Background Methods & Examples HSRL Data Conclusions & Future Work Generalized Significance in Scale Space: The GS3 Package Daniel V. Samarov Statistical Engineering Division Information Technology Laboratory National Institute of Standards and Technology July 21, 2010 1
Background Methods & Examples HSRL Data Conclusions & Future Work Table of Contents Background 1 Methods & Examples 2 Local Polynomial Regression Scale Space & RODEO Generalized Scale Space & d > 1 Algorithm Speed HSRL Data 3 Conclusions & Future Work 4 2
Background Methods & Examples HSRL Data Conclusions & Future Work Green House Gas (GHG) Emission Measurement NIST developing technology & standards for remote sensing of GHG’s 3
Background Methods & Examples HSRL Data Conclusions & Future Work Green House Gas (GHG) Emission Measurement NIST developing technology & standards for remote sensing of GHG’s DIAL for distributed sources DI fferential A bsorbtion L IDAR Range resolved, column integrated measurements 3
Background Methods & Examples HSRL Data Conclusions & Future Work Green House Gas (GHG) Emission Measurement NIST developing technology & standards for remote sensing of GHG’s DIAL for distributed sources DI fferential A bsorbtion L IDAR Range resolved, column integrated measurements 3
Background Methods & Examples HSRL Data Conclusions & Future Work HSRL DIAL technology not quite ready for primetime. Collaboration w/ NASA 4
Background Methods & Examples HSRL Data Conclusions & Future Work HSRL DIAL technology not quite ready for primetime. Collaboration w/ NASA HSRL H igh S pectral R esolution L IDAR Similar technology/data Validation of Calipso satellite measurements 4
Background Methods & Examples HSRL Data Conclusions & Future Work HSRL DIAL technology not quite ready for primetime. Collaboration w/ NASA HSRL H igh S pectral R esolution L IDAR Similar technology/data Validation of Calipso satellite measurements Hair et al. (2008) Data graciously provided by NASA Langley Research Center 4
Background Methods & Examples HSRL Data Conclusions & Future Work Challenges associated w/ HSRL & DIAL data 5
Background Methods & Examples HSRL Data Conclusions & Future Work Challenges associated w/ HSRL & DIAL data Highly variable 5
Background Methods & Examples HSRL Data Conclusions & Future Work Challenges associated w/ HSRL & DIAL data Highly variable Subtle local structure Hair et al. (2008) 5
Background Methods & Examples HSRL Data Conclusions & Future Work Challenges associated w/ HSRL & DIAL data Highly variable Subtle local structure Hair et al. (2008) Large ( ∼ 300 × 30 , 000) 5
Background Methods & Examples HSRL Data Conclusions & Future Work Challenges associated w/ HSRL & DIAL data Highly variable Subtle local structure Hair et al. (2008) Large ( ∼ 300 × 30 , 000) Goals Estimate concentration (derivative) Calculate uncertainty 5
Background Local Polynomial Regression Methods & Examples Scale Space & RODEO HSRL Data Generalized Scale Space & d > 1 Conclusions & Future Work Algorithm Speed Local Polynomial Regression (LPR) Natural choice for derivative estimation: LPR (Fan & Gijbels (1995)) 6
Background Local Polynomial Regression Methods & Examples Scale Space & RODEO HSRL Data Generalized Scale Space & d > 1 Conclusions & Future Work Algorithm Speed Local Polynomial Regression (LPR) Natural choice for derivative estimation: LPR (Fan & Gijbels (1995)) Pros Provides derivative estimate Locally adaptive Many other appealing properties 6
Background Local Polynomial Regression Methods & Examples Scale Space & RODEO HSRL Data Generalized Scale Space & d > 1 Conclusions & Future Work Algorithm Speed Local Polynomial Regression (LPR) Natural choice for derivative estimation: LPR (Fan & Gijbels (1995)) Pros Provides derivative estimate Locally adaptive Many other appealing properties Cons Challenge in 2( > )-d: bandwidth choice (in particular local ) Speed Exploratory analysis 6
Background Local Polynomial Regression Methods & Examples Scale Space & RODEO HSRL Data Generalized Scale Space & d > 1 Conclusions & Future Work Algorithm Speed Local Polynomial Regression (LPR) Natural choice for derivative estimation: LPR (Fan & Gijbels (1995)) Pros Provides derivative estimate Locally adaptive Many other appealing properties Cons Challenge in 2( > )-d: bandwidth choice (in particular local ) Speed Exploratory analysis The GS3 package provides a solution 6
