Dynamic Bayesian Influenza Forecasting James Gattiker - - PowerPoint PPT Presentation

Slide 1

Managed by Triad National Security, LLC for the U.S. Department of Energy’s NNSA

Dynamic Bayesian Influenza Forecasting

James Gattiker

gatt@lanl.gov

David Osthus

Los Alamos National Laboratory Statistical Sciences Group

ISDA 2019 January 23rd

Slide 2

Managed by Triad National Security, LLC for the U.S. Department of Energy’s NNSA

Acknowledgments

Talk based on: Osthus, Dave, James Gattiker, Reid Priedhorsky, and Sara Y. Del Valle. "Dynamic Bayesian influenza forecasting in the United States with hierarchical discrepancy." Bayesian Analysis (2018).

Slide 3

Managed by Triad National Security, LLC for the U.S. Department of Energy’s NNSA

Acknowledgments

Dave Osthus is the Investigator for this work in methodology and applied modeling Thanks, Dave:

Talk based on: Osthus, Dave, James Gattiker, Reid Priedhorsky, and Sara Y. Del Valle. "Dynamic Bayesian influenza forecasting in the United States with hierarchical discrepancy." Bayesian Analysis (2018).

Slide 4

Managed by Triad National Security, LLC for the U.S. Department of Energy’s NNSA

Statistics and Uncertainty Quantification

Statistics:

• Model trends & patterns in data for prediction and discovery
• Statistical model inherently represents uncertainty

Uncertainty Quantification (one thread: statistical)

• Use data and domain models for prediction and discovery

– often, domain models are computational simulations – domain models are a different type of information from

• bservations or priors
• As a statistical model, includes uncertainty in inferences

– including: parameters, model inadequacy, data uncertainty Uncertainty Quantification (another thread: numerical analysis)

• Concept based in solution methods for ODE systems
• Extend to unknown parameters and linking to data
• Make inferences, including uncertainty

Slide 5

Managed by Triad National Security, LLC for the U.S. Department of Energy’s NNSA

Uncertainty Quantification

Key Advances in Statistical UQ in early 2000’s *

• Based on work in response surface methods for approximating

the output of computationally expensive models

• Markov-Chain Monte Carlo for inverse computations for Bayesian

modeling

• Using the statistical concept of a “discrepancy” between model

and data.

* Kennedy and O’Hagan, Bayesian Calibration of Computer Models, JRSS-B, 2001 Observed data System model Discrepancy

Slide 6

Managed by Triad National Security, LLC for the U.S. Department of Energy’s NNSA

Uncertainty Quantification: Discrepancy

Observed data System model Discrepancy

Discrepancy considerations

• Capture the potential shortfalls of the model
• Consistent with the domain being modeled
• Discrepancy and parameter bias trade off
• Justify in extrapolation

Slide 7

Managed by Triad National Security, LLC for the U.S. Department of Energy’s NNSA

Overview

• Flu surveillance, data and forecasting objectives
• Model inadequacy and discrepancy
• Dynamic Bayesian (DB) model with hierarchical

discrepancy

• Forecasting results and conclusions

Slide 8

Managed by Triad National Security, LLC for the U.S. Department of Energy’s NNSA

Flu surveillance

• Centers for Disease Control and Prevention (CDC)

Outpatient Influenza-like Illness Surveillance Network (ILINet) monitors influenza-like illness (ILI) levels in the U.S.

• Thousands of providers participate in ILINet across the U.S.
• ILI is

1. Temperature ≥ 100 ºF 2. A sore throat or cough 3. No other known cause

ILI ≠ Influenza

Slide 9

Managed by Triad National Security, LLC for the U.S. Department of Energy’s NNSA

ILI data

• ILI is released weekly by the CDC
• ILI has predictable and unpredictable features

2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 2017/18 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 1997/98 through 2016/17 0.01 0.02 0.03 0.04 0.05 0.06 0.07 O N D J F M A M

