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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

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SLIDE 1

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

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SLIDE 2

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

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SLIDE 3

Introduction

−100 −95 −90 −85 −80 −75 −70 25 30 35 40 45 Longitude Latitude 20 30 40 50 60 70 80 90

(a) Observed ozone concentration

  • Sparse locations
  • Missing data
  • Essentially, true value

−100 −95 −90 −85 −80 −75 −70 25 30 35 40 45 Longitude Latitude 20 30 40 50 60 70 80 90

(b) Predicted ozone concentration

  • Large spatial domains
  • No missingness
  • Calibration concerns

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

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SLIDE 4

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

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SLIDE 5

Downscaler: main idea

−100 −95 −90 −85 −80 −75 −70 25 30 35 40 45 Longitude Latitude 20 30 40 50 60 70 80 90

(a) Observed ozone concentration

−100 −95 −90 −85 −80 −75 −70 25 30 35 40 45 Longitude Latitude 20 30 40 50 60 70 80 90

(b) Predicted ozone concentration

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

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SLIDE 6

Downscaler: main idea

−95.0 −94.5 −94.0 −93.5 −93.0 38.0 38.5 39.0 39.5 40.0 Longitude Latitude 20 30 40 50 60 70 80 90

(a) Observed ozone concentration

−95.0 −94.5 −94.0 −93.5 −93.0 38.0 38.5 39.0 39.5 40.0 Longitude Latitude 20 30 40 50 60 70 80 90

(b) Predicted ozone concentration

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

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SLIDE 7

Downscaler: main idea

−95.0 −94.5 −94.0 −93.5 −93.0 38.0 38.5 39.0 39.5 40.0 Longitude Latitude 20 30 40 50 60 70 80 90

(a) Observed ozone concentration

−95.0 −94.5 −94.0 −93.5 −93.0 38.0 38.5 39.0 39.5 40.0 Longitude Latitude 20 30 40 50 60 70 80 90

(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

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SLIDE 8

Downscaler

  • Time t is fixed. Y (s) observation at point s, x(B) numerical

model output at grid cell B. For s in B: Y (s) = ˜ β0(s)+ ˜ β1(s)x(B)+ε(s) ε(s) ind ∼ N(0,τ2) with ˜ βi(s) = βi +βi(s), i=0,1.

  • β0(s) and β1(s) correlated mean-zero GP.

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

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SLIDE 9

Downscaler

  • Time t is fixed. Y (s) observation at point s, x(B) numerical

model output at grid cell B. For s in B: Y (s) = ˜ β0(s)+ ˜ β1(s)x(B)+ε(s) ε(s) ind ∼ N(0,τ2) with ˜ βi(s) = βi +βi(s), i=0,1.

  • β0(s) and β1(s) correlated mean-zero GP.
  • Extension to space-time: For s in B and for each t

Y (s,t) = ˜ β0(s,t)+˜ β1(s,t)x(B,t)+ε(s,t) ε(s,t) ind ∼ N(0,τ2) with ˜ βi(s,t) = βi,t +βi(s,t), i=0,1.

  • Temporal dependence in βi,t and βi(s,t): dynamic or

independent

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

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SLIDE 10

Downscaler

  • Driven by true station data rather than uncalibrated model
  • utput
  • 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

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SLIDE 11

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

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SLIDE 12

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

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Static setting

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

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A MEM for outputs from computer model

  • Downscaler: Y (s) = ˜

β0(s)+ ˜ β1(s)x(B)+ε(s)

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

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SLIDE 15

A MEM for outputs from computer model

  • Downscaler: Y (s) = ˜

β0(s)+ ˜ β1(s)x(B)+ε(s)

  • Model the numerical model output as a stochastic process

x(B) = ˜ V (B)+η(B) η(B) ind ∼ N(0,σ2) with ˜ V (B) = µ+V (B) and V (B) ∼ ICAR(ξ2,W ).

