Optimal representation of bivariate density functions by level sets - - PowerPoint PPT Presentation

Introduction One density Several densities Conclusions References

Optimal representation of bivariate density functions by level sets

Pedro Delicado

Universitat Polit` ecnica de Catalunya, Barcelona, Spain

Philippe Vieu

Universit´ e Paul Sabatier, Toulouse, France 7th Journ´ es Statistiques du Sud Barcelona, June 2014

Optimal level sets for bivariate densities 1/52 Pedro Delicado and Philippe Vieu

Introduction One density Several densities Conclusions References

Aim of the talk

• Bivariate density functions are

usually represented by a few level sets, for instance those with probability content equal to .25, .5 and .75.

• In this work we deal with

choosing which level sets provide the best graphical representation of a single bivariate density, according to certain optimality criteria.

x y f(x,y)

(a)

x y f(x,y)

(b)

−3 −2 −1 1 2 3 −3 −2 −1 1 2 3 (c)

−3 −2 −1 1 2 3 −3 −2 −1 1 2 3 (d)

Optimal level sets for bivariate densities 2/52 Pedro Delicado and Philippe Vieu

Introduction One density Several densities Conclusions References

Outline

1

Introduction

2

Optimal level sets for a single density Optimality based on distances between density level sets Optimality based on distances between bivariate densities Difficulties with managing several densities

3

Optimal representation of several densities by level sets Representation with a single level set Representation with more than a level set Case of estimated densities Some Monte Carlo experiments

4

Conclusions

Optimal level sets for bivariate densities 3/52 Pedro Delicado and Philippe Vieu

Introduction One density Several densities Conclusions References

1

Introduction

2

Optimal level sets for a single density Optimality based on distances between density level sets Optimality based on distances between bivariate densities Difficulties with managing several densities

3

Optimal representation of several densities by level sets Representation with a single level set Representation with more than a level set Case of estimated densities Some Monte Carlo experiments

4

Conclusions

Optimal level sets for bivariate densities 4/52 Pedro Delicado and Philippe Vieu

Introduction One density Several densities Conclusions References

Introduction

• Let f be a bivariate probability density function.
• For α ∈]0, 1[, the density level set with probability content α is

Cα = {x ∈ R2 : f (x) ≥ γα}, where γα is such that

• Cα f (x)dx = α.
• A standard way to graphically represent the bivariate density f is by

drawing in the same graphic density level sets corresponding to several values α1, . . . , αJ, or just their boundaries.

• Problem: Given a bivariate density function f (respectively, N

densities f1, . . . , fN) choose α1, . . . , αJ defining the best (in a sense to be specified) graphical representation of f (resp., .f1, . . . , fN).

Optimal level sets for bivariate densities 5/52 Pedro Delicado and Philippe Vieu

Introduction One density Several densities Conclusions References

Why is it worth choosing α1, . . . , αJ carefully?

• A usual way to represent a bivariate density function is by plotting

J = 3 of its density level sets, those corresponding to α = 1/4, 1/2 and 3/4 (by analogy with the univariate boxplots).

• Bowman and Azzalini (1997) call these plots ’sliceplots’.
• A relevant question is to know whether the choice of α = 1/4, 1/2

and 3/4 is sensible or maybe there exists an alternative better choice.

Optimal level sets for bivariate densities 6/52 Pedro Delicado and Philippe Vieu

Introduction One density Several densities Conclusions References

Why is it worth choosing α1, . . . , αJ carefully? (Cont.)

• An additional important reason: Representing each of N bivariate

density functions by a unique density level set Cα (J = 1) allows us to draw in the same graphic more than one bivariate density function.

• So it is worthwhile to make a good choice of α.
• This kind of graphics is helpful in different situations, as the

following examples illustrate.

Optimal level sets for bivariate densities 7/52 Pedro Delicado and Philippe Vieu

Introduction One density Several densities Conclusions References

Example: Aircraft data (Bowman and Azzalini 1997)

• Bowman and Azzalini (1997) study six characteristics of 709 aircraft

designs from periods 1914-1935, 1936-1955 and 1956-1984.

• They obtain the first two principal components (identified as “size”

and “speed adjusted by size”, respectively) and represent their joint density using only a level plot (α = 0.75) for each period.

• A single graphic

summarizes the way aircraft designs evolved over the last century.

