Using Synthetic Data to Test Downscaling Methods
John Lanzante (GFDL/NOAA)
CONCEPTS
Testing Downscaling:
Like Product Testing
My Product
STEP1: Recruit Test Subjects (Gather Data)CONCEPTS
STEP2: Feed Cereal For Several Decades (Apply Downscaling Method)
CONCEPTS
STEP3: How are subjects affected? How well did downscaling do?
Not so clear – Need more subjects? Need more data?
Real-world data may be limited? Can we generate synthetic data to fill the void?
CONCEPTS
STEP 4a: Snowmen most affected? Generate a new sample.
CONCEPTS
STEP 4b: Snow-women affected differently? Generate a new sample.
REALISTIC EXAMPLESCASE 1 – Linearity:
Simplest downscaling – linear regression
REALISTIC EXAMPLESCASE 1 – Strong Nonlinearity:
Simplest downscaling – linear regression
REALISTIC EXAMPLESSUMMARY CASE 1 – Nonlinearity:
Hard to test nonlinearity in real-world data ? (if we are just entering “non-linear regime”)Simulate various degrees of nonlinearityCompare linear & nonlinear downscaling methodsDetermine amount of degradationDetermine time in future when degradation becomes “too large”
REALISTIC EXAMPLESCASE 2 – Coastal Error:
Downscaling error maximizes along coastline
REALISTIC EXAMPLESCASE 2 – Coastal Error:
Obs gridpoint Entirely landModel gridpoint Partly land, partly ocean
REALISTIC EXAMPLESCASE 2 – Coastal Error:
Land more detail (extremes) than Ocean (damped)Missing peaks & troughs unrecoverable
REALISTIC EXAMPLES
SUMMARY CASE 2 – Costal Error:Simulate land & ocean pointsDownscale land from mixture (land + sea)Vary the proportions of the mixtureIs coastal effect due to mixture/mismatch?
SYNTHETIC DATA MODELOne Particular Synthetic Data Model: O= Observations M= Model y= year d= day Red = free parameter (user selects the value)
Oy d = Ōy + O’y d Yearly mean + AR1
O’y d = rlag1 * O’
y d-1 + ay d AR1
fvar = varŌ / varO [ varO = varŌ + varO]
My d = Oy d + by d corr = correlation(O,M)
a ~ N(0,vara) Proper choice of a & b b ~ N(0,varb) yields desired rlag1 & corr
SYNTHETIC DATA MODEL
STEP 1: Generate Base Time Series rlag1 day-to-day persistence fvar interannual vs. day-to-day variability corr strength of relation: model vs. obsSTEP 2: Historical Adjustment meanOBS characteristics of the distribution
meanMODEL
varOBS
varMODEL
STEP 3: Future Adjustment meanOBS characteristics of the distribution meanMODEL
varOBS
varMODEL
SYNTHETIC DATA MODEL
OUR APPLICATIONS OF THIS MODEL:Downscaling (just getting started)
No results yetApplied successfully to several related
issues (cross-validation, exceedance statistics, testing two distributions)
SUMMARYREAL-WORLD COMPLICATIONS:
Results may not be clear-cut: Sample size too small? Multiple factors may contribute?
Some conditions more interesting?
SOLUTION – GENERATE SYNTHETIC DATA: Advantages of Synthetic Data:
Unlimited sample size (enhance signal/noise)Change one factor at a time
Prescribe exact conditionsVary factor over a wide range (“turn the knob”)
Can extend outside the range of historical data Turn knob “all the way” for unambiguous results
A CAUTIONARY NOTE
No “One Size Fits All”:No single “best” synthetic data modelMust possess appropriate real-world characteristicsAbility to vary the relevant factors
Possible Models For Future Development:Skewed data (transform Gaussian data nonlinearly?)Precipitation (discrete Markov + bounded distribution?)
Model occurrence & amount separately?Multivariate model?
THE END
REALISTIC EXAMPLESCASE 1 – Weak Nonlinearity:
Simplest downscaling – linear regression
SUPPLEMENTAL
Causes of Nonlinearity?At highest T – model soil becomes excessively dry – T becomes excessiveOther possibilities: Water Vapor, Clouds, Sea-Ice, etc.
REALISTIC EXAMPLESCASE 2 – Coastal Error:
Land More extremesOcean Damped
REALISTIC EXAMPLESCASE 2 – Coastal Error:
X/Y Plot: Land (model) vs. Land (obs)
REALISTIC EXAMPLESCASE 2 – Coastal Error:
X/Y Plot: Ocean (model) vs. Land (obs)
SYNTHETIC DATA MODEL
STEP 4:Fit downscaling model to historical sample
STEP 5:Test downscaling in historical & future samples
OUR APPLICATIONS OF THIS MODEL:No results to show todayDownscaling (just getting started)Guidance in the use of cross-validationBiases in exceedance statisticsTesting difference between 2 distributions
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