Abstract. Predictive spatial modelling is an important task in natural hazard assessment and regionalisation of geomorphic processes or landforms. Logistic regression is a multivariate statistical approach frequently used in predictive modelling; it can be conducted stepwise in order to select from a number of candidate independent variables those that lead to the best model. In our case study on a debris flow susceptibility model, we investigate the sensitivity of model selection and quality to different sample sizes in light of the following problem: on the one hand, a sample has to be large enough to cover the variability of geofactors within the study area, and to yield stable and reproducible results; on the other hand, the sample must not be too large, because a large sample is likely to violate the assumption of independent observations due to spatial autocorrelation. Using stepwise model selection with 1000 random samples for a number of sample sizes between n = 50 and n = 5000, we investigate the inclusion and exclusion of geofactors and the diversity of the resulting models as a function of sample size; the multiplicity of different models is assessed using numerical indices borrowed from information theory and biodiversity research. Model diversity decreases with increasing sample size and reaches either a local minimum or a plateau; even larger sample sizes do not further reduce it, and they approach the upper limit of sample size given, in this study, by the autocorrelation range of the spatial data sets. In this way, an optimised sample size can be derived from an exploratory analysis. Model uncertainty due to sampling and model selection, and its predictive ability, are explored statistically and spatially through the example of 100 models estimated in one study area and validated in a neighbouring area: depending on the study area and on sample size, the predicted probabilities for debris flow release differed, on average, by 7 to 23 percentage points. In view of these results, we argue that researchers applying model selection should explore the behaviour of the model selection for different sample sizes, and that consensus models created from a number of random samples should be given preference over models relying on a single sample.
We show a Cameron-Martin theorem for Slepian processes Wt :and Bs is Brownian motion. More exactly, we determine the class of functions F for which a density of F (t) + Wt with respect to Wt exists. Moreover, we prove an explicit formula for this density. p-Slepian processes are closely related to Slepian processes. p-Slepian processes play a prominent role among others in scan statistics and in testing for parameter constancy when data are taken from a moving window.
Abstract. Predictive spatial modelling is an important task in natural hazard assessment and regionalisation of geomorphic processes or landforms. Logistic regression is a multivariate statistical approach frequently used in predictive modelling; it can be conducted stepwise in order to select from a number of candidate independent variables those that lead to the best model. In our case study on a debris flow susceptibility model, we investigate the sensitivity of model selection and quality to different sample sizes in light of the following problem: on the one hand, a sample has to be large enough to cover the variability of geofactors within the study area, and to yield stable results; on the other hand, the sample must not be too large, because a large sample is likely to violate the assumption of independent observations due to spatial autocorrelation. Using stepwise model selection with 1000 random samples for a number of sample sizes between n = 50 and n = 5000, we investigate the inclusion and exclusion of geofactors and the diversity of the resulting models as a function of sample size; the multiplicity of different models is assessed using numerical indices borrowed from information theory and biodiversity research. Model diversity decreases with increasing sample size and reaches either a local minimum or a plateau; even larger sample sizes do not further reduce it, and approach the upper limit of sample size given, in this study, by the autocorrelation range of the spatial datasets. In this way, an optimised sample size can be derived from an exploratory analysis. Model uncertainty due to sampling and model selection, and its predictive ability, are explored statistically and spatially through the example of 100 models estimated in one study area and validated in a neighbouring area: depending on the study area and on sample size, the predicted probabilities for debris flow release differed, on average, by 7 to 23 percentage points. In view of these results, we argue that researchers applying model selection should explore the behaviour of the model selection for different sample sizes, and that consensus models created from a number of random samples should be given preference over models relying on a single sample.
a b s t r a c tWe consider regression models with multiple correlated responses for each design point. Under the null hypothesis, a linear regression is assumed. For the least-squares residuals of this linear regression, we establish the limit of the partial sums. This limit is a projection on a certain subspace of the reproducing Kernel Hilbert space of a multivariate Brownian motion. Based on this limit, we propose a significance test of Kolmogorov-Smirnov type to test the null hypothesis and show that this result can be used to study a change-point problem in the case of linear profile data (panel data). We compare our proposed method, which does not rely on any distributional assumptions, with the likelihood ratio test in a simulation study.
defined as centered, stationary Gaussian process with continuous sample paths and covariancewhere Bt is standard Brownian motion, is a (q, d)-Slepian-process. In this paper we prove an analytical formula for the boundary crossing probabilityaffine function. This formula can be used as approximation for the boundary crossing probability of an arbitrary boundary by approximating the boundary function by piecewise affine functions.
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