ML19011A449

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Workshop 4 Estimation 2019-01-18
ML19011A449
Person / Time
Issue date: 01/16/2019
From:
Office of Nuclear Regulatory Research
To:
Nathan Siu 415-0744
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ML19011A416 List:
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Download: ML19011A449 (9)
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Bayesian Estimation Workshop 4 1

Learning Objectives

  • Additional practice constructing an informative prior distribution (see also Workshop 2)
  • Practice using conjugate likelihood-prior pairs
  • Exposure to solution techniques for non-conjugate pairs Important:

a) The workshop problems can be performed as group exercises.

b) The purpose is to exercise the modeling thought process, not to get the right answer.

2

Weather-Related Loss of Power in the Boston Area Weather-related losses of offsite power can be important in NPP PRA. This workshop uses real data and simple models for grid losses in the Boston area to provide practice with the mechanics of Bayesian estimation.

Self-rating

i. How many years have you lived in the Boston area?

ii. Do you have direct experience with weather-related losses of the power grid?

3

Problem #1 - Frequency of Weather-Related Loss of Grid (WRLOG)

A. Is the Poisson model a good model for the occurrence of WRLOG? Why or why not?

B. Assume WRLOG can be adequately characterized by a Poisson distribution with frequency WR.

i. What is your best guess for WR?

ii. What are reasonable upper and lower bounds for WR?

iii. Assuming that your state of knowledge can be roughly characterized by an exponential distribution,* develop appropriate values for the parameter where 1 1

= 1 =

  • The exponential is likely an unsatisfactory form for realistic analysis but more complicated forms require numerical analysis.

4

Problem #1 (cont.) Date Years Since Last Event 2/8/2013 0.76 C. Consider the given data for WRLOG in 10/29/2012 1.00 Eastern Massachusetts. The data are from 1/1/2001 through 6/30/2014.* 10/29/2011 0.17

i. The data do not include the August 14, 8/28/2011 0.63 2003 Northeast Blackout which affected 1/12/2011 2.08 numerous U.S. and Canadian plants, but 12/12/2008 0.51 did not extend to the Boston area. For the purpose of a PRA, should it? 6/10/2008 0.60 ii. What are the mean and standard deviation 11/3/2007 0.55 for the posterior distribution for WR? (Note 4/14/2007 0.70 that the posterior distribution will be a gamma distribution.) 8/2/2006 1.95 8/20/2004 3.64**
  • Jordan Wirfs-Brock, Data: Explore 15 Years of Power

Problem #2 - Duration of Date Duration (hr)

Weather-Related Loss of Grid 2/8/2013 79 (WRLOG) 10/29/2012 28 10/29/2011 216 A. Is the Poisson model a good model for the duration of WRLOG? Why or why not? 8/28/2011 0 B. Assume WRLOG duration can be 1/12/2011 8 characterized by a lognormal distribution 12/12/2008 239 with parameters µWR and WR. 6/10/2008 10

i. How would you develop a prior distribution 11/3/2007 12 for µWR and WR?

4/14/2007 2 ii. Given the data for WRLOG recovery times, how would you update this prior 8/2/2006 6 distribution? 8/20/2004 6 6

Conjugate Likelihood-Prior Pairs

  • Binomial Likelihood, Beta Prior

- Likelihood Function

, = 1

- Prior Distribution

+ 1 0 , = 1 1 0 =

+

- Posterior Distribution

+ 1 1 1 , = 1 1 =

+

= + = + ( )

7

Conjugate Likelihood-Prior Pairs

  • Poisson Likelihood, Gamma Prior

- Likelihood Function

, =

- Prior Distribution 1

0 , 0 = 0 =

- Posterior Distribution 1

1 , 1 = 1 =

8

Lognormal Distribution Characteristics Characteristic Formula 1 1 2 pdf 2 2

1 Mean + 2 2

Variance 2 2 1 5th Percentile 1.6448 50th Percentile (median) 95th percentile +1.6448 2

Most Likely (mode)

Range Factor (95th/50th) 1.6448 9

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