Statistical Models of Spatial Processes Based on Local-Interaction - PowerPoint PPT Presentation

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++++++++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++++ ● ● ● ● ● ● ● +++++++++++++++++++++++++++++++++++ ● ● ● ● ● ● Isotropic Anisotropic +++++++++++++++++++++++++++++++++++++++ +++++++++++++++++++++++++++++++++++++ ● ● ● ● ●● ● ● ● ● ++++++++++++++++++++++++++++++++++++++++ ● ● ● ● ● ●● ● ++++++++++++++++++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++++++++++++++ ● ●●● ● ● ●● ● ● ● ● ● ● ● ● +++++++++++++++++++++++++++++++++++++++++++++++ ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ++++++++++++++++++++++++++++++++++++++++++++++++ ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ++++++++++++++++++++++++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++++++++++++++++++++++++ ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ++++++++++++++++++++++++++++++++++++++++++++++++++++++ ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● +++++++++++++++++++++++++++++++++++++++++++++++++++++++ +++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ● ● ●● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● 4e+06 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ● ● ● ● ● ● 10 ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ● ● ● ● ● ● ● ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ● ● ● ● ● ● ● ● ● ● +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ● ● ● ● ● ● ● +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ● ●● ● ● ● ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ● ● ● ● ● ● ● ● ● ● ● ● ● ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ● ● ● ● ● ● ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ● ● ● ● ● ● ● ● ● ● ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ● ● ● ● ● ● ● ● ● ● ● ● +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ● ● ● ● ● ● ● ● ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ●● ● ● ● +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ● +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● 8 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● 3e+06 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 6 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● OneDomainIso OneDomainAniso ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ● ● ● ● ● ● ● ●● ● ● ● ● ● ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 2e+06 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ● ● ● ● ● ● ● ● ● ● ● ● ● ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ● ● ● ● ● ● +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ● ● ● ● ●● ● ● ● ● ● ● ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ● ● ● ● ● ● ● ● +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ● ● ● ●● ● ● ● ● ● ● ● ● +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ● ● ● ● ● 4 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ● ● ● ● ● ● ● ● ● ● +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ● ● ● ● ● ● ● ● ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ● ● ●● ● ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ● ● ●● ● ● +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++++++++++++++++++++++++ +++++++++++++++++++++++++++++ ● +++ ● 2 3e+06 4e+06 5e+06 6e+06 7e+06 0 Radioactivity dose rate (nSv/h) using background measurements and simulated accident (Bundesamt f¨ ur Strahlenschutz). Spiliopoulos et al., Computers & Geosciences , 37, 320-330 (2011). Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 5/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio Uncertainty Quantification in Spatial Processes Directions of Further Research Non-Gaussian, non-stationary models of spatial variability Flexible and efficient spatiotemporal statistical models Incorporation of physical laws (PDEs, SPDEs) in statistical models Developing connections between spatial modelling and statistical physics Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 6/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio Data Reconstruction Problem From a sample ( X 1 , . . . , X N ) of a process X ( s ) , s ∈ R d at points ( s 1 , . . . , s N ) : Estimate missing data X ( z p ) , p = 1 , . . . , P , i.e., ˆ X ( z p ) = Φ( X 1 , . . . , X N ) Sampling: Missing data Interpolated maps Training set GDR (nSv/h) 5 x 10 6 140 5 4 120 Y (m) 3 100 2 80 1 0 60 −2 0 2 4 X (m) 5 x 10 Right panel: (a) NNI training (b) NNI prediction (c) SSRF SpartFit (d) SSRF Spart Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 7/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio Data Reconstruction Problem From a sample ( X 1 , . . . , X N ) of a process X ( s ) , s ∈ R d