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

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SLIDE 16

A MEM for outputs from computer model

  • Downscaler: Y (s) = ˜

β0(s)+ ˜ β1(s)x(B)+ε(s)

  • Model the numerical model output as a stochastic process

x(B) = ˜ V (B)+η(B) η(B) ind ∼ N(0,σ2) with ˜ V (B) = µ+V (B) and V (B) ∼ ICAR(ξ2,W ). For s ∈ B: Y (s) = ˜ β0(s)+β1 ˜ V (B)+ε(s) ε(s) ind ∼ N(0,τ2) with ˜ β0(s) = β0 +β0(s) and β0(s) mean-zero GP with exponential covariance function.

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

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SLIDE 17

A MEM for outputs from computer model

  • Downscaler: Y (s) = ˜

β0(s)+ ˜ β1(s)x(B)+ε(s)

  • Model the numerical model output as a stochastic process

x(B) = ˜ V (B)+η(B) η(B) ind ∼ N(0,σ2) with ˜ V (B) = µ+V (B) and V (B) ∼ ICAR(ξ2,W ). For s ∈ B: Y (s) = ˜ β0(s)+β1 ˜ V (B)+ε(s) ε(s) ind ∼ N(0,τ2) with ˜ β0(s) = β0 +β0(s) and β0(s) mean-zero GP with exponential covariance function.

  • Then: X MEM

← − ˜ V downscaler − → Y

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

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A MEM for outputs from computer model

−100 −95 −90 −85 −80 −75 −70 25 30 35 40 45 Longitude Latitude 20 30 40 50 60 70 80 90

(a) Observed ozone concentration

−100 −95 −90 −85 −80 −75 −70 25 30 35 40 45 Longitude Latitude 20 30 40 50 60 70 80 90

(b) Predicted ozone concentration

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

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SLIDE 19

A MEM for outputs from computer model

−95.0 −94.5 −94.0 −93.5 −93.0 38.0 38.5 39.0 39.5 40.0 Longitude Latitude 20 30 40 50 60 70 80 90

(a) Observed ozone concentration

−95.0 −94.5 −94.0 −93.5 −93.0 38.0 38.5 39.0 39.5 40.0 Longitude Latitude 20 30 40 50 60 70 80 90

(b) Predicted ozone concentration

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

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SLIDE 20

A MEM for outputs from computer model

−95.0 −94.5 −94.0 −93.5 −93.0 38.0 38.5 39.0 39.5 40.0 Longitude Latitude 20 30 40 50 60 70 80 90

(a) Observed ozone concentration

−95.0 −94.5 −94.0 −93.5 −93.0 38.0 38.5 39.0 39.5 40.0 Longitude Latitude 20 30 40 50 60 70 80 90

(b) Predicted ozone concentration

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

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SLIDE 21

A MEM for outputs from computer model

−95.0 −94.5 −94.0 −93.5 −93.0 38.0 38.5 39.0 39.5 40.0 Longitude Latitude 20 30 40 50 60 70 80 90

(a) Observed ozone concentration

−95.0 −94.5 −94.0 −93.5 −93.0 38.0 38.5 39.0 39.5 40.0 Longitude Latitude 20 30 40 50 60 70 80 90

(b) Predicted ozone concentration

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

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SLIDE 22

A MEM for outputs from computer model

−95.0 −94.5 −94.0 −93.5 −93.0 38.0 38.5 39.0 39.5 40.0 Longitude Latitude 20 30 40 50 60 70 80 90

(a) Observed ozone concentration

−95.0 −94.5 −94.0 −93.5 −93.0 38.0 38.5 39.0 39.5 40.0 Longitude Latitude 20 30 40 50 60 70 80 90 1/5 1/5 1/5 1/5 1/5

(b) Predicted ozone concentration

To each point s in the domain S with observation Y (s) we associate the latent Gaussian Markov Random Field ˜ V (B), where B is such that s ∈ B and ˜ V (B) also associated with x(B) Y (s) x(B⋆),B⋆ ∈ δB

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

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SLIDE 23

Smoother with spatially-varying weights

  • Downscaler: Y (s) = ˜

β0(s)+ ˜ β1(s)x(B)+ε(s)

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

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SLIDE 24

Smoother with spatially-varying weights

  • Downscaler: Y (s) = ˜

β0(s)+ ˜ β1(s)x(B)+ε(s)

  • The numerical model output, {x(B)}, is NOT modeled as a

stochastic process. For s ∈ B : Y (s) = ˜ β0(s)+β1˜ x(s)+ε(s) ε(s) ind ∼ N(0,τ2) with ˜ x(s) = ∑

g k=1 wk(s)x(Bk), and wk(s) random and

spatially-varying.

