Fusing space-time data under measurement error for computer model - PowerPoint PPT Presentation

Fusing space-time data under measurement error for computer model output Veronica J. Berrocal ( vjb2@stat.duke.edu ) SAMSI joint work with Alan E. Gelfand and David M. Holland Veronica J. Berrocal Fusing space-time data under measurement error

Introduction • In many environmental disciplines data come from two sources: monitoring networks and numerical models • Numerical models are deterministic mathematical models used to predict environmental spatio-temporal processes • Describe the underlying physical and chemical processes via partial differential equations • Equations solved via numerical methods by discretizing space and time • Predictions are given in terms of averages over grid cells Veronica J. Berrocal Fusing space-time data under measurement error

Introduction 45 45 40 40 Latitude Latitude 35 35 30 30 25 25 − 100 − 95 − 90 − 85 − 80 − 75 − 70 − 100 − 95 − 90 − 85 − 80 − 75 − 70 Longitude Longitude 20 30 40 50 60 70 80 90 20 30 40 50 60 70 80 90 (a) Observed ozone concentration (b) Predicted ozone concentration • Sparse locations • Large spatial domains • Missing data • No missingness • Essentially, true value • Calibration concerns Veronica J. Berrocal Fusing space-time data under measurement error

Our goal • Fuse the two sources of data • Address the following issues • Spatial scale of outputs from numerical models • Outputs from numerical models are given in terms of predictions over grid cells, but predictions at points are more useful • Calibration of numerical models • Correct outputs from numerical models • Problem: Comparing averages over grid cells with points Veronica J. Berrocal Fusing space-time data under measurement error

Downscaler: main idea 45 45 40 40 Latitude Latitude 35 35 30 30 25 25 − 100 − 95 − 90 − 85 − 80 − 75 − 70 − 100 − 95 − 90 − 85 − 80 − 75 − 70 Longitude Longitude 20 30 40 50 60 70 80 90 20 30 40 50 60 70 80 90 (a) Observed ozone concentration (b) Predicted ozone concentration Veronica J. Berrocal Fusing space-time data under measurement error

Downscaler: main idea 40.0 40.0 39.5 39.5 Latitude Latitude 39.0 39.0 38.5 38.5 38.0 38.0 − 95.0 − 94.5 − 94.0 − 93.5 − 93.0 − 95.0 − 94.5 − 94.0 − 93.5 − 93.0 Longitude Longitude 20 30 40 50 60 70 80 90 20 30 40 50 60 70 80 90 (a) Observed ozone concentration (b) Predicted ozone concentration Veronica J. Berrocal Fusing space-time data under measurement error

Downscaler: main idea 40.0 40.0 39.5 39.5 Latitude Latitude 39.0 39.0 38.5 38.5 38.0 38.0 − 95.0 − 94.5 − 94.0 − 93.5 − 93.0 − 95.0 − 94.5 − 94.0 − 93.5 − 93.0 Longitude Longitude 20 30 40 50 60 70 80 90 20 30 40 50 60 70 80 90 (a) Observed ozone concentration (b) Predicted ozone concentration To each point s in the domain S with observation Y ( s ) we associate the numerical model output at grid cell B , x ( B ), where B is such that s ∈ B : Y ( s ) ⇐ ⇒ x ( B ) Veronica J. Berrocal Fusing space-time data under measurement error

Downscaler • Time t is fixed. Y ( s ) observation at point s , x ( B ) numerical model output at grid cell B . For s in B : ε ( s ) ind Y ( s ) = ˜ β 0 ( s )+ ˜ ∼ N (0 , τ 2 ) β 1 ( s ) x ( B )+ ε ( s ) with ˜ β i ( s ) = β i + β i ( s ), i=0,1. • β 0 ( s ) and β 1 ( s ) correlated mean-zero GP. Veronica J. Berrocal Fusing space-time data under measurement error

Downscaler • Time t is fixed. Y ( s ) observation at point s , x ( B ) numerical model output at grid cell B . For s in B : ε ( s ) ind Y ( s ) = ˜ β 0 ( s )+ ˜ ∼ N (0 , τ 2 ) β 1 ( s ) x ( B )+ ε ( s ) with ˜ β i ( s ) = β i + β i ( s ), i=0,1. • β 0 ( s ) and β 1 ( s ) correlated mean-zero GP. • Extension to space-time: For s in B and for each t ε ( s , t ) ind Y ( s , t ) = ˜ β 0 ( s , t )+˜ ∼ N (0 , τ 2 ) β 1 ( s , t ) x ( B , t )+ ε ( s , t ) with ˜ β i ( s , t ) = β i , t + β i ( s , t ), i=0,1. • Temporal dependence in β i , t and β i ( s , t ): dynamic or independent Veronica J. Berrocal Fusing space-time data under measurement error

Downscaler • Driven by true station data rather than uncalibrated model output • Computationally feasible also for large spatial domains • Allows local calibration of the numerical model output • Endows the spatial process Y ( s ) with a non-stationary covariance structure • Straightforward prediction at an unmonitored sites • Better predictive performance than other methods (geostatistical and model-based) Veronica J. Berrocal Fusing space-time data under measurement error

