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Hierarchical Models and Longitudinal Data
Published in Gary L. Rosner, Purushottam W. Laud, Wesley O. Johnson, Bayesian Thinking in Biostatistics, 2021
Gary L. Rosner, Purushottam W. Laud, Wesley O. Johnson
All that remains is to specify a Gaussian process that gives the Wi an autoregressive structure. We accomplish this structure by specifying a covariance “function” for the processes, . We use a covariance function from the literature that specifies
Bayesian Disease Mapping Models
Published in Andrew B. Lawson, Using R for Bayesian Spatial and Spatio-Temporal Health Modeling, 2021
where β is a non-zero mean level of the process, and is a zero mean Gaussian process with, for example, a powered exponential correlation function defined for the distance between the i th and j th locations as and variance σ2. Other forms of covariance function can be specified. One popular example is the Matérn class defined for the distance () as
Computer-Aided Diagnosis Systems for Prostate Cancer Detection
Published in Ayman El-Baz, Gyan Pareek, Jasjit S. Suri, Prostate Cancer Imaging, 2018
Guillaume Lemaître, Robert Martí, Fabrice Meriaudeau
where is the covariance function the testing sample , and is the covariance function of training-testing samples and . Then, the function is squashed using the sigmoid function and the probability of the class membership is defined such that
Spatio-temporal parse network-based trajectory modeling on the dynamics of criminal justice system
Published in Journal of Applied Statistics, 2022
Han Yu, Shanhe Jiang, Hong Huang
To expand the capability of GBTM, a partition-and-group framework is proposed to accommodate space-based groups of trajectories as well as time-based groups of trajectories. One unique quality of such a study that differentiates ST data from other non-ST data in the classical statistical analysis literature is the presence of structural dependencies among the objects induced by the important and indispensable spatial and temporal dimensions of the real world. Since spatial or temporal dependency is due to the unknown or unobserved latent variables, it is more interesting in modeling the covariance function appropriately to offset the effects of the unobserved variables and get more valid estimate of a key explanatory variable effect on a phenomenon. The spatio-temporal models include the two components, a systematic component with available explanatory variables and the spatial and temporal correlation component, and how the two components interact across processes and scales of variability. Such spatio-temporal studies enable us to predict big gaps of unknown values at unmeasured locations, identify unusual regions, forecast values at future times, and produce maps.
Impact of missing data on the prediction of random fields
Published in Journal of Applied Statistics, 2020
Abdelghani Hamaz, Ouerdia Arezki, Farida Achemine
Assuming that X is only observed at a finite number of points D. The basic idea of Kriging is to predict the value of a function at a given point by computing a weighted average of the known values of the function in the neighborhood of the point. Kriging belongs to the family of linear least squares estimation methods, and estimates values at unobserved location as a weighted average of the neighboring observed values. The determination of unknown weights requires specification of a parametric model for the covariance structure with few parameters. In practice, the model for the covariance is selected from a list of available models and then its parameters are estimated from the observed data 9] and might suffer from the problem of model misspecification. Some work has been done by using the nonparametric methods of estimating a multivariate covariance function, cf. [10,13]. Another drawback of this method is that the assumption of similar dependence structure at all the points may not hold true when the data is highly irregular. Hence, developing a prediction method which makes minimal assumptions on the covariance structure and its data driven is a major challenge for prediction on random fields.
Bias induced by adaptive dose-finding designs
Published in Journal of Applied Statistics, 2020
Start with the definition of the covariance function for two random variables U and V, with 16]) yields U,V and U/V, and using (2), we arrive at