Confidence regions for level curves and a limit theorem for the maxima of Gaussian random fields
| dc.contributor.author | French, Joshua, author | |
| dc.contributor.author | Davis, Richard A., advisor | |
| dc.date.accessioned | 2024-03-13T19:53:46Z | |
| dc.date.available | 2024-03-13T19:53:46Z | |
| dc.date.issued | 2009 | |
| dc.description.abstract | One of the most common display tools used to represent spatial data is the contour plot. Informally, a contour plot is created by taking a "slice" of a three-dimensional surface at a certain level of the response variable and projecting the slice onto the two-dimensional coordinate-plane. The "slice" at each level is known as a level curve. | |
| dc.description.abstract | Consider a Gaussian random field {Z(s) : s ϵ D}, where s is the location in the continuous two-dimensional region of interest D ⊂ R2. The level curve for the process Z at level u is defined to be Iu = {s : Z(s) = u}. From the observed data z(s1),...,z(sN), one can predict Ẑ(s) for any location s ϵ D using kriging or some other procedure, and then construct the estimated level curve Îu = {s : Ẑ(s) = u] as an approximation of Iu. We present two methods for constructing confidence regions for the level curves of a contour plot. | |
| dc.description.abstract | The first method proposed is an extension of Lindgren and Rychlik (1995) and Wameling (2003b) based on level crossings. A series of rectangular confidence regions are constructed along Îu which should individually intersect the true level curve with high confidence. The boxes extend in directions perpendicular to the estimated level curve and the widths of the boxes are chosen so that the edge of each box touches the neighboring box and there are no gaps between the boxes along the estimated level curve. The heights of the boxes are chosen by simulating realizations of Z(s)|z(sN),...,z(sN), and then taking the appropriate quantiles of the set of nearest level crossings for the realizations. The heights of the boxes give insight into the approximate distance between the estimated level curve and the true level curve. | |
| dc.description.abstract | The second method constructs a confidence region for Iu through hypothesis testing, adjusting the critical value to control the simultaneous Type I error rate. Our goal is to construct a confidence region S for the true level curve such that P(Iu ⊆ S) ≥ 1-α. Instead of finding S directly, we adopt a different approach and find a set R which does not intersect Iu with high confidence, so that P({R∩Iu} = 0) ≥ 1-α. Consequently, the set S = Rc will satisfy our goal since P(Iu ⊆ Rc) ≥ 1-α. The region R is constructed by testing H0 : Z(s) = u versus Ha : Z(s) ≠ u, and taking R to be the union of all s for which we conclude that Z(s) ≠ u. Using kriging, we construct a test statistic which has a standard normal distribution. The critical value is adjusted to control the simultaneous Type I error rate through empirical simulation of the test statistic. | |
| dc.description.abstract | We conclude by introducing a limit theorem for the distribution of the maxima of a triangular sequence of stationary Gaussian random fields on an nxn lattice. The result is an extension of the work presented by Hsing et al. (1996) to two dimensions. The result was motivated by the desire to control the simultaneous Type I error rate of hypothesis tests at locations on an n x n lattice where the test statistics are Gaussian and correlated. Under certain dependence and limiting conditions we show that the maximum of the random fields exhibits extremal clustering in the limit. Consideration is then given to the use of this result in approximating P(max1<ij<n Zitj < u), where {Zi,j} is a stationary Gaussian random field on a square nxn lattice of equally spaced locations. | |
| dc.format.medium | born digital | |
| dc.format.medium | doctoral dissertations | |
| dc.identifier | ETDF_French_2009_3385155.pdf | |
| dc.identifier.uri | https://hdl.handle.net/10217/237741 | |
| dc.language | English | |
| dc.language.iso | eng | |
| dc.publisher | Colorado State University. Libraries | |
| dc.relation.ispartof | 2000-2019 | |
| dc.rights | Copyright and other restrictions may apply. User is responsible for compliance with all applicable laws. For information about copyright law, please see https://libguides.colostate.edu/copyright. | |
| dc.rights.license | Per the terms of a contractual agreement, all use of this item is limited to the non-commercial use of Colorado State University and its authorized users. | |
| dc.subject | confidence regions | |
| dc.subject | Gaussian random fields | |
| dc.subject | level curves | |
| dc.subject | limit theorem | |
| dc.subject | stochastic process | |
| dc.subject | statistics | |
| dc.title | Confidence regions for level curves and a limit theorem for the maxima of Gaussian random fields | |
| dc.type | Text | |
| dcterms.rights.dpla | This Item is protected by copyright and/or related rights (https://rightsstatements.org/vocab/InC/1.0/). You are free to use this Item in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you need to obtain permission from the rights-holder(s). | |
| thesis.degree.discipline | Statistics | |
| thesis.degree.grantor | Colorado State University | |
| thesis.degree.level | Doctoral | |
| thesis.degree.name | Doctor of Philosophy (Ph.D.) |
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