Detecting mixtures in multivariate extremes S.H.A. Tendijck - - PowerPoint PPT Presentation

Detecting mixtures in multivariate extremes

S.H.A. Tendijck

Lancaster University January 31, 2020

Motivating application

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Motivating application

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Two types of waves

Swell versus wind waves:

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Contents

1 Crash course in underlying theory 2 My model

Overview

1 Crash course in underlying theory 2 My model

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Univariate extremes

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5 10 x 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

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Univariate extremes

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5 10 x 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

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Univariate extremes

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5 10 x 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

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Univariate extremes

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5 10 x 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

5 10

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Multivariate extremes

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Multivariate extremes

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Multivariate extremes

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Multivariate extremes

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Conditional extremes

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Conditional extremes

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Conditional extremes

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Conditional extremes

Heffernan-Tawn model: Y = αX + Z for (X,Y ) on standard margins and Z some residual distribution, independent of X.

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Conditional extremes

Heffernan-Tawn model: Y ∣(X > u) = αX + Z for (X,Y ) on standard margins and Z some residual distribution, independent of X.

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Conditional extremes

Heffernan-Tawn model: Y ∣(X > u) = αX + X βZ for (X,Y ) on standard margins and Z some residual distribution, independent of X.

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Conditional extremes

Heffernan-Tawn model: Y ∣(X > u) = αX + X βZ for (X,Y ) on standard margins and Z some residual distribution, independent of X. Pros:

• It can capture both asymptotic dependence (α = 1) and asymptotic

independence (α < 1);

• Many bivariate distributions follow this structure asymptotically;
• Extends well to multivariate distributions.

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Conditional extremes

Heffernan-Tawn model: Y ∣(X > u) = αX + X βZ for (X,Y ) on standard margins and Z some residual distribution, independent of X. Pros:

• It can capture both asymptotic dependence (α = 1) and asymptotic

independence (α < 1);

• Many bivariate distributions follow this structure asymptotically;
• Extends well to multivariate distributions.

Cons:

• It doesn’t capture mixture structures;
• Data needs to be on standard margins;
• Inconsistent in modelling X∣Y and Y ∣X when both are large.

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Mixtures in extremes

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XL

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YL

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Mixtures in extremes

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XL

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YL

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Mixtures in extremes

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YL

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Mixtures in extremes

The Heffernan-Tawn model extends to Y ∣(X > u) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ α1X + X β1Z1 with probability p; α2X + X β2Z2 with probability 1 − p.

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Mixtures in extremes

The Heffernan-Tawn model extends to Y ∣(X > u) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ α1X + X β1Z1 with probability p; α2X + X β2Z2 with probability 1 − p. What do we want:

• Fit the model;
• Estimate the number of mixture components;
• Estimate the mixture probabilities.

Methods:

1 Quantile-Regression model; 2 Fitting a Heffernan-Tawn mixture model directly.

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Quantile Regression

How do we estimate the 90% conditional quantile of Y given X?

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X

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Y

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Quantile Regression

How do we estimate the 90% conditional quantile of Y given X?

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X

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Y

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Quantile Regression

How do we estimate the 90% conditional quantile of Y given X?

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X

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Y

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Quantile Regression

How do we estimate the 90% conditional quantile of Y given X?

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X

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Y

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Quantile Regression

How do we estimate the 90% conditional quantile of Y given X?

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Y

Minimise the L1 distance to the line, while keeping 90% below.

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Overview

1 Crash course in underlying theory 2 My model

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My model

We assume the Heffernan-Tawn model holds, i.e., Y ∣(X > u) = αX + X βZ. Our quantile regression model is given by qτ(x) = αx + xβz.

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My model

We assume the Heffernan-Tawn model holds, i.e., Y ∣(X > u) = αX + X βZ. Our quantile regression model is given by qτ(x) = c + αx + xβz.

