Bagplots, boxplots and detection for functional data 1

Bagplots, boxplots and outlier detection for functional data

Han Lin Shang & Rob J Hyndman

Business & Economic Forecasting Unit Bagplots, boxplots and outlier detection for functional data 2 Outline

1 Introduction

2 Functional bagplot and HDR boxplot

3 Outlier detection

4 Conclusions Bagplots, boxplots and outlier detection for functional data Introduction 3 Outline

1 Introduction

2 Functional bagplot and HDR boxplot

3 Outlier detection

4 Conclusions Bagplots, boxplots and outlier detection for functional data Introduction 4 French male mortality rates France: male death rates (1899−2003) 0 −2 −4 Log death rate −6 −8

0 20 40 60 80 100

Age Bagplots, boxplots and outlier detection for functional data Introduction 4 French male mortality rates France: male death rates (1899−2003) 0

War years −2 −4 Log death rate −6 −8

0 20 40 60 80 100

Age Bagplots, boxplots and outlier detection for functional data Introduction 4 French male mortality rates France: male death rates (1899−2003) 0

War years −2 −4 Log death rate −6

Aims −8 1 “Boxplots” for functional data

2 0 20 40Tools for60 detecting80 outliers100 in

functionalAge data µ(x) is curve {φk (x)} are principal components {zi,k } are PC scores

1 Apply a robust principal component algorithm n−1 X yi (xi ) = µ(x) + zi,k φk (x) k=1

2 Plot zi,2 vs zi,1

¯ Each point in scatterplot represents one curve. ¯ show up in bivariate score space.

Bagplots, boxplots and outlier detection for functional data Introduction 5 Robust principal components

Let {yi (x)}, i = 1,..., n, be a set of curves. ¯ Each point in scatterplot represents one curve. ¯ Outliers show up in bivariate score space.

µ(x) is mean curve {φk (x)} are principal components {zi,k } are PC scores

2 Plot zi,2 vs zi,1

Bagplots, boxplots and outlier detection for functional data Introduction 5 Robust principal components

Let {yi (x)}, i = 1,..., n, be a set of curves.

1 Apply a robust principal component algorithm n−1 X yi (xi ) = µ(x) + zi,k φk (x) k=1 ¯ Each point in scatterplot represents one curve. ¯ Outliers show up in bivariate score space.

{φk (x)} are principal components {zi,k } are PC scores

2 Plot zi,2 vs zi,1

Bagplots, boxplots and outlier detection for functional data Introduction 5 Robust principal components

Let {yi (x)}, i = 1,..., n, be a set of curves.

1 Apply a robust principal component algorithm n−1 X yi (xi ) = µ(x) + zi,k φk (x) k=1 µ(x) is mean curve ¯ Each point in scatterplot represents one curve. ¯ Outliers show up in bivariate score space.

{zi,k } are PC scores

2 Plot zi,2 vs zi,1

Bagplots, boxplots and outlier detection for functional data Introduction 5 Robust principal components

Let {yi (x)}, i = 1,..., n, be a set of curves.

1 Apply a robust principal component algorithm n−1 X yi (xi ) = µ(x) + zi,k φk (x) k=1 µ(x) is mean curve {φk (x)} are principal components ¯ Each point in scatterplot represents one curve. ¯ Outliers show up in bivariate score space.

2 Plot zi,2 vs zi,1

Bagplots, boxplots and outlier detection for functional data Introduction 5 Robust principal components

Let {yi (x)}, i = 1,..., n, be a set of curves.

1 Apply a robust principal component algorithm n−1 X yi (xi ) = µ(x) + zi,k φk (x) k=1 µ(x) is mean curve {φk (x)} are principal components {zi,k } are PC scores ¯ Each point in scatterplot represents one curve. ¯ Outliers show up in bivariate score space.

Bagplots, boxplots and outlier detection for functional data Introduction 5 Robust principal components

Let {yi (x)}, i = 1,..., n, be a set of curves.

1 Apply a robust principal component algorithm n−1 X yi (xi ) = µ(x) + zi,k φk (x) k=1 µ(x) is mean curve {φk (x)} are principal components {zi,k } are PC scores

2 Plot zi,2 vs zi,1 ¯ Each point in scatterplot represents one curve. ¯ Outliers show up in bivariate score space.

Bagplots, boxplots and outlier detection for functional data Introduction 5 Robust principal components

Let {yi (x)}, i = 1,..., n, be a set of curves.