−4 −2 2 4 6 −2 −1 1 2 3 Comp.1 Comp.2

7 5 75 7 5

Period: 1914 − 1935 Period: 1936 − 1955 Period: 1956 − 1984

Optimal level sets for bivariate densities 8/52 Pedro Delicado and Philippe Vieu

Introduction One density Several densities Conclusions References

Dynamic representation of many density functions

• Consider the case when the number of bivariate density functions to

be represented is large.

• Assume that they are sorted according to the time they were
• bserved and that the elapsed time between two consecutive

densities is short.

• A convenient way to represent them is by an animated graphic,

where each image corresponds to the graphic of each bivariate density.

• In this case it is appropriate to represent each density by a few (3,

for instance) density level sets.

• The animated graphic is showing how the level sets evolve over time.

Optimal level sets for bivariate densities 9/52 Pedro Delicado and Philippe Vieu

Introduction One density Several densities Conclusions References

Example: Aircraft data. Animated graphic.

Optimal level sets for bivariate densities 10/52 Pedro Delicado and Philippe Vieu

−4 −2 2 4 6 −2 −1 1 2 3 Comp.1 Comp.2 Period: 1914 − 1935

2 5 2 5 50 7 5

Introduction One density Several densities Conclusions References

Showing results of FPCA for bivariate densities

• Assume that a functional principal component analysis (FPCA) is

performed from a sample of bivariate densities f1, . . . , fN.

• In FPCA for one-dimensional functions it is standard to graphically

represent the principal functions by superimposing in the same plot three functions: the mean function and the mean function plus (and minus) the principal function (multiplied by a constant).

• In order to do a similar graphic when dealing with bivariate density

functions we need a way to represent three such functions in the same graphic.

• The use of a level set for representing each of them is a simple and

effective way to do it.

Optimal level sets for bivariate densities 11/52 Pedro Delicado and Philippe Vieu

Introduction One density Several densities Conclusions References

1

Introduction

2

Optimal level sets for a single density Optimality based on distances between density level sets Optimality based on distances between bivariate densities Difficulties with managing several densities

3

Optimal representation of several densities by level sets Representation with a single level set Representation with more than a level set Case of estimated densities Some Monte Carlo experiments

4

Conclusions

Optimal level sets for bivariate densities 12/52 Pedro Delicado and Philippe Vieu

Introduction One density Several densities Conclusions References

Optimal level sets for a single density

• We consider first the problem of representing only one density by

some of its density level sets.

• We assume that J has been fixed in advance and we want to do the

best choice of α1, . . . , αJ.

• There is not a unique way for specifying what best could mean.
• We examine two possibilities:

1 Choosing the J density level sets that best represent the whole family

• f level sets {Cα : α ∈]0, 1[} in the sense that each non-plotted Cα is

close to the nearest level among those that are plotted: Cα1, . . . , CαJ .

2 Each collection of level sets Cα1, . . . , CαJ defines in a natural way a

piecewise uniform bivariate density function. We propose to minimize in α1, . . . , αJ the distance between this piecewise uniform density and the density that we want to represent by Cα1, . . . , CαJ .

Optimal level sets for bivariate densities 13/52 Pedro Delicado and Philippe Vieu

Introduction One density Several densities Conclusions References Distances between level sets

1

Introduction

2

Optimal level sets for a single density Optimality based on distances between density level sets Optimality based on distances between bivariate densities Difficulties with managing several densities

3

Optimal representation of several densities by level sets Representation with a single level set Representation with more than a level set Case of estimated densities Some Monte Carlo experiments

4

Conclusions

Optimal level sets for bivariate densities 14/52 Pedro Delicado and Philippe Vieu

Introduction One density Several densities Conclusions References Distances between level sets

Optimality based on distances between density level sets

• We consider the following distances between sets A, B ⊆ R2:

dλ(A, B) =

• A∆B

dx = λ(A∆B), df (A, B) =

• A∆B

f (x)dx = µf (A∆B), where ∆ denotes the symmetric difference between sets, λ is the Lebesgue measure in R2 and µf is the probability measure in R2 having f as a density function.

• Other distances between sets could be used alternatively (e.g.,

Hausdorff’s distance).

Optimal level sets for bivariate densities 15/52 Pedro Delicado and Philippe Vieu

Introduction One density Several densities Conclusions References Distances between level sets

Our proposal

• We propose to choose values α1, . . . , αJ by solving this minimization

problem: min

0<α1<···<αJ<1

1 d(Cu, Cαj(u))du (1) where d is either dλ or df , and j(u) is such that d(Cu, Cαj(u)) = min

j=1...J d(Cu, Cαj),

that is, Cαj(u) is the closest set to Cu among the sets Cα1, . . . , Cαj.

