Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks
Depth in Multivariate Statistics
Ignacio Cascos
Departamento de Estadística Universidad Carlos III de Madrid
Getafe, January 2011 Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks
Outline
Motivation
Main definitions Depth function and depth-trimmed region Data depths
Algorithms Halfspaces and convex hulls (depth functions) Circular sequence (depth-trimmed regions)
Applications Bagplot, DDplot, volume statistics, L-statistics, stochastic orders, quality control, risk measures
Parameter depth & regression Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks
Boxplot & Bagplot (Rousseeuw, Ruts & Tukey, 1999) Decathlon @ Athens 2004: long jump & 100m Boxplot long jump
6.6 6.8 7.0 7.2 7.4 7.6 7.8 8.0
Boxplot 100m
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10.6 10.8 11.0 11.2 Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks
Boxplot & Bagplot (Rousseeuw, Ruts & Tukey, 1999) Decathlon @ Athens 2004: long jump & 100m 11.2 11.0 10.8 10.6
6.6 6.8 7.0 7.2 7.4 7.6 7.8 8.0 Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks
PPplot & DDplot Liu, Parelius & Singh (1999) Histogram 40 30 20 Frequency 10 0
−2 −1 0 1 2 3
x Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks
PPplot & DDplot Liu, Parelius & Singh (1999)
PP plot
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0.0 0.2 0.4 0.6 0.8 1.0
(Fn(xi), F(xi)) Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks
PPplot & DDplot Liu, Parelius & Singh (1999) DD plot
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0.1 ● ●●●● ●● ●● ●● ●●●● ● ● ●●● ●●● ●●●●●●●●● ● ● ●●●● 0.0
0.0 0.1 0.2 0.3 0.4 0.5
(Dn(xi), D(xi))
Dn(x) = min{Fn(x), 1 − Fn(x)} ; D(x) = min{F(x), 1 − F(x)} Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks
PPplot & DDplot Liu, Parelius & Singh (1999)
Data cloud
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−2 −1 0 1 2 Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks
PPplot & DDplot Liu, Parelius & Singh (1999)
DD plot
● ● ● ● ● ●● ●●● 0.4 ● ● ● ● ● ● ●● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● 0.3 ● ● ● ● ● ● ●●● ● ● ● ●● ● ●●● ●● ● ●● ● ●●●● ● ● ●●●● ● ●●●● ● ● ● ●●●●●●
0.2 ● ● ●● ●●●● ●● ●● ●● ●●●● ●●● ●● ●● ●●●● ● ●● ●●● ●● ● ●●● ●●●● ● ● ● ●● ● ● ●●●● ●● ●●●●●●●● ●● ● ●● 0.1 ● ●●● ●●● ●●●●● ●●●●● ●● ●● ●●● ●●●●● ●●●●● ●●●●●● ●●●●● ●●●●● ●●●●●● ●●●● ●● 0.0
0.0 0.1 0.2 0.3 0.4
(Dn(xi), D(xi)) Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks
Convex hull peeling Barnett (1976)
For a given multivariate data set S.
Step 0. Set i = 1. Step 1. Find the convex hull of the data set S. Ci is the set of its extreme points.
Step 2. Delete Ci from the data set, S = S \ Ci. Step 3. While there are some points left in the data set S go back to Step 1, with i = i + 1. Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks
Convex hull peeling Barnett (1976)
For a given multivariate data set S.
Step 0. Set i = 1. Step 1. Find the convex hull of the data set S. Ci is the set of its extreme points.
Step 2. Delete Ci from the data set, S = S \ Ci. Step 3. While there are some points left in the data set S go back to Step 1, with i = i + 1. Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks
Convex hull peeling Barnett (1976)
For a given multivariate data set S.
Step 0. Set i = 1. Step 1. Find the convex hull of the data set S. Ci is the set of its extreme points.
Step 2. Delete Ci from the data set, S = S \ Ci. Step 3. While there are some points left in the data set S go back to Step 1, with i = i + 1. Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks
Convex hull peeling Barnett (1976) 11.2 11.0 10.8 10.6
6.6 6.8 7.0 7.2 7.4 7.6 7.8 8.0 Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks
Convex hull peeling Barnett (1976)
Peeling depth If m is the level of the centermost (deepest) layer, the peeling depth of a point from Ci is given by i/m. Problem There is no distributional counterpart of the convex hull peeling. Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks
Convex hull peeling Barnett (1976)
Peeling depth If m is the level of the centermost (deepest) layer, the peeling depth of a point from Ci is given by i/m. Problem There is no distributional counterpart of the convex hull peeling. Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks
Outline
Motivation
Main definitions Depth function and depth-trimmed region Data depths
Algorithms Halfspaces and convex hulls (depth functions) Circular sequence (depth-trimmed regions)
Applications Bagplot, DDplot, volume statistics, L-statistics, stochastic orders, quality control, risk measures
Parameter depth & regression Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks
Depth function Liu (1990), Zuo & Serfling (2000), Dyckerhoff (2004)
A depth function, D(x; P) (or D(x) shortly), satisfies:
D1 Affine invariance. D(Ax + b; PAX+b) = D(x; PX) for every d×d d nonsingular A ∈ R and b ∈ R ; D2 Vanishes at infinity. D(x; P) −→ 0 if kxk → ∞ ; d D3 Upper semicontinuity. {x ∈ R : D(x; P) ≥ α} is closed ; D4 Monotonicity relative to deepest point. D(x; P) ≤ D(θ + λ(x − θ); P) for θ = arg maxxD(x; P) and 0 ≤ λ ≤ 1 ; D4’ Quasiconcavity. D(λx + (1 − λ)y; P) ≥ min{D(x; P), D(y; P)} for 0 ≤ λ ≤ 1 . Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks
Depth function Liu (1990), Zuo & Serfling (2000), Dyckerhoff (2004)
A depth function, D(x; P) (or D(x) shortly), satisfies:
D1 Affine invariance. D(Ax + b; PAX+b) = D(x; PX) for every d×d d nonsingular A ∈ R and b ∈ R ; D2 Vanishes at infinity. D(x; P) −→ 0 if kxk → ∞ ; d D3 Upper semicontinuity. {x ∈ R : D(x; P) ≥ α} is closed ; D4 Monotonicity relative to deepest point. D(x; P) ≤ D(θ + λ(x − θ); P) for θ = arg maxxD(x; P) and 0 ≤ λ ≤ 1 ; D4’ Quasiconcavity. D(λx + (1 − λ)y; P) ≥ min{D(x; P), D(y; P)} for 0 ≤ λ ≤ 1 . Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks
Depth function Liu (1990), Zuo & Serfling (2000), Dyckerhoff (2004)
A depth function, D(x; P) (or D(x) shortly), satisfies:
D1 Affine invariance. D(Ax + b; PAX+b) = D(x; PX) for every d×d d nonsingular A ∈ R and b ∈ R ; D2 Vanishes at infinity. D(x; P) −→ 0 if kxk → ∞ ; d D3 Upper semicontinuity. {x ∈ R : D(x; P) ≥ α} is closed ; D4 Monotonicity relative to deepest point. D(x; P) ≤ D(θ + λ(x − θ); P) for θ = arg maxxD(x; P) and 0 ≤ λ ≤ 1 ; D4’ Quasiconcavity. D(λx + (1 − λ)y; P) ≥ min{D(x; P), D(y; P)} for 0 ≤ λ ≤ 1 . Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks
Depth function Liu (1990), Zuo & Serfling (2000), Dyckerhoff (2004)
A depth function, D(x; P) (or D(x) shortly), satisfies:
D1 Affine invariance. D(Ax + b; PAX+b) = D(x; PX) for every d×d d nonsingular A ∈ R and b ∈ R ; D2 Vanishes at infinity. D(x; P) −→ 0 if kxk → ∞ ; d D3 Upper semicontinuity. {x ∈ R : D(x; P) ≥ α} is closed ; D4 Monotonicity relative to deepest point. D(x; P) ≤ D(θ + λ(x − θ); P) for θ = arg maxxD(x; P) and 0 ≤ λ ≤ 1 ; D4’ Quasiconcavity. D(λx + (1 − λ)y; P) ≥ min{D(x; P), D(y; P)} for 0 ≤ λ ≤ 1 . Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks
Depth function Liu (1990), Zuo & Serfling (2000), Dyckerhoff (2004)
A depth function, D(x; P) (or D(x) shortly), satisfies:
D1 Affine invariance. D(Ax + b; PAX+b) = D(x; PX) for every d×d d nonsingular A ∈ R and b ∈ R ; D2 Vanishes at infinity. D(x; P) −→ 0 if kxk → ∞ ; d D3 Upper semicontinuity. {x ∈ R : D(x; P) ≥ α} is closed ; D4 Monotonicity relative to deepest point. D(x; P) ≤ D(θ + λ(x − θ); P) for θ = arg maxxD(x; P) and 0 ≤ λ ≤ 1 ; D4’ Quasiconcavity. D(λx + (1 − λ)y; P) ≥ min{D(x; P), D(y; P)} for 0 ≤ λ ≤ 1 . Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks
Depth function Liu (1990), Zuo & Serfling (2000), Dyckerhoff (2004)
A depth function, D(x; P) (or D(x) shortly), satisfies:
D1 Affine invariance. D(Ax + b; PAX+b) = D(x; PX) for every d×d d nonsingular A ∈ R and b ∈ R ; D2 Vanishes at infinity. D(x; P) −→ 0 if kxk → ∞ ; d D3 Upper semicontinuity. {x ∈ R : D(x; P) ≥ α} is closed ; D4 Monotonicity relative to deepest point. D(x; P) ≤ D(θ + λ(x − θ); P) for θ = arg maxxD(x; P) and 0 ≤ λ ≤ 1 ; D4’ Quasiconcavity. D(λx + (1 − λ)y; P) ≥ min{D(x; P), D(y; P)} for 0 ≤ λ ≤ 1 . Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks
Depth-trimmed or central regions Dyckerhoff (2004)
The sets α d D (P) = {x ∈ R : D(x; P) ≥ α} are nested, Dβ(P) ⊆ Dα(P) if α ≤ β, and further: α α R1 Affine equivariance. D (PAX+b) = {Ax + b : x ∈ D (PX)} for d×d d every nonsingular A ∈ R and b ∈ R ; R2 Bounded. Dα(P) is bounded ; R3 Closed. Dα(P) is closed ; R4 Starshaped. If x ∈ Dα(P) for all α, then each Dα(P) is starshaped wrt x ; R4’ Convex. Dα(P) is convex . Conversely D(x; P) = sup{α : x ∈ Dα(P)} . Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks
Depth-trimmed or central regions Dyckerhoff (2004)
The sets α d D (P) = {x ∈ R : D(x; P) ≥ α} are nested, Dβ(P) ⊆ Dα(P) if α ≤ β, and further: α α R1 Affine equivariance. D (PAX+b) = {Ax + b : x ∈ D (PX)} for d×d d every nonsingular A ∈ R and b ∈ R ; R2 Bounded. Dα(P) is bounded ; R3 Closed. Dα(P) is closed ; R4 Starshaped. If x ∈ Dα(P) for all α, then each Dα(P) is starshaped wrt x ; R4’ Convex. Dα(P) is convex . Conversely D(x; P) = sup{α : x ∈ Dα(P)} . Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks
Depth-trimmed or central regions Dyckerhoff (2004)
The sets α d D (P) = {x ∈ R : D(x; P) ≥ α} are nested, Dβ(P) ⊆ Dα(P) if α ≤ β, and further: α α R1 Affine equivariance. D (PAX+b) = {Ax + b : x ∈ D (PX)} for d×d d every nonsingular A ∈ R and b ∈ R ; R2 Bounded. Dα(P) is bounded ; R3 Closed. Dα(P) is closed ; R4 Starshaped. If x ∈ Dα(P) for all α, then each Dα(P) is starshaped wrt x ; R4’ Convex. Dα(P) is convex . Conversely D(x; P) = sup{α : x ∈ Dα(P)} . Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks
Outline
Motivation
Main definitions Depth function and depth-trimmed region Data depths
Algorithms Halfspaces and convex hulls (depth functions) Circular sequence (depth-trimmed regions)
Applications Bagplot, DDplot, volume statistics, L-statistics, stochastic orders, quality control, risk measures
Parameter depth & regression Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks
Decathlon @ Athens 2004: long jump vs. 100m Raw data 11.2 11.0 10.8 10.6
6.6 6.8 7.0 7.2 7.4 7.6 7.8 8.0 −1 HDn(x) = n min #{i : hXi, ui ≥ hx, ui} u∈Rd Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks
Halfspace depth (Tukey, 1975) Decathlon @ Athens 2004: long jump vs. 100m 11.2 11.0 10.8 10.6
6.6 6.8 7.0 7.2 7.4 7.6 7.8 8.0 −1 HDn(x) = n min #{i : hXi, ui ≥ hx, ui} u∈Rd Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks
Halfspace depth (Tukey, 1975) Decathlon @ Athens 2004: long jump vs. 100m 11.2 11.0 10.8 10.6
6.6 6.8 7.0 7.2 7.4 7.6 7.8 8.0 −1 HDn(x) = n min #{i : hXi, ui ≥ hx, ui} u∈Rd Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks
Halfspace depth (Tukey, 1975) Decathlon @ Athens 2004: long jump vs. 100m 11.2 11.0 10.8 10.6
6.6 6.8 7.0 7.2 7.4 7.6 7.8 8.0 −1 HDn(x) = n min #{i : hXi, ui ≥ hx, ui} u∈Rd Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks
Halfspace depth Tukey (1975), Rousseeuw & Ruts (1999) Population halfspace depth
HD(x; P) = inf{P(H): x ∈ H closed halfspace} ;
• Univariate HD(x; P) = min{F(x), 1 − F(x)} ; • Satisfies Properties R1–R4 and R4’ .
Halfspace trimming
\ HDα(P) = {H : H closed halfspace P(H) > 1 − α} ;
α − • Univariate HD (P) = [qX (α), qX(1 − α)] . Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks
Halfspace depth Tukey (1975), Rousseeuw & Ruts (1999) Population halfspace depth
HD(x; P) = inf{P(H): x ∈ H closed halfspace} ;
• Univariate HD(x; P) = min{F(x), 1 − F(x)} ; • Satisfies Properties R1–R4 and R4’ .