at points ( s 1 , . . . , s N ) : Estimate missing data X ( z p ) , p = 1 , . . . , P , i.e., ˆ X ( z p ) = Φ( X 1 , . . . , X N ) Interpolated maps Sampling: Missing data 550 550 500 500 50 50 450 450 400 400 100 100 350 350 300 300 250 250 150 150 200 200 100 200 300 100 200 300 x x Daily ozone (June 2007, NASA Earth Science DGC interpolation based on 10 simulations (by M. ˇ Data) measurements Zukoviˇ c) Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 7/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio Preliminaries X ( s , ω ) ∈ R , s ∈ R d is a random function or spatial random field Wide-sense (second-order) Stationarity: 1 Constant mean: E [ X ( s )] = m x 2 Two-point covariance: G ( r ) = E [ X ( s ) X ( s + r )] − m 2 x Statistical Isotropy: G ( r ) = G ( � r � ) Intrinsic Stationarity: 1 Constant mean: E [ X ( s + r ) − X ( s )] = 0 2 Semivariogram (structure) function: � [ X ( s ) − X ( s + r )] 2 � γ ( r ) = 1 2 E Second-order stationarity: G ( r ) = σ 2 x − γ ( r ) Yaglom, Correlation Theory of Stationary and Related Random Functions , Springer, 1987. Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 8/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio The Classical Geostatistical Approach The semivariogram function Scattered data (SIC 2004) Semivariogram structure function γ ( r ) = 1 � { X ( s ) − X ( s + r ) } 2 � 2 E Empirical semivariogram (MoM) N 1 �� 2 � � γ ( r k ) = ˆ X ( s i ) − X ( s j ) ϑ ij ( r k ) , 2 n ( r k ) i , j = 1 ( k = 1 , . . . , N c ) � 1 , s i − s j ∈ B ( r k ) ϑ ij ( r k ) = . 0 , s i − s j �∈ B ( r k ) G. Matheron, Trait´ e de g´ eostatistique applique´ e , Editions Technip, France, 1962-63 Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 9/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio The Classical Geostatistical Approach The semivariogram function Scattered data (SIC 2004) Semivariogram structure function Training set GDR (nSv/h) 5 x 10 6 γ ( r ) = 1 � { X ( s ) − X ( s + r ) } 2 � 140 2 E 5 120 4 Empirical semivariogram (MoM) Y (m) 3 100 2 N 1 �� 2 � � γ ( r k ) = ˆ X ( s i ) − X ( s j ) ϑ ij ( r k ) , 80 1 2 n ( r k ) i , j = 1 0 ( k = 1 , . . . , N c ) 60 −2 0 2 4 � 1 , s i − s j ∈ B ( r k ) X (m) 5 x 10 ϑ ij ( r k ) = . 0 , s i − s j �∈ B ( r k ) G. Matheron, Trait´ e de g´ eostatistique applique´ e , Editions Technip, France, 1962-63 Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 9/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio The Classical Geostatistical Approach Numerical complexity of ˆ γ ( r k ) WLS fit to theoretical variogram estimation is O ( N 2 ) (spherical model) 450 400 Empirical semivariogram 350 300 250 200 150 100 50 0 1 2 3 4 5 lag distance (m) 5 x 10 G. Matheron, Trait´ e de g´ eostatistique applique´ e , Editions Technip, France, 1962-63 Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 9/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio The Classical Geostatistical Approach Numerical complexity of ˆ γ ( r k ) WLS fit to theoretical variogram estimation is O ( N 2 ) (spherical model) 450 450 400 400 Empirical semivariogram 350 350 Semivariograms 300 300 250 250 200 Empirical 200 Model 150 150 100 100 50 50 0 1 2 3 4 5 0 1 2 3 4 5 lag distance (m) 5 x 10 lag distance (m) 5 x 10 G. Matheron, Trait´ e de g´ eostatistique applique´ e , Editions Technip, France, 1962-63 Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 9/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio The Classical Geostatistical Approach Ordinary Kriging is a Best Linear Unbiased Estimator (BLUE) m ˆ � X ( s ) = λ j X ( s j ) , s j ∈ B ( s ; r c ) , B : search neighborhood j = 1 Best : minimum (ensemble) mean square estimation error � �� 2 � �� ˆ { λ 1 , λ 2 , . . . , λ m } = arg min E X ( s ) − X ( s ) � λ 1 + λ 2 + . . . λ m = 1 � λ 1 ,λ 2 ,...,λ m Spatial weights follow from the linear system:  γ ( s 1 − s 1 ) γ ( s 1 − s 2 ) . . . γ ( s 1 − s m ) 1  λ 1 γ ( s 1 − s )     γ ( s 2 − s 1 ) γ ( s 2 − s 2 ) . . . γ ( s 2 − s m ) 1 γ ( s 2 − s )   λ 2  . . . . .       . . . . .   .   .  . . . . . . = .       . .       .       . γ ( s m − s ) λ m       γ ( s m − s 1 ) γ ( s m − s 2 ) . γ ( s m − s m ) 1   µ 1 1 1 . . . 