[PPT] - Investigating bias in return level estimates due to the use of a PowerPoint Presentation

SLIDE 1

Investigating bias in return level estimates due to the use

f a stopping rule.

Callum Barltrop1, Anna Maria Barlow1

1STOR-i, Lancaster University

August 31, 2017

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SLIDE 2

Overview

1

December 2015 Floods.

2

Motivation behind project.

3

Theory.

4

Methodology.

5

Results.

6

Discussion.

7

Bibliography.

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SLIDE 3

December 2015 Floods

On the 3rd of December, 2015, Lancashire and Cumbria were hit by the wrath of Storm Desmond, causing an estimated 400-500 million pounds in damages. (http://pwc.blogs.com/press_room/2015/ 12/updated-estimates-on-cost-of-storm-desmond-pwc.html)

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SLIDE 4

December 2015 Floods

On the 3rd of December, 2015, Lancashire and Cumbria were hit by the wrath of Storm Desmond, causing an estimated 400-500 million pounds in damages. (http://pwc.blogs.com/press_room/2015/ 12/updated-estimates-on-cost-of-storm-desmond-pwc.html) After these floods, there was a lot of interest from the Government and insurance companies in re-evaluating the river models.

Figure: December 2015 Floods in Lancaster. Picture: M. Yates Photography

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SLIDE 5

December 2015 Floods

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December 2015 Floods

The inclusion of such large events can often lead to significant changes within the model estimates.

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December 2015 Floods

The inclusion of such large events can often lead to significant changes within the model estimates. This could potentially cause significant bias.

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SLIDE 8

December 2015 Floods

The inclusion of such large events can often lead to significant changes within the model estimates. This could potentially cause significant bias. This can cause big issues for governments and insurance companies alike.

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SLIDE 9

December 2015 Floods

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SLIDE 10

December 2015 Floods

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1960

1970 1980 1990 2000 2010 200 400 600 800 1000 1200 1400

Plot of Annual River Flow Maximua (m3 s) against Year.

River Lune at Caton. Year of measurement. Annual River Flow Maximua. December 2015 Floods

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December 2015 Floods

1970 1980 1990 2000 2010 500 1000 1500 2000

Plot of updating 200 year return levels* against year of measurement.

95% confidence intervals obtained from bootstrapping Year of measurement. Updated 200 year return level.

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*Estimates obtained using standard likelihood method.

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SLIDE 12

Motivation behind project

Two problems arose from this scenario:

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SLIDE 13

Motivation behind project

Two problems arose from this scenario: When collecting data, when is an appropriate point to stop and build a model?

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SLIDE 14

Motivation behind project

Two problems arose from this scenario: When collecting data, when is an appropriate point to stop and build a model? For datasets with large final values, is there a better method for calculating the model estimates given the information known about the final value?

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SLIDE 15

Motivation behind project

Two problems arose from this scenario: When collecting data, when is an appropriate point to stop and build a model? For datasets with large final values, is there a better method for calculating the model estimates given the information known about the final value? During this project, I attempted to begin to address these problems.

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SLIDE 16

Theory.

Generalised Extreme Value Distribution. In the context of weather events, it is normally the extreme values that we are interested in. Normally, data is blocked into sequences and the maxima of each block is taken (for example, annual maxima) which can be fitted to the generalized extreme value distribution: G(z) = exp

−
1 + ξ

z − µ σ −1/ξ , defined on the set {z : 1 + ξ(z − µ)/σ) > 0}, where the parameters satisfy −∞ < µ < ∞, σ > 0 and −∞ < ξ < ∞.

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SLIDE 17

0.0

0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

Probability Plot

Empirical Model

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300

400 500 600 700 200 400 600 800

Quantile Plot

Model Empirical 1e−01 1e+00 1e+01 1e+02 1e+03 200 400 600 800 1000 Return Period Return Level

Return Level Plot

Density Plot

z f(z) 200 400 600 800 0.000 0.002 0.004

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Figure: Caton fit diagnostics before 2015 floods.

