Málstofa í stærðfræði

### Speaker: Óli Páll Geirsson

### Title: Weak solution to stochastic partial differential equations and applications in spatial statistics

Location: Room **V-147** in building VR-II on the UI campus

Time: **Friday** March 21, 2014, at **12:00 to 13:00**.

#### Abstract:

The literature on spatial statistics is rich and recognized, where Gaussian fields play a dominant role in statistical modeling. Gaussian fields (GFs) are both practical and readily interpretable due to the flexible structure of the parameterization of the GF. One of the most commonly used covariance model in the Gaussian field class is the Matérn covariance model which has gained recognition within statistical climatology in recent years.

Although Gaussian fields suit well from both analytic and practical point of view, they become computationally demanding as data sets get larger as the covariance matrices tend to be fully populated. Gaussian fields can be approximated with Gaussian Markov random fields (GMRF), which increases the speed of computation significantly. Even though GMRFs have very good computational properties, using them for involved spatial models has not been feasible as there has been no good way to parametrize the precision matrix of a GMRF to achieve a predefined behaviour in terms of correlation between two sites and to control marginal variances.

In recent work it has been shown that using an approximate stochastic weak solution to (linear) stochastic partial differential equations (SPDEs), it is possible to provide an explicit link, for any triangulation over a spatial domain of interest, between GFs and GMRFs. The consequence is that we can take the best from the both worlds and do the modelling using GFs but do the computations using GMRFs. The approximate is then used to construct a GMRF representation of the desired GF on the triangulated mesh. This allows for continuous spatial predictions by choosing appropriate basis function for the approximation.

#### Note that the time, date and location is different from the usual one.

Málstofa í stærðfræði

### Speaker: Óli Páll Geirsson

### Title: Weak solution to stochastic partial differential equations and applications in spatial statistics

Location: Room **V-147** in building VR-II on the UI campus

Time: **Friday** March 21, 2014, at **12:00 to 13:00**.

#### Abstract:

The literature on spatial statistics is rich and recognized, where Gaussian fields play a dominant role in statistical modeling. Gaussian fields (GFs) are both practical and readily interpretable due to the flexible structure of the parameterization of the GF. One of the most commonly used covariance model in the Gaussian field class is the Matérn covariance model which has gained recognition within statistical climatology in recent years.

Although Gaussian fields suit well from both analytic and practical point of view, they become computationally demanding as data sets get larger as the covariance matrices tend to be fully populated. Gaussian fields can be approximated with Gaussian Markov random fields (GMRF), which increases the speed of computation significantly. Even though GMRFs have very good computational properties, using them for involved spatial models has not been feasible as there has been no good way to parametrize the precision matrix of a GMRF to achieve a predefined behaviour in terms of correlation between two sites and to control marginal variances.

In recent work it has been shown that using an approximate stochastic weak solution to (linear) stochastic partial differential equations (SPDEs), it is possible to provide an explicit link, for any triangulation over a spatial domain of interest, between GFs and GMRFs. The consequence is that we can take the best from the both worlds and do the modelling using GFs but do the computations using GMRFs. The approximate is then used to construct a GMRF representation of the desired GF on the triangulated mesh. This allows for continuous spatial predictions by choosing appropriate basis function for the approximation.