Investigating bias in return level estimates due to the use of a stopping rule. Callum Barltrop 1 , Anna Maria Barlow 1 1 STOR-i, Lancaster University August 31, 2017 Callum Barltrop 1 , Anna Maria Barlow 1 Investigating bias in return level estimates due to the use of a stopping rule. 1 / 30
Overview December 2015 Floods. 1 Motivation behind project. 2 Theory. 3 Methodology. 4 Results. 5 Discussion. 6 Bibliography. 7 Callum Barltrop 1 , Anna Maria Barlow 1 Investigating bias in return level estimates due to the use of a stopping rule. 2 / 30
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 ) Callum Barltrop 1 , Anna Maria Barlow 1 Investigating bias in return level estimates due to the use of a stopping rule. 3 / 30
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 Callum Barltrop 1 , Anna Maria Barlow 1 Investigating bias in return level estimates due to the use of a stopping rule. 3 / 30
December 2015 Floods Callum Barltrop 1 , Anna Maria Barlow 1 Investigating bias in return level estimates due to the use of a stopping rule. 4 / 30
December 2015 Floods The inclusion of such large events can often lead to significant changes within the model estimates. Callum Barltrop 1 , Anna Maria Barlow 1 Investigating bias in return level estimates due to the use of a stopping rule. 4 / 30
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. Callum Barltrop 1 , Anna Maria Barlow 1 Investigating bias in return level estimates due to the use of a stopping rule. 4 / 30
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. Callum Barltrop 1 , Anna Maria Barlow 1 Investigating bias in return level estimates due to the use of a stopping rule. 4 / 30
December 2015 Floods Callum Barltrop 1 , Anna Maria Barlow 1 Investigating bias in return level estimates due to the use of a stopping rule. 5 / 30
December 2015 Floods Plot of Annual River Flow Maximua ( m 3 s ) against Year. 1400 ● 1200 December 2015 Floods Annual River Flow Maximua. 1000 800 ● ● ● 600 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 400 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 200 ● 1960 1970 1980 1990 2000 2010 Year of measurement. River Lune at Caton. Callum Barltrop 1 , Anna Maria Barlow 1 Investigating bias in return level estimates due to the use of a stopping rule. 6 / 30
December 2015 Floods Plot of updating 200 year return levels* against year of measurement. 2000 ● Updated 200 year return level. 1500 ● ● ● ● ● ● ● 1000 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 500 ● 0 *Estimates obtained using standard likelihood method. 1970 1980 1990 2000 2010 Year of measurement. 95% confidence intervals obtained from bootstrapping Callum Barltrop 1 , Anna Maria Barlow 1 Investigating bias in return level estimates due to the use of a stopping rule. 7 / 30
Motivation behind project Two problems arose from this scenario: Callum Barltrop 1 , Anna Maria Barlow 1 Investigating bias in return level estimates due to the use of a stopping rule. 8 / 30
Motivation behind project Two problems arose from this scenario: When collecting data, when is an appropriate point to stop and build a model? Callum Barltrop 1 , Anna Maria Barlow 1 Investigating bias in return level estimates due to the use of a stopping rule. 8 / 30
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? Callum Barltrop 1 , Anna Maria Barlow 1 Investigating bias in return level estimates due to the use of a stopping rule. 8 / 30
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. Callum Barltrop 1 , Anna Maria Barlow 1 Investigating bias in return level estimates due to the use of a stopping rule. 8 / 30
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: � �� − 1 /ξ � � � z − µ G ( z ) = exp − 1 + ξ , σ defined on the set { z : 1 + ξ ( z − µ ) /σ ) > 0 } , where the parameters satisfy −∞ < µ < ∞ , σ > 0 and −∞ < ξ < ∞ . Callum Barltrop 1 , Anna Maria Barlow 1 Investigating bias in return level estimates due to the use of a stopping rule. 9 / 30
Probability Plot Quantile Plot 1.0 800 ●● ● ● ● ● ●● ● ● 0.8 ● ●● ● ●● ● ● ● ● ● ● ● ● 600 0.6 ● ● Empirical Model ● ● ●●●●● ● ● ● ● ● ● ● ● ● ● 0.4 ●● ● ● ● ● ● ● 400 ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.2 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 200 0.0 ● ● ● 0.0 0.2 0.4 0.6 0.8 1.0 300 400 500 600 700 Empirical Model Return Level Plot Density Plot 1000 0.004 800 ● Return Level ● f(z) 600 ● 0.002 ●● ● ● ● ● ● ● ● ● ● ● ● ● ● 400 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.000 200 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 1e−01 1e+00 1e+01 1e+02 1e+03 200 400 600 800 Return Period z Figure: Caton fit diagnostics before 2015 floods. Callum Barltrop 1 , Anna Maria Barlow 1 Investigating bias in return level estimates due to the use of a stopping rule. 10 / 30
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. Callum Barltrop 1 , Anna Maria Barlow 1 Investigating bias in return level estimates due to the use of a stopping rule. 11 / 30
Methodology. Stopping Rules. During this project, two different stopping rules were investigated: 1) Stop when X n > c where c is a critical value set before gathering data. 2) Stop when X n > m where m = max { x n − k , x n − k +1 , ..., x n − 1 } and 1 ≤ k < n − 1. Callum Barltrop 1 , Anna Maria Barlow 1 Investigating bias in return level estimates due to the use of a stopping rule. 12 / 30
Methodology. Likelihood Functions. Several likelihood functions were tested using these stopping rules: n � n − 1 � f ( x n ) � � L 1 = f ( x i ) L 2 = f ( x i ) · P ( X > p ) i =1 i =1 � n − 1 � n − 1 f ( x i ) f ( x n ) f ( x i ) � � L 3 = L 4 = · F ( c ) P ( X > c ) F ( c ) i =1 i =1 � n − 1 � n − 1 � � L 5 = f ( x i ) · f ( p ) L 6 = f ( x i ) i =1 i =1 Where p = c or m , depending on the rule used. Callum Barltrop 1 , Anna Maria Barlow 1 Investigating bias in return level estimates due to the use of a stopping rule. 13 / 30
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. Callum Barltrop 1 , Anna Maria Barlow 1 Investigating bias in return level estimates due to the use of a stopping rule. 14 / 30
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