• This problem is related with the continuous k-median problem.

Optimal level sets for bivariate densities 16/52 Pedro Delicado and Philippe Vieu

Introduction One density Several densities Conclusions References Distances between level sets

Theorem

For d = df the optimal solution of problem (1) is αf

j = 2j−1 2J , j = 1 . . . J.

For the first values of J the optimal αf

j are:

J αf

j , j = 1, . . . , J

1 1/2 2 1/4, 3/4 3 1/6, 1/2, 5/6 Assume now that the support of f , say C1, is compact. For d = dλ the

• ptimal solution of problem (1) is αλ

j , j = 1, . . . , J, such that

λ(Cαλ

j ) = 2j − 1

2J λ(C1), j = 1, . . . , J.

Optimal level sets for bivariate densities 17/52 Pedro Delicado and Philippe Vieu

Introduction One density Several densities Conclusions References Distances between level sets

Example: Aircraft data.

αf

1 = 0.5

α1 = 0.75

−4 −2 2 4 6 −2 −1 1 2 3 Comp.1 Comp.2

50 50 50 5 50 50

Period: 1914 − 1935 Period: 1936 − 1955 Period: 1956 − 1984 −4 −2 2 4 6 −2 −1 1 2 3 Comp.1 Comp.2

7 5 75 7 5

Period: 1914 − 1935 Period: 1936 − 1955 Period: 1956 − 1984

Optimal level sets for bivariate densities 18/52 Pedro Delicado and Philippe Vieu

Introduction One density Several densities Conclusions References Distances between level sets

Plug-in density level set estimators

• In practice it is unusual to know the bivariate density f .
• Commonly we observe n i.i.d. data coming from f and we define an

estimator fn of f based on these data (e.g., kernel estimator).

• Then the level sets finally plotted are not those of f but those of fn.
• They are known as plug-in density level set estimators:

Cα,n = {x ∈ R2 : fn(x) ≥ γα,n}, with

• Cα,n fn(x)dx = α.
• There is a vast literature on the convergence of the plug-in density

level estimating sets Cα,n to the density level set Cα of f , when fn is a kernel density estimator of f and α is fixed.

• Ba´

ıllo, Cuesta-Albertos, and Cuevas (2001), Ba´ ıllo (2003), Cadre (2006).

Optimal level sets for bivariate densities 19/52 Pedro Delicado and Philippe Vieu

Introduction One density Several densities Conclusions References Distances between densities

1

Introduction

2

Optimal level sets for a single density Optimality based on distances between density level sets Optimality based on distances between bivariate densities Difficulties with managing several densities

3

Optimal representation of several densities by level sets Representation with a single level set Representation with more than a level set Case of estimated densities Some Monte Carlo experiments

4

Conclusions

Optimal level sets for bivariate densities 20/52 Pedro Delicado and Philippe Vieu

Introduction One density Several densities Conclusions References Distances between densities

Optimality based on distances between bivariate densities

• When we plot J density level sets in a graphic we are trying to

represent a bivariate probability density function f .

• Therefore it is desirable that the graphic is as close as possible (in

certain sense) to the target density f .

• A natural way to measure closeness between a graphic of density

level sets and a density function is to realize that in fact such a graphic defines itself a bivariate density.

• Then we propose to use distance measures between bivariate

densities.

Optimal level sets for bivariate densities 21/52 Pedro Delicado and Philippe Vieu

Introduction One density Several densities Conclusions References Distances between densities

x y f ( x , y ) 0.1 0.25 0.5 . 7 5 . 9 5 −3 −2 −1 1 2 3 −3 −2 −1 1 2 3 x y f ( x , y ) x y f ( x , y ) . 1 0.1 0.25 0.5 . 7 5 . 9 5 −3 −2 −1 1 2 3 −3 −2 −1 1 2 3 x y f ( x , y )

Optimal level sets for bivariate densities 22/52 Pedro Delicado and Philippe Vieu

Introduction One density Several densities Conclusions References Distances between densities

Choosing J ≥ 1 level sets

• Let us assume that the support of f is a known compact set C1.
• Let Θ = {θ = (α1, . . . , αJ) ∈ RJ : 0 ≤ α1 ≤ · · · ≤ αJ ≤ 1}. For

θ ∈ Θ define gf ,θ =

J

• j=0

αj+1 − αj λ(Cαj+1) − λ(Cαj)ICαj+1\Cαj (x), where α0 = 0, αJ+1 = 1 and Cα0 = ∅.