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

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SLIDE 25

Smoother with spatially-varying weights

  • Downscaler: Y (s) = ˜

β0(s)+ ˜ β1(s)x(B)+ε(s)

  • The numerical model output, {x(B)}, is NOT modeled as a

stochastic process. For s ∈ B : Y (s) = ˜ β0(s)+β1˜ x(s)+ε(s) ε(s) ind ∼ N(0,τ2) with ˜ x(s) = ∑

g k=1 wk(s)x(Bk), and wk(s) random and

spatially-varying. Let rk, k = 1,...,g be the centroids of the numerical model grid cells Bk, then wk(s) :=

K (s −rk;ψ)·exp(Q(rk))

∑

g l=1K (s −rl;ψ)·exp(Q(rl))

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

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SLIDE 26

Smoother with spatially-varying weights

  • Downscaler: Y (s) = ˜

β0(s)+ ˜ β1(s)x(B)+ε(s)

  • The numerical model output, {x(B)}, is NOT modeled as a

stochastic process. For s ∈ B : Y (s) = ˜ β0(s)+β1˜ x(s)+ε(s) ε(s) ind ∼ N(0,τ2) with ˜ x(s) = ∑

g k=1 wk(s)x(Bk), and wk(s) random and

spatially-varying. Let rk, k = 1,...,g be the centroids of the numerical model grid cells Bk, then wk(s) :=

K (s −rk;ψ)·exp(Q(rk))

∑

g l=1K (s −rl;ψ)·exp(Q(rl))

  • Q(·) mean-zero GP with variance σ2

Q, exponential covariance

function.

  • K (s −rk;ψ) = exp(−ψs −rk).

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

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SLIDE 27

Smoother with spatially-varying weights

−100 −95 −90 −85 −80 −75 −70 25 30 35 40 45 Longitude Latitude 20 30 40 50 60 70 80 90

(a) Observed ozone concentration

−100 −95 −90 −85 −80 −75 −70 25 30 35 40 45 Longitude Latitude 20 30 40 50 60 70 80 90

(b) Predicted ozone concentration

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

slide-28
SLIDE 28

Smoother with spatially-varying weights

−95.0 −94.5 −94.0 −93.5 −93.0 38.0 38.5 39.0 39.5 40.0 Longitude Latitude 20 30 40 50 60 70 80 90

(a) Observed ozone concentration

−95.0 −94.5 −94.0 −93.5 −93.0 38.0 38.5 39.0 39.5 40.0 Longitude Latitude 20 30 40 50 60 70 80 90

(b) Predicted ozone concentration

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

slide-29
SLIDE 29

Smoother with spatially-varying weights

−95.0 −94.5 −94.0 −93.5 −93.0 38.0 38.5 39.0 39.5 40.0 Longitude Latitude 20 30 40 50 60 70 80 90

(a) Observed ozone concentration

−95.0 −94.5 −94.0 −93.5 −93.0 38.0 38.5 39.0 39.5 40.0 Longitude Latitude 20 30 40 50 60 70 80 90

(b) Predicted ozone concentration

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

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SLIDE 30

Smoother with spatially-varying weights

−95.0 −94.5 −94.0 −93.5 −93.0 38.0 38.5 39.0 39.5 40.0 Longitude Latitude 20 30 40 50 60 70 80 90

(a) Observed ozone concentration

−95.0 −94.5 −94.0 −93.5 −93.0 38.0 38.5 39.0 39.5 40.0 Longitude Latitude 20 30 40 50 60 70 80 90

