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Applying False Discovery Rate (FDR) Control in Detecting Future Climate Changes. ZongBo Shang SIParCS Program, IMAGe , NCAR August 4, 2009. North American Regional Climate Change Assessment Program (NARCCAP) Predicted Changes in Future Winter Temperature ( °C).
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Applying False Discovery Rate (FDR) Control in Detecting Future Climate Changes ZongBo Shang SIParCS Program, IMAGe, NCAR August 4, 2009
North American Regional Climate Change Assessment Program (NARCCAP)Predicted Changes in Future Winter Temperature ( °C) Note: This figure shows the difference between the mean of future (2040 – 2069 ) winter temperature vs. current (1970 – 1999) winter temperature.
Can We Trust What We See? Note: Those two figures show the means of 10 replicate random fields that are generated from the same Matèrn semi-variogram model, but with different random seeds.
False Discovery Rate (FDR) Control • FDR controls the expected proportion of incorrectly rejected null hypotheses (type I errors) among the rejected null hypotheses. • Less conservative than Bonferroni procedures, with greater power than Familywise Error Rate (FWER) control, at a cost of increasing the likelihood of obtaining type I errors. Applications of FDR in Functional Magnetic Resonance Imaging Applications of FDR in Genes Expression and Microarray
Definition of False Discovery Rate Let Q = V / (V + S) define the proportion of errors committed by falsely rejecting null hypotheses. Notice Q is an unobservable random variable. Define the FDR to be the expectation of Q:
False Discovery Rates for Spatial Signals • Testing on clusters rather than individual locations • Procedure 1: Weighted Benjamini & Hochberg (BH) procedure • Procedure 2: Weighted two-stage procedure • Procedure 3: Hierarchical testing procedure • Testing stage: control FDR on clusters • Trimming stage: control FDR on selected points Reference: Benjamini, Y. and Heller, R. 2007. False discovery rates for spatial signals. Journal of the American Statistical Association. 102:1272-1281.
Simulation Studies • 1. Random Fields • 2. Random Field Block • 3. Random Field Gradient
Simulation Study I: Two Random Fields Note: Those two figures show the means of 10 replicate random fields that are generated from the same Matèrn semi-variogram model, but with different random seeds.
Simulation Study 1: Pointwise vs. False Discover Rate Control
Simulation Study II: Pre-defined Block Trend 4 -10 -2 2 10 -4
Simulation Study II: Average of 10 Replicates 4 -10 -2 2 10 -4 Random Field (Matèrn, σ = 0.4) Random Field (Matèrn, σ = 0.4) + Block Trends
Simulation Study II: Pointwise vs. False Discover Rate Control
Study III: Average of 10 Replicates Random Field (Matèrn, σ = 2) Random Field (Matèrn, σ = 2) + Gradient Trends
Simulation Study III: Pointwise vs. False Discover Rate Control
Applying FDR Control for Detecting Future Climate Changes • Download climate datasets from NARCCAP program • Calculate seasonal average • Construct clusters from EPA Eco-regions • Conduct two-sample t test on temperature/precipitation • Pointwise p-values and corresponding z scores • Build semi-variogram model to estimate spatial autocorrelation • Calculate z score and p-value by cluster • Reject clusters based on FDR control
http://www.epa.gov/wed/pages/ecoregions/na_eco.htm GIS: Vector Dataset, Lambert Equal-Area Projection
H0: Future Winter Temperature Increase by 3 ˚C 61 regions rejected at q=0.25 level 56 regions rejected at q=0.1 level 54 regions rejected at q=0.05 level 51 regions rejected at q=0.01 level
FDR Tests on Winter Temperature H0: Winter Temperature ↑ 1 ˚C H0: Winter Temperature ↑ 2 ˚C H0: Winter Temperature ↑ 3 ˚C H0: Winter Temperature ↑ 4 ˚C H0: Winter Temperature ↑ 5 ˚C H0: Winter Temperature ↑ 6 ˚C
FDR Tests on Winter Precipitation H0: Winter Prec ↓ 20 Kg/ m² H0: ↓ 10 Kg/ m² H0: ↑ 10 Kg/ m² H0: ↑ 20 Kg/ m² H0: Winter Prec ↑ 30 Kg/ m² H0: ↑ 50 Kg/ m² H0: ↑ 75 Kg/ m² H0: ↑ 100 Kg/ m²
Acknowledgement • Dr. Steve Sain, IMAGe, NCAR • Drs. Douglas Nychka, Tim Hoar, IMAGe, NCAR • Dr. Armin Schwartzman, Harvard University • University of Wyoming • SIParCS, IMAGe, NCAR • NARCCAP