• Observe that the function gf ,θ is the density function of a piecewise

uniform distribution.

• Let D be a distance function between bivariate density functions (Lp

norm, Hellinger distance, symmetric Kullback-Leibler divergence...)

• We propose to choose the best θ by solving the problem:

min

θ∈Θ D(f , gf ,θ).

(2)

Optimal level sets for bivariate densities 23/52 Pedro Delicado and Philippe Vieu

Introduction One density Several densities Conclusions References Distances between densities

Example (J = 1, 2, 3, 4)

• Bivariate normal truncated to [−3.035, 3.035] × [−3.035, 3.035] .
• In this case, for J=1,2,3, the optimal α values are ordered as follows,

according to the used distance, α∗

j,L2 < α∗ j,L1 < α∗ j,Hellinger ≈ α∗ j,K-L ≈ α∗ j,Sym. K-L < α∗ j,L2logs.

• For J = 4 all the distances lead to similar optimal α values, except

L2 norm between logs, that gives larger values. Symmetric L2 norm L1 norm Kullback-Leibler

• f logs

J α∗

j , j = 1, . . . , J

α∗

j , j = 1, . . . , J

α∗

j , j = 1, . . . , J

1 0.75 0.84 0.95 2 0.55, 0.88 0.67, 0.94 0.77, 0.95 3 0.43, 0.73, 0.93 0.51, 0.80, 0.95 0.63, 0.86, 0.95 4 0.32, 0.61, 0.82, 0.95 0.25, 0.61, 0.83, 0.95 0.53, 0.78, 0.89, 0.95

Optimal level sets for bivariate densities 24/52 Pedro Delicado and Philippe Vieu

Introduction One density Several densities Conclusions References Distances between densities

Dealing with estimated densities (J=1)

• Consider the more realistic case of having an unknown density

function f .

• Let {fn}n be a sequence of density functions approximating f (e.g.,

fn is a nonparametric estimation of f from a sample of size n).

• Let α∗ be the solution of (2) for J = 1: min0<α<1 D(f , gf ,α).
• Let ˆ

αn be the solution of the following minimization problem: min

0<α<1 D(fn, gfn,α).

(3)

• Under what conditions is ˆ

αn converging to α∗?

Optimal level sets for bivariate densities 25/52 Pedro Delicado and Philippe Vieu

Introduction One density Several densities Conclusions References Distances between densities

Assumptions

(MD1) D(fn, f ) → 0 as n → ∞. (MD2) D(α) = D(f , gf ,α) has a unique minimum α∗. (MD3) For all α0 in [0, 1] and {αr} ⊂ [0, 1] we have that lim

r→∞ D(f , gf ,αr ) = D(f , gf ,α0) implies

lim

r→∞ αr = α0.

(LS) Assumption (MD1) implies that limn→∞ supα∈[0,1] D(gfn,α, gf ,α) = 0.

• (MD1), (MD2) and (MD3) are standard assumptions in minimum distance

parametric estimation.

• For the case of fn being kernel estimators with bandwidth h = h(n), there are

known conditions on h(n) that imply (MD1) almost surely (see, e.g., Devroye and Gy¨

• rfi 1985, for the case of L1 distance).
• They also verify (LS) when in (2) and (3) supα∈[0,1] is replaced by maxα1,...,αK ,

the maximum over a finite thin grid (see, e.g., Ba´ ıllo (2003) as a starting point).

Optimal level sets for bivariate densities 26/52 Pedro Delicado and Philippe Vieu

Introduction One density Several densities Conclusions References Distances between densities

Theorem

Let f be a bivariate density function with compact support C1 and let fn be a sequence of bivariate density functions with support C1. Let D be a distance between density functions for which assumptions (MD1), (MD2), (MD3) and (LS) are verified. Let α∗ be the solution of problem (2) for J = 1 and let {ˆ αn}n be a sequence of solutions for problem (3). Then lim

n→∞ ˆ

αn = α∗.

