Tempered Stable and Pareto Distributions: Predictions Under Uncertainty
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1 Tempered Stable and Pareto Distributions: Predictions Under Uncertainty Robert L Wolpert & Kerrie L Mengersen Duke University & Queensland University of Technology Session 2A: Foundations I Thu 1:50 2:15pm Frontiers of Statistical Decision Making & Bayesian Analysis
2 Motivation What s at stake Modeling Power Laws Tempering Uncertainty Parameters Predictions Conclusions Discussion Future Work
3 Caribbean Island of Montserrat
4 Soufrière Hills Volcano View from MVO at sunset, 18 April 2007
5 Where is Montserrat?
6 What s been happening since 1995? Annual Rate e+04 5e+05 5e+06 5e+07 5e+08 PF Volume (m 3 ) Seems kind of linear, on log-log scale... remember that.
7 Pyroclastic Flow Volumes PF s exceeding 10 4 m 3 : Daily, km runout PF s exceeding 10 5 m 3 : Fortnightly, km runout PF s exceeding 10 6 m 3 : Quarterly, km runout PF s exceeding 10 7 m 3 : Yearly, km runout PF s exceeding 10 8 m 3 : Just one, unknown runout PF s exceeding 10 9 m 3 : None YET, but...
8 Pyroclastic Flow Volumes PF s exceeding 10 4 m 3 : Daily, km runout PF s exceeding 10 5 m 3 : Fortnightly, km runout PF s exceeding 10 6 m 3 : Quarterly, km runout PF s exceeding 10 7 m 3 : Yearly, km runout PF s exceeding 10 8 m 3 : Just one, unknown runout PF s exceeding 10 9 m 3 : None YET, but...
9 Pyroclastic Flow Volumes PF s exceeding 10 4 m 3 : Daily, km runout PF s exceeding 10 5 m 3 : Fortnightly, km runout PF s exceeding 10 6 m 3 : Quarterly, km runout PF s exceeding 10 7 m 3 : Yearly, km runout PF s exceeding 10 8 m 3 : Just one, unknown runout PF s exceeding 10 9 m 3 : None YET, but...
10 Flow Volume Context How much is 10 9 m 3 of flow? V Cube Width Depth in 1km 2 Radius 1m deep m (32.8 ) 1mm 18 m m (328 ) 1m 564 m m (0.6 mi) 1km 17.8 km m (1.3 mi) 10 km 56 km Moral: Flows bigger than 10 9 m 3 spell trouble. Flows bigger than m 3 are unrealistic. Flows bigger than m 3 cause mass extinction.
11 Flow Volume Context How much is 10 9 m 3 of flow? V Cube Width Depth in 1km 2 Radius 1m deep m (32.8 ) 1mm 18 m m (328 ) 1m 564 m m (0.6 mi) 1km 17.8 km m (1.3 mi) 10 km 56 km Montserrat is 16 km long 10 km wide. Moral: Flows bigger than 10 9 m 3 spell trouble. Flows bigger than m 3 are unrealistic. Flows bigger than m 3 cause mass extinction.
12 Flow Volume Context How much is 10 9 m 3 of flow? V Cube Width Depth in 1km 2 Radius 1m deep m (32.8 ) 1mm 18 m m (328 ) 1m 564 m m (0.6 mi) 1km 17.8 km m (1.3 mi) 10 km 56 km Montserrat is 16 km long 10 km wide. Largest volcanoes in recorded history: Tambora, Indonesia 1815 AD m 3 Baekdu Mt., China 979 AD m 3 Moral: Flows bigger than 10 9 m 3 spell trouble. Flows bigger than m 3 are unrealistic. Flows bigger than m 3 cause mass extinction.
