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Spatial sampling and prediction
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
2012 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis discusses two aspects of spatial statistics: sampling and prediction. In spatial statistics, we observe some phenomena in space. Space is typically of two or three dimensions, but can be of higher dimension. Questions in mind could be; What is the total amount of gold in a gold-mine? How much precipitation could we expect in a specific unobserved location? What is the total tree volume in a forest area? In spatial sampling the aim is to estimate global quantities, such as population totals, based on samples of locations (papers III and IV). In spatial prediction the aim is to estimate local quantities, such as the value at a single unobserved location, with a measure of uncertainty (papers I, II and V).

In papers III and IV, we propose sampling designs for selecting representative probability samples in presence of auxiliary variables. If the phenomena under study have clear trends in the auxiliary space, estimation of population quantities can be improved by using representative samples. Such samples also enable estimation of population quantities in subspaces and are especially needed for multi-purpose surveys, when several target variables are of interest.

In papers I and II, the objective is to construct valid prediction intervals for the value at a new location, given observed data. Prediction intervals typically rely on the kriging predictor having a Gaussian distribution. In paper I, we show that the distribution of the kriging predictor can be far from Gaussian, even asymptotically. This motivated us to propose a semiparametric method that does not require distributional assumptions. Prediction intervals are constructed from the plug-in ordinary kriging predictor. In paper V, we consider prediction in the presence of left-censoring, where observations falling below a minimum detection limit are not fully recorded. We review existing methods and propose a semi-naive method. The semi-naive method is compared to one model-based method and two naive methods, all based on variants of the kriging predictor.

Place, publisher, year, edition, pages
Umeå: Umeå universitet , 2012. , 42 p.
Keyword [en]
Auxiliary variables, Censoring, Inclusion probabilities, Kriging, Local pivotal method, Minimum detection limit, Prediction intervals, Representative sample, Spatial process, Spatial sampling, Semiparametric bootstrap
National Category
Probability Theory and Statistics
Research subject
Mathematical Statistics
Identifiers
URN: urn:nbn:se:umu:diva-53286ISBN: 978-91-7459-373-0 (print)OAI: oai:DiVA.org:umu-53286DiVA: diva2:511130
Public defence
2012-04-12, MIT-huset, MA 121, Umeå universitet, Umeå, 10:15 (English)
Opponent
Supervisors
Available from: 2012-03-22 Created: 2012-03-20 Last updated: 2012-03-20Bibliographically approved
List of papers
1. Kriging prediction intervals based on semiparametric bootstrap
Open this publication in new window or tab >>Kriging prediction intervals based on semiparametric bootstrap
2010 (English)In: Mathematical Geosciences, ISSN 1874-8961, Vol. 42, no 8, 985-1000 p.Article in journal (Refereed) Published
Abstract [en]

Kriging is a widely used method for prediction, which, given observations of a (spatial) process, yields the best linear unbiased predictor of the process at a new location. The construction of corresponding prediction intervals typically relies on Gaussian assumptions. Here we show that the distribution of kriging predictors for non-Gaussian processes may be far from Gaussian, even asymptotically. This emphasizes the need for other ways to construct prediction intervals. We propose a semiparametric bootstrap method with focus on the ordinary kriging predictor. No distributional assumptions about the data generating process are needed. A simulation study for Gaussian as well as lognormal processes shows that the semiparametric bootstrap method works well. For the lognormal process we see significant improvement in coverage probability compared to traditional methods relying on Gaussian assumptions.

Place, publisher, year, edition, pages
Springer Verlag, 2010
National Category
Computational Mathematics Probability Theory and Statistics
Research subject
Mathematical Statistics
Identifiers
urn:nbn:se:umu:diva-37310 (URN)10.1007/s11004-010-9302-9 (DOI)000283087600006 ()
Available from: 2010-10-26 Created: 2010-10-26 Last updated: 2012-03-20Bibliographically approved
2. Construction of kriging prediction intervals for non-Gaussian spatial processes
Open this publication in new window or tab >>Construction of kriging prediction intervals for non-Gaussian spatial processes
(English)Manuscript (preprint) (Other academic)
Abstract [en]

In this article, we compare three methods to construct prediction intervals for the value of a stationary process, based on plug-in ordinary kriging predictors. Ordinary kriging is a widely used method for prediction that, given observations of a (spatial) process, forms the best linear unbiased predictor of the process at a new location. Construction of prediction intervals for the value of interest based on ordinary kriging predictors typically rely on Gaussian assumptions. Special attention is here given to non-Gaussian processes, where construction of such intervals is less straightforward.  Methods based on asymptotic normality, Gaussian transformations and semiparametric bootstrap are compared on simulated and real data. The study suggests that the semiparametric method (that does not rely on distributional assumptions) is robust and is to be recommended for non-Gaussian processes. For practitioners the semiparametric method is an attractive alternative since the method can be used without spcifying a link function or making distributional assumptions.

