financial Time Series ◽  
Domain Of Attraction ◽  
Real Data ◽  
Autoregressive Processes ◽  
Financial Time ◽  
Causal Representation ◽  
Heavy Tailed ◽  
Non Gaussian

Noncausal autoregressive models with heavy-tailed errors generate locally explosive processes and, therefore, provide a convenient framework for modelling bubbles in economic and financial time series. We investigate the probability properties of mixed causal-noncausal autoregressive processes, assuming the errors follow a stable non-Gaussian distribution. Extending the study of the noncausal AR(1) model by Gouriéroux and Zakoian (2017), we show that the conditional distribution in direct time is lighter-tailed than the errors distribution, and we emphasize the presence of ARCH effects in a causal representation of the process. Under the assumption that the errors belong to the domain of attraction of a stable distribution, we show that a causal AR representation with non-i.i.d. errors can be consistently estimated by classical least-squares. We derive a portmanteau test to check the validity of the estimated AR representation and propose a method based on extreme residuals clustering to determine whether the AR generating process is causal, noncausal, or mixed. An empirical study on simulated and real data illustrates the potential usefulness of the results.

scholarly journals A New Right-Skewed Upside Down Bathtub Shaped Heavy-tailed Distribution and its Applications

2021 ◽  
Vol 19 (1) ◽  
Author(s):  
Sandeep Kumar Maurya ◽  
Sanjay K Singh ◽  
Umesh Singh
Keyword(s):  
Maximum Likelihood ◽  
Real Data ◽  
Statistical Properties ◽  
New Right ◽  
Data Sets ◽  
Proposed Model ◽  
Heavy Tailed Distribution ◽  
Heavy Tailed

A one parameter right skewed, upside down bathtub type, heavy-tailed distribution is derived. Various statistical properties and maximum likelihood approaches for estimation purpose are studied. Five different real data sets with four different models are considered to illustrate the suitability of the proposed model.

Inference for stationary random fields given Poisson samples

1986 ◽  
Vol 18 (2) ◽  
pp. 406-422 ◽  
Author(s):  
Alan F. Karr
Keyword(s):  
Random Field ◽  
Poisson Process ◽  
Random Fields ◽  
Covariance Function ◽  
Mean Squared Error ◽  
Compact Sets ◽  
Mild Conditions ◽  
Minimum Mean Squared Error ◽  
Squared Error ◽  
The Mean

Given a d-dimensional random field and a Poisson process independent of it, suppose that it is possible to observe only the location of each point of the Poisson process and the value of the random field at that (randomly located) point. Non-parametric estimators of the mean and covariance function of the random field—based on observation over compact sets of single realizations of the Poisson samples—are constructed. Under fairly mild conditions these estimators are consistent (in various senses) as the set of observation becomes unbounded in a suitable manner. The state estimation problem of minimum mean-squared error reconstruction of unobserved values of the random field is also examined.

Simulated annealing wavelet estimation via fourth‐order cumulant matching

Geophysics ◽  
1996 ◽  
Vol 61 (6) ◽  
pp. 1939-1948 ◽  
Author(s):  
Danilo R. Velis ◽  
Tadeusz J. Ulrych
Keyword(s):  
Simulated Annealing ◽  
Fourth Order ◽  
Simulated Annealing Algorithm ◽  
Numerical Solutions ◽  
Mean Squared Error ◽  
Real Data ◽  
Hybrid Strategy ◽  
Minimum Mean Squared Error ◽  
Highly Nonlinear ◽  
Fourth Order Cumulant

The fourth‐order cumulant matching method has been developed recently for estimating a mixed‐phase wavelet from a convolutional process. Matching between the trace cumulant and the wavelet moment is done in a minimum mean‐squared error sense under the assumption of a non‐Gaussian, stationary, and statistically independent reflectivity series. This leads to a highly nonlinear optimization problem, usually solved by techniques that require a certain degree of linearization, and that invariably converge to the minimum closest to the initial model. Alternatively, we propose a hybrid strategy that makes use of a simulated annealing algorithm to provide reliability of the numerical solutions by reducing the risk of being trapped in local minima. Beyond the numerical aspect, the reliability of the derived wavelets depends strongly on the amount of data available. However, by using a multidimensional taper to smooth the trace cumulant, we show that the method can be used even in a trace‐by‐trace implementation, which is very important from the point of view of stationarity and consistency. We demonstrate the viability of the method under several reflectivity models. Finally, we illustrate the hybrid strategy using marine and field real data examples. The consistency of the results is very encouraging because the improved cumulant matching strategy we describe can be effectively used with a limited amount of data.

scholarly journals Joint influence of measurement errors and randomized response technique on mean estimation under stratified double sampling

2021 ◽  
Vol 5 (1) ◽  
pp. 192-199
Author(s):  
Ronald Onyango ◽  
◽  
Brian Oduor ◽  
Francis Odundo ◽  
◽  
...  
Keyword(s):  
Measurement Errors ◽  
Mean Squared Error ◽  
Numerical Study ◽  
Real Data ◽  
Auxiliary Variable ◽  
Randomized Response Technique ◽  
Randomized Response ◽  
Minimum Mean Squared Error ◽  
Squared Error ◽  
Joint Influence

The present study proposes a generalized mean estimator for a sensitive variable using a non-sensitive auxiliary variable in the presence of measurement errors based on the Randomized Response Technique (RRT). Expressions for the bias and mean squared error for the proposed estimator are correctly derived up to the first order of approximation. Furthermore, the optimum conditions and minimum mean squared error for the proposed estimator are determined. The efficiency of the proposed estimator is studied both theoretically and numerically using simulated and real data sets. The numerical study reveals that the use of the Randomized Response Technique (RRT) in a survey contaminated with measurement errors increases the variances and mean squared errors of estimators of the finite population mean.

scholarly journals Scaling limits for a random boxes model

2019 ◽  
Vol 51 (03) ◽  
pp. 773-801
Author(s):  
F. Aurzada ◽  
S. Schwinn
Keyword(s):  
Random Fields ◽  
Random Measure ◽  
Statistical Properties ◽  
Scaling Limits ◽  
Poisson Random Measure ◽  
Scaling Behaviour ◽  
The Mean ◽  
Heavy Tailed

AbstractWe consider random rectangles in $\mathbb{R}^2$ that are distributed according to a Poisson random measure, i.e. independently and uniformly scattered in the plane. The distributions of the length and the width of the rectangles are heavy tailed with different parameters. We investigate the scaling behaviour of the related random fields as the intensity of the random measure grows to infinity while the mean edge lengths tend to zero. We characterise the arising scaling regimes, identify the limiting random fields, and give statistical properties of these limits.

Inference for stationary random fields given Poisson samples

1986 ◽  
Vol 18 (02) ◽  
pp. 406-422 ◽  
Author(s):  
Alan F. Karr
Keyword(s):  
Random Field ◽  
Poisson Process ◽  
Random Fields ◽  
Covariance Function ◽  
Mean Squared Error ◽  
Compact Sets ◽  
Mild Conditions ◽  
Minimum Mean Squared Error ◽  
Squared Error ◽  
The Mean

Given a d-dimensional random field and a Poisson process independent of it, suppose that it is possible to observe only the location of each point of the Poisson process and the value of the random field at that (randomly located) point. Non-parametric estimators of the mean and covariance function of the random field—based on observation over compact sets of single realizations of the Poisson samples—are constructed. Under fairly mild conditions these estimators are consistent (in various senses) as the set of observation becomes unbounded in a suitable manner. The state estimation problem of minimum mean-squared error reconstruction of unobserved values of the random field is also examined.

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