ILI

1997 through 2017 US Flu Seasons

Slide 10

Managed by Triad National Security, LLC for the U.S. Department of Energy’s NNSA

CDC forecasting objectives

• Annual CDC flu forecasting challenge; started in 2013/14
• Probabilistically forecast the peak intensity (PI), peak timing (PT),
• nset, and short term forecasts (1 to 4 weeks ahead) each week

throughout the flu season

• Primary interest is forecast accuracy

Slide 11

Managed by Triad National Security, LLC for the U.S. Department of Energy’s NNSA

Mechanistic model for infectious disease transmission

SIR model assumptions

• closed population
• homogeneous mixing
• time-invariant disease

transmission and recovery rates

Slide 12

Managed by Triad National Security, LLC for the U.S. Department of Energy’s NNSA

Model inadequacy

• 2011

2012 2013 2014 2015

1 5 10 15 20 25 30 35 1 5 10 15 20 25 30 35 1 5 10 15 20 25 30 35 1 5 10 15 20 25 30 35 1 5 10 15 20 25 30 35 0.01 0.02 0.03 0.02 0.04 0.06 0.01 0.02 0.03 0.04 0.02 0.04 0.06 0.010 0.015 0.020

Week ILI

• −0.01

0.00 0.01 0.02 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35

Week Residuals

• Systematic deviation between ILI and best version of SIR model results in

discrepancy (i.e., structured residuals)

SIR ILI

Slide 13

Managed by Triad National Security, LLC for the U.S. Department of Energy’s NNSA

Where are we? Where are we going?

• ILI exhibits commonalities across flu seasons
• SIR model can capture general “shape” of ILI trajectory, but not more

nuanced structure, resulting in discrepancy

• Discrepancy appears to share commonalities across seasons
• Develop a statistical model that

1. Leverages information across flu seasons 2. Exploits common discrepancy across flu seasons 3. Strikes a balance between model structure (e.g., mechanistic models) and model flexibility (e.g., empirical models) 4. Probabilistically forecast ILI

Slide 14

Managed by Triad National Security, LLC for the U.S. Department of Energy’s NNSA

[data, process, disc] =[data|process, disc]× [disc|process]× [process]

Bayesian hierarchical model for ILI forecasting

• Data: ILI data, indexed by week (t) and flu season (j)
• Process: The disease transmission model – motivated by the

SIR model

• Discrepancy: Flexible function capturing the systematic

deviations between the data and the process

Slide 15

Managed by Triad National Security, LLC for the U.S. Department of Energy’s NNSA

Process model

[process]

• Let Ij,t represent the infectious proportion of the population motivated by

the SIR model for flu season j and week t.

• We model Ij,t as the solution to the SIR model:
• SIR model numerically approximated with the fourth order Runge-Kutta

method (RK4)

• Given initial conditions (S0,j, I0,j, R0,j) and parameter values (βj, γj), Ij,t is

completely determined

Slide 16

Managed by Triad National Security, LLC for the U.S. Department of Energy’s NNSA

Process model (cont.)

• Question: How do we model the initial conditions and parameters?
• Answer: We fit a truncated multivariate Normal distribution to the initial

conditions and parameter estimates of the historical flu seasons

• 0.0

0.4 0.8 1.2 1.6 0.000 0.005 0.010

Ij,0 bj

• 0.6

0.7 0.8 0.000 0.005 0.010

Ij,0 rj

• 0.6

0.7 0.8 0.0 0.4 0.8 1.2 1.6

bj rj

2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2015/16 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 2012/13 0.00 0.02 0.04 0.06 0.08 1 5 10 15 20 25 30 35

Week Infectiousness

[process]

ρj = γj/βj

Slide 17

Managed by Triad National Security, LLC for the U.S. Department of Energy’s NNSA

Discrepancy model

[disc|process]

• We model discrepancy (ψ) at week t
• f flu season j as the sum of two

terms:

ψj,t = µt + δj,t

• μt captures the week-specific

discrepancy common to all flu seasons

• δj,t captures the week and flu

season-specific discrepancy unaccounted for by μt.