1 0 G. Matheron, Trait´ e de g´ eostatistique applique´ e , Editions Technip, France, 1962-63 Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 9/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio The Classical Geostatistical Approach OK Problems Kriging radius Computational cost of 5 x 10 inverting covariance 6 matrix ∝ O ( N 3 ) 5 Empirical solution: use of kriging search radius 4 Implicit assumption: Y (m) 3 covariance truncation 2 Optimality conditions: Gaussian data and 1 knowledge of true 0 semivariogram function −2 0 2 4 X (m) 5 x 10 G. Matheron, Trait´ e de g´ eostatistique applique´ e , Editions Technip, France, 1962-63 Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 9/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio The Classical Geostatistical Approach OK Problems Kriging radius Computational cost of inverting covariance matrix ∝ O ( N 3 ) Empirical solution: use of kriging search radius Implicit assumption: covariance truncation Optimality conditions: Gaussian data and knowledge of true semivariogram function G. Matheron, Trait´ e de g´ eostatistique applique´ e , Editions Technip, France, 1962-63 Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 9/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio Motivation for SSRF In spatial statistics the “key players” are the data and the covariance model In statistical field theories the key players are spatial interactions and their coupling strengths (Gaussian, Landau - Ginzburg, Ising models) Idea: To use interaction-based models of spatial dependence inspired from statistical physics, i.e., Spartan Spatial Random Fields (SSRFs) Motivation: Flexible parametrization of spatial dependence & Efficient interpolation/simulation 1 No variograms? 2 Fast spatial prediction algorithms (no kriging system?) Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 10/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio The Fluctuation-Gradient-Curvature (FGC) SSRF Gibbs probability density function (PDF) f [ X ( s )] = e −H [ X ( s )] , H [ X ( s )] : energy functional , Z � D X ( s ) e −H [ X ( s )] Z: partition function Z = FGC energy functional - for simplicity assume E [ X ( s )] = 0 1 � [ X ( s )] 2 + η 1 ξ 2 [ ∇ X ( s )] 2 + ξ 4 � � � 2 � ∇ 2 X ( s ) H fgc [ X ( s )] = d s 2 η 0 ξ d D Properties: Gaussian, zero-mean, stationary, isotropic SRF Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 11/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio The Fluctuation-Gradient-Curvature (FGC) SSRF Gibbs probability density function (PDF) f [ X ( s )] = e −H [ X ( s )] , H [ X ( s )] : energy functional , Z � D X ( s ) e −H [ X ( s )] Z: partition function Z = FGC energy functional - for simplicity assume E [ X ( s )] = 0 1 � [ X ( s )] 2 + η 1 ξ 2 [ ∇ X ( s )] 2 + ξ 4 � � � 2 � ∇ 2 X ( s ) H fgc [ X ( s )] = d s 2 η 0 ξ d D Properties: Gaussian, zero-mean, stationary, isotropic SRF FGC-SSRF Coefficients η 0 : scale, η 1 : stiffness, ξ : characteristic length; k c : spectral cutoff Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 11/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio FGC-SSRF Energy Functional on Hypercubic Lattice Grid structure = ⇒ Gauss-Markov random fields N � d � 2 � X ( s n + a i ˆ 1 e i ) − X ( s n ) � � X 2 ( s n ) + η 1 ξ 2 H fgc [ X ( s )] ∝ 2 η 0 ξ d a i n = 1 i = 1 d � 2 � � X ( s n + a i ˆ e i ) − 2 X ( s n ) + X ( s n − a i ˆ e i ) + ξ 4 � a 2 i i = 1 e i , i = 1 , . . . , d : unit vectors in lattice directions ˆ 10 9 a i : lattice steps 8 7 6 5 H fgc [ X ( s )] = α 0 S 0 + α 1 S G + α 2 S c 4 3 2 Rue and Held, Gaussian Markov Random Fields: Theory 1 and Applications , Chapman and Hall/CRC, 2005 0 0 2 4 6 8 10 Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 12/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio FGC-SSRF Covariance Function & Spectral Density Covariance: G ( r ) = E [ X ( s ) X ( s + r )] . Fourier transform pair: � 1 � d r e −  k · r G ( r ) , d k e  k · r ˜ ˜ G ( k ) = G ( r ) = G ( k ) . ( 2 π ) d Covariance spectral density: ✶ k c ≥ κ ( κ ) η 0 ξ d ˜ G ( k ) = 1 + η 1 κ 2 ξ 2 + κ 4 ξ 4 , κ = � k � , ✶ B ( · ) : indicator function , Permissibility conditions (Bochner’s theorem) : For any k c : For finite k c : � η 0 > 0 , ξ > 0 , η 1 > − 2 | η 1 |− ∆ η 1 < − 2, if k c ξ < 2 � η 2 ∆ = 1 − 4 Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 13/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio FGC-SSRF Covariance Function & Spectral Density SPD: Positive stiffness SPD: Negative stiffness Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 13/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio FGC-SSRF Covariance Function & Spectral Density Covariance ( d = 2 ) : Positive Covariance ( d = 2 ) : Negative stiffness stiffness Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 13/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio Spectral Representation of the Covariance Function Spectral Representation (Inverse Hankel transform For isotropic covariance functions the following holds: � k c η 0 ξ d � r � d κ κ d / 2 J d / 2 − 1 ( κ � r � ) G ( r ) = 1 + η 1 ( κξ ) 2 + ( κξ ) 4 ( 2 π � r � ) d / 2 0 J d / 2 − 1 ( � r � ) : Bessel