Background Local Polynomial Regression Methods & Examples Scale Space & RODEO HSRL Data Generalized Scale Space & d > 1 Conclusions & Future Work Algorithm Speed Scale Space Scale space (Chaudhuri & Marron (2000)) a good starting point. Consider the following example: 7
Background Local Polynomial Regression Methods & Examples Scale Space & RODEO HSRL Data Generalized Scale Space & d > 1 Conclusions & Future Work Algorithm Speed Scale Space Scale space (Chaudhuri & Marron (2000)) a good starting point. Consider the following example: Many instances where a fit desired 7
Background Local Polynomial Regression Methods & Examples Scale Space & RODEO HSRL Data Generalized Scale Space & d > 1 Conclusions & Future Work Algorithm Speed Scale Space Scale space (Chaudhuri & Marron (2000)) a good starting point. Consider the following example: Many instances where a fit desired 7
Background Local Polynomial Regression Methods & Examples Scale Space & RODEO HSRL Data Generalized Scale Space & d > 1 Conclusions & Future Work Algorithm Speed Scale Space Scale space (Chaudhuri & Marron (2000)) a good starting point. Consider the following example: Many instances where a fit desired However, good practice to look at multiple smooths 7
Background Local Polynomial Regression Methods & Examples Scale Space & RODEO HSRL Data Generalized Scale Space & d > 1 Conclusions & Future Work Algorithm Speed Scale Space Scale space (Chaudhuri & Marron (2000)) a good starting point. Consider the following example: Many instances where a fit desired However, good practice to look at multiple smooths Scale space studies a“family”of smooths 7
Background Local Polynomial Regression Methods & Examples Scale Space & RODEO HSRL Data Generalized Scale Space & d > 1 Conclusions & Future Work Algorithm Speed RODEO RODEO (Wasserman & Lafferty (2008)) greedy algorithm for traversing “scale space surface” Algorithm 8
Background Local Polynomial Regression Methods & Examples Scale Space & RODEO HSRL Data Generalized Scale Space & d > 1 Conclusions & Future Work Algorithm Speed RODEO RODEO (Wasserman & Lafferty (2008)) greedy algorithm for traversing “scale space surface” Algorithm At a point, stay? 8
Background Local Polynomial Regression Methods & Examples Scale Space & RODEO HSRL Data Generalized Scale Space & d > 1 Conclusions & Future Work Algorithm Speed RODEO RODEO (Wasserman & Lafferty (2008)) greedy algorithm for traversing “scale space surface” Algorithm At a point, stay? or move? 8
Background Local Polynomial Regression Methods & Examples Scale Space & RODEO HSRL Data Generalized Scale Space & d > 1 Conclusions & Future Work Algorithm Speed RODEO RODEO (Wasserman & Lafferty (2008)) greedy algorithm for traversing “scale space surface” Algorithm At a point, stay? or move? i.e. change from ˆ m h ( x ) to m h ( x ) significant? ˆ 8
Background Local Polynomial Regression Methods & Examples Scale Space & RODEO HSRL Data Generalized Scale Space & d > 1 Conclusions & Future Work Algorithm Speed RODEO RODEO (Wasserman & Lafferty (2008)) greedy algorithm for traversing “scale space surface” Algorithm At a point, stay? or move? i.e. change from ˆ m h ( x ) to m h ( x ) significant? ˆ Z = ∂ ˆ m h ( x ) , test ∂ h � | Z | > 2 log( n )Var( Z ) 8
Background Local Polynomial Regression Methods & Examples Scale Space & RODEO HSRL Data Generalized Scale Space & d > 1 Conclusions & Future Work Algorithm Speed RODEO RODEO (Wasserman & Lafferty (2008)) greedy algorithm for traversing “scale space surface” Algorithm At a point, stay? or move? i.e. change from ˆ m h ( x ) to m h ( x ) significant? ˆ Z = ∂ ˆ m h ( x ) , test ∂ h � | Z | > 2 log( n )Var( Z ) NB: Var( Z ) ∼ σ 2 σ 2 unknown population parameter 8
Background Local Polynomial Regression Methods & Examples Scale Space & RODEO HSRL Data Generalized Scale Space & d > 1 Conclusions & Future Work Algorithm Speed Illustration of RODEO in Scale Space For σ = 0 . 025 9
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