1 Apply a robust principal component algorithm n−1 X yi (xi ) = µ(x) + zi,k φk (x) k=1 µ(x) is mean curve {φk (x)} are principal components {zi,k } are PC scores

2 Plot zi,2 vs zi,1 ¯ Outliers show up in bivariate score space.

Bagplots, boxplots and outlier detection for functional data Introduction 5 Robust principal components

Let {yi (x)}, i = 1,..., n, be a set of curves.

1 Apply a robust principal component algorithm n−1 X yi (xi ) = µ(x) + zi,k φk (x) k=1 µ(x) is mean curve {φk (x)} are principal components {zi,k } are PC scores

2 Plot zi,2 vs zi,1

¯ Each point in scatterplot represents one curve. Bagplots, boxplots and outlier detection for functional data Introduction 5 Robust principal components

Let {yi (x)}, i = 1,..., n, be a set of curves.

1 Apply a robust principal component algorithm n−1 X yi (xi ) = µ(x) + zi,k φk (x) k=1 µ(x) is mean curve {φk (x)} are principal components {zi,k } are PC scores

2 Plot zi,2 vs zi,1

¯ Each point in scatterplot represents one curve. ¯ Outliers show up in bivariate score space. Bagplots, boxplots and outlier detection for functional data Introduction 6 Robust principal components Scatterplot of first two PC scores ● 4 ●

● 3 ● ● ●

● 2 ●

● ●● ●● ● ● ●●● ● ●●● ● ● ● ●●● ● ● ●

1 ●● ● PC score 2 ●●● ● ● ● ● ● ●● ● ● ●

0 ● ● ● ●● ●●●●● ● ● ● ● ● ●● ●●● ●●●●●● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ●● ● −1 ●● ●

−10 −5 0 5 10 15

PC score 1 Bagplots, boxplots and outlier detection for functional data Introduction 6 Robust principal components Scatterplot of first two PC scores 1914● 4 1915●

1916● 3 1944● 1918● 1917● 1943● 2 1940● ● ●● ●● ● ● ●●● ● ●●● ● ● ● ●●● ● ● ●

1 ●● ● PC score 2 1919 ●●● 1942● ● ● ● 1945 ● ●● ● ● ●

0 ● ● ● 1941●● ●●●●● ● ● ● ● ● ●● ●●● ●●●●●● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ●● ● −1 ●● ●

−10 −5 0 5 10 15

PC score 1 Bagplots, boxplots and outlier detection for functional data Functional bagplot and HDR boxplot 7 Outline

1 Introduction

2 Functional bagplot and HDR boxplot

3 Outlier detection

4 Conclusions Bagplots, boxplots and outlier detection for functional data Functional bagplot and HDR boxplot 8 Functional bagplot Bivariate bagplot due to Rousseeuw et al. (1999). Rank points by halfspace location depth. Display , 50% convex hull and outer convex hull (with 99% coverage if bivariate normal). 5

1914●● 4 1915●●

1916●● 3 ●● 1944 1918●● 1917●● 1943●● 2 1940●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●

PC score 2 ● ● ● ● ● ● 1 1919● ● ● ●● ● ● 1942 ● ● ● ● ● ● ● ● ● ● ●● ● ● ● 0 ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ●● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●

−1 ● ●

−10 −5 0 5 10 15

PC score 1 Bagplots, boxplots and outlier detection for functional data Functional bagplot and HDR boxplot 8 Functional bagplot Bivariate bagplot due to Rousseeuw et al. (1999). Rank points by halfspace location depth. Display median, 50% convex hull and outer convex hull (with 99% coverage if bivariate normal). Boundaries contain all curves inside bags. 95% CI for median curve also shown. 5 0

1914●● 4 1915●● −2 1916●● 3 ●● 1944 1918●● 1917●●

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1943 −4 2 1940●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●

PC score 2 ● ● ● ● ● ● 1 ●

1919 Log death rate ● ● ● ● ● ● 1942 −6 ● ● ● ● ● ● ● ● ● ● ●● ● ● ● 0 ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ●● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● −8 ● ● ● ● ●● ● ●

−1 ● ●

−10 −5 0 5 10 15 0 20 40 60 80 100

PC score 1 Age Bagplots, boxplots and outlier detection for functional data Functional bagplot and HDR boxplot 8 Functional bagplot 0 −2 −4 Log death rate −6 −8