function of the first kind of order d / 2 − 1 For k c → ∞ the integral exists for d ≤ 3 I. J. Schoenberg, “Metric spaces and completely monotone functions,” Ann. Math. , 39(4), 811841, 1938 Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 14/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio Spectral Representation of the Covariance Function Unlimited band covariance function d = 1, k c → ∞ G ( h ) = η 0 � cos ( h β 1 ) + sin ( h β 1 ) � 4 e − h β 2 , | η 1 | < 2 β 2 β 1 ( 1 + h ) G ( h ) = η 0 , η 1 = 2 4 e h � e − h ω 1 − e − h ω 2 G ( h ) = η 0 � , η 1 > 2 2 ∆ ω 1 ω 2 h = | r | /ξ : normalized lag, � 1 / 2 � 1 / 2 � � | 2 ∓ η 1 | | η 1 ∓ ∆ | 1 ∆ = | η 2 β 1 , 2 = , ω 1 , 2 = , 1 − 4 | 2 4 2 Hristopulos and Elogne, IEEE Trans. Information Theory , 53(12), 4667–4679, 2007 Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 14/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio Spectral Representation of the Covariance Function 1 η 1 =−1.9 0.8 η 1 =−1 η 1 =1 0.6 η 1 =16 0.4 Correlation 0.2 0 −0.2 −0.4 −0.6 −0.8 0 1 2 3 4 5 Distance lag Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 14/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio Spectral Representation of the Covariance Function Unlimited band covariance function d = 3, k c → ∞ e − h β 2 � sin ( h β 1 ) � G ( h ) = η 0 , | η 1 | < 2 ∆ h G ( h ) = η 0 4 e − h , η 1 = 2 � e − h ω 1 − e − h ω 2 � 1 G ( h ) = , η 1 > 2 2 ∆ h h = � r � /ξ, � 1 / 2 � 1 / 2 � � | 2 ∓ η 1 | | η 1 ∓ ∆ | 1 ∆ = | η 2 β 1 , 2 = , ω 1 , 2 = , 1 − 4 | 2 4 2 Hristopulos and Elogne, IEEE Trans. Information Theory , 53(12), 4667–4679, 2007 Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 14/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio Spectral Representation of the Covariance Function 1.2 η 1 =−1 η 1 =2 1 η 1 =8 η 1 =16 0.8 Correlation 0.6 0.4 0.2 0 −0.2 0 0.2 0.4 0.6 0.8 1 Distance lag Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 14/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio FGC-SSRF Realizations d = 1 η 1 = − 1 . 999 η 1 = − 1 Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 15/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio FGC-SSRF Realizations d = 1 η 1 = 0 . 15 η 1 = 15 Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 15/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio FGC-SSRF Length Scales Definitions FGC-SSRF Integral Range 1 Integral range: 35 30 η 1 =−1.8 � 1 / d �� η 1 =1 d r G ( r ) ℓ c . η 1 =2 = 25 η 1 =4 G ( 0 ) A 2 ( η 1 ,k c ξ ) 20 � 1 / d = A d σ − 2 / d � ˜ G ( 0 ) 15 10 2 Correlation length: 5 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 � 1 / 2 d r r 2 G ( r ) �� k c ξ r c . = � d r G ( r ) Hristopulos and ˇ Zukoviˇ c, Stoch. Env. � Res. Risk Assess. , 25, 2511 (2011) �� d 2 ˜ � G ( k ) / dk 2 � � � � = = | η 1 | ξ � � � 2 ˜ � � G ( k ) � � k = 0 Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 16/43 2-D Integral range

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio FGC-SSRF Length Scales Definitions FGC-SSRF Integral Range Ratio of Integral Range to Characteristic Length 1 Integral range: 6 d=2 5.5 � 1 / d �� d r G ( r ) ℓ c . 5 = 4.5 G ( 0 ) A 2 ( η 1 ) 4 � 1 / d = A d σ − 2 / d � 3.5 ˜ G ( 0 ) 3 2.5 2 2 Correlation length: 1.5 0 2 4 6 8 10 12 14 16 18 20 Shape parameter η 1 � 1 / 2 d r r 2 G ( r ) �� r c . = � Hristopulos and ˇ d r G ( r ) Zukoviˇ c, Stoch. Env. Res. Risk Assess. , 25, 2511 (2011) � �� d 2 ˜ � G ( k ) / dk 2 � � � � = = | η 1 | ξ � � � 2 ˜ � � G ( k ) � � k = 0 Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 16/43 2-D Integral range

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio FGC-SSRF Parameter Inference The FGC-SSRF joint PDF belongs to the exponential family, i.e., f [ X ( s )] ∝ e −H [ X ( s )] , where H [ X ( s )] ∝ S 0 + α 1 S G + α 2 S C We use a modified method of moments : the PDF is determined by means of ˆ S 0 , ˆ S G , ˆ S C which are sample-based estimators of E [ S 0 ] , E [ S G ] , E [ S C ] Lattice estimators are based on sample averages, e.g., N d � 2 S G = 1 � X ( s n + a i ˆ e i ) − X ( s n ) ˆ � � N a i n = 1 i = 1 For irregularly sampled data we adapt kernel averaging methods (Watson-Nadaraya estimators) Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 17/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio The Nadaraya-Watson Kernel Average Definition We use kernel functions K ( r ) , r ∈ R d such that: (i) K ( r ) ∈ R and � K ( r ) ≥ 0, (ii) d r K ( r ) = 1 and (iii) K ( r ) = K ( − r ) . If K ( r ) is a kernel function, then K h = h − d K ( r / h ) is also a kernel function. Kernel averages of field values and two-point distances K h ( s i − s j ) u i , j ( h ) = � N � N j = 1 , j � = i K h ( s i − s j ) i = 1 � N � N � Φ( X i , X j ) � = j = 1 , j � = i Φ( X i , X j ) u i , j ( h ) i = 1 � N � N j = 1 , j � = i � s i − s j � β u i , j ( h ) �� s i − s j � β � = i = 1 Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 18/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio “Typical” Step and Kernel Bandwidth Selection 1 Find the Delaunay triangulation of the sampling network: computational cost is O ( N logN ) α using the ℓ 2 − norm of the near-neighbor distances 2 Estimate ˆ (Delaunay triangle edges) of the sampling points � 1 / 2 � � N 0 1 i = 1 ∆ 2 α = ˆ i N 0 3 Determine a uniform bandwidth by solving consistency relations, e.g., for curvature constraint �� s i − s j � 4 � h = ˆ α 4 4 Adaptive bandwidth estimation is also possible Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 19/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio “Typical” Step and Kernel Bandwidth Selection Example 1 - Sampling pattern Uniform step calculation � 1 / 2 1 � � N 0 1 i = 1 ∆ 2 α = ˆ N 0 i 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 19/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio “Typical” Step and Kernel Bandwidth Selection Example 1 - Sampling pattern Uniform step calculation 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 � 1 / 2 � � N 0 1 i = 1 ∆ 2 α = ˆ i N 0 Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 19/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio FGC-SSRF Interpolation FGC Spartan predictor Given the sample ( X 1 , . . . , X N ) , where X i = X ( s i ) , estimate ˆ X ( z p ) at point z p ∋ ( s 1 , . . . , s N ) Local interactions ⇒ Efficient prediction: Estimate ˆ X ( z p ) maximizes the joint PDF, i.e., minimizes the FGC energy functional over D ∪ z p An explicit expression follows for the predictor N ˆ � X ( z p ) = φ i ( z p ) X ( s i ) i = 1 The linear weights φ i ( z p ) depend on network geometry and the kernel function Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 20/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio “Spartan” Interpolators Use of simplifying assumptions Various interpolation methods can be formulated based on the FGC-SSRF energy function. We use the following terms: Spartan or FGC-SSRF: An interpolator based on the FGC energy functional General Uniform Bandwidth Spartan (GUBS): Explicit summations over the sampling points and uniform kernel bandwidth Asymptotic General Uniform Bandwidth Spartan (AGUBS): Uniform kernel bandwidth and discrete summations replaced by integrals Asymptotic Locally Adaptive Spartan (ALAS): Locally tuned kernel bandwidth that adapts to sampling density variations Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 21/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio Spartan (FGC-SSRF) Interpolation Equations of FGC General Uniform Bandwidth Spartan predictor � � N Linear coefficients: φ i ( z p ) = ν ( s i , z p ) i = 1 ν ( s i , z p ) Network weights: ν ( s i , z p ) = � 4 q = 1 b q u i , p ( h q ) + ( N + 1 ) − 2 K hq ( s i − s j ) Normalized kernel weights: u i , j ( h q ) = � N � N j = 1 , j � = i K hq ( s i − s j ) i = 1 ξ 2 ξ 4 ξ 4 ξ 4 FGC - SSRF coeffs: b 1 = β d η 1 α 2 , b 2 = δ d g 1 α 4 , b 3 = − ζ d g 2 α 4 , b 4 = − β d α 4 ˆ ˆ ˆ ˆ Curvature bias corrections: g 1 , g 2 = 1 + o ( 1 ) √ √ √ α ≈ h 1 α ≈ h 2 4 Step - bandwidth relations: ˆ B 2 , ˆ B 4 , h 3 = 2 h 2 , h 4 = 2 h 2 � R 0 ds K ( s ) s d + p − 1 �� R 0 ds K ( s ) s d − 1 Kernel moment ratio: B p = Hristopulos and Elogne (2009), IEEE TSP , 57(9), 3475. Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 22/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio Numerical Complexity of FGC-SSRF interpolation Numerical complexity of Ordinary kriging (OK) vs Spartan 1 OK: ∝ O ( N 3 + P N 2 ) or (using search radius) ∝ O ( P M 3 ) 2 FGC (GUBS): ∝ O ( N 2 + P N ) or (asymptotic GUBS) ∝ O ( P N ) Comparison of CPU times: ALAS vs OK Table: Gaussian RF is sampled at N + P randomly selected points inside a square domain. P = 1000 points are removed and used for prediction. 1 Method N = 500 N = 700 N = 1000 N = 2000 N = 3000 Spartan 1.5 sec 2.1 sec 2.8 sec 5.5 sec 7.3 sec Kriging 398.7 sec 821.1 sec 1737.2 sec 7588.3 sec 18250.1 sec 6.64 min 13.68 min 28.95 min 2.10 hr 5.06 hr 1 In Matlab � environment running under Windows XP , on a laptop with AMD Turion Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop processor, clock speed 1.6 GHz, 960 MB RAM Spatial Processes Based on Local-Interaction Energy Functionals 23/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio Time-series Interpolation � ∞ [ X ( t )] 2 + η 1 ξ 2 [ ˙ � X ( t )] 2 + ξ 4 [¨ X ( t )] 2 � 1 1 H fgc [ X ; θ ] = −∞ dt 2 η 0 ξ 2 H fgc [ X ; θ ] = 1 2 X ( t i ) J x ( t i , t j ; θ ) X ( t j ) � � ξ 2 α 2 J 1 ( t i , t j ) + ξ 4 1 3 J x ( t i , t j ; θ ) = J 0 ( t i , t j ) + η 1 α 4 J 2 ( t i , t j ) η 0 ξ Precision matrix Interpolation 120 Aerosol concentration [ µ g/m 3 ] 100 1 − 2 1 0 · · · 0   80 1 − 1 0 · · · 0   − 2 5 − 4 1 · · · 0 60 − 1 2 − 1 · · · 0   1 − 4 6 − 4 1 0     40     ... ... ... J 1 = , J 2 =     ... ... ... ...     