(Proof based on minimum distance estimation results; see Parr and Schucany 1982 and Cao, Cuevas, and Fraiman 1995)

Optimal level sets for bivariate densities 27/52 Pedro Delicado and Philippe Vieu

Introduction One density Several densities Conclusions References Difficulties with managing several densities

1

Introduction

2

Optimal level sets for a single density Optimality based on distances between density level sets Optimality based on distances between bivariate densities Difficulties with managing several densities

3

Optimal representation of several densities by level sets Representation with a single level set Representation with more than a level set Case of estimated densities Some Monte Carlo experiments

4

Conclusions

Optimal level sets for bivariate densities 28/52 Pedro Delicado and Philippe Vieu

Introduction One density Several densities Conclusions References Difficulties with managing several densities

Several densities

• Consider now the case of N density functions, f 1, . . . , f N, to be

represented each of them by J level sets, having common probability contents α1 ≤ · · · ≤ αJ.

• Let θ∗

N be the solution of the minimization problem

min

θ∈Θ N

• i=1

D(f i, gf i,θ).

• For i = 1, . . . , N, let {f i

n}n be a sequence of density functions

approaching f i.

• Let ˆ

θN

n be the solution of the minimization problem

min

θ∈Θ N

• i=1

D(f i

n, gf i

n ,θ).

• An extension of Theorem 2 would guarantee that ˆ

θN

n goes to θ∗ N.

Optimal level sets for bivariate densities 29/52 Pedro Delicado and Philippe Vieu

Introduction One density Several densities Conclusions References Difficulties with managing several densities

Example: Aircraft data, N = 3, J = 1.

L1 L2 L2 norm Kullback- Symmetric D: norm norm Hellinger

• f logs

Leibler K-L ˆ α3

n:

0.81 0.72 0.90 0.95 0.88 0.95 αf

1 = 0.5

α1 = 0.75 ˆ α3

n = 0.9

−4 −2 2 4 6 −2 −1 1 2 3 Comp.1 Comp.2

Period: 1914 − 1935 Period: 1936 − 1955 Period: 1956 − 1984 −4 −2 2 4 6 −2 −1 1 2 3 Comp.1 Comp.2

Period: 1914 − 1935 Period: 1936 − 1955 Period: 1956 − 1984 −4 −2 2 4 6 −2 −1 1 2 3 Comp.1 Comp.2

Period: 1914 − 1935 Period: 1936 − 1955 Period: 1956 − 1984

Optimal level sets for bivariate densities 30/52 Pedro Delicado and Philippe Vieu

Introduction One density Several densities Conclusions References Difficulties with managing several densities

Difficulties with managing several densities

• Using this approach for N densities: the common θ = (α1, . . . , αJ)

that we are seeking tries only to provide a good individual representation of the densities involved: it does not attempt to highlight the differences between them. An exemple:

. 5 . 1 . 1 . 2 5 0.5 . 7 5 0.95

−3 −2 −1 1 2 3 −3 −2 −1 1 2 3

. 5 . 1 . 1 . 2 5 0.5 . 7 5 0.95

−3 −2 −1 1 2 3 −3 −2 −1 1 2 3

Optimal level sets for bivariate densities 31/52 Pedro Delicado and Philippe Vieu

Introduction One density Several densities Conclusions References

1

Introduction

2

Optimal level sets for a single density Optimality based on distances between density level sets Optimality based on distances between bivariate densities Difficulties with managing several densities

3

Optimal representation of several densities by level sets Representation with a single level set Representation with more than a level set Case of estimated densities Some Monte Carlo experiments

4

Conclusions

Optimal level sets for bivariate densities 32/52 Pedro Delicado and Philippe Vieu

Introduction One density Several densities Conclusions References

Representation of several densities by level sets

• Let us consider N bivariate density functions, f 1, . . . , f N.
• It could be the case that they form a random sample corresponding

to observations of a common stochastic process.

• Given D, a distance function between bivariate density functions, we

define the distance matrix D =

• dij = D(f i, f j)
• i,j=1,...,N

reflecting the inter-distances between any pair of density functions f i and f j in the previous list.

Optimal level sets for bivariate densities 33/52 Pedro Delicado and Philippe Vieu

Introduction One density Several densities Conclusions References A single level set

1

Introduction

2

Optimal level sets for a single density Optimality based on distances between density level sets Optimality based on distances between bivariate densities Difficulties with managing several densities

3

Optimal representation of several densities by level sets Representation with a single level set Representation with more than a level set Case of estimated densities Some Monte Carlo experiments

4

Conclusions

Optimal level sets for bivariate densities 34/52 Pedro Delicado and Philippe Vieu

Introduction One density Several densities Conclusions References A single level set

Representation with a single level set

• Let α ∈ [0, 1] and let C i

α be the level set of fi with probability

content α, for i = 1, . . . , N.