National Category
Probability Theory and Statistics
Research subject
Mathematical Statistics
Identifiers
urn:nbn:se:umu:diva-53277 (URN)
Available from: 2012-03-20 Created: 2012-03-19 Last updated: 2012-03-20Bibliographically approved
3. Spatially balanced sampling through the pivotal method
Open this publication in new window or tab >>Spatially balanced sampling through the pivotal method
2012 (English)In: Biometrics, ISSN 0006-341X, E-ISSN 1541-0420, Vol. 68, no 2, 514-520 p.Article in journal (Refereed) Published
Abstract [en]

A simple method to select a spatially balanced sample using equal or unequal inclusion probabilities is presented. For populations with spatial trends in the variables of interest, the estimation can be much improved by selecting samples that are well spread over the population. The method can be used for any number of dimensions and can hence also select spatially balanced samples in a space spanned by several auxiliary variables. Analysis and examples indicate that the suggested method achieves a high degree of spatial balance and is therefore efficient for populations with trends.

Place, publisher, year, edition, pages
John Wiley & Sons, 2012
Keyword
Generalized random-tessellation stratified, Pivotal method, Spatial sampling, Unequal probability sampling
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:umu:diva-50402 (URN)10.1111/j.1541-0420.2011.01699.x (DOI)
Available from: 2011-12-07 Created: 2011-12-07 Last updated: 2017-12-08Bibliographically approved
4. How to Select Representative Samples
Open this publication in new window or tab >>How to Select Representative Samples
2014 (English)In: Scandinavian Journal of Statistics, ISSN 0303-6898, E-ISSN 1467-9469, Vol. 41, no 2, 277-290 p.Article in journal (Refereed) Published
Abstract [en]

We give a formal definition of a representative sample, but roughly speaking, it is a scaled-down version of the population, capturing its characteristics. New methods for selecting representative probability samples in the presence of auxiliary variables are introduced. Representative samples are needed for multipurpose surveys, when several target variables are of interest. Such samples also enable estimation of parameters in subspaces and improved estimation of target variable distributions. We describe how two recently proposed sampling designs can be used to produce representative samples. Both designs use distance between population units when producing a sample. We propose a distance function that can calculate distances between units in general auxiliary spaces. We also propose a variance estimator for the commonly used Horvitz–Thompson estimator. Real data as well as illustrative examples show that representative samples are obtained and that the variance of the Horvitz–Thompson estimator is reduced compared with simple random sampling.

Keyword
auxiliary variables, local pivotal method, spatially correlated Poisson sampling
National Category
Probability Theory and Statistics
Research subject
Mathematical Statistics
Identifiers
urn:nbn:se:umu:diva-53279 (URN)10.1111/sjos.12016 (DOI)000335388400001 ()
Available from: 2012-03-20 Created: 2012-03-19 Last updated: 2017-12-07Bibliographically approved
5. Spatial prediction in the presence of left-censoring
Open this publication in new window or tab >>Spatial prediction in the presence of left-censoring
2014 (English)In: Computational Statistics & Data Analysis, ISSN 0167-9473, E-ISSN 1872-7352, Vol. 74, 125-141 p.Article in journal (Other academic) Published
Abstract [en]

Environmental (spatial) monitoring of different variables often involves left-censored observations falling below the minimum detection limit (MDL) of the instruments used to quantify them. Several methods to predict the variables at new locations given left-censored observations of a stationary spatial process are compared. The methods use versions of kriging predictors, being the best linear unbiased predictors minimizing the mean squared prediction errors. A semi-naive method that determines imputed values at censored locations in an iterative algorithm together with variogram estimation is proposed. It is compared with a computationally intensive method relying on Gaussian assumptions, as well as with two distribution-free methods that impute the MDL or MDL divided by two at the locations with censored values. Their predictive performance is compared in a simulation study for both Gaussian and non-Gaussian processes and discussed in relation to the complexity of the methods from a user’s perspective. The method relying on Gaussian assumptions performs, as expected, best not only for Gaussian processes, but also for other processes with symmetric marginal distributions. Some of the (semi-)naive methods also work well for these cases. For processes with skewed marginal distributions (semi-)naive methods work better. The main differences in predictive performance arise for small true values. For large true values no difference between methods is apparent.

Place, publisher, year, edition, pages
Elsevier, 2014
Keyword
kriging, left-censoring, minimum detection limit, prediction, spatial process
National Category
Probability Theory and Statistics
Research subject
Mathematical Statistics
Identifiers
urn:nbn:se:umu:diva-53278 (URN)10.1016/j.csda.2014.01.004 (DOI)000333781500010 ()
Note

Originally published in dissertation in manuscript form.

Available from: 2012-03-20 Created: 2012-03-19 Last updated: 2017-12-07Bibliographically approved

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Schelin, Lina

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