20     0 · · · − 1 2 − 1     0 0 50 100 150 200 250 300 350 0 · · · 1 − 4 5 − 2   time [ α = 40 min.] 0 · · · 0 − 1 1 0 · · · 0 1 − 2 1 × : Validation • : Training • : Estimates Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 24/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio Time-series Interpolation J x ( θ ; t l , z p ) ˆ � X ( z p ) = − J x ( θ ; z p , z p ) X ( t l ) , p = 1 , ..., P t l ∈ V ( z p ) Spartan vs. Kriging Comparison Spartan-Kriging performance comparison on aerosol concentration data. Model parameters are inferred by MLE, using training set of 121 points. Statistics are calculated on validation set of 232 points. Category ( i , j ) includes points in the training set with i nearest and j next-nearest neighbors (2,2) (1,2) (0,2) (0,1) (0,0) (1,1) (1,0) (2,0) (2,1) MAE SP 1.75 2.67 3.85 5.30 8.15 2.88 3.24 2.05 2.03 OK 1.80 2.68 3.87 5.32 8.10 2.90 3.25 2.06 2.06 MARE [%] SP 4.3 6.6 9.3 12.5 19.1 6.8 7.5 4.5 4.9 OK 4.4 6.7 9.4 12.6 19.0 6.9 7.6 4.5 5.0 MRE [%] SP − 0.4 − 1.1 − 1.4 − 2.6 − 5.2 − 0.9 − 0.9 − 0.2 − 0.8 OK − 0.4 − 0.9 − 1.3 − 2.5 − 5.0 − 0.8 − 0.8 − 0.2 − 0.6 RMSE SP 2.15 3.72 5.14 7.44 11.04 4.23 4.65 2.92 2.93 OK 2.19 3.73 5.12 7.46 10.97 4.25 4.65 2.92 2.98 ˇ Zukoviˇ c & Hristopulos, Atmospheric Environment , 42 (2008) 7669-7678 Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 24/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio Unconditional Simulations Spectral Methods Regular lattice: Unstructured lattice: N M � � 1 / 2 � � � 2 ˜ X ( s n ) = IFFT u ( k i ) G ( k i ) , � X ( s n ) ≈ σ cos ( k p · s n + ϕ p ) , N M p = 1 u ( k i ) ∼ N ( 0 , 1 ) φ p ∼ U ( 0 , 2 π ) , PDF ( � k � ) ∝ ˜ G ( k ) η 0 = 1 , η 1 = 0 . 2 , ξ = 5 η 0 = 3 , η 1 = − 0 . 2 , ξ = 50 Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 25/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio Unconditional Simulations 3 D Simulations η 0 = 1 , η 1 = 0 . 2 , ξ = 5 η 0 = 3 , η 1 = − 0 . 2 , ξ = 20 Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 25/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio Conditional Simulations � � Z ( s ) − ˆ ˆ Spectral simulation + SSRF interpolation: Z c ( s ) = Z u ( s ) + Z u ( s ) . Conditional simulations Mean & StD of 1000 states; 300 × 300 grid T cpu ≈ 20s per state 600 Mean based on 1000 Simulations 400 160 150 200 600 140 0 500 0 100 200 0 100 200 130 140 120 400 600 110 400 300 100 200 200 120 90 0 80 0 100 200 0 100 200 100 70 0 600 60 100 −50 0 50 100 150 200 250 400 200 0 Standard Deviation based on 1000 Simulations 0 100 200 0 100 200 80 22 600 20 600 18 500 400 16 200 60 400 14 0 12 300 0 100 200 0 100 200 10 200 8 600 40 6 100 400 4 200 0 2 0 0 20 −50 0 50 100 150 200 250 0 100 200 0 100 200 Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 26/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio Karhunen-Lo` eve Expansions of SSRFs Karhunen-Lo` eve Theorem 1 A second-order X ( s ) with continuous covariance covariance G ( s , s ′ ) can be expanded on a closed and bounded domain D as: ∞ √ � X ( s ) = m x ( s ) + λ m c m ψ m ( s ) . m = 1 The convergence is uniform on D . 2 The λ m and ψ m ( s ) are respectively, eigenvalues and eigenfunctions of the covariance operator, that satisfy the Fredholm integral equation � d s ′ G ( s , s ′ ) ψ m ( s ′ ) = λ m ψ m ( s ′ ) . D 3 The c m are zero-mean, uncorrelated random variables , i.e, E [ c m ] = 0 and E [ c m c n ] = δ n , m , ∀ n , m ∈ N . Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 27/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio Karhunen-Lo` eve Expansions of SSRFs η 0 = 2 , η 1 = − 1 . 5 , ξ = 5 η 0 = 2 , η 1 = 1 . 5 , ξ = 5 η 0 = 2 , η 1 = 0 , ξ = 5 η 0 = 2 , η 1 = 15 , ξ = 5 Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 27/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio Karhunen-Lo` eve Expansions of SSRFs SRRF Variance Evolution versus number of ordered eigenvalues 0.26 η 1 =−1.5 0.24 η 1 =0 0.22 η 1 =1.5 0.2 η 1 =−15 0.18 σ 0.16 0.14 0.12 0.1 0.08 0.06 10 20 30 40 50 60 70 80 90 100 Mode number Figure: SSRF standard deviation of K-L simulation at s 0 = ( 25 , 25 ) on 100 × 100 square domain with pinned boundaries. SSRF parameters: η 0 = 2, ξ = 5 and η 1 = ( − 1 . 5 , 0 , 1 . 5 , 15 ) T . Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 27/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio Brief Description of Spatial Interpolation Comparison (SIC) 2004 Data 1 Two data sets of radioactivity gamma dose rates (GDR) over Germany. GDR is measured in nanoSievert per hour (nSv/h) 2 First data set: “Routine” measurements of background radiation 3 Second data set: Simulated radioactivity release; a few hot spots 4 Data are available at: http://www.ai-geostats.org/bin/ view/AI_GEOSTATS/AI_GEOSTATSData Elogne, Hristopulos, Varouchakis, Stoch. Environ. Res. Risk Assess. (2008) 22:633-646 Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 28/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio Brief Description of Spatial Interpolation Comparison (SIC) 2004 Data Locations of training (+) and Histogram of routine training prediction (dots) sets data and best-fit normal PDF 600 500 400 y (km) 300 200 100 0 0 100 200 x (km) Elogne, Hristopulos, Varouchakis, Stoch. Environ. Res. Risk Assess. (2008) 22:633-646 Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 28/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio Brief Description of Spatial Interpolation Comparison (SIC) 2004 Data Locations of training (+) and Bubble plot of emergency prediction (dots) sets training data 600 600 1400 500 500 1200 400 1000 400 y (km) y (km) 300 800 300 200 600 200 100 400 100 0 200 0 0 100 200 0 100 200 x (km) x (km) Elogne, Hristopulos, Varouchakis, Stoch. Environ. Res. Risk Assess. (2008) 22:633-646 Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 28/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio Comparison of “Spartan” & Ordinary Kriging Cross validation - SIC 2004 Background radiation data OK versus True Spartan - adaptive vs True 180 180 160 160 140 140 True values True values 120 120 100 100 80 80 60 60 60 80 100 120 140 160 180 60 80 100 120 140 160 180 OK ALAS Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 29/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio Comparison of “Spartan” & Ordinary Kriging Ordinary kriging Spartan - adaptive OK ALAS 140 140 600 600 130 130 500 500 120 120 400 400 110 110 y (km) y (km) 300 300 100 100 90 90 200 200 80 80 100 100 70 70 0 0 0 100 200 0 100 200 x (km) x (km) Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 29/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio Comparison of “Spartan” & Ordinary Kriging Cross validation - SIC 2004 emergency radiation data OK versus True Spartan - adaptive vs True 1400 1400 1200 1200 1000 1000 True values True values 800 800 600 600 400 400 200 200 500 1000 1500 500 1000 1500 OK ALAS Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 30/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio Comparison of “Spartan” & Ordinary Kriging OK contour plots ALAS contour plots OK ALAS 1200 1000 600 600 900 1000 500 500 800 700 800 400 400 600 y (km) y (km) 300 600 300 500 200 200 400 400 300 100 100 200 200 0 0 100 0 0 100 200 0 100 200 x (km) x (km) Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 30/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio Comparison of “Spartan” & Ordinary Kriging OK contour plots - zoom on ALAS contour plots - zoom on spreading plume spreading plume Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 30/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio Cross validation - SIC 2004 data set Table: ME: Mean error; MAE: Mean absolute error; RMSE: Root mean square error; RMSRE: Root mean square relative error; RS: Spearman rank correlation coefficient. δ x min = ˆ x min − x min ; δ x max = ˆ x max − x max . ME MAE MARE RMSE RMSRE RS δ x min δ x max SIC 2004 Background data set OK -1.30 9.08 0.09 12.42 0.12 0.77 12.55 -54.61 ALAS 2 -1.38 9.35 0.09 12.74 0.12 0.75 1.20 -44.79 SIC 2004 Emergency data set OK 0.80 21.91 0.16 77.83 0.43 0.73 1.20 -900.77 ALAS 3 0.34 20.74 0.15 74.43 0.33 0.78 1.20 -1045.41 2 with quadratic kernel 3 with tricubic kernel Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 31/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio Brief Introduction to Anisotropy Simulated anisotropic spatial Schematic of isotropy restoring random field transformation Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 32/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio The Covariance Tensor Identity (CTI) Method The covariance tensor identity (Swerling, 1962) Assume an anisotropic, differentiable, second-order stationary spatial random field (SRF) X ( s ) with a covariance function G ( r ) . Then: = − ∂ 2 G ( r ) � ∂ X ( s ) ∂ X ( s ) � � Q i , j ≡ E � ∂ s i ∂ s j ∂ r i ∂ r j � r =( 0 , 0 ) Expression for the covariance Hessian