Variations in MIK Class Means Jeremy Vincent Variations

Variations in MIK Class Means Jeremy Vincent

Variations in MIK Class Means Jeremy Vincent

Outline • Motivation to investigate class mean • Data • Observations of class mean behaviour: Ø Ø Comparison to global class mean Spatial variation Dependence Estimation error • Conclusions • Future Work 3

Motivation and Setup • The MIK estimator assumes a constant class mean, : • No direct access to the conditional class mean in MIK. • Need MG framework to calculate exact conditional probabilities and class mean values. Motivating question: How does a constant class mean impact MIK estimation? 4

Data Generation • • 5 Simulate Gaussian grid (unconditional). Split into four sample-grid patterns to capture a range of data configurations. Estimate under MG framework and assess sampling of space Convert to log-normal distribution using the transform:

Global Class Mean Calculation • Global class mean ( 6 ): Mean of class quantiles (declustered global distribution).

Conditional Class Mean Calculation • Conditional class mean ( 7 ): The mean of the quantiles of the conditional distribution within a class

Distribution of Conditional Class Means: Gaussian Data • Local class mean ( ): The mean of the conditional class means (vertical, black lines) • Note the divergence between 8 and (vertical, teal lines) from the middle class to the upper tail

Distribution of Conditional Class Means: Log-Normal Data 9

Global Class Mean vs Conditional Class Mean Log-Normal Data 10

Partial Dependence of Conditional Class Mean 11

MIK Estimation Error 12

MIK Estimation Error 13

Estimation Error: Log-Normal Data 5 thresholds 14 10 thresholds 15 thresholds

MIK Error: Impact on Contained Metal 15

Conclusions • Yes, the class mean varies! • Class mean depends on the estimated by kriging (LN: ~ 55/45). • The class mean varies greatest in the distribution tails, which translates into greatest estimation error. • Estimation using the correct mean vs. the global mean suggests a bias due to poor estimation in the tails. • Further confirmation of bias is needed. 16

Future Work • Cross-validation study comparing traditional MIK implementation to confirm bias. • Investigate the consistency of MIK class probabilities under a MG framework. 17

Thank you! Questions? 18

Stochastic Gradient Boosting: Model Fitting (Gaussian Data) 19

Stochastic Gradient Boosting: Model Fitting (Log-Normal Data) 20

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Motivation and Setup • First, look to answer some other questions: • By how much does the class mean vary? • Upon which factors does the class mean depend? • Where are the class mean differences greatest? 23

Data Generation • Remove asymmetry by multiplying by -1 (still valid • Decimate data (random sampling used) 24 pairs)

Global Class Mean vs Conditional Class Mean Gaussian Data 25 Log-Normal Data

Partial Dependence of Conditional Class Mean 26

Estimation Error: Gaussian Data 5 thresholds 27 10 thresholds 15 thresholds