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SLIDE 18

Theory.

Stopping rules. A stopping rule is a criterion to decide when to stop gathering data and build a model. This rule can be based on many things but most commonly is related to what has already been observed. Return Levels. In the context of extreme values, it is common to talk about x-year return

levels. This is the value that the model would be expected to exceed once

every x-years.

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SLIDE 19

Methodology.

Stopping Rules. During this project, two different stopping rules were investigated: 1) Stop when Xn > c where c is a critical value set before gathering data. 2) Stop when Xn > m where m = max{xn−k, xn−k+1, ..., xn−1} and 1 ≤ k < n − 1.

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SLIDE 20

Methodology.

Likelihood Functions. Several likelihood functions were tested using these stopping rules:

L1 =

n

i=1

f (xi) L3 = n−1

i=1

f (xi) F(c)

·

f (xn) P(X > c) L5 = n−1

i=1

f (xi)

· f (p)

L2 = n−1

i=1

f (xi)

·

f (xn) P(X > p) L4 =

n−1

i=1

f (xi) F(c) L6 =

n−1

i=1

f (xi)

Where p = c or m, depending on the rule used.

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SLIDE 21

Methodology.

Tested the stopping rules and likelihood functions on simulated exponential and extreme value data. Applied the rules and functions to the raw data from Caton to see how they altered the model estimates.

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SLIDE 22

Results.

Stopping rule 1 Simulated data from an Exp(1) distribution with stopping rules andused functions to obtain estimates of 1/λ. Simulated from a GEV(0, 1, −0.1) distribution to obtain parameter estimates for µ, σ and ξ. Calculated the bias for these values

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SLIDE 23

Results.

2

4 6 8 −0.3 −0.2 −0.1 0.0 0.1 0.2 0.3

Plot of average against critical values, stopping rule 1.

10000 samples taken per critical value. Critical value used in stopping rule. Average bias over 10000 samples.

Regular Likelihood

Scaled Likelihood 1 Scaled Likelihood 2 Scaled Likelihood 3 Likelihood including critical Likelihood removing final value

Figure: Bias plot for exponential distrubtion using stopping rule 1.

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SLIDE 24

Results.

2

4 6 8 −0.1 0.0 0.1 0.2 0.3

Plot of variance against critical values, stopping rule 1.

10000 samples taken per critical value. Critical value used in stopping rule. Variance over 10000 samples.

Regular Likelihood

Scaled Likelihood 1 Scaled Likelihood 2 Scaled Likelihood 3 Likelihood including critical Likelihood removing final value

2

4 6 8 −0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6

Plot of RMSE against critical values, stopping rule 1.

10000 samples taken per critical value. Critical value used in stopping rule. RMSE over 10000 samples.

Regular Likelihood

Scaled Likelihood 1 Scaled Likelihood 2 Scaled Likelihood 3 Likelihood including critical Likelihood removing final value

2

4 6 8 0.90 0.92 0.94 0.96 0.98 1.00

Plot of coverage against critical values, stopping rule 1.

10000 samples taken per critical value. Critical value used in stopping rule. Coverage over 10000 samples.

Regular Likelihood

Scaled Likelihood 1 Scaled Likelihood 2 Scaled Likelihood 3 Likelihood including critical Likelihood removing final value

Figure: Variance, RMSE and coverage plots for exponential distrubtion using stopping rule 1.

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SLIDE 25

Results.

1

2 3 4 5 6 −1 1 2 3 4 5

Plot of average bias in mu against critical value.

Critical value. Average bias in mu.

Regular Likelihood

Scaled Likelihood 1 Scaled Likelihood 2 Scaled Likelihood 3 Likelihood including critical Likelihood removing final value

1

2 3 4 5 6 −1 1 2 3 4 5

Plot of average bias in sigma against critical value.