Halfspace trimming
\ HDα(P) = {H : H closed halfspace P(H) > 1 − α} ;
α − • Univariate HD (P) = [qX (α), qX(1 − α)] . Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks
Halfspace depth Tukey (1975), Rousseeuw & Ruts (1999) Population halfspace depth
HD(x; P) = inf{P(H): x ∈ H closed halfspace} ;
• Univariate HD(x; P) = min{F(x), 1 − F(x)} ; • Satisfies Properties R1–R4 and R4’ .
Halfspace trimming
\ HDα(P) = {H : H closed halfspace P(H) > 1 − α} ;
α − • Univariate HD (P) = [qX (α), qX(1 − α)] . Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks
Halfspace depth Tukey (1975), Rousseeuw & Ruts (1999) Population halfspace depth
HD(x; P) = inf{P(H): x ∈ H closed halfspace} ;
• Univariate HD(x; P) = min{F(x), 1 − F(x)} ; • Satisfies Properties R1–R4 and R4’ .
Halfspace trimming
\ HDα(P) = {H : H closed halfspace P(H) > 1 − α} ;
α − • Univariate HD (P) = [qX (α), qX(1 − α)] . Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks
Halfspace depth Tukey (1975), Rousseeuw & Ruts (1999) Population halfspace depth
HD(x; P) = inf{P(H): x ∈ H closed halfspace} ;
• Univariate HD(x; P) = min{F(x), 1 − F(x)} ; • Satisfies Properties R1–R4 and R4’ .
Halfspace trimming
\ HDα(P) = {H : H closed halfspace P(H) > 1 − α} ;
α − • Univariate HD (P) = [qX (α), qX(1 − α)] . Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks
Simplicial depth (Liu, 1990) Decathlon @ Athens 2004: long jump vs. 100m 11.2 11.0 10.8 10.6
6.6 6.8 7.0 7.2 7.4 7.6 7.8 8.0
−1 n X SD (x) = (x ∈ co{X , X ,..., X }) n d + 1 I i1 i2 id+1 1≤i1 Simplicial depth (Liu, 1990) Decathlon @ Athens 2004: long jump vs. 100m 11.2 11.0 10.8 10.6 6.6 6.8 7.0 7.2 7.4 7.6 7.8 8.0 −1 n X SD (x) = (x ∈ co{X , X ,..., X }) n d + 1 I i1 i2 id+1 1≤i1 Simplicial depth (Liu, 1990) Decathlon @ Athens 2004: long jump vs. 100m 11.2 11.0 10.8 10.6 6.6 6.8 7.0 7.2 7.4 7.6 7.8 8.0 Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Simplicial depth Liu (1990) Population simplicial depth SD(x; P) = Pr(x ∈ co{X1, X2,..., Xd+1}) where X1,..., Xd+1 are independent with distribution P . • Univariate SD(x; P) = 2F(x)(1 − F(x)) ; • Satisfies R1–R3. Also R4 on absolutely continuous and angularly symmetric distributions. Generalizations • Averaging open and closed convex hull. • Convex hull of k 6= d + 1 observations. • Average number of independent observations needed to contain a fixed point inside their convex hull. Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Simplicial depth Liu (1990) Population simplicial depth SD(x; P) = Pr(x ∈ co{X1, X2,..., Xd+1}) where X1,..., Xd+1 are independent with distribution P . • Univariate SD(x; P) = 2F(x)(1 − F(x)) ; • Satisfies R1–R3. Also R4 on absolutely continuous and angularly symmetric distributions. Generalizations • Averaging open and closed convex hull. • Convex hull of k 6= d + 1 observations. • Average number of independent observations needed to contain a fixed point inside their convex hull. Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Oja depth !−1 EVol (co{x, X ,..., X }) OD(x; P) = 1 + d 1 d pdet(Σ) Mahalanobis depth −1 MhD(x; P) = 1 + (x − µ)0Σ−1(x − µ) Projection depth !−1 hx, ui − Me(hX, ui) PD(x; P) = 1 + sup d MEDA(hX, ui) u∈R Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Oja depth !−1 EVol (co{x, X ,..., X }) OD(x; P) = 1 + d 1 d pdet(Σ) Mahalanobis depth −1 MhD(x; P) = 1 + (x − µ)0Σ−1(x − µ) Projection depth !−1 hx, ui − Me(hX, ui) PD(x; P) = 1 + sup d MEDA(hX, ui) u∈R Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Oja depth !−1 EVol (co{x, X ,..., X }) OD(x; P) = 1 + d 1 d pdet(Σ) Mahalanobis depth −1 MhD(x; P) = 1 + (x − µ)0Σ−1(x − µ) Projection depth !−1 hx, ui − Me(hX, ui) PD(x; P) = 1 + sup d MEDA(hX, ui) u∈R Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Zonoid trimming (Koshevoy & Mosler, 1997) Decathlon @ Athens 2004: long jump vs. 100m ● ● ● ● 11.2 ● ● ● ● ● ● ● 11.0 ● ● ● ● ● ● ● ● 10.8 10.6 ● 6.6 6.8 7.0 7.2 7.4 7.6 7.8 8.0 ( k ) k/n 1 X ZD = co X : π permutation of {1,..., n} n k π(i) i=1 Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Zonoid trimming (Koshevoy & Mosler, 1997) Decathlon @ Athens 2004: long jump vs. 100m ● ● ● ● 11.2 ● ● ● ● ● ● ● 11.0 ● ● ● ● ● ● ● ● 10.8 10.6 ● 6.6 6.8 7.0 7.2 7.4 7.6 7.8 8.0 ( k ) k/n 1 X ZD = co X : π permutation of {1,..., n} n k π(i) i=1 Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Zonoid trimming (Koshevoy & Mosler, 1997) Decathlon @ Athens 2004: long jump vs. 100m 11.2 11.0 ● 10.8 10.6 6.6 6.8 7.0 7.2 7.4 7.6 7.8 8.0 Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Zonoid trimming Koshevoy & Mosler (1997) Population zonoid trimmed regions Z Z α n d −1 o ZD (P) = xg(x)dP(x): g : R 7→ [0, α ], g(x)dP(x) = 1 • Univariate Z α Z 1 α 1 1 ZD (P) = qX(t)dt , qX(t)dt α 0 α 1−α • Properties R1–R4 and R4’, ZD1(P) = {EX}. • Zonoid depth ZD(x; P) = sup{α : x ∈ ZDα(P)}. Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Expected convex hull trimming (Cascos, 2007) Minkowski addition A ⊕ B = {a + b : a ∈ A, b ∈ B} 1.0 0.5 0.0 −0.5 −1.0 −1.0 −0.5 0.0 0.5 1.0 co{(−1/3, −1/2), (1/3, 1/2)} ⊕ co{(−2/3, 1/4), (2/3, −1/4)} Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Expected convex hull trimming (Cascos, 2007) Minkowski addition A ⊕ B = {a + b : a ∈ A, b ∈ B} 1.0 0.5 0.0 −0.5 −1.0 −1.0 −0.5 0.0 0.5 1.0 co{(−1/3, −1/2), (1/3, 1/2)} ⊕ co{(−2/3, 1/4), (2/3, −1/4)} Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Expected convex hull trimming (Cascos, 2007) Toy example • 5 points in the plane • mean value of the 5 points • segments joining each pair of the 5 points • Minkowski addition 5 of the 2 segments 5−1 weighted by 2 Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Expected convex hull trimming (Cascos, 2007) Toy example • 5 points in the plane • mean value of the 5 points • segments joining ● each pair of the 5 points • Minkowski addition 5 of the 2 segments 5−1 weighted by 2 Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Expected convex hull trimming (Cascos, 2007) Toy example • 5 points in the plane • mean value of the 5 points • segments joining each pair of the 5 points • Minkowski addition 5 of the 2 segments 5−1 weighted by 2 Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Expected convex hull trimming (Cascos, 2007) Toy example • 5 points in the plane • mean value of the 5 points • segments joining each pair of the 5 points • Minkowski addition 5 of the 2 segments 5−1 weighted by 2 Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Expected convex hull trimming (Cascos, 2007) Decathlon @ Athens 2004: long jump vs. 100m 11.2 11.0 ● 10.8 10.6 6.6 6.8 7.0 7.2 7.4 7.6 7.8 8.0 −1 n M CDk = co{X ,..., X } n k i1 ik 1≤i1<... Expected convex hull trimming (Cascos, 2007) Decathlon @ Athens 2004: long jump vs. 100m 11.2 11.0 ● 10.8 10.6 6.6 6.8 7.0 7.2 7.4 7.6 7.8 8.0 −1 n M CDk = co{X ,..., X } n k i1 ik 1≤i1<... Expected convex hull trimming Cascos (2007) Population expected convex hull 1/k CD (P) = Eco{X1, X2,..., Xk} , where E is the selection expectation and X1, X2,. . . are independent observations drawn from P. • Univariate 1/k CD (P) = [E min{X1,..., Xk} , E max{X1,..., Xk}] • Properties R1–R4 and R4’, CD1(P) = {EX}. Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Outline Motivation Main definitions Depth function and depth-trimmed region Data depths Algorithms Halfspaces and convex hulls (depth functions) Circular sequence (depth-trimmed regions) Applications Bagplot, DDplot, volume statistics, L-statistics, stochastic orders, quality control, risk measures Parameter depth & regression Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Halfspace depth Rousseeuw & Ruts (1996) • Aim of the algorithm: Compute bivariate halfspace depth HDn(x) Step 1. For each data point Xi, compute the angle of the line through x and Xi with horizontal axis. Step 2. Sort the angles as α1 ≤ α2 ≤ ... ≤ αn. Step 3. Let di be the number of angles in (αi, αi + π]. HDn(x) = min di/n Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Halfspace depth Rousseeuw & Ruts (1996) • Aim of the algorithm: Compute bivariate halfspace depth HDn(x) Step 1. For each data point Xi, compute the angle of the line through x and Xi with horizontal axis. Step 2. Sort the angles as α1 ≤ α2 ≤ ... ≤ αn. Step 3. Let di be the number of angles in (αi, αi + π]. HDn(x) = min di/n Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Halfspace depth Rousseeuw & Ruts (1996) • Aim of the algorithm: Compute bivariate halfspace depth HDn(x) Step 1. For each data point Xi, compute the angle of the line through x and Xi with horizontal axis. Step 2. Sort the angles as α1 ≤ α2 ≤ ... ≤ αn. Step 3. Let di be the number of angles in (αi, αi + π]. HDn(x) = min di/n Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Halfspace depth Rousseeuw & Ruts (1996) • Aim of the algorithm: Compute bivariate halfspace depth HDn(x) Step 1. For each data point Xi, compute the angle of the line through x and Xi with horizontal axis. Step 2. Sort the angles as α1 ≤ α2 ≤ ... ≤ αn. Step 3. Let di be the number of angles in (αi, αi + π]. HDn(x) = min di/n Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Halfspace depth Rousseeuw & Ruts (1996) • Aim of the algorithm: Compute bivariate halfspace depth HDn(x) Step 1. For each data point Xi, compute the angle of the line through x and Xi with horizontal axis. Step 2. Sort the angles as α1 ≤ α2 ≤ ... ≤ αn. Step 3. Let di be the number of angles in (αi, αi + π]. HDn(x) = min di/n Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Simplicial depth Halfspaces and convex hulls A fixed point x is contained in the convex hull of a finite set of points iff there is no open halfspace with x in its boundary containing all the points from the set. • The bivariate simplicial depth can be computed in a similar way. • R package DEPTH, Massé & Plante (2009) Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Simplicial depth Halfspaces and convex hulls A fixed point x is contained in the convex hull of a finite set of points iff there is no open halfspace with x in its boundary containing all the points from the set. • The bivariate simplicial depth can be computed in a similar way. • R package DEPTH, Massé & Plante (2009) Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Simplicial depth Halfspaces and convex hulls A fixed point x is contained in the convex hull of a finite set of points iff there is no open halfspace with x in its boundary containing all the points from the set. • The bivariate simplicial depth can be computed in a similar way. • R package DEPTH, Massé & Plante (2009) Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Outline Motivation Main definitions Depth function and depth-trimmed region Data depths Algorithms Halfspaces and convex hulls (depth functions) Circular sequence (depth-trimmed regions) Applications Bagplot, DDplot, volume statistics, L-statistics, stochastic orders, quality control, risk measures Parameter depth & regression Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Circular sequence algorithm for expected convex hull Cascos (2007) n k X CDk = co (j − 1)(k−1)X : π permutation of {1,..., n} n n(k) π(j) j=k • Aim of the algorithm: Finding the extreme points of the expected convex hull trimmed region of a fixed level k for a set of points in the plane. • X and Y contain the x and y coordinates of a set of points. • ORD and RANK are initialized with the values from 1 to n. k • EXT1 and EXT2 will store extreme points of CDn. Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Circular sequence algorithm for expected convex hull Cascos (2007) n k X CDk = co (j − 1)(k−1)X : π permutation of {1,..., n} n n(k) π(j) j=k • Aim of the algorithm: Finding the extreme points of the expected convex hull trimmed region of a fixed level k for a set of points in the plane. • X and Y contain the x and y coordinates of a set of points. • ORD and RANK are initialized with the values from 1 to n. k • EXT1 and EXT2 will store extreme points of CDn. Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Circular sequence algorithm for expected convex hull Cascos (2007) n k X CDk = co (j − 1)(k−1)X : π permutation of {1,..., n} n n(k) π(j) j=k • Aim of the algorithm: Finding the extreme points of the expected convex hull trimmed region of a fixed level k for a set of points in the plane. • X and Y contain the x and y coordinates of a set of points. • ORD and RANK are initialized with the values from 1 to n. k • EXT1 and EXT2 will store extreme points of CDn. Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Circular sequence algorithm for expected convex hull Cascos (2007) n k X CDk = co (j − 1)(k−1)X : π permutation of {1,..., n} n n(k) π(j) j=k • Aim of the algorithm: Finding the extreme points of the expected convex hull trimmed region of a fixed level k for a set of points in the plane. • X and Y contain the x and y coordinates of a set of points. • ORD and RANK are initialized with the values from 1 to n. k • EXT1 and EXT2 will store extreme points of CDn. Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Circular sequence Step 1. Order points as X[i] ≤ X[i + 1] and if X[i] = X[i + 1], then Y[i] > Y[i + 1]. Step 2. Create array ANGLE with the angle determined by the line through the points (X[i], Y[i]) and (X[j], Y[j]). Sort ANGLE. Step 3. Find entries of ANGLE with the same value (collinear points). Step 4. Reverse the order of each set of collinear points in the arrays ORD and RANK. Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Circular sequence Step 1. Order points as X[i] ≤ X[i + 1] and if X[i] = X[i + 1], then Y[i] > Y[i + 1]. Step 2. Create array ANGLE with the angle determined by the line through the points (X[i], Y[i]) and (X[j], Y[j]). Sort ANGLE. Step 3. Find entries of ANGLE with the same value (collinear points). Step 4. Reverse the order of each set of collinear points in the arrays ORD and RANK. Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Circular sequence Step 1. Order points as X[i] ≤ X[i + 1] and if X[i] = X[i + 1], then Y[i] > Y[i + 1]. Step 2. Create array ANGLE with the angle determined by the line through the points (X[i], Y[i]) and (X[j], Y[j]). Sort ANGLE. Step 3. Find entries of ANGLE with the same value (collinear points). Step 4. Reverse the order of each set of collinear points in the arrays ORD and RANK. Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Circular sequence Step 1. Order points as X[i] ≤ X[i + 1] and if X[i] = X[i + 1], then Y[i] > Y[i + 1]. Step 2. Create array ANGLE with the angle determined by the line through the points (X[i], Y[i]) and (X[j], Y[j]). Sort ANGLE. Step 3. Find entries of ANGLE with the same value (collinear points). Step 4. Reverse the order of each set of collinear points in the arrays ORD and RANK. Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Circular sequence Step 5. If any RANK[i] ≥ k is modified in Step 4, Pn (k−1) j=k(j − 1) (X[ORD[j]], Y[ORD[j]]) is appended at the end of EXT1. If any RANK[i] ≤ n − k is modified in Step 4, Pn (k−1) j=k(j − 1) (X[ORD[n + 1 − j]], Y[ORD[n + 1 − j]]) is appended at the end of EXT2. Step 6. If any element is left in array ANGLE continue with Step 3. Step 7. Multiply EXT1 and EXT2 by k/n(k). The sorting of Step 2 has complexity O(n2 log n) and it determines the overall complexity. Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Circular sequence Step 5. If any RANK[i] ≥ k is modified in Step 4, Pn (k−1) j=k(j − 1) (X[ORD[j]], Y[ORD[j]]) is appended at the end of EXT1. If any RANK[i] ≤ n − k is modified in Step 4, Pn (k−1) j=k(j − 1) (X[ORD[n + 1 − j]], Y[ORD[n + 1 − j]]) is appended at the end of EXT2. Step 6. If any element is left in array ANGLE continue with Step 3. Step 7. Multiply EXT1 and EXT2 by k/n(k). The sorting of Step 2 has complexity O(n2 log n) and it determines the overall complexity. Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Circular sequence Step 5. If any RANK[i] ≥ k is modified in Step 4, Pn (k−1) j=k(j − 1) (X[ORD[j]], Y[ORD[j]]) is appended at the end of EXT1. If any RANK[i] ≤ n − k is modified in Step 4, Pn (k−1) j=k(j − 1) (X[ORD[n + 1 − j]], Y[ORD[n + 1 − j]]) is appended at the end of EXT2. Step 6. If any element is left in array ANGLE continue with Step 3. Step 7. Multiply EXT1 and EXT2 by k/n(k). The sorting of Step 2 has complexity O(n2 log n) and it determines the overall complexity. Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Circular sequence Step 5. If any RANK[i] ≥ k is modified in Step 4, Pn (k−1) j=k(j − 1) (X[ORD[j]], Y[ORD[j]]) is appended at the end of EXT1. If any RANK[i] ≤ n − k is modified in Step 4, Pn (k−1) j=k(j − 1) (X[ORD[n + 1 − j]], Y[ORD[n + 1 − j]]) is appended at the end of EXT2. Step 6. If any element is left in array ANGLE continue with Step 3. Step 7. Multiply EXT1 and EXT2 by k/n(k). The sorting of Step 2 has complexity O(n2 log n) and it determines the overall complexity. Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Outline Motivation Main definitions Depth function and depth-trimmed region Data depths Algorithms Halfspaces and convex hulls (depth functions) Circular sequence (depth-trimmed regions) Applications Bagplot, DDplot, volume statistics, L-statistics, stochastic orders, quality control, risk measures Parameter depth & regression Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Bagplot Rousseeuw, Ruts & Tukey (1999) Center. Center of gravity of halfspace deepest region. Bag. Halfspace trimmed region with deeper half observations. If not exact, interpolate radially from center. Fence. (unplotted) Enlarge the bag radially from the center by a factor of 3. Outliers. Points out of the fence. R package APLPACK, Wolf (2007) Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Bagplot Rousseeuw, Ruts & Tukey (1999) Center. Center of gravity of halfspace deepest region. Bag. Halfspace trimmed region with deeper half observations. If not exact, interpolate radially from center. Fence. (unplotted) Enlarge the bag radially from the center by a factor of 3. Outliers. Points out of the fence. R package APLPACK, Wolf (2007) Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Bagplot Rousseeuw, Ruts & Tukey (1999) Center. Center of gravity of halfspace deepest region. Bag. Halfspace trimmed region with deeper half observations. If not exact, interpolate radially from center. Fence. (unplotted) Enlarge the bag radially from the center by a factor of 3. Outliers. Points out of the fence. R package APLPACK, Wolf (2007) Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Bagplot Rousseeuw, Ruts & Tukey (1999) Center. Center of gravity of halfspace deepest region. Bag. Halfspace trimmed region with deeper half observations. If not exact, interpolate radially from center. Fence. (unplotted) Enlarge the bag radially from the center by a factor of 3. Outliers. Points out of the fence. R package APLPACK, Wolf (2007) Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Boxplot & Bagplot (Rousseeuw, Ruts & Tukey, 1999) Decathlon @ Athens 2004: long jump & 100m 11.2 11.0 10.8 10.6 6.6 6.8 7.0 7.2 7.4 7.6 7.8 8.0 Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks DDplot, Multivariate goodness-of-fit Liu, Parelius & Singh (1999) A data set and a population distribution Plot the empirical depth of every point of the data set versus its population depth. Two data sets Take every point from each of the two data sets and plot its empirical depth wrt the first data set versus the empirical depth wrt the second. Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks DDplot, Multivariate goodness-of-fit Liu, Parelius & Singh (1999) A data set and a population distribution Plot the empirical depth of every point of the data set versus its population depth. Two data sets Take every point from each of the two data sets and plot its empirical depth wrt the first data set versus the empirical depth wrt the second. Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks DDplot, Multivariate goodness-of-fit Liu, Parelius & Singh (1999) ● ●●● ● 0.25 ●● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.20 ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ●●● ● ●● ●● ● ● ●● 0.15 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ●● ● ● ● ● ●● 0.10 ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ●● ● ● ●● ● ● ● ● ● ●● ● ● ● ●●● ● ●● ●● ● ● ●●● ● ● ● ● ●● ●● ● ● ● 0.05 ● ● ● ● ●● ●● ● ● ●●●●●● ●● ●●● ●● ● ● ●●● ●●● ● ● ● ●●●●●● ● ● ● ● ● ●● ●● ●● ●● ● ● ●●●●●●● ● ●● ● ●●●●●●●● 0.00 0.00 0.05 0.10 0.15 0.20 0.25 Two data sets drawn from the same distribution. Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks DDplot, Multivariate goodness-of-fit Liu, Parelius & Singh (1999) ●● ●● ● ● 0.25 ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● 0.20 ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● 0.15 ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.10 ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● 0.05 ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ●●●●●●●●●●●●●●●●●●●●●●●●●● ● ●●●●●●●●●● ●● ●● ●●● ●● ● ● 0.00 0.00 0.05 0.10 0.15 0.20 0.25 Location shift. Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks DDplot, Multivariate goodness-of-fit Liu, Parelius & Singh (1999) ●● ● ● ● ●● ● ● ● ● ● ● ● 0.25 ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ●● ● ●● ●● ● ● ●● ● ● ● ●● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● 0.20 ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ●●● ● ● ● ● ● ● 0.15 ● ● ●● ● ● ●● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ●●●●● ● ● ● ●● ● ●● ●● ●●● ● ●● ●● 0.10 ●● ●●●●● ● ●● ● ●●● ● ●● ●●●●● ● ●● ●● ● ●● ●● ● ● ● ●●● 0.05 ● ●●● ●● ● ● ● ●●● ● ●● ●● ● ● ● 0.00 0.00 0.05 0.10 0.15 0.20 0.25 Different scale parameter. Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Volume statistics Scatter estimates (Zuo & Serfling, 2000 & Oja 1983) The volume of affine equivariant (R1) depth trimmed regions is a statistic of scatter α α Vold(D (PAX+b)) = | det(A)|Vold(D (PX)) Expected convex hull & Gini mean difference 1/2 Vol1(CD (P)) = E|X1 − X2| = 2M1(X) Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Volume statistics Scatter estimates (Zuo & Serfling, 2000 & Oja 1983) The volume of affine equivariant (R1) depth trimmed regions is a statistic of scatter α α Vold(D (PAX+b)) = | det(A)|Vold(D (PX)) Expected convex hull & Gini mean difference 1/2 Vol1(CD (P)) = E|X1 − X2| = 2M1(X) Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Depth-weighted L-statistics Classical construction P i XiW(Dn(Xi)) L = P i W(Dn(Xi)) • Set-valued? • Multivariate L-moments? Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Depth-weighted L-statistics Classical construction P i XiW(Dn(Xi)) L = P i W(Dn(Xi)) • Set-valued? • Multivariate L-moments? Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Stochastic orderings Scatter (Zuo & Serfling, 2000) and dispersion (Massé & Theodorescu, 1994) α α • X ≤sc Y iff Vold(HD (PX)) ≤ Vold(HD (PY )) for all α. • X ≤disp Y iff for all α < β α β α β Vold(HD (PX))−Vold(HD (PX)) ≤ Vold(HD (PY ))−Vold(HD (PY )) . Variability - convex orderings (Koshevoy & Mosler, 1998) d • Ef ◦ l(X) ≤ Ef ◦ l(Y) for all l : R 7→ R linear and f : R 7→ R α α convex iff ZD (PX) ⊆ ZD (PY ) for all α. • Usually denoted by X ≤lcx Y. Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Stochastic orderings Scatter (Zuo & Serfling, 2000) and dispersion (Massé & Theodorescu, 1994) α α • X ≤sc Y iff Vold(HD (PX)) ≤ Vold(HD (PY )) for all α. • X ≤disp Y iff for all α < β α β α β Vold(HD (PX))−Vold(HD (PX)) ≤ Vold(HD (PY ))−Vold(HD (PY )) . Variability - convex orderings (Koshevoy & Mosler, 1998) d • Ef ◦ l(X) ≤ Ef ◦ l(Y) for all l : R 7→ R linear and f : R 7→ R α α convex iff ZD (PX) ⊆ ZD (PY ) for all α. • Usually denoted by X ≤lcx Y. Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Quality Control Control Chart ● ● 74.02 ● ● ● ● ● ● ● 74.01 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 74.00 ● ● ● ● ● ● ● ● ● ● ● 73.99 0 10 20 30 40 Depth-based Multivariate range rP(x) = P({y : D(y; P) ≤ D(x; P)}) Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Quality Control Control Chart ● ● 74.02 ● ● ● ● ● ● ● 74.01 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 74.00 ● ● ● ● ● ● ● ● ● ● ● 73.99 0 10 20 30 40 Depth-based Multivariate range rP(x) = P({y : D(y; P) ≤ D(x; P)}) Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Quality Control Liu (1995) Multivariate Control Charts • Control Chart for r instead of X-Chart. r (X ) = #{Y : D (Y) ≤ D (X )}/m . Pˆm i j m m i • Control Chart for Q (average range) instead of X-Chart. n 1 X Q(Pˆm; X1,..., Xk) = rˆ (Xi) . k Pm i=1 • Control Chart for S (accumulated range) instead of CUSUM Chart. n X 1 S(Pˆm; X1,..., Xn) = rˆ (Xi) − . Pm 2 i=1 Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Quality Control Liu (1995) Multivariate Control Charts • Control Chart for r instead of X-Chart. r (X ) = #{Y : D (Y) ≤ D (X )}/m . Pˆm i j m m i • Control Chart for Q (average range) instead of X-Chart. n 1 X Q(Pˆm; X1,..., Xk) = rˆ (Xi) . k Pm i=1 • Control Chart for S (accumulated range) instead of CUSUM Chart. n X 1 S(Pˆm; X1,..., Xn) = rˆ (Xi) − . Pm 2 i=1 Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Quality Control Liu (1995) Multivariate Control Charts • Control Chart for r instead of X-Chart. r (X ) = #{Y : D (Y) ≤ D (X )}/m . Pˆm i j m m i • Control Chart for Q (average range) instead of X-Chart. n 1 X Q(Pˆm; X1,..., Xk) = rˆ (Xi) . k Pm i=1 • Control Chart for S (accumulated range) instead of CUSUM Chart. n X 1 S(Pˆm; X1,..., Xn) = rˆ (Xi) − . Pm 2 i=1 Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Risk measurement A risky portfolio is modeled as a random variable X that represents a financial gain. Classical univariate risk measures • Value at Risk: V@Rα(X) = −qX(α) • Expected Shortfall: Z α 1 α ESα(X) = − qX(t)dt = − min ZD (PX) α 0 • Expected Minimum: 1/n EM1/n(X) = −E min{X1,..., Xn} = − min CD (PX) Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Classical definition of coherent risk measure Artzner, Delbaen, Eber and Heath (1999) ∞ ρ(X) ∈ R is the risk associated with X ∈ L1 (essentially bounded) 1. Cash-invariance. ρ(X + y) = ρ(X) − y for y ∈ R. (Translation equivariance) 2. Monotonicity. If X ≥ Y a.s., then ρ(X) ≤ ρ(Y). 3. Homogeneity. ρ(tX) = tρ(X) if t > 0. ρ is coherent if further 4. Subadditivity. ρ(X + Y) ≤ ρ(X) + ρ(Y) (ES and EM) Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Classical definition of coherent risk measure Artzner, Delbaen, Eber and Heath (1999) ∞ ρ(X) ∈ R is the risk associated with X ∈ L1 (essentially bounded) 1. Cash-invariance. ρ(X + y) = ρ(X) − y for y ∈ R. (Translation equivariance) 2. Monotonicity. If X ≥ Y a.s., then ρ(X) ≤ ρ(Y). 