• For a pair of density functions fi and fj we define

δα,d

ij

= d(C i

α, C j α)

where d is a distance function between sets.

• We can also use the density functions corresponding to piecewise

uniform distributions to define δα,D

ij

= D(gf i,α, gf j,α), where D is a distance between bivariate density functions.

• Let δα be the N × N distance matrix whose elements are δα

ij

computed as δα,d

ij

• r as δα,D

ij

.

Optimal level sets for bivariate densities 35/52 Pedro Delicado and Philippe Vieu

Introduction One density Several densities Conclusions References A single level set

• Our objective is to choose α in order to have distance matrices

D =

• dij = D(f i, f j)
• i,j=1,...,N between densities

and δα =

• δα

ij

• i,j=1,...,N between level sets

as similar as possible.

• It is an idea similar to the Generalised Multidimensional Scaling

(Bronstein, Bronstein, and Kimmel 2006) where the objective is to find configurations in two different metric spaces with similar distance matrices.

Optimal level sets for bivariate densities 36/52 Pedro Delicado and Philippe Vieu

Introduction One density Several densities Conclusions References A single level set

Our proposal

• Let S(δα, D) be the Spearman’s rank correlation coefficient of δα

ij

and Dij, with i > j.

• It is the Pearson’s correlation coefficient between r α

ij and Rij, i < j,

r α

ij : the rank of δα ij among the elements δα.

Rij: the rank of Dij among the elements of D.

• Our proposal is to look for the value of α that solves the following
• ptimisation problem:

max

α∈[0,1] S(δα, D)2.

• S(δα, D)2 can be computed as the coefficient of determination R2
• f the simple linear regression of Rij, i > j, against r α

ij , i > j.

Optimal level sets for bivariate densities 37/52 Pedro Delicado and Philippe Vieu

Introduction One density Several densities Conclusions References More than a level set

1

Introduction

2

Optimal level sets for a single density Optimality based on distances between density level sets Optimality based on distances between bivariate densities Difficulties with managing several densities

3

Optimal representation of several densities by level sets Representation with a single level set Representation with more than a level set Case of estimated densities Some Monte Carlo experiments

4

Conclusions

Optimal level sets for bivariate densities 38/52 Pedro Delicado and Philippe Vieu

Introduction One density Several densities Conclusions References More than a level set

Representation with more than a level set

• Now we use J level sets for representing each density fi,

i = 1, . . . , N.

• Let 0 ≤ α1 ≤ · · · ≤ αJ ≤ 1 be J probability contents, for a given J.
• For a given bivariate density function f we define

Cf

θ = (C f α1, . . . , C f α1), the family of level sets of f with probability

contents given by the elements of θ = (α1, . . . , αJ).

• Consider the multiple linear regression where the response is the set
• f ranks Rij, i < j, defined before, with J explanatory variables, the

ranks r αh

ij

• f the J distances δαh

ij , αh ∈ θ.

• Our proposal: to solve the optimisation problem

max

θ∈Θ R2 S(θ),

where R2

S(θ) is the corresponding coefficient of determination.

Optimal level sets for bivariate densities 39/52 Pedro Delicado and Philippe Vieu

Introduction One density Several densities Conclusions References Estimated densities

1

Introduction

2

Optimal level sets for a single density Optimality based on distances between density level sets Optimality based on distances between bivariate densities Difficulties with managing several densities

3

Optimal representation of several densities by level sets Representation with a single level set Representation with more than a level set Case of estimated densities Some Monte Carlo experiments

4

Conclusions

Optimal level sets for bivariate densities 40/52 Pedro Delicado and Philippe Vieu

Introduction One density Several densities Conclusions References Estimated densities

Estimated densities (J = 1)

• We assume that for i = 1, . . . , N, there is a sequence {f i

n}n of

density functions approaching fi as n goes to infinity.

• Consider the following version of the previous problem, where δα

and D, defined from fi, i = 1, . . . , N, has been substituted by their counterparts, say δα

n and Dn, defined from f i n, i = 1, . . . , N:

max

α∈[0,1] S(δα n , Dn)2.

• Additional assumption:

(LS2) Assumption (MD1) implies that limn→∞ supα∈[0,1] d(C fn

α , C f α) = 0 almost surely.

