Statistics, data, and deterministic models NRCSE.
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Transcript of Statistics, data, and deterministic models NRCSE.
Statistics, data, and deterministic models
NRCSE
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
Assessing the SARMAP model
60 days of hourly observations at 32 sites in Sacramento region
Hourly model runs for three “episodes”
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
Transformation
Heterogeneous variability–mean and variance positively related
Square root transformation
All modeling now on square root scale–approximately normal
Diurnal cycle
Temporal dependence
Spatial dependence
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{ }∑
Looking at an episode
Afternoon comparison
Nighttime comparison
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
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
Estimated model bias
The multiplicatice bias is taken spatially constant (= 0.5). The additive bias E(M+A+) is spatially distributed.
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
Crossvalidation
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)
Some applications
Data assimilation
Satellite tracking
Precipitation measurement
Combination of data on different scales
Image analysis
Agricultural field trials
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
Predictions
NCAR-GSP (IMAGe)
Theme for 2005: Data Assimilation in the Geosciences
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)
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
Latest ENSO forecast
Latest forecast with data
fore
cast
dat
a
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
Comparison to climatology and persistence