1000 samples per critical value. Critical value. Average bias in sigma.

Regular Likelihood

Scaled Likelihood 1 Scaled Likelihood 2 Scaled Likelihood 3 Likelihood including critical Likelihood removing final value

1

2 3 4 5 6 −1 1 2 3 4 5

Plot of average bias in xi against critical value.

Critical value. Average bias in xi.

Regular Likelihood

Scaled Likelihood 1 Scaled Likelihood 2 Scaled Likelihood 3 Likelihood including critical Likelihood removing final value

Figure: Bias plots for GEV distrubtion using stopping rule 1.

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SLIDE 26

Results.

1

2 3 4 5 6 −0.5 0.0 0.5 1.0 1.5 2.0 Variance in mu against critical value. Critical value. Variance in mu.

Regular Likelihood

Scaled Likelihood 1 Scaled Likelihood 2 Scaled Likelihood 3 Likelihood including critical Likelihood removing final value

1

2 3 4 5 6 −0.5 0.0 0.5 1.0 1.5 2.0 Variance in sigma against critical value. 1000 samples per critical value. Critical value. Variance in sigma.

Regular Likelihood

Scaled Likelihood 1 Scaled Likelihood 2 Scaled Likelihood 3 Likelihood including critical Likelihood removing final value

1

2 3 4 5 6 2 4 6 8 10 Variance in xi against critical value. Critical value. Variance in xi.

Regular Likelihood

Scaled Likelihood 1 Scaled Likelihood 2 Scaled Likelihood 3 Likelihood including critical Likelihood removing final value

1

2 3 4 5 6 −0.5 0.0 0.5 1.0 1.5 2.0 RMSE in mu against critical value. Critical value. RMSE in mu.

Regular Likelihood

Scaled Likelihood 1 Scaled Likelihood 2 Scaled Likelihood 3 Likelihood including critical Likelihood removing final value

1

2 3 4 5 6 −0.5 0.0 0.5 1.0 1.5 2.0 RMSE sigma against critical value. 1000 samples per critical value. Critical value. RMSE in sigma.

Regular Likelihood

Scaled Likelihood 1 Scaled Likelihood 2 Scaled Likelihood 3 Likelihood including critical Likelihood removing final value

1

2 3 4 5 6 1 2 3 4 5 RMSE in xi against critical value. Critical value. RMSE in xi.

Regular Likelihood

Scaled Likelihood 1 Scaled Likelihood 2 Scaled Likelihood 3 Likelihood including critical Likelihood removing final value

Figure: Variance and RMSE plots for GEV distrubtion using stopping rule 1.

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SLIDE 27

Results.

Stopping Rule 2.

5

10 15 −0.2 −0.1 0.0 0.1 0.2

Plot of average bias against k value, stopping rule 2.

10000 samples taken per k value, 20 initial sample points used per simulation. K value used in stopping rule. Average bias over 10000 samples.

Regular Likelihood

Scaled Likelihood 1 Scaled Likelihood 2 Scaled Likelihood 3 Likelihood including maximum Likelihood removing final value

Figure: Bias plot for exponential distrubtion using stopping rule 2.

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SLIDE 28

Results.

5

10 15 −0.05 0.00 0.05 0.10

Plot of variance against k value, stopping rule 2.

10000 samples taken per k value, 20 initial sample points used per simulation. K value used in stopping rule. Variance over 10000 samples.

Regular Likelihood

Scaled Likelihood 1 Scaled Likelihood 2 Scaled Likelihood 3 Likelihood including maximum Likelihood removing final value

5

10 15 −0.1 0.0 0.1 0.2 0.3 0.4

Plot of RMSE against k value, stopping rule 2.

10000 samples taken per k value, 20 initial sample points used per simulation. K value used in stopping rule. RMSE over 10000 samples.

Regular Likelihood

Scaled Likelihood 1 Scaled Likelihood 2 Scaled Likelihood 3 Likelihood including maximum Likelihood removing final value

5

10 15 0.90 0.92 0.94 0.96 0.98 1.00

Plot of coverage against k value, stopping rule 2.