3. Homogeneity. ρ(tX) = tρ(X) if t > 0. ρ is coherent if further 4. Subadditivity. ρ(X + Y) ≤ ρ(X) + ρ(Y) (ES and EM) Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Classical definition of coherent risk measure Artzner, Delbaen, Eber and Heath (1999) ∞ ρ(X) ∈ R is the risk associated with X ∈ L1 (essentially bounded) 1. Cash-invariance. ρ(X + y) = ρ(X) − y for y ∈ R. (Translation equivariance) 2. Monotonicity. If X ≥ Y a.s., then ρ(X) ≤ ρ(Y). 3. Homogeneity. ρ(tX) = tρ(X) if t > 0. ρ is coherent if further 4. Subadditivity. ρ(X + Y) ≤ ρ(X) + ρ(Y) (ES and EM) Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Classical definition of coherent risk measure Artzner, Delbaen, Eber and Heath (1999) ∞ ρ(X) ∈ R is the risk associated with X ∈ L1 (essentially bounded) 1. Cash-invariance. ρ(X + y) = ρ(X) − y for y ∈ R. (Translation equivariance) 2. Monotonicity. If X ≥ Y a.s., then ρ(X) ≤ ρ(Y). 3. Homogeneity. ρ(tX) = tρ(X) if t > 0. ρ is coherent if further 4. Subadditivity. ρ(X + Y) ≤ ρ(X) + ρ(Y) (ES and EM) Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Classical definition of coherent risk measure Artzner, Delbaen, Eber and Heath (1999) ∞ ρ(X) ∈ R is the risk associated with X ∈ L1 (essentially bounded) 1. Cash-invariance. ρ(X + y) = ρ(X) − y for y ∈ R. (Translation equivariance) 2. Monotonicity. If X ≥ Y a.s., then ρ(X) ≤ ρ(Y). 3. Homogeneity. ρ(tX) = tρ(X) if t > 0. ρ is coherent if further 4. Subadditivity. ρ(X + Y) ≤ ρ(X) + ρ(Y) (ES and EM) Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Multivariate risk measures Cascos & Molchanov (2007) d A risky portfolio is modeled as a random vector X in R Marginalisation d If ρ(X) is vector-valued (ρ(X) ∈ R ) and subadditive, it is marginalized, i.e. ρ(X1, X2,..., Xd) = (ρ1(X1), ρ2(X2), . . . , ρd(Xd)) . Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Family of (set-valued) risks Operations with sets: Minkowski addition: A ⊕ B = {a + b : a ∈ A, b ∈ B} Multiplication by scalar:( λ ∈ R): λA = {λa : a ∈ A} The family of risks G is composed by all closed and convex d d subsets of R that are upper (i.e. A ⊕ R+ = A whenever A ∈ G). G is the family of all risks. It is possible to operate with them (Minkowski addition and multiplication by scalar) and compare them (inclusion). d On deterministic portfolios ρ(x) = −x + R+ . Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Family of (set-valued) risks Operations with sets: Minkowski addition: A ⊕ B = {a + b : a ∈ A, b ∈ B} Multiplication by scalar:( λ ∈ R): λA = {λa : a ∈ A} The family of risks G is composed by all closed and convex d d subsets of R that are upper (i.e. A ⊕ R+ = A whenever A ∈ G). G is the family of all risks. It is possible to operate with them (Minkowski addition and multiplication by scalar) and compare them (inclusion). d On deterministic portfolios ρ(x) = −x + R+ . Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Family of (set-valued) risks Operations with sets: Minkowski addition: A ⊕ B = {a + b : a ∈ A, b ∈ B} Multiplication by scalar:( λ ∈ R): λA = {λa : a ∈ A} The family of risks G is composed by all closed and convex d d subsets of R that are upper (i.e. A ⊕ R+ = A whenever A ∈ G). G is the family of all risks. It is possible to operate with them (Minkowski addition and multiplication by scalar) and compare them (inclusion). d On deterministic portfolios ρ(x) = −x + R+ . Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Family of (set-valued) risks Operations with sets: Minkowski addition: A ⊕ B = {a + b : a ∈ A, b ∈ B} Multiplication by scalar:( λ ∈ R): λA = {λa : a ∈ A} The family of risks G is composed by all closed and convex d d subsets of R that are upper (i.e. A ⊕ R+ = A whenever A ∈ G). G is the family of all risks. It is possible to operate with them (Minkowski addition and multiplication by scalar) and compare them (inclusion). d On deterministic portfolios ρ(x) = −x + R+ . Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Set-valued risks 5 4 3 2 1 0 −1 −2 −2 −1 0 1 2 3 4 5 Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Set-valued risks 5 4 3 2 1 0 −1 −2 −2 −1 0 1 2 3 4 5 Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Set-valued risks 5 4 3 2 1 0 −1 −2 −2 −1 0 1 2 3 4 5 Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks General definition of a set-valued risk measures Jouini, Meddeb and Touzi (2004) d Given a random vector X, its risk SR(X) ∈ G and SR(0) = R+. d 1. Translation equivariance. SR(X − y) = SR(X) + y for y ∈ R . 2. Monotonicity. If X ≥ Y a.s., then SR(X) ⊇ SR(Y). 3. Homogeneity. SR(tX) = tSR(X) if t > 0. 4. Coherent if further SR(X) ⊕ SR(Y) ⊆ SR(X + Y) . Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks General definition of a set-valued risk measures Jouini, Meddeb and Touzi (2004) d Given a random vector X, its risk SR(X) ∈ G and SR(0) = R+. d 1. Translation equivariance. SR(X − y) = SR(X) + y for y ∈ R . 2. Monotonicity. If X ≥ Y a.s., then SR(X) ⊇ SR(Y). 3. Homogeneity. SR(tX) = tSR(X) if t > 0. 4. Coherent if further SR(X) ⊕ SR(Y) ⊆ SR(X + Y) . Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks General definition of a set-valued risk measures Jouini, Meddeb and Touzi (2004) d Given a random vector X, its risk SR(X) ∈ G and SR(0) = R+. d 1. Translation equivariance. SR(X − y) = SR(X) + y for y ∈ R . 2. Monotonicity. If X ≥ Y a.s., then SR(X) ⊇ SR(Y). 3. Homogeneity. SR(tX) = tSR(X) if t > 0. 4. Coherent if further SR(X) ⊕ SR(Y) ⊆ SR(X + Y) . Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks General definition of a set-valued risk measures Jouini, Meddeb and Touzi (2004) d Given a random vector X, its risk SR(X) ∈ G and SR(0) = R+. d 1. Translation equivariance. SR(X − y) = SR(X) + y for y ∈ R . 2. Monotonicity. If X ≥ Y a.s., then SR(X) ⊇ SR(Y). 3. Homogeneity. SR(tX) = tSR(X) if t > 0. 4. Coherent if further SR(X) ⊕ SR(Y) ⊆ SR(X + Y) . Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks General definition of a set-valued risk measures Jouini, Meddeb and Touzi (2004) d Given a random vector X, its risk SR(X) ∈ G and SR(0) = R+. d 1. Translation equivariance. SR(X − y) = SR(X) + y for y ∈ R . 2. Monotonicity. If X ≥ Y a.s., then SR(X) ⊇ SR(Y). 3. Homogeneity. SR(tX) = tSR(X) if t > 0. 4. Coherent if further SR(X) ⊕ SR(Y) ⊆ SR(X + Y) . Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Further properties of depth-trimmed regions R5 Monotonicity. (HD, ZD, CD) If X ≥ Y a.s., then α α d D (X) ⊆ D (Y) ⊕ R+ , R6 Subadditivity. (ZD, CD) Dα(X + Y) ⊆ Dα(X) ⊕ Dα(Y) . Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Further properties of depth-trimmed regions R5 Monotonicity. (HD, ZD, CD) If X ≥ Y a.s., then α α d D (X) ⊆ D (Y) ⊕ R+ , R6 Subadditivity. (ZD, CD) Dα(X + Y) ⊆ Dα(X) ⊕ Dα(Y) . Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Acceptance Acceptable risks & risk value The set of acceptable risks is constituted by those whose depth-trimmed region lies in the first quadrant, ∞ α d Aα = {X ∈ Ld : D (PX) ⊆ R+} The risk measure is obtained from the acceptance set as d SRα(X) = {x ∈ R : X + x ∈ Aα} d α α d = R+ (−D (PX)) = − min D (PX) + R+ Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Zonoid trimming 0.5 ZD (PX), where X follows a 2-dimensional standard Gaussian distribution 4 3 2 1 0 −1 −2 −3 −3 −2 −1 0 1 2 3 4 Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Risk from zonoid trimming 0.5 d SR0.5(X) = − min ZD (PX) + R+, where X follows a 2-dimensional standard Gaussian distribution 4 3 2 1 0 −1 ● −2 −3 −3 −2 −1 0 1 2 3 4 Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Set-valued risks graphically • Cone of acceptable deterministic 1.5 d 1.0 portfolios, K ⊃ R+ • Trimmed region 0.5 1/2 CD (PX) 0.0 • Cone and region −0.5 • Set-valued risk −1.0 SR (X) 1/2 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Set-valued risks graphically • Cone of acceptable deterministic 1.5 d 1.0 portfolios, K ⊃ R+ • Trimmed region 0.5 1/2 CD (PX) 0.0 • Cone and region −0.5 • Set-valued risk −1.0 SR (X) 1/2 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Set-valued risks graphically • Cone of acceptable deterministic 1.5 d 1.0 portfolios, K ⊃ R+ • Trimmed region 0.5 1/2 CD (PX) 0.0 • Cone and region −0.5 • Set-valued risk −1.0 SR (X) 1/2 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Set-valued risks graphically • Cone of acceptable deterministic 1.5 d 1.0 portfolios, K ⊃ R+ • Trimmed region 0.5 1/2 CD (PX) 0.0 • Cone and region −0.5 • Set-valued risk −1.0 SR (X) 1/2 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Non-marginalised set-valued risks 0.9 ZD (PX), where X follows a 2-dimensional standard Gaussian d distribution, K0.9 = R+ 4 3 2 1 0 −1 −2 −3 −3 −2 −1 0 1 2 3 4 Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Non-marginalised set-valued risks 0.5 ZD (PX), where X follows a 2-dimensional standard Gaussian d distribution, K0.5 ⊃ R+ 4 3 2 1 0 −1 −2 −3 −3 −2 −1 0 1 2 3 4 Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Non-marginalised set-valued risks 0.9 \ 0.5 K0.9 (−ZD (PX)) K0.5 (−ZD (PX)) is also a set-valued risk measure 4 3 2 1 0 −1 −2 −3 −3 −2 −1 0 1 2 3 4 Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Outline Motivation Main definitions Depth function and depth-trimmed region Data depths Algorithms Halfspaces and convex hulls (depth functions) Circular sequence (depth-trimmed regions) Applications Bagplot, DDplot, volume statistics, L-statistics, stochastic orders, quality control, risk measures Parameter depth & regression Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Parameter depth Mizera (2002), Mizera & Müller (2004) • Nonfit: Element from a parameter space that is not a suitable for a parameter wrt a sample • Depth: Fraction of points to be deleted from a sample in order to make our candidate a nonfit Halfspace depth • Nonfit: point out of the convex hull of the sample Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Parameter depth Mizera (2002), Mizera & Müller (2004) • Nonfit: Element from a parameter space that is not a suitable for a parameter wrt a sample • Depth: Fraction of points to be deleted from a sample in order to make our candidate a nonfit Halfspace depth • Nonfit: point out of the convex hull of the sample Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Regression depth Rousseeuw & Hubert (1999) • Problem: Univariate regression • Parameter space: lines on R2 or R2 itself • Nonfit: line that does not contain any data point and whose residuals change sign, at most, once Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Location-scale depth (Cascos & López-Díaz, 2011) what for? Joint (X, S) control charts Control Chart ● ● 74.02 ● ● ● ● ● ● ● 74.01 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 74.00 ● ● ● ● ● ● ● ● ● ● ● 73.99 0 10 20 30 40 Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Location-scale depth (Cascos & López-Díaz, 2011) what for? Joint (X, S) control charts joint control chart 0.03 0.02 0.01 0.00 −0.01 73.97 73.98 73.99 74.00 74.01 74.02 74.03 Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Location-scale depth (Cascos & López-Díaz, 2011) what for? Joint (X, S) control charts joint control chart 0.03 0.02 0.01 0.00 −0.01 73.97 73.98 73.99 74.00 74.01 74.02 74.03 Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Location-scale depth (Cascos & López-Díaz, 2011) what for? Joint (X, S) control charts Control Chart 1.0 ● ● ● ● ● ● ● ● ● 0.8 ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.6 ● ● ● ● depth ● ● 0.4 ● ● ● ● ● ● ● 0.2 ● ● ● ● 0.0 0 10 20 30 40 Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Concluding remarks • Data depth • Specific depth functions and depth-trimmed regions • Applications (emphasis on risk measurement) • Parameter depth (location-scale depth) Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Concluding remarks • Data depth • Specific depth functions and depth-trimmed regions • Applications (emphasis on risk measurement) • Parameter depth (location-scale depth) Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Concluding remarks • Data depth • Specific depth functions and depth-trimmed regions • Applications (emphasis on risk measurement) • Parameter depth (location-scale depth) Motivation Main definitions Algorithms Applications Parameter depth & regression Concluding remarks Concluding remarks • Data depth • Specific depth functions and depth-trimmed regions • Applications (emphasis on risk measurement) • Parameter depth (location-scale depth) Appendix References from the speakerI Cascos, I. The expected convex hull trimmed regions of a sample. Comput. Statist. 22, 557–569 (2007) Cascos I. & Molchanov, I. Multivariate risks and depth-trimmed regions. Finance Stoch. 11, 373–397 (2007) Cascos I. Data depth: multivariate statistics and geometry. In: W.S. Kendall and I. Molchanov Eds. New Perspectives in Stochastic Geometry, Oxford University Press, 398–423 (2010) Appendix References from the speakerII Cascos I. & López-Díaz, M. Trimmed regions induced by parameters of a probability. In preparation Cascos I. & López-López, A. Data depth in Multivariate Statistics. In preparation for BEIO Appendix Thank you!!