0 20 40 60 80 100

Age Bagplots, boxplots and outlier detection for functional data Functional bagplot and HDR boxplot 9 Functional HDR boxplot Bivariate HDR boxplot due to Hyndman (1996). Rank points by value of kernel density estimate. Display , 50% and (usually) 99% highest density regions (HDRs) and mode. 5

1914●● 4

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● 2 ●

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PC score 2 ● ● ● ● ● ● 1 1919● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● 0 ● ● ● ● ● ● ● ●o ● ● ●● ● ● ●● ● ●● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●

−1 ● ●

−10 −5 0 5 10 15

PC score 1 Bagplots, boxplots and outlier detection for functional data Functional bagplot and HDR boxplot 9 Functional HDR boxplot Bivariate HDR boxplot due to Hyndman (1996). Rank points by value of kernel density estimate. Display mode, 50% and (usually) 99% highest density regions (HDRs) and mode. 5 91% outer region 1914●● 4 1915●●

1916●● 3 ●● 1944 1918●● 1917●● 1943●● 2 1940●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●

PC score 2 ● ● ● ● ● ● 1 1919● ● ● ●● ● ● 1942 ● ● ● ● ● ● ● ● ● ● ●● ● ● ● 0 ● ● ● ● ● ● ● o ● ● ● ●● ● ● ●● ● ●● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●

−1 ● ●

−10 −5 0 5 10 15

PC score 1 Bagplots, boxplots and outlier detection for functional data Functional bagplot and HDR boxplot 9 Functional HDR boxplot Bivariate HDR boxplot due to Hyndman (1996). Rank points by value of kernel density estimate. Display mode, 50% and (usually) 99% highest density regions (HDRs) and mode. Boundaries contain all curves inside HDRs. 5 0 91% outer region 1914●● 4 1915●● −2 1916●● 3 ●● 1944 1918●● 1917●●

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1943 −4 2 1940●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●

PC score 2 ● ● ● ● ● ● 1 ●

1919 Log death rate ● ● ● ● ● ● 1942 −6 ● ● ● ● ● ● ● ● ● ● ●● ● ● ● 0 ● ● ● ● ● ● ● o ● ● ● ●● ● ● ●● ● ●● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● −8 ● ● ● ● ●● ● ●

−1 ● ●

−10 −5 0 5 10 15 0 20 40 60 80 100

PC score 1 Age Bagplots, boxplots and outlier detection for functional data Functional bagplot and HDR boxplot 9 Functional HDR boxplot 0 −2 −4 Log death rate −6 −8

0 20 40 60 80 100

Age Bagplots, boxplots and outlier detection for functional data Outlier detection 10 Outline

1 Introduction

2 Functional bagplot and HDR boxplot

3 Outlier detection

4 Conclusions Disadvantages Computationally intensive. Ignores shape outliers. If trimmed mean is used and there is no outlier, C will be downward biased.

Bagplots, boxplots and outlier detection for functional data Outlier detection 11 Outlier detection: existing methods

Likelihood ratio method Febrero et al. (2007) find curve that maximizes LRT . If LRT > C, then curve is considered outlier. C is computed via smoothed bootstrap. Process continues until no more outliers. Bagplots, boxplots and outlier detection for functional data Outlier detection 11 Outlier detection: existing methods

Likelihood ratio method Febrero et al. (2007) find curve that maximizes LRT statistic. If LRT > C, then curve is considered outlier. C is computed via smoothed bootstrap. Process continues until no more outliers. Disadvantages Computationally intensive. Ignores shape outliers. If trimmed mean is used and there is no outlier, C will be downward biased. √ Curve is outlier if vi > s + λ s, where s = median(v1, ··· , vt) and λ is tuning parameter. Disadvantages Depends on K and λ. If K large, outliers modelled by higher components.

Bagplots, boxplots and outlier detection for functional data Outlier detection 12 Outlier detection: existing methods

Integrated squared error method Hyndman & Ullah (2007) proposed the use of

K 2 Z  X  vi = yˆi (x) − µ(x) − zi,k φk (x) dx x k=1

where zi,k and (robust) PC scores and φk (x) are PCs. Disadvantages Depends on K and λ. If K large, outliers modelled by higher components.