13 Flow Volume Context How much is 10 9 m 3 of flow? V Cube Width Depth in 1km 2 Radius 1m deep m (32.8 ) 1mm 18 m m (328 ) 1m 564 m m (0.6 mi) 1km 17.8 km m (1.3 mi) 10 km 56 km Montserrat is 16 km long 10 km wide. Moral: Flows bigger than 10 9 m 3 spell trouble. Flows bigger than m 3 are unrealistic. Flows bigger than m 3 cause mass extinction.
14 Flow Volume Context How much is 10 9 m 3 of flow? V Cube Width Depth in 1km 2 Radius 1m deep m (32.8 ) 1mm 18 m m (328 ) 1m 564 m m (0.6 mi) 1km 17.8 km m (1.3 mi) 10 km 56 km Montserrat is 16 km long 10 km wide. Moral: Flows bigger than 10 9 m 3 spell trouble. Flows bigger than m 3 are unrealistic. Flows bigger than m 3 cause mass extinction.
15 Flow Volume Context How much is 10 9 m 3 of flow? V Cube Width Depth in 1km 2 Radius 1m deep m (32.8 ) 1mm 18 m m (328 ) 1m 564 m m (0.6 mi) 1km 17.8 km m (1.3 mi) 10 km 56 km Montserrat is 16 km long 10 km wide. Moral: Flows bigger than 10 9 m 3 spell trouble. Flows bigger than m 3 are unrealistic. Flows bigger than m 3 cause mass extinction.
16 How big is bad? V= V= V= V=10 9 V= V= Kinda depends which way it goes... but Ψ = or Ψ = 10 9 seems bad.
17 Power Laws Remember the nearly-linear Freq. vs. Volume log-log plot? For some slope α and intercept c, log ( P[V > v ]) αlog(v) + c, v > ǫ After exponentiation, P[V > v] = (v/ǫ) α, v > ǫ. This is the Pareto Pa(α,ǫ) distribution! Which is kind of bad news.
18 Power Laws Remember the nearly-linear Freq. vs. Volume log-log plot? For some slope α and intercept c, log ( P[V > v ]) αlog(v) + c, v > ǫ After exponentiation, P[V > v] = (v/ǫ) α, v > ǫ. This is the Pareto Pa(α,ǫ) distribution! Which is kind of bad news.
19 Power Laws Remember the nearly-linear Freq. vs. Volume log-log plot? For some slope α and intercept c, log ( P[V > v ]) αlog(v) + c, v > ǫ After exponentiation, P[V > v] = (v/ǫ) α, v > ǫ. This is the Pareto Pa(α,ǫ) distribution! Which is kind of bad news.
20 The Pareto Distribution The slope seems to be in the α 0.64 or so for 0 < α < 1, the Pareto has: Heavy tails; Infinite mean E[V ] = ; Infinite variance E[V 2 ] = ; No Central Limit Theorem for sums aggregate flow is approx. α-stable, not Gaussian.
21 Frequency of Large Flows MLE and reference Bayes estimates give α 0.64, and λ ǫ 22 flows/year exceeding ǫ = m 3. In twenty years we should expect about: 16.3 flows exceeding 10 7 m flows exceeding 10 8 m flows exceeding 10 9 m flows exceeding m flows exceeding m flows exceeding m 3 Suggest about 18% chance of V > in two decades, and about 1% chance of V > 10 12! Which are unrealistically high (we think). An idea
22 Frequency of Large Flows MLE and reference Bayes estimates give α 0.64, and λ ǫ 22 flows/year exceeding ǫ = m 3. In twenty years we should expect about: 16.3 flows exceeding 10 7 m flows exceeding 10 8 m flows exceeding 10 9 m flows exceeding m flows exceeding m flows exceeding m 3 Suggest about 18% chance of V > in two decades, and about 1% chance of V > 10 12! Which are unrealistically high (we think). An idea
23 Frequency of Large Flows MLE and reference Bayes estimates give α 0.64, and λ ǫ 22 flows/year exceeding ǫ = m 3. In twenty years we should expect about: 16.3 flows exceeding 10 7 m flows exceeding 10 8 m flows exceeding 10 9 m flows exceeding m flows exceeding m flows exceeding m 3 Suggest about 18% chance of V > in two decades, and about 1% chance of V > 10 12! Which are unrealistically high (we think). An idea
24 Frequency of Large Flows MLE and reference Bayes estimates give α 0.64, and λ ǫ 22 flows/year exceeding ǫ = m 3. In twenty years we should expect about: 16.3 flows exceeding 10 7 m flows exceeding 10 8 m flows exceeding 10 9 m flows exceeding m flows exceeding m flows exceeding m 3 Suggest about 18% chance of V > in two decades, and about 1% chance of V > 10 12! Which are unrealistically high (we think). An idea
25 Tempered Pareto Models There s a slight suggestion of a downturn in the Freq-vs-Vol log-log plot; perhaps a better model is Tempered Pareto, with P[V > v] = (v/ǫ) α e β(v ǫ), v > ǫ : Tempered Pareto TP(α = 0.43, ε = 10 4, κ = ) Distribution log 10 Prob [V > v] Note Downturn log 10 PF volume v (m 3 ) See the drop-off near 10 8 m 3 or 10 9 m 3? That s Good news!