2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 −0.01 0.00 0.01 0.02 1 5 10 15 20 25 30 35

Week

yj,t

−0.01 0.00 0.01 0.02 1 5 10 15 20 25 30 35

Week

µt

2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 −0.01 0.00 0.01 0.02 1 5 10 15 20 25 30 35

Week

dj,t

Slide 18

Managed by Triad National Security, LLC for the U.S. Department of Energy’s NNSA

µT ∼ N(0, σ2

µT )

µt|µt+1 ∼ N(µt+1, σ2

µ)

Reverse Random Walk Reverse Random Walk

δj,T = −logit(Ij,T ) δj,t|δj,t+1 ∼ N(αjδj,t+1, σ2

δ,j)

Discrepancy model (cont.)

[disc|process] ψj,t = µt + δj,t

2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 −0.01 0.00 0.01 0.02 1 5 10 15 20 25 30 35

Week

yj,t

−0.01 0.00 0.01 0.02 1 5 10 15 20 25 30 35

Week

µt

2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 2010/11 −0.01 0.00 0.01 0.02 1 5 10 15 20 25 30 35

Week

dj,t

αj ∈ [0, 1]

Slide 19

Managed by Triad National Security, LLC for the U.S. Department of Energy’s NNSA

Data model

[data|process, disc]

E(yj,t|πj,t) = πj,t

yj,t|πj,t ∼ Beta(λπj,t, λ(1 − πj,t))

• The data model is a relatively simple model, as most of the intricate

structure comes from the process and discrepancy models

• We model ILI on week t from flu season j (yj,t) as

logit(πj,t) = logit(Ij,t) + µt + δj,t

• Let πj,t be the sum of the process and the discrepancy for flu season j of

week t, representing the true but unobserved state of ILI

Slide 20

Managed by Triad National Security, LLC for the U.S. Department of Energy’s NNSA

Posterior distributions

Posterior samples generated with Markov chain Monte Carlo (MCMC) methods, executed by calling the software JAGS (Just Another Gibbs Sampler)

[future data|data] [process|data]

[disc|data]

[unobservables|observables]

95% posterior predictive interval

Slide 21

Managed by Triad National Security, LLC for the U.S. Department of Energy’s NNSA

CDC flu forecasting challenge

ln(.04 + .33 + .02) = -0.94 ln(.00 + .99 + .00) = -0.01

• ● ● ● ● ● ● ●
• ●
• ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
• ● ● ● ● ● ● ●
• ●
• ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
• ● ● ● ● ● ● ●
• ●
• ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
• ● ● ● ● ● ● ●
• ●
• ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
• 18

15 12 9 1 4 7 10 13 16 19 22 25 28 31 34 0.00 0.02 0.04 0.06 0.08 0.00 0.02 0.04 0.06 0.08 0.00 0.02 0.04 0.06 0.08 0.00 0.02 0.04 0.06 0.08

Week ILI

18 15 12 9 6 9 12 15 18 21 24 27 30 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00

PT probability

• Forecasts are scored with a modified log scoring rule (0 is best score)

ln(.12 + .41 + .02) = -0.60 ln(.00 + .65 + .00) = -0.43

Modified Log Score

• bserved

unobserved 95% posterior predictive interval Peak timing (x) and intensity (y) Peak timing, +/- 1 week

2014/15 Flu Season

Slide 22

Managed by Triad National Security, LLC for the U.S. Department of Energy’s NNSA

3 1 1 2 1 4 1 1 5 2 1 1 4 6 3 1

2015 − 2016 2016 − 2017 −10 −9 −8 −7 −6 −5 −4 −3 −2 −1 −10 −9 −8 −7 −6 −5 −4 −3 −2 −1 logpi logpt logonset log1wk log2wk log3wk log4wk logscore

log score

Comparison to CDC challengers

Rank 14 models 30 models

Slide 23

Managed by Triad National Security, LLC for the U.S. Department of Energy’s NNSA

Modified Log Score (0 is best) Target DB model: no hierarchical discrepancy DB model: hierarchical discrepancy Onset