Bagplots, boxplots and outlier detection for functional data Outlier detection 12 Outlier detection: existing methods

Integrated squared error method Hyndman & Ullah (2007) proposed the use of

K 2 Z  X  vi = yˆi (x) − µ(x) − zi,k φk (x) dx x k=1

where zi,k and (robust) PC scores and φk (x) are PCs. √ Curve is outlier if vi > s + λ s, where s = median(v1, ··· , vt) and λ is tuning parameter. Disadvantages Depends on K and λ. If K large, outliers modelled by higher components.

Bagplots, boxplots and outlier detection for functional data Outlier detection 12 Outlier detection: existing methods

Integrated squared error method Hyndman & Ullah (2007) proposed the use of

K 2 Z  X  vi = yˆi (x) − µ(x) − zi,k φk (x) dx x k=1

where zi,k and (robust) PC scores and φk (x) are PCs. √ Curve is outlier if vi > s + λ s, where s = median(v1, ··· , vt) and λ is tuning parameter. Bagplots, boxplots and outlier detection for functional data Outlier detection 12 Outlier detection: existing methods

Integrated squared error method Hyndman & Ullah (2007) proposed the use of

K 2 Z  X  vi = yˆi (x) − µ(x) − zi,k φk (x) dx x k=1

where zi,k and (robust) PC scores and φk (x) are PCs. √ Curve is outlier if vi > s + λ s, where s = median(v1, ··· , vt) and λ is tuning parameter. Disadvantages Depends on K and λ. If K large, outliers modelled by higher components. Method Sensitivity Specificity Time (s) Likelihood ratio 0% 100% 18.8 Integrated squared error 50% 94% 3.4 Bagplot 83% 98% 0.6 91% HDR boxplot 83% 98% 0.3

Bagplots, boxplots and outlier detection for functional data Outlier detection 13 Outlier detection: comparison

French male mortality data set Based on historical information, the outliers are expected to be 1914–1919 & 1940–1945.

Method Outliers detected Likelihood ratio — Integrated squared error 1914–1918, 1940, 1943–1944 Bagplot 1914–1919, 1940, 1942–1944 91% HDR boxplot 1914–1919, 1940, 1942–1944 Bagplots, boxplots and outlier detection for functional data Outlier detection 13 Outlier detection: comparison

French male mortality data set Based on historical information, the outliers are expected to be 1914–1919 & 1940–1945.

Method Outliers detected Likelihood ratio — Integrated squared error 1914–1918, 1940, 1943–1944 Bagplot 1914–1919, 1940, 1942–1944 91% HDR boxplot 1914–1919, 1940, 1942–1944

Method Sensitivity Specificity Time (s) Likelihood ratio 0% 100% 18.8 Integrated squared error 50% 94% 3.4 Bagplot 83% 98% 0.6 91% HDR boxplot 83% 98% 0.3 Bagplots, boxplots and outlier detection for functional data Outlier detection 14 Outlier detection: comparison

Simulation

yi (x) = ai sin(x) + bi cos(x), 0 < x < 2π ai , bi ∼ Unif(0, 0.1) with probability 99% ai , bi ∼ Unif(0.1, 0.108) with probability 1% 0.15 0.10

0.05 Outliers shown y 0.00 in black −0.05 −0.10 −0.15 0 1 2 3 4 5 6

x Bagplots, boxplots and outlier detection for functional data Outlier detection 15 Outlier detection: comparison Scatterplot of first two PC scores ● ●●● ●

1.0 ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ●● ● ●● ● ● ● ● ●● ● ● ● ● ● ●● ● ●●●● ●●● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ●● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●●● ● ●● ●● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ●● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● 0.5 ●● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ●● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ●● ● ● ● ●● ● ● ● ●●● ●●● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●●● ●● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ●●●● ● ●● ●● ● ● ● ●●● ● ●● ●● ● ● ● ●● ● ● ●● ● ● ● ● ●●● ● ● ● ●●● ● ●●● 0.0 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ●●● ● ●● ● ● ● ●●● ● ● ● ● ●● ● ● ● ● ● ● ● ● PC score 2 ●● ● ● ● ●●● ● ● ● ● ●● ● ● ●● ●● ● ●● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ●●● ●● ●● ●● ●●● ●● ● ● ● ●● ● ●●● ● ● ● ● ● ● ● ●● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ●● ● ●● ●● ●● ●● ● ● ● ● ● ● ● ●● ●● ● ● ●● ● ●●● ● ●●● ● ● ●● ● ● ● ● ●● ●● ●●● ● ● ● ●●● ●●● ● ●● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ●● ●● ●● ●●● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ●● ●● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● −0.5 ● ● ● ● ●● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●●● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ●●● ● ●● ● ● ● ● ●● ●● ● ● ● ● ●● ● ● ●● ● ● ●● ● ● ●●● ● ●