1 2 3 94 95 96 97 498 499 500 w_6669 w_6670 w_6671 w_6672 w_6673 w_6674 w_6675 w_6676 w_6677 w_6678 w_6679 w_6680 w_6681 w_6682 w_6683 w_6684 w_6685 w_6686 w_6687 w_6688 w_6857 w_6858 w_6859 w_6860 w_6861 w_6862 w_6863 w_6864 w_6865 w_6866 w_6867 w_6868 w_6869 w_6870 w_6871 w_6872 w_6873 w_6874 w_6875 w_6876 w_7045 w_7046 w_7047 w_7048 w_7049 w_7050 w_7051 w_7052 w_7053 w_7054 w_7055 w_7056 w_7057 w_7058 w_7059 w_7060 w_7061 w_7062 w_7063 w_7064 w_7233 w_7234 w_7235 w_7236 w_7237 w_7238 w_7239 w_7240 w_7241 w_7242 w_7243 w_7244 w_7245 w_7246 w_7247 w_7248 w_7249 w_7250 w_7251 w_7252 w_7421 w_7422 w_7423 w_7424 w_7425 w_7426 w_7427 w_7428 w_7429 w_7430 w_7431 w_7432 w_7433 w_7434 w_7435 w_7436 w_7437 w_7438 w_7439 w_7440 w_7609 w_7610 w_7611 w_7612 w_7613 w_7614 w_7615 w_7616 w_7617 w_7618 w_7619 w_7620 w_7621 w_7622 w_7623 w_7624 w_7625 w_7626 w_7627 w_7628 w_7797 w_7798 w_7799 w_7800 w_7801 w_7802 w_7803 w_7804 w_7805 w_7806 w_7807 w_7808 w_7809 w_7810 w_7811 w_7812 w_7813 w_7814 w_7815 w_7816 w_7985 w_7986 w_7987 w_7988 w_7989 w_7990 w_7991 w_7992 w_7993 w_7994 w_7995 w_7996 w_7997 w_7998 w_7999 w_8000 w_8001 w_8002 w_8003 w_8004 w_8173 w_8174 w_8175 w_8176 w_8177 w_8178 w_8179 w_8180 w_8181 w_8182 w_8183 w_8184 w_8185 w_8186 w_8187 w_8188 w_8189 w_8190 w_8191 w_8192 w_8361 w_8362 w_8363 w_8364 w_8365 w_8366 w_8367 w_8368 w_8369 w_8370 w_8371 w_8372 w_8373 w_8374 w_8375 w_8376 w_8377 w_8378 w_8379 w_8380 w_8549 w_8550 w_8551 w_8552 w_8553 w_8554 w_8555 w_8556 w_8557 w_8558 w_8559 w_8560 w_8561 w_8562 w_8563 w_8564 w_8565 w_8566 w_8567 w_8568 w_8737 w_8738 w_8739 w_8740 w_8741 w_8742 w_8743 w_8744 w_8745 w_8746 w_8747 w_8748 w_8749 w_8750 w_8751 w_8752 w_8753 w_8754 w_8755 w_8756 w_8925 w_8926 w_8927 w_8928 w_8929 w_8930 w_8931 w_8932 w_8933 w_8934 w_8935 w_8936 w_8937 w_8938 w_8939 w_8940 w_8941 w_8942 w_8943 w_8944 w_9113 w_9114 w_9115 w_9116 w_9117 w_9118 w_9119 w_9120 w_9121 w_9122 w_9123 w_9124 w_9125 w_9126 w_9127 w_9128 w_9129 w_9130 w_9131 w_9132 w 9301 w_9302 w_9303 w_9304 w_9305 w_9306 w_9307 w_9308 w_9309 w_9310 w_9311 w_9312 w_9313 w_9314 w_9315 w_9316 w_9317 w_9318 w_9319 w_9320 w 948 w_949 w_94 w_94 w_9 w_ w_ w_ w w

(b) Predicted ozone concentration

To each point s in the domain S with observation Y (s) we associate ˜ x(s), where ˜ x(s)= ∑

g k=1 wk(s)x(Bk)

Y (s) − → ˜ x(s)

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

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SLIDE 31

Spatio-temporal setting

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

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A MEM for outputs from computer model

  • Y (s,t) observations at s and time t, x(B,t) numerical model
  • utput at grid cell B and time t

x(B,t) = ˜ V (B,t)+η(B,t) η(B,t) ind ∼ N(0,σ2) with ˜ V (B,t) = µt +V (B,t) and Vt = {V (B,t)}.

