Statistics, data, and deterministic models NRCSE.
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Transcript of Statistics, data, and deterministic models NRCSE.
![Page 1: Statistics, data, and deterministic models NRCSE.](https://reader036.fdocuments.net/reader036/viewer/2022081516/56649d5f5503460f94a3fa5d/html5/thumbnails/1.jpg)
Statistics, data, and deterministic models
NRCSE
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Some issues in model assessment
Spatiotemporal misalignmentGrid boxes vs observations
Types of errorMeasurement error and bias
Model error
Approximation error
Manipulate data or model output?
Two case studies:SARMAP – kriging
MODELS-3 – Bayesian melding
Other uses of Bayesian hierarchical models
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Assessing the SARMAP model
60 days of hourly observations at 32 sites in Sacramento region
Hourly model runs for three “episodes”
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Task
Estimate from data the ozone level at x’s in a grid square. Use sum to estimate integral over grid square.
Issues:Transformation
Diurnal cycle
Temporal dependence
Spatial dependence
Space-time interaction
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Transformation
Heterogeneous variability–mean and variance positively related
Square root transformation
All modeling now on square root scale–approximately normal
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Diurnal cycle
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Temporal dependence
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Spatial dependence
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Estimating a grid square average
Estimate using
(not averages of squares of kriging estimates on the square root scale)
Vt (s) = Zt (s)
Vt (s) =μ t (s) +Wt (s)Wt (s) =α1(s)Wt−1(s) + α2 (s)Wt−2 (s) + Yt (s)
1
AVt
2 (s)dsA∫
1
ME Vt (s j )
2 data from 1,..., t{ }∑
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Looking at an episode
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Afternoon comparison
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Nighttime comparison
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A Bayesian approach
SARMAP study spatially data richIf spatially sparse data, how estimate grid squares?
P = Z + M + A + O = ()Z + B + E
P = process model outputO = observationsZ = truth
Calculate (Z |P,O) for prediction
Calculate (O |P, = 1, M = S = A = 0) for model assessment
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CASTNet and Models-3
CASTNet is a dry deposition network
Models-3 sophisticated air quality model
Average fluxes on 36x36 km2 grid
Weekly data and hourly output
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Estimated model bias
The multiplicatice bias is taken spatially constant (= 0.5). The additive bias E(M+A+) is spatially distributed.
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Assessing model fit
Predict CASTNet observation Oi from posterior mean of prediction using Models-3 output Pi and remaining observations O-I.
Average length of 90% credible intervals is 7 ppb
Average length using only Models-3 is 3.5 ppb
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Crossvalidation
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The Bayesian hierarchical approach
Three levels of modelling:
Data model:
f(data | process, parameters)
Process model:
f(process | parameters)
Parameter model:
f(parameters)
Use Bayes’ theorem to compute posterior
f(process, parameters | data)
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Some applications
Data assimilation
Satellite tracking
Precipitation measurement
Combination of data on different scales
Image analysis
Agricultural field trials
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Application to Models-3
where (I) are samples from the posterior distribution of
(Z(s0 ) P,O) ∝ f(Z(s0 ) P,O,)f( P,O)d∫≈
1M
f(Z(s0 P,O,(i) )i=1
M
∑
ME IL NC IN FL MI
CASTNet 0.15 3.29 0.90 3.14 0.57 1.02
Models-3 0.33 3.33 5.32 9.59 0.52 1.04
Adjusted
Models-3
0.12 2.88 1.09 3.12 0.44 1.01
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Predictions
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NCAR-GSP (IMAGe)
Theme for 2005: Data Assimilation in the Geosciences
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ENSO project
El Niño/Southern Oscillation is driven by surface temperature in tropical Pacific
Data 2ox2o monthly SST anomalies at 2261 locations; zonal 10m wind
Previous work indicates EOFs of SST may develop in a Markovian fashion
Forecast 7 months ahead uses data from Jan 70 through latest available.
Cressie-Wikle-Berliner (http://www.stat.ohio-state.edu/ ~sses/collab_enso.php)
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Model
Data model:
Process model:
Parameter model:
The current state is a mixture over three regimes (determined by SOI), with mixing probabilities that depend on the wind statistic
Standardize by subtracting climatology (monthly average 1971-2000)
Zt =Φat + νt
EOFs
a t+τ =μ t +Htat + ηt+τ
Ht =H(It, J t )regimes
winds
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Latest ENSO forecast
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Latest forecast with data
fore
cast
dat
a
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Relative performance
Performance measure for anomalies:
ave((forecast - data)2} over all pixels in Niño3.4-region
Relative Performance of Forecast A relative to Forecast B is
RP(A,B)=log(Perf B / Perf A)
RP(A,B)>0 indicates A better than B
Persistence: Predict using data 7 months ago
Climatology: Predict using 0
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Comparison to climatology and persistence