10000 samples taken per k value, 20 initial sample points used per simulation. K value used in stopping rule. Coverage over 10000 samples.

Regular Likelihood

Scaled Likelihood 1 Likelihood including maximum Likelihood removing final value

Figure: Variance, RMSE and coverage plots for exponential distrubtion using stopping rule 2.

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SLIDE 29

Results.

5

10 15 −0.2 −0.1 0.0 0.1 0.2

Plot of average bias in mu against k value.

K value. Average bias in mu.

Regular Likelihood

Scaled Likelihood 1 Scaled Likelihood 2 Scaled Likelihood 3 Likelihood including critical Likelihood removing final value

5

10 15 −0.2 −0.1 0.0 0.1 0.2

Plot of average bias in sigma against k value.

1000 samples per k value. K value. Average bias in sigma.

Regular Likelihood

Scaled Likelihood 1 Scaled Likelihood 2 Scaled Likelihood 3 Likelihood including critical Likelihood removing final value

5

10 15 −0.2 −0.1 0.0 0.1 0.2

Plot of average bias in xi against k value.

K value. Average bias in xi.

Regular Likelihood

Scaled Likelihood 1 Scaled Likelihood 2 Scaled Likelihood 3 Likelihood including critical Likelihood removing final value

Figure: Bias plots for GEV distrubtion using stopping rule 2.

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Results.

5

10 15 −0.10 −0.05 0.00 0.05 0.10 Variance in mu against k value. K value. Variance in mu.

Regular Likelihood

Scaled Likelihood 1 Scaled Likelihood 2 Scaled Likelihood 3 Likelihood including critical Likelihood removing final value

5

10 15 −0.10 −0.05 0.00 0.05 0.10 Variance in sigma against k value. 1000 samples per k value. K value. Variance in sigma.

Regular Likelihood

Scaled Likelihood 1 Scaled Likelihood 2 Scaled Likelihood 3 Likelihood including critical Likelihood removing final value

5

10 15 −0.10 −0.05 0.00 0.05 0.10 Variance in xi against k value. K value. Variance in xi.

Regular Likelihood

Scaled Likelihood 1 Scaled Likelihood 2 Scaled Likelihood 3 Likelihood including critical Likelihood removing final value

5

10 15 −0.1 0.0 0.1 0.2 0.3 0.4 RMSE in mu against k value. K value. RMSE in mu.

Regular Likelihood

Scaled Likelihood 1 Scaled Likelihood 2 Scaled Likelihood 3 Likelihood including critical Likelihood removing final value

5

10 15 −0.1 0.0 0.1 0.2 0.3 0.4 RMSE sigma against k value. 1000 samples per k value. K value. RMSE in sigma.

Regular Likelihood

Scaled Likelihood 1 Scaled Likelihood 2 Scaled Likelihood 3 Likelihood including critical Likelihood removing final value

5

10 15 −0.1 0.0 0.1 0.2 0.3 0.4 RMSE in xi against k value. K value. RMSE in xi.

Regular Likelihood

Scaled Likelihood 1 Scaled Likelihood 2 Scaled Likelihood 3 Likelihood including critical Likelihood removing final value

Figure: Variance and RMSE plots for GEV distrubtion using stopping rule 2.

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Results.

−1

1 2 3 4 5 6 −5 5 10 15 20 25 30

Plot of average 200 year return level bias against critical value.

1000 samples per critical value. Critical value. Average 200 year return level bias.

Regular Likelihood

Scaled Likelihood 1 Scaled Likelihood 2 Scaled Likelihood 3 Likelihood including critical Likelihood removing final value

5

10 15 −2 −1 1 2

Plot of average 200 year return level bias against k value.

1000 samples per k value, 20 initial values per sample. K value. Average 200 year return level bias.