• 0.45
• 0.14

Peak Timing

• 0.70
• 0.35

Peak Intensity

• 1.25
• 0.92

1 week ahead

• 0.46
• 0.28

2 week ahead

• 0.66
• 0.40

3 week ahead

• 0.83
• 0.50

4 week ahead

• 0.90
• 0.58

Overall

• 0.75
• 0.45

The effect of hierarchical discrepancy

• A version of the non-hierarchical discrepancy model participated in the

2016/17 flu challenge, allowing us to isolate the effect of hierarchical discrepancy on forecasts

Slide 24

Managed by Triad National Security, LLC for the U.S. Department of Energy’s NNSA

Summary

• The SIR disease transmission model is inadequate to

capture all features of the ILI data-generating process

• Discrepancy accounts for the systematic deviations

between the disease transmission model and the ILI data

• By exploiting and hierarchically modeling common

discrepancy structure across flu seasons, we were able to improve ILI forecasts

• Discrepancy is an important modeling component that

should be considered in modeling and forecasting

Slide 25

Managed by Triad National Security, LLC for the U.S. Department of Energy’s NNSA

2019 Forecasts: regional

• 1997

1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2010 2011 2012 2013 2014 2015 2016 2017

• ● ● ● ● ● ●
• ● ●
• 2001

2002 2003 2004 2005 2006 2007 2010 2011 2012 2013 2014 2015 2016 2017

• ● ●
• ● ● ●
• ● ●
• 2001

2002 2003 2004 2005 2006 2007 2010 2011 2012 2013 2014 2015 2016 2017

• ● ● ● ● ● ● ● ● ●
• ● ●
• 2001

2002 2003 2004 2005 2006 2007 2010 2011 2012 2013 2014 2015 2016 2017

• ● ● ● ● ● ●
• ●
• ●
• ●
• 2001

2002 2003 2004 2005 2006 2007 2010 2011 2012 2013 2014 2015 2016 2017

• ● ● ● ● ● ● ● ● ●
• 2001

2002 2003 2004 2005 2006 2007 2010 2011 2012 2013 2014 2015 2016 2017

• ● ● ● ● ●
• ● ● ●
• ● ●
• 2001

2002 2003 2004 2005 2006 2007 2010 2011 2012 2013 2014 2015 2016 2017

• ● ● ● ● ● ●
• 2001

2002 2003 2004 2005 2006 2007 2010 2011 2012 2013 2014 2015 2016 2017

• ● ●
• ● ●
• ● ● ●
• 2001

2002 2003 2004 2005 2006 2007 2010 2011 2012 2013 2014 2015 2016 2017

• ● ● ● ● ● ● ● ● ● ● ●
• ● ●
• 2001

2002 2003 2004 2005 2006 2007 2010 2011 2012 2013 2014 2015 2016 2017

• ● ● ● ● ● ●
• ● ●
• ●
• 2001

2002 2003 2004 2005 2006 2007 2010 2011 2012 2013 2014 2015 2016 2017

• ● ● ●
• ●
• ● ●
• hhs8

hhs9 hhs10 hhs4 hhs5 hhs6 hhs7 nat hhs1 hhs2 hhs3 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 11 12 13 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 10

wILI

DBMplus−EW02−2019 80% and 95% Predictive Intervals

Slide 26

Managed by Triad National Security, LLC for the U.S. Department of Energy’s NNSA

Questions?

CDC Flu Forecasting webpage: predict.cdc.gov (FluSight)

Slide 27

Managed by Triad National Security, LLC for the U.S. Department of Energy’s NNSA

Los Alamos National Laboratory is looking for Excellent Statisticians

Contribute to cutting-edge scientific applications of national importance in a multidisciplinary, collaborative environment. Develop innovative methods to address the challenges of real-world data.

30 colleagues, early career to senior professionals

See your work having an impact!

ASA Fellows

stat.lanl.gov

Journal editorial board members Conference founders Over 40 articles with 100+ citations

Work with internationally recognized statisticians and scientists

A network of international collaborators

LA-UR-18-27149