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

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SLIDE 33

A MEM for outputs from computer model

  • Y (s,t) observations at s and time t, x(B,t) numerical model
  • utput at grid cell B and time t

x(B,t) = ˜ V (B,t)+η(B,t) η(B,t) ind ∼ N(0,σ2) with ˜ V (B,t) = µt +V (B,t) and Vt = {V (B,t)}. For s ∈ B: Y (s,t) = ˜ β0(s,t)+β1,t ˜ V (B,t)+ε(s,t) ε(s,t) ind ∼ N(0,τ2) with ˜ β0(s,t) = β0,t +β0(s,t).

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

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SLIDE 34

A MEM for outputs from computer model

  • Y (s,t) observations at s and time t, x(B,t) numerical model
  • utput at grid cell B and time t

x(B,t) = ˜ V (B,t)+η(B,t) η(B,t) ind ∼ N(0,σ2) with ˜ V (B,t) = µt +V (B,t) and Vt = {V (B,t)}. For s ∈ B: Y (s,t) = ˜ β0(s,t)+β1,t ˜ V (B,t)+ε(s,t) ε(s,t) ind ∼ N(0,τ2) with ˜ β0(s,t) = β0,t +β0(s,t).

  • Temporal dependence in βi,t, i = 0,1, β0(s,t) and Vt:

either (i) independent in time, or (ii) dynamic.

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

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SLIDE 35

Smoother with spatially-varying weights

  • Y (s,t) observations at s and time t, x(B,t) numerical model
  • utput at grid cell B and time t

For s ∈ B : Y (s,t) = ˜ β0(s,t)+β1,t˜ x(s,t)+ε(s,t) ε(s,t) ind ∼ N(0,τ2) with ˜ β0(s,t) = β0,t +β0(s,t), ˜ x(s,t) = ∑

g k=1 wk,t(s)x(Bk,t), and

wk,t(s) random and varying both in space and time.

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

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SLIDE 36

Smoother with spatially-varying weights

  • Y (s,t) observations at s and time t, x(B,t) numerical model
  • utput at grid cell B and time t

For s ∈ B : Y (s,t) = ˜ β0(s,t)+β1,t˜ x(s,t)+ε(s,t) ε(s,t) ind ∼ N(0,τ2) with ˜ β0(s,t) = β0,t +β0(s,t), ˜ x(s,t) = ∑

g k=1 wk,t(s)x(Bk,t), and

wk,t(s) random and varying both in space and time. wk,t(s) :=

K (s −rk;ψ)·exp(Q(rk,t))

∑

g l=1K (s −rl;ψ)·exp(Q(rl,t))

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

slide-37
SLIDE 37

Smoother with spatially-varying weights

  • Y (s,t) observations at s and time t, x(B,t) numerical model
  • utput at grid cell B and time t

For s ∈ B : Y (s,t) = ˜ β0(s,t)+β1,t˜ x(s,t)+ε(s,t) ε(s,t) ind ∼ N(0,τ2) with ˜ β0(s,t) = β0,t +β0(s,t), ˜ x(s,t) = ∑

g k=1 wk,t(s)x(Bk,t), and

wk,t(s) random and varying both in space and time. wk,t(s) :=

K (s −rk;ψ)·exp(Q(rk,t))

∑

g l=1K (s −rl;ψ)·exp(Q(rl,t))

  • Temporal dependence in βi,t, i = 0,1, β0(s,t) and Q(s,t):

either (i) independent in time, or (ii) dynamic.

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

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SLIDE 38

Application

−100 −95 −90 −85 −80 −75 −70 25 30 35 40 45 50 Longitude Latitude Training sites Validation sites

  • Daily maximum 8-hour ozone

concentration (ppb): observations and CMAQ model output.

  • Model output on 12-km grid cells

(g=40,044).

  • Period: June 1- August 31, 2001
  • 800 monitoring sites: 400 training sites

and 400 validation sites

  • Transformation to square root scale;

predictions were back-transformed

  • Time-varying parameters: independent

in time

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

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SLIDE 39

Results

  • Assessed predictive performance at the 400 validation sites for

the period June 1 - August 31, 2001.