Extending the downscaler • Improve the input to the downscaler provided by the numerical model output • accounting for uncertainty in the numerical model output • accounting for uncertainty in the association x ( B ) �− → Y ( s ) with s in B Veronica J. Berrocal Fusing space-time data under measurement error

Extending the downscaler • Improve the input to the downscaler provided by the numerical model output • accounting for uncertainty in the numerical model output • accounting for uncertainty in the association x ( B ) �− → Y ( s ) with s in B • Will consider two possibilities: • a Measurement Error Model (MEM) • a Smoother with spatially-varying weights Veronica J. Berrocal Fusing space-time data under measurement error

Static setting Veronica J. Berrocal Fusing space-time data under measurement error

A MEM for outputs from computer model • Downscaler: Y ( s ) = ˜ β 0 ( s )+ ˜ β 1 ( s ) x ( B )+ ε ( s ) Veronica J. Berrocal Fusing space-time data under measurement error

A MEM for outputs from computer model • Downscaler: Y ( s ) = ˜ β 0 ( s )+ ˜ β 1 ( s ) x ( B )+ ε ( s ) • Model the numerical model output as a stochastic process η ( B ) ind x ( B ) = ˜ ∼ N (0 , σ 2 ) V ( B )+ η ( B ) with ˜ V ( B ) = µ + V ( B ) and V ( B ) ∼ ICAR ( ξ 2 , W ). Veronica J. Berrocal Fusing space-time data under measurement error

A MEM for outputs from computer model • Downscaler: Y ( s ) = ˜ β 0 ( s )+ ˜ β 1 ( s ) x ( B )+ ε ( s ) • Model the numerical model output as a stochastic process η ( B ) ind x ( B ) = ˜ ∼ N (0 , σ 2 ) V ( B )+ η ( B ) with ˜ V ( B ) = µ + V ( B ) and V ( B ) ∼ ICAR ( ξ 2 , W ). For s ∈ B : ε ( s ) ind Y ( s ) = ˜ β 0 ( s )+ β 1 ˜ ∼ N (0 , τ 2 ) V ( B )+ ε ( s ) with ˜ β 0 ( s ) = β 0 + β 0 ( s ) and β 0 ( s ) mean-zero GP with exponential covariance function. Veronica J. Berrocal Fusing space-time data under measurement error

A MEM for outputs from computer model • Downscaler: Y ( s ) = ˜ β 0 ( s )+ ˜ β 1 ( s ) x ( B )+ ε ( s ) • Model the numerical model output as a stochastic process η ( B ) ind x ( B ) = ˜ ∼ N (0 , σ 2 ) V ( B )+ η ( B ) with ˜ V ( B ) = µ + V ( B ) and V ( B ) ∼ ICAR ( ξ 2 , W ). For s ∈ B : ε ( s ) ind Y ( s ) = ˜ β 0 ( s )+ β 1 ˜ ∼ N (0 , τ 2 ) V ( B )+ ε ( s ) with ˜ β 0 ( s ) = β 0 + β 0 ( s ) and β 0 ( s ) mean-zero GP with exponential covariance function. • Then: X MEM V downscaler − ˜ ← − → Y Veronica J. Berrocal Fusing space-time data under measurement error

A MEM for outputs from computer model 45 45 40 40 Latitude Latitude 35 35 30 30 25 25 − 100 − 95 − 90 − 85 − 80 − 75 − 70 − 100 − 95 − 90 − 85 − 80 − 75 − 70 Longitude Longitude 20 30 40 50 60 70 80 90 20 30 40 50 60 70 80 90 (a) Observed ozone concentration (b) Predicted ozone concentration Veronica J. Berrocal Fusing space-time data under measurement error

A MEM for outputs from computer model 40.0 40.0 39.5 39.5 Latitude Latitude 39.0 39.0 38.5 38.5 38.0 38.0 − 95.0 − 94.5 − 94.0 − 93.5 − 93.0 − 95.0 − 94.5 − 94.0 − 93.5 − 93.0 Longitude Longitude 20 30 40 50 60 70 80 90 20 30 40 50 60 70 80 90 (a) Observed ozone concentration (b) Predicted ozone concentration Veronica J. Berrocal Fusing space-time data under measurement error

A MEM for outputs from computer model 40.0 40.0 39.5 39.5 Latitude Latitude 39.0 39.0 38.5 38.5 38.0 38.0 − 95.0 − 94.5 − 94.0 − 93.5 − 93.0 − 95.0 − 94.5 − 94.0 − 93.5 − 93.0 Longitude Longitude 20 30 40 50 60 70 80 90 20 30 40 50 60 70 80 90 (a) Observed ozone concentration (b) Predicted ozone concentration Veronica J. Berrocal Fusing space-time data under measurement error

A MEM for outputs from computer model 40.0 40.0 39.5 39.5 Latitude Latitude 39.0 39.0 38.5 38.5 38.0 38.0 − 95.0 − 94.5 − 94.0 − 93.5 − 93.0 − 95.0 − 94.5 − 94.0 − 93.5 − 93.0 Longitude Longitude 20 30 40 50 60 70 80 90 20 30 40 50 60 70 80 90 (a) Observed ozone concentration (b) Predicted ozone concentration Veronica J. Berrocal Fusing space-time data under measurement error