26 Tempered Pareto Models There s a slight suggestion of a downturn in the Freq-vs-Vol log-log plot; perhaps a better model is Tempered Pareto, with P[V > v] = (v/ǫ) α e β(v ǫ), v > ǫ : Tempered Pareto TP(α = 0.43, ε = 10 4, κ = ) Distribution log 10 Prob [V > v] Note Downturn log 10 PF volume v (m 3 ) See the drop-off near 10 8 m 3 or 10 9 m 3? That s Good news!
27 Uncertainty MLE and reference Bayesian posterior means (± SD s) for the three parameters of this model are: Shape α 0.64 ± 0.04 Rate λ ± 1.3 /year Bend β ± /m 3 Note how uncertain β is... only 1.5 SD above zero. How does this affect risk predictions? How does Pareto (β = 0) model differ from Tempered Pareto (β > 0)?
28 Uncertainty MLE and reference Bayesian posterior means (± SD s) for the three parameters of this model are: Shape α 0.64 ± 0.04 Rate λ ± 1.3 /year Bend β ± /m 3 Note how uncertain β is... only 1.5 SD above zero. How does this affect risk predictions? How does Pareto (β = 0) model differ from Tempered Pareto (β > 0)?
29 Uncertainty MLE and reference Bayesian posterior means (± SD s) for the three parameters of this model are: Shape α 0.64 ± 0.04 Rate λ ± 1.3 /year Bend β ± /m 3 Note how uncertain β is... only 1.5 SD above zero. How does this affect risk predictions? How does Pareto (β = 0) model differ from Tempered Pareto (β > 0)?
30 Uncertainty MLE and reference Bayesian posterior means (± SD s) for the three parameters of this model are: Shape α 0.64 ± 0.04 Rate λ ± 1.3 /year Bend β ± /m 3 Note how uncertain β is... only 1.5 SD above zero. How does this affect risk predictions? How does Pareto (β = 0) model differ from Tempered Pareto (β > 0)?
31 Risk for Fifty Years of V > 10 8 m 3 Risk Comparisons, Tempered vs. Untempered Pareto, for Ψ 10 8 m 3 P[ Catastrophe in t years Data ] Pareto Model (Untempered) Pareto Model (Tempered) Time t (years)
32 Risk for Fifty Years of V > m 3 Risk Comparisons, Tempered vs. Untempered Pareto, for Ψ m 3 P[ Catastrophe in t years Data ] Pareto Model (Untempered) Pareto Model (Tempered) Time t (years)
33 Why the wide uncertainty? Annual Rate e+04 5e+05 5e+06 5e+07 5e+08 PF Volume (m 3 )
34 How does that affect estimated risk? P[ Catastrophe in t years Data ] Risk for Ψ m 3 Tempered Pareto model: Eα = 0.63, Eβ 1 = m Time t (years)
35 What if there is no tempering? P[ Catastrophe in t years Data ] Risk for Ψ m 3 Untempered Pareto model: Eα = Time t (years)
36 A little more extreme Truncated Pareto P[V > v] = v α L α ǫ α L α ǫ v L : Log 10 Prob [V > v] Log 10 PF volume V (m 3 ) Now the drop-off approaching 10 9 m 3 is fast!