Coverage

  • Avg. length

Method MSPE MAPE 95% PI 95% PI Downscaler 59.9 5.6 94.5% 31.6 MEM 49.3 5.1 95.0% 29.1 Smoother 48.1 5.0 95.0% 28.8

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

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SLIDE 40

Results

  • Downscaler: x(B,t) −

→ Y (s,t)

  • MEM: ˜

V (B,t) − → Y (s,t)

  • Smoother: ˜

x(s,t) − → Y (s,t)

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

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SLIDE 41

Results

  • Downscaler: x(B,t) −

→ Y (s,t)

  • MEM: ˜

V (B,t) − → Y (s,t)

  • Smoother: ˜

x(s,t) − → Y (s,t)

  • Computed the correlation between Y (s,t) and: (i) x(B,t) and the

posterior predictive mean of (ii) ˜ V (B,t) and (iii) ˜ x(s,t).

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

slide-42
SLIDE 42

Results

  • Downscaler: x(B,t) −

→ Y (s,t)

  • MEM: ˜

V (B,t) − → Y (s,t)

  • Smoother: ˜

x(s,t) − → Y (s,t)

  • Computed the correlation between Y (s,t) and: (i) x(B,t) and the

posterior predictive mean of (ii) ˜ V (B,t) and (iii) ˜ x(s,t). Posterior mean of Posterior mean Day x(B,t) ˜ V (B,t)

  • f ˜

x(s,t) July 4, 2001 0.55 0.62 0.63 July 20, 2001 0.75 0.79 0.80 August 9, 2001 0.78 0.84 0.85

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

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SLIDE 43

Results

  • Posterior mean of the random and spatially and temporally- varying

weights wk,t(s) for July 4, 2001 at four sites.

−100 −95 −90 −85 −80 −75 −70 25 30 35 40 45 Longitude Latitude

s1 s2 s3 s4

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

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SLIDE 44

Results

  • Posterior mean of the random and spatially and temporally- varying

weights wk,t(s) for July 4, 2001 at four sites.

−75.0 −74.5 −74.0 −73.5 40.5 41.0 41.5 42.0 42.5 Longitude Latitude 0.00 0.05 0.10 0.15 −83.5 −83.0 −82.5 −82.0 −81.5 29.5 30.0 30.5 31.0 Longitude Latitude 0.00 0.05 0.10 0.15 −93.0 −92.5 −92.0 −91.5 34.0 34.5 35.0 35.5 Longitude Latitude 0.00 0.05 0.10 0.15 −86.5 −86.0 −85.5 −85.0 −84.5 41.5 42.0 42.5 43.0 Longitude Latitude 0.00 0.05 0.10 0.15

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

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SLIDE 45

Results

  • Posterior mean of the random and spatially and temporally- varying

weights wk,t(s) for July 4, 2001, July 20, and August 9, 2001 at two sites.

−83.5 −83.0 −82.5 −82.0 −81.5 29.5 30.0 30.5 31.0 Longitude Latitude 0.00 0.05 0.10 0.15 −83.5 −83.0 −82.5 −82.0 −81.5 29.5 30.0 30.5 31.0 Longitude Latitude 0.00 0.05 0.10 0.15 −83.5 −83.0 −82.5 −82.0 −81.5 29.5 30.0 30.5 31.0 Longitude Latitude 0.00 0.05 0.10 0.15 −93.0 −92.5 −92.0 −91.5 34.0 34.5 35.0 35.5 Longitude Latitude 0.00 0.05 0.10 0.15 −93.0 −92.5 −92.0 −91.5 34.0 34.5 35.0 35.5 Longitude Latitude 0.00 0.05 0.10 0.15 −93.0 −92.5 −92.0 −91.5 34.0 34.5 35.0 35.5 Longitude Latitude 0.00 0.05 0.10 0.15

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

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SLIDE 46

Conclusions

  • Presented two methods to extend the downscaler model to fuse

numerical model ouput and monitoring data

  • Both are computationally feasibile even for large computer model
  • utput (smoother + predictive processes (Banerjee et al., JRSS B,

2008))

  • Smoother with spatially-varying weights yielded better predictions

than other methods

  • Clear directionality in the weights of the smoother model

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