Regular Likelihood

Scaled Likelihood 1 Scaled Likelihood 2 Scaled Likelihood 3 Likelihood including critical Likelihood removing final value

Figure: 200 year return level bias plots.

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SLIDE 32

Results.

Applied to the data from Caton...

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SLIDE 33

Results.

Applied to the data from Caton...

100 200 300 400 500 600 800 1000 1200 1400

Plot of return level against return year, stopping rule 1.

Critical Value = 1125.65 Return Year. Return Level. Predicted model before floods Regular Likelihood Scaled Likelihood 1 Scaled Likelihood 2 Scaled Likelihood 3 Likelihood including critical Likelihood removing final value

Figure: Return levels plots for data at Caton, stopping rule 1.

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SLIDE 34

Results.

100 200 300 400 500 600 800 1000 1200 1400

Plot of return level against return year, stopping rule 2.

K value = 19 Return Year. Return Level. Predicted model before floods Regular Likelihood Scaled Likelihood 1 Scaled Likelihood 2 Scaled Likelihood 3 Likelihood including maximum Likelihood removing final value

Figure: Return levels plots for data at Caton, stopping rule 2.

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SLIDE 35

Discussion.

Conclusions.

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Discussion.

Conclusions. Both stopping rules gave unbiased parameter estimators for sufficiently large sample sizes and chosen k/c value.

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Discussion.

Conclusions. Both stopping rules gave unbiased parameter estimators for sufficiently large sample sizes and chosen k/c value. For the first stopping rule, the standard likelihood function was more biased at higher critical values than some modified functions.

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SLIDE 38

Discussion.

Conclusions. Both stopping rules gave unbiased parameter estimators for sufficiently large sample sizes and chosen k/c value. For the first stopping rule, the standard likelihood function was more biased at higher critical values than some modified functions. The second stopping rule gave some unbiased estimators depending

n the likelihood function and k value used.

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SLIDE 39

Discussion.

Conclusions. Both stopping rules gave unbiased parameter estimators for sufficiently large sample sizes and chosen k/c value. For the first stopping rule, the standard likelihood function was more biased at higher critical values than some modified functions. The second stopping rule gave some unbiased estimators depending

n the likelihood function and k value used.

The standard likelihood function may not be appropriate to use when modelling datasets with large final values.

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SLIDE 40

Discussion.

Further Work.

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Discussion.

Further Work. Investigate the theory behind the likelihood functions and stopping rules.

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SLIDE 42

Discussion.

Further Work. Investigate the theory behind the likelihood functions and stopping rules. Test out more stopping rules and develop more similar likelihood functions.

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SLIDE 43

Discussion.

Further Work. Investigate the theory behind the likelihood functions and stopping rules. Test out more stopping rules and develop more similar likelihood functions. Test rules and functions on more real world data to see accuracy in predicting future events (e.g. on the recent Hurricane from the US).

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SLIDE 44

Discussion.

Further Work. Investigate the theory behind the likelihood functions and stopping rules. Test out more stopping rules and develop more similar likelihood functions. Test rules and functions on more real world data to see accuracy in predicting future events (e.g. on the recent Hurricane from the US). Evaluate more metrics to determine whether the modified likelihood functions are more accurate than the standard function.

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SLIDE 45

Discussion.

Further Work. Investigate the theory behind the likelihood functions and stopping rules. Test out more stopping rules and develop more similar likelihood functions. Test rules and functions on more real world data to see accuracy in predicting future events (e.g. on the recent Hurricane from the US). Evaluate more metrics to determine whether the modified likelihood functions are more accurate than the standard function. Expand the stopping rules to multivariate extreme value models.

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SLIDE 46

Bibliography.

1 Coles, S. (2001). An Introduction to Statistical Modeling of

Extreme Values Classical Extreme Value Theory and Models.

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SLIDE 47

Thank you all for listening and thank you to Anna for being a fantastic supervisor! :) Any questions?

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