Optimal level sets for bivariate densities 41/52 Pedro Delicado and Philippe Vieu

Introduction One density Several densities Conclusions References Estimated densities

Theorem

Let D and d be distances between density functions and level sets, respectively, for which assumptions (MD1) and (LS2) are verified for any density function fi and sequence {f i

n}n, i = 1, . . . , N. Assume that there

are no ties in distances: for i < j and k < l, with (i, j) = (k, l), D(fi, fj) = D(fk, fl), inf

0<α<1 |d(C fi α, C fj α) − d(C fk α , C fl α)| > 0.

Let α∗ be the solution to problem maxα∈[0,1] S(δα, D)2. Let ˆ αn be the solution to problem maxα∈[0,1] S(δα

n , Dn)2.

Then lim

n→∞ ˆ

αn = α∗ almost surely.

Optimal level sets for bivariate densities 42/52 Pedro Delicado and Philippe Vieu

Introduction One density Several densities Conclusions References Monte Carlo experiments

1

Introduction

2

Optimal level sets for a single density Optimality based on distances between density level sets Optimality based on distances between bivariate densities Difficulties with managing several densities

3

Optimal representation of several densities by level sets Representation with a single level set Representation with more than a level set Case of estimated densities Some Monte Carlo experiments

4

Conclusions

Optimal level sets for bivariate densities 43/52 Pedro Delicado and Philippe Vieu

Introduction One density Several densities Conclusions References Monte Carlo experiments

Monte Carlo experiments

We consider sets of N = 50 bivariate density functions fi, i = 1, . . . , N, such that fi is a mixture of three bivariate normal densities, truncated at the square [−3.035, 3.035] × [−3.035, 3.035]: f (x, y) =

2

• j=0

ηjφ2(x, y; µ1,j, µ2,j, σ2

j I2),

where I2 is the identity matrix of size 2 and φ2(x, y; µ1, µ2, Σ) is the density function of a bivariate normal centred at (µ1, µ2) with variance matrix Σ, evaluated at (x, y) ∈ R2. (µ1,0, µ2,0) = (0, 0), (µ1,j, µ2,j) = ρj(cos θj, sin θj), j = 1, 2 θj ∼ U(0, 2π), ρj ∼ U(rj − .1, rj + .1) (rj is a fixed value in {0, 1, 2}).

Optimal level sets for bivariate densities 44/52 Pedro Delicado and Philippe Vieu

Introduction One density Several densities Conclusions References Monte Carlo experiments

Three cases

−3 −2 −1 1 2 3 −3 −2 −1 1 2 3

−3 −2 −1 1 2 3 −3 −2 −1 1 2 3

−3 −2 −1 1 2 3 −3 −2 −1 1 2 3 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Case 1 alpha coefficient of determination 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Case 2 alpha coefficient of determination 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Case 3 alpha coefficient of determination

Resuslts for J = 1. R2

S(α) as a function of α. L1 and distance in probability.

Optimal level sets for bivariate densities 45/52 Pedro Delicado and Philippe Vieu

Introduction One density Several densities Conclusions References Monte Carlo experiments

Case Parameters J α∗

1

α∗

2

α∗

3

α∗

4

R2

S(θ)

1 r1 = 0 r2 = 1 1 0.244 0.8753 σ1 = 1 σ2 = 0.25 2 0.209 0.340 0.8960 η1 = 0.05 η2 = 0.05 3 0.197 0.290 0.431 0.9017 4 0.212 0.404 0.595 0.787 0.9254 2 r1 = 2 r2 = 1 1 0.571 0.4840 σ1 = 0.4 σ2 = 0.25 2 0.228 0.763 0.7572 η1 = 0.025 η2 = 0.05 3 0.199 0.622 0.800 0.8326 4 0.191 0.407 0.614 0.845 0.8446 3 r1 = 2 r2 = 0 1 0.780 0.6638 σ1 = 0.4 σ2 = 1 2 0.409 0.816 0.8312 η1 = 0.025 η2 = 0.05 3 0.230 0.653 0.869 0.8366 4 0.184 0.831 0.837 0.860 0.8600

Optimal level sets for bivariate densities 46/52 Pedro Delicado and Philippe Vieu

Introduction One density Several densities Conclusions References Monte Carlo experiments

Comparing methods designed for N = 1 or N > 1 when there are N > 1 densities. Case 1

−2 −1 1 2 −2 −1 1 2 (a) aplha: 0.75 −2 −1 1 2 −2 −1 1 2 (b) aplha: 0.244

Optimal level sets for bivariate densities 47/52 Pedro Delicado and Philippe Vieu