−0.5 0.0 0.5

PC score 1 Method Sensitivity Specificity Time (s) Likelihood ratio 0% 100% 28.5 Integrated squared error 0% 100% 18.8 Bagplot 0% 100% 7.3 99% HDR boxplot 100% 100% 6.9

Bagplots, boxplots and outlier detection for functional data Outlier detection 16 Outlier detection: comparison

Simulation

Method Outliers detected Likelihood ratio — Integrated squared error — Bagplot — 99% HDR boxplot All Bagplots, boxplots and outlier detection for functional data Outlier detection 16 Outlier detection: comparison

Simulation

Method Outliers detected Likelihood ratio — Integrated squared error — Bagplot — 99% HDR boxplot All

Method Sensitivity Specificity Time (s) Likelihood ratio 0% 100% 28.5 Integrated squared error 0% 100% 18.8 Bagplot 0% 100% 7.3 99% HDR boxplot 100% 100% 6.9 Bagplots, boxplots and outlier detection for functional data Conclusions 17 Outline

1 Introduction

2 Functional bagplot and HDR boxplot

3 Outlier detection

4 Conclusions Functional HDR boxplot more flexible but coverage probability needs tuning. Functional HDR boxplot can detect bimodality and inliers. Existing depth method performs poorly and ignores shape outliers. Existing ISE method often misses outliers.

¯ Paper and R code: www.robhyndman.info ¯ Comments to: [email protected]

Bagplots, boxplots and outlier detection for functional data Conclusions 18 Conclusions

Functional bagplot highly robust but sometimes misses outliers. Functional HDR boxplot can detect bimodality and inliers. Existing depth method performs poorly and ignores shape outliers. Existing ISE method often misses outliers.

¯ Paper and R code: www.robhyndman.info ¯ Comments to: [email protected]

Bagplots, boxplots and outlier detection for functional data Conclusions 18 Conclusions

Functional bagplot highly robust but sometimes misses outliers. Functional HDR boxplot more flexible but coverage probability needs tuning. Existing depth method performs poorly and ignores shape outliers. Existing ISE method often misses outliers.

¯ Paper and R code: www.robhyndman.info ¯ Comments to: [email protected]

Bagplots, boxplots and outlier detection for functional data Conclusions 18 Conclusions

Functional bagplot highly robust but sometimes misses outliers. Functional HDR boxplot more flexible but coverage probability needs tuning. Functional HDR boxplot can detect bimodality and inliers. Existing ISE method often misses outliers.

¯ Paper and R code: www.robhyndman.info ¯ Comments to: [email protected]

Bagplots, boxplots and outlier detection for functional data Conclusions 18 Conclusions

Functional bagplot highly robust but sometimes misses outliers. Functional HDR boxplot more flexible but coverage probability needs tuning. Functional HDR boxplot can detect bimodality and inliers. Existing depth method performs poorly and ignores shape outliers. ¯ Paper and R code: www.robhyndman.info ¯ Comments to: [email protected]

Bagplots, boxplots and outlier detection for functional data Conclusions 18 Conclusions

Functional bagplot highly robust but sometimes misses outliers. Functional HDR boxplot more flexible but coverage probability needs tuning. Functional HDR boxplot can detect bimodality and inliers. Existing depth method performs poorly and ignores shape outliers. Existing ISE method often misses outliers. ¯ Paper and R code: www.robhyndman.info ¯ Comments to: [email protected]

Bagplots, boxplots and outlier detection for functional data Conclusions 18 Conclusions

Functional bagplot highly robust but sometimes misses outliers. Functional HDR boxplot more flexible but coverage probability needs tuning. Functional HDR boxplot can detect bimodality and inliers. Existing depth method performs poorly and ignores shape outliers. Existing ISE method often misses outliers. Bagplots, boxplots and outlier detection for functional data Conclusions 18 Conclusions

Functional bagplot highly robust but sometimes misses outliers. Functional HDR boxplot more flexible but coverage probability needs tuning. Functional HDR boxplot can detect bimodality and inliers. Existing depth method performs poorly and ignores shape outliers. Existing ISE method often misses outliers.

¯ Paper and R code: www.robhyndman.info ¯ Comments to: [email protected]