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My model

We assume the Heffernan-Tawn model holds, i.e., Y ∣(X > u) = αX + X βZ. Our quantile regression model is given by qτ(x) = c + αx + xβz. For stability, we fit simultaneously for τ = 0.05,0.15,...,0.95. We get 13 estimated parameters: (ˆ α, ˆ β, ˆ c, ˆ z1,..., ˆ z10).

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My model

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X

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Y Logistic Model

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My model

We assume a mixture HT model holds, i.e., Y ∣(X > u) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ α1X + X β1Z1 with probability 1 − p, α2X + X β2Z2 with probability p. where α1 > α2.

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My model

We assume a mixture HT model holds, i.e., Y ∣(X > u) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ α1X + X β1Z1 with probability 1 − p, α2X + X β2Z2 with probability p. where α1 > α2. Our quantile regression model is given by qτ(x) ∼ ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ c1 + α1x + xβ1z if τ > p, c2 + α2x + xβ2z if τ < p,

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My model

We assume a mixture HT model holds, i.e., Y ∣(X > u) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ α1X + X β1Z1 with probability 1 − p, α2X + X β2Z2 with probability p. where α1 > α2. Our quantile regression model is given by qτ(x) ∼ ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ c1 + α1x + xβ1z if τ > p, c2 + α2x + xβ2z if τ < p, For stability, we fit simultaneously for τ = 0.05,0.15,...,0.95. We get 17 estimated parameters: (ˆ p, ˆ α1, ˆ α2, ˆ β1, ˆ β2, ˆ c1, ˆ c2, ˆ z1, ..., ˆ z10).

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My model

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X

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Y Asymmetric Logistic Model

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Estimating the number of components

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2 4 6 8 10 Y Best fit with 1 mixture(s)

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Estimating the number of components

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2 4 6 8 10 Y Best fit with 1 mixture(s) 2 3 4 5 6 7 8 9 10 11 X

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2 4 6 8 10 12 Y Best fit with 2 mixture(s)

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Estimating the number of components

2 3 4 5 6 7 8 9 10 11 X

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2 4 6 8 10 Y Best fit with 1 mixture(s) 2 3 4 5 6 7 8 9 10 11 X

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2 4 6 8 10 12 Y Best fit with 2 mixture(s) 2 3 4 5 6 7 8 9 10 11 X

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2 4 6 8 10 12 Y Best fit with 3 mixture(s)

Question: How can we compare?

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Estimating the number of components

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2 4 6 8 10 Y Best fit with 1 mixture(s) 2 3 4 5 6 7 8 9 10 11 X

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2 4 6 8 10 12 Y Best fit with 2 mixture(s) 2 3 4 5 6 7 8 9 10 11 X

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2 4 6 8 10 12 Y Best fit with 3 mixture(s)

Question: How can we compare? Method: 10-fold cross-validation.

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Estimating the number of components

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Number of components

1.188 1.19 1.192 1.194 1.196 1.198 1.2 1.202 1.204 1.206

Cross-Validation Statistics

104

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Estimating the number of components

1.185 1.19 1.195 1.2 1.205 1.21

CV statistics density

104 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 9 8 7 6 5 4 3 2 1

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Assessing the model fit

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Assessing the model fit

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Assessing the model fit

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Assessing the model fit

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Assessing the model fit

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Method ˆ p 95% confidence interval Simulation Quantile Regression

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Assessing the model fit

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Method ˆ p 95% confidence interval Simulation 1.11 ⋅ 1e − 5 (1.00, 1.21) ⋅ 1e − 5 Quantile Regression 0.90 ⋅ 1e − 5 ??

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Problems

Is this method already perfect?