matrix (CHM) in 2 D � Q 11 cos 2 θ + R 2 sin 2 θ = σ 2 x ζ 2 sin θ cos θ ( 1 − R 2 ) � � � Q 12 R 2 cos 2 θ + sin 2 θ Q 11 Q 12 ξ 2 sin θ cos θ ( 1 − R 2 ) 1 Hristopulos (2002). Stoch. Environ. Res. Risk Assess. , 16(1), 43-62 Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 33/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio Explicit CTI Solution Relation between CHM and ( R , θ ) = R 2 + tan 2 θ q diag := Q 22 1 + R 2 tan 2 θ Q 11 = tan θ ( 1 − R 2 ) q off := Q 21 1 + R 2 tan 2 θ Q 11 Solution of nonlinear equations 2 ˆ θ = 1 � q off � − π 4 , π � � 2 arctan , θ ∈ 1 − ˆ q diag 4 1 − ˆ q diag R 2 = 1 + q diag ) cos 2 θ, R ∈ [ 0 , ∞ ) . ˆ q diag − ( 1 + ˆ Chorti and Hristopulos, IEEE Trans. Signal Proc. 56(10), 4738-4751, 2008 Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 34/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio Sample-based Estimation of CHM On regular grids the CHM is estimated using discretization of the first-order partial derivatives, e.g. on square grids of step a : N X ( s k + a � X ( s k + a � Q ij = 1 e i ) − X ( s k ) e j ) − X ( s k ) ˆ � N a a k = 1 where � e i ,� e j are the unit vectors in the respective directions For scattered data, estimation of partial derivatives is based on : 1 Interpolation of scattered data on background grid followed by finite differencing 2 or, estimation of derivatives using Savitzky-Golay (SG) polynomial filters 3 or, SG filtering is applied directly to the scattered data Interpolation on background grid is performed using the bilinear, bicubic and biharmonic spline methods Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 35/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio Outline of Steps Involved in Joint PDF Approximation 1 Approximate the JPDF f ˆ Q 22 with multivariate Gaussian Q 11 , ˆ Q 12 , ˆ (Central Limit Theorem + short-range correlations). 2 Evaluate the sequence of transformations: f ˆ Q 22 → f ˆ q off → f ˆ Q 11 , ˆ Q 12 , ˆ q diag , ˆ θ, ˆ R 3 Neglecting covariance-specific corrections in C ov Q a model independent approximation of the JPDF is obtained Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 36/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio Detection of Anisotropy Change in SIC2004 Data GDR Histogram: routine GDR Histogram: emergency (background) data (simulated release) data Histogram of cluster 1 Histogram of cluster 1 200 40 150 30 Frequency Frequency 100 20 10 50 0 0 60 80 100 120 140 160 0 500 1000 1500 field field Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 37/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio Anisotropy Estimates of GDR data - SIC 2004 Anisotropy statistics for “background” and “emergency” cases CTI estimates ˆ ˆ Anisotropy statistics R θ − 36 ◦ Background radioactivity 1.2 − 39 ◦ Simulated accidental release 0.68 Is the difference significant? Comparison of confidence regions for ˆ R , ˆ θ can provide an answer Petrakis and Hristopulos, arxiv: 1203.5010v1, 2012 Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 38/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio Anisotropy Estimates of GDR data - SIC 2004 Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Figure: Comparison of the JPDFs for the normal and emergency data sets Spatial Processes Based on Local-Interaction Energy Functionals 38/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio Non-Gaussian distributions: Discrete “Spin” & Gradient-Curvature Models Interpolation of missing lattice data: Definitions Define a series of discretization thresholds t q , q = 1 , . . . , N c Threshold-dependent sample S ( q ) ⊂ D & prediction domains P ( q ) ⊂ D Spin assignments: s ( q ) = 1 × sign ( X i − t q ) i Sample correlation energy E ( q ) ( S ( q ) ) = � s ( q ) s ( q ) � � i , j � , i = 1 , . . . , N i j Total correlation energy E ( q , m ) ( D ) = � s ( q , m ) s ( q , m ) � � i , j � , i = 1 , . . . , N + P i j Define cost functional (m: state index) � 2 U ( q , m ) = � 1 − E ( q , m ) ( D ) / E ( q ) ( S ( q ) ) E ( q ) ( S ( q ) ) � = 0 , U ( q , m ) = E ( q , m ) ( D ) 2 , E ( q ) ( S ( q ) ) = 0 . Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 39/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio Non-Gaussian distributions: Discrete “Spin” & Gradient-Curvature Models Schematic of spin state Schematic of interacting pairs assignment Figure: Majority rule assignment of initial spin values using adaptive stencil. NaN: sites of missing data Figure: Sample sites (solid circles) and nearest-neighbor pairs (solid circles linked with bold lines) Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 39/43

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