37 A little more extreme Truncated Pareto Annual Rate e+04 5e+05 5e+06 5e+07 5e+08 PF Volume (m 3 )
38 A little more extreme Truncated Pareto P[ Catastrophe in t years Data ] Risk for Ψ 10 8 m 3 Truncated Zipf model: Eα = 0.63, EL = m 3 Prior L Ga(1, 1e 09) Time t (years)
39 Morals It s hard to tell about tail behaviour (extremes) from data; Clearer picture of tempering would help working to elicit expert opinion about physical limits Other model assumptions might bear closer scrutiny.
40 Morals It s hard to tell about tail behaviour (extremes) from data; Clearer picture of tempering would help working to elicit expert opinion about physical limits Other model assumptions might bear closer scrutiny.
41 Morals It s hard to tell about tail behaviour (extremes) from data; Clearer picture of tempering would help working to elicit expert opinion about physical limits Other model assumptions might bear closer scrutiny.
42 Morals It s hard to tell about tail behaviour (extremes) from data; Clearer picture of tempering would help working to elicit expert opinion about physical limits Other model assumptions might bear closer scrutiny.
43 Model Strengths Inspired by data feature (Flow vs. Frequency linearity on log-log scale); Accommodates wide range of censoring; Simulations from model appear similar to real data; Missing observations, irregular space and time are okay.
44 Model Shortcomings Assumes Stationarity, but patterns change over decades Assumes Independence, but patterns exist over short periods Employs Pareto, whose extreme tails are too flat after all, the m 3 volume of the earth is an absolute limit!
45 Open Questions How much does false Stationarity assumption matter? Can we make a dynamic time-varying version? How much does false Independence assumption matter? Should we construct cluster models? Do we need to? How much does false Pareto assumption matter? Does it help to temper the tails like P[V > v] = (v/ǫ) α e β(v ǫ), v > ǫ?
46 Open Questions How much does false Stationarity assumption matter? Can we make a dynamic time-varying version? How much does false Independence assumption matter? Should we construct cluster models? Do we need to? How much does false Pareto assumption matter? Does it help to temper the tails like P[V > v] = (v/ǫ) α e β(v ǫ), v > ǫ?
47 Open Questions How much does false Stationarity assumption matter? Can we make a dynamic time-varying version? How much does false Independence assumption matter? Should we construct cluster models? Do we need to? How much does false Pareto assumption matter? Does it help to temper the tails like P[V > v] = (v/ǫ) α e β(v ǫ), v > ǫ? or P[V > v] (v/ǫ) α 1 {v<l}, v > ǫ?
48 Ongoing Work & Extensions Studying possible variable rate of eruption process; Studying possible dependence between angle φ and flow volume V ; Refining dataset and accumulating additional data; Improving our censorship models for complex data. Group trip to Montserrat to see data acquisition first-hand!
49 Ongoing Work & Extensions Studying possible variable rate of eruption process; Studying possible dependence between angle φ and flow volume V ; Refining dataset and accumulating additional data; Improving our censorship models for complex data. Group trip to Montserrat to see data acquisition first-hand!
50 A Collaborative Effort
51 Thanks! More details (references, this talk in.pdf, related work) are available from Froggy website or on request from Special thanks to Kerrie Mengersen, collaborator in ongoing work on tempering extremes in a wide area of applications. Thanks to SAMSI and to NSF/FRG and, especially, to Jim Berger!
52 Collaboration is such hard work...
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