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Problems

Is this method already perfect? No, there are just a couple of minor issues:

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Problems

Is this method already perfect? No, there are just a couple of minor issues:

1 Cross-Validation statistics are not necessarily convex;

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Problems

Is this method already perfect? No, there are just a couple of minor issues:

1 Cross-Validation statistics are not necessarily convex; 2 Not trivial how to fit this framework into a Bayesian setting;

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Problems

Is this method already perfect? No, there are just a couple of minor issues:

1 Cross-Validation statistics are not necessarily convex; 2 Not trivial how to fit this framework into a Bayesian setting; 3 Not trivial how to quantify uncertainty;

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Problems

Is this method already perfect? No, there are just a couple of minor issues:

1 Cross-Validation statistics are not necessarily convex; 2 Not trivial how to fit this framework into a Bayesian setting; 3 Not trivial how to quantify uncertainty; 4 Overfitting is not really penalised;

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Problems

Is this method already perfect? No, there are just a couple of minor issues:

1 Cross-Validation statistics are not necessarily convex; 2 Not trivial how to fit this framework into a Bayesian setting; 3 Not trivial how to quantify uncertainty; 4 Overfitting is not really penalised; 5 It is not clear how to efficiently extend it to d-dimensions;

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Problems

Is this method already perfect? No, there are just a couple of minor issues:

1 Cross-Validation statistics are not necessarily convex; 2 Not trivial how to fit this framework into a Bayesian setting; 3 Not trivial how to quantify uncertainty; 4 Overfitting is not really penalised; 5 It is not clear how to efficiently extend it to d-dimensions; 6 It is not very quick;

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Problems

Is this method already perfect? No, there are just a couple of minor issues:

1 Cross-Validation statistics are not necessarily convex; 2 Not trivial how to fit this framework into a Bayesian setting; 3 Not trivial how to quantify uncertainty; 4 Overfitting is not really penalised; 5 It is not clear how to efficiently extend it to d-dimensions; 6 It is not very quick; 7 It is only suited for picking up mixtures that fan out;

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Problems

Is this method already perfect? No, there are just a couple of minor issues:

1 Cross-Validation statistics are not necessarily convex; 2 Not trivial how to fit this framework into a Bayesian setting; 3 Not trivial how to quantify uncertainty; 4 Overfitting is not really penalised; 5 It is not clear how to efficiently extend it to d-dimensions; 6 It is not very quick; 7 It is only suited for picking up mixtures that fan out; 8 Standard tests for testing number of mixture components based on

the likelihood is not (directly) applicable;

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Problems

Is this method already perfect? No, there are just a couple of minor issues:

1 Cross-Validation statistics are not necessarily convex; 2 Not trivial how to fit this framework into a Bayesian setting; 3 Not trivial how to quantify uncertainty; 4 Overfitting is not really penalised; 5 It is not clear how to efficiently extend it to d-dimensions; 6 It is not very quick; 7 It is only suited for picking up mixtures that fan out; 8 Standard tests for testing number of mixture components based on

the likelihood is not (directly) applicable;

9 Fitting a Heffernan-Tawn mixture model using Dirichlet processes

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Detecting mixtures in multivariate extremes

S.H.A. Tendijck

Lancaster University January 31, 2020

Estimating the change point

How to estimate the change point using only 10 quantile levels? For example, assume that it is optimal to split up the quantile levels according to: {0.05,0.15,...,0.45},{0.55,0.65,...,0.95} We have then estimates (α1,β1,c1) for the first set, and (α2,β2,c2) for the second set. Define Q = {qz ∶ qz(x) = c1 + α1x + xβ1z, or qz(x) = c2 + α2x + xβ2z, z ∈ R}. Pick p = 0.5;

1 Choose the q ∈ Q such that q represents best the pth quantile of Y

given X;

2 Find out to which of the models the fitted regression model belongs; 3 Next iteration: estimate the change point in [0.45,0.5] or

[0.5,0.55].

4 Repeat

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Estimating the change point

0.42 0.44 0.46 0.48 0.5 0.52 0.54 0.56 0.58 0.6 quantile 20 40 60 80 100 120 140 Change point estimate | 1 change point Meth1 Meth2

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