Introduction One density Several densities Conclusions References

1

Introduction

2

Optimal level sets for a single density Optimality based on distances between density level sets Optimality based on distances between bivariate densities Difficulties with managing several densities

3

Optimal representation of several densities by level sets Representation with a single level set Representation with more than a level set Case of estimated densities Some Monte Carlo experiments

4

Conclusions

Optimal level sets for bivariate densities 48/52 Pedro Delicado and Philippe Vieu

Introduction One density Several densities Conclusions References

Conclusions I

• Level sets are known to be nice graphical tools for visualising

bivariate density functions.

• Usually, levels are fixed arbitrarily (most usual choices being those

with probability contents 0.25, 0.5 and 0.75), but changing the levels may lead to different conclusions from the same data.

• We try to respond to two natural questions:
• How can levels be chosen?
• Can such a choice be made in some data-driven way to take into

account both the specificity of the available data and the kind of statistical problem one has to deal with?

Optimal level sets for bivariate densities 49/52 Pedro Delicado and Philippe Vieu

Introduction One density Several densities Conclusions References

Conclusions II

• We study separately two scenarios.
• In both cases we show that our proposals provide good theoretical

properties and ease of implementation as well as a satisfactory practical finite sample performance.

• In particular, emphasis is given to the fact that our selected levels

can detect information that standard level choices (like those with probability contents 0.25, 0.5 and 0.75) may hide.

Optimal level sets for bivariate densities 50/52 Pedro Delicado and Philippe Vieu

Introduction One density Several densities Conclusions References

Conclusions III

• In the first scenario we deal with the case when there is only one

density function to be represented.

• Our proposals here are based on the minimum distance between sets
• r between density functions.
• Our main findings are the following.
• We have presented a quick method to choose the J optimal

probability contents that does not depend on the specific density to be represented (for instance, for J = 3 they are 1/6, 1/2 and 5/6).

• In general, higher values for probability contents α are obtained when

using methods that depend on the density to be represented.

• They also depend on the specific distance in use (distance L1 leading

to not so high values of α).

Optimal level sets for bivariate densities 51/52 Pedro Delicado and Philippe Vieu

Introduction One density Several densities Conclusions References

Conclusions IV

• In the second scenario we consider the case of many densities to be

represented.

• Our approach here is related with generalised MDS.
• Several possibilities have been analized and, finally, our proposal

consists on maximizing the coefficient of determination of a linear regression involving the ranks of distances between the densities to be represented and the ranks of the distances between their level sets.

• As practical advice, we recommend to solve the maximization

problem using L1 distance between density functions and distance in probability between level sets.

• Moreover using only one level set for each density function usually

gives good results when representing many densities in the same graphic.

Optimal level sets for bivariate densities 52/52 Pedro Delicado and Philippe Vieu

Introduction One density Several densities Conclusions References

Thank you for your attention!

Optimal level sets for bivariate densities 53/52 Pedro Delicado and Philippe Vieu

Introduction One density Several densities Conclusions References Ba´ ıllo, A. (2003). Total error in a plug-in estimator of level sets. Statistics & Probability Letters 65(4), 411–417. Ba´ ıllo, A., J. Cuesta-Albertos, and A. Cuevas (2001). Convergence rates in nonparametric estimation of level sets. Statistics & Probability Letters 53(1), 27–35. Bowman, A. and A. Azzalini (1997). Applied Smoothing Techniques for Data Analysis. Oxford: Oxford University Press. Bronstein, A., M. Bronstein, and R. Kimmel (2006). Generalized multidimensional scaling: a framework for isometry-invariant partial surface matching. Proceedings of the National Academy of Sciences of the United States of America 103(5), 1168–1172. Cadre, B. (2006). Kernel estimation of density level sets. Journal of Multivariate Analysis 97(4), 999–1023. Cao, R., A. Cuevas, and R. Fraiman (1995). Minimum distance density-based estimation. Computational Statistics & Data Analysis 20(6), 611–631. Devroye, L. and L. Gy¨

• rfi (1985).

Nonparametric Density Estimation: The L1-View. New York: Wiley. Parr, W. and W. Schucany (1982). Minimum distance estimation and components of goodness-of-fit statistics. Journal of the Royal Statistical Society. Series B, 178–189. Optimal level sets for bivariate densities 53/52 Pedro Delicado and Philippe Vieu