MAT269_slides-04.pdf7896546879878797897897

matematica aplicada

1. MAT-269: Estadstica descriptiva
multivariada
Felipe Osorio
fosorios.mat.utfsm.cl
Departamento de Matematica, UTFSM
1 / 27

2. Datos multivariados
Datos multivariados:
Tenemos una muestra aleatoriax1; : : : ;xndonde para cada observacion se ha medido
p2variables xi= (xi1; : : : ; xip)
>
es vector
p-dimensional.
Podemos disponer la informacion en una
1
X=
0
B
B
B
@
x11x12: : : x1p
x21x22: : : x2p
.
.
.
.
.
.
.
.
.
xn1xn2: : : xnp
1
C
C
C
A
=
0
B
B
B
@
x
>
1
x
>
2
.
.
.
x
>
n
1
C
C
C
A
:
Observacion:
Por simplicidad asumiremos quex1; : : : ;xnson variables aleatorias IID desde
Fp(;)(conFpcomun).
1
X= (xij), parai= 1; : : : ; n;j= 1; : : : ; p.
2 / 27

3. Datos de Flores Iris (Anderson, 1935; Fisher, 1936)
IrisIrisIris
3 / 27

4. Datos de Flores Iris (Anderson, 1935; Fisher, 1936)Edgar Anderson's Iris Data
Scatter Plot Matrix
Sepal.Length
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Sepal.Width
3.5
4.0
4.5
3.54.04.5
2.0
2.5
3.0
2.02.53.0
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Petal.Length4
5
6
7
4567
1
2
3
4
1234 ll
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Petal.Width
1.5
2.0
2.5
1.52.02.5
0.0
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1.0
0.00.51.0
4 / 27

5. Datos de Flores Iris (Anderson, 1935; Fisher, 1936)
Datos observados:
Mediciones (cm) del largo y ancho de los
ores desde 3 especies de
Objetivo:
IObtener una funcion que permita
IUsando las medidas de una or,
Caractersticas del problema:
IEl analisis exploratorio revela una separacion evidente en.
ITecnicas mas renadas permiten identicar las 3 especies, p.ej.:
IAnalisis discriminante,
ITecnicas de clasicacion
IAprendizaje de maquina
5 / 27

6. Conjunto de datos
> iris
S e p a l . L e n g t h S e p a l . W i d t h P e t a l . L e n g t h P e t a l . W i d t h S p e c i e s
1 5.1 3.5 1.4 0.2 s e t o s a
2 4.9 3.0 1.4 0.2 s e t o s a
3 4.7 3.2 1.3 0.2 s e t o s a
4 4.6 3.1 1.5 0.2 s e t o s a
5 5.0 3.6 1.4 0.2 s e t o s a
6 5.4 3.9 1.7 0.4 s e t o s a
7 4.6 3.4 1.4 0.3 s e t o s a
8 5.0 3.4 1.5 0.2 s e t o s a
9 4.4 2.9 1.4 0.2 s e t o s a
10 4.9 3.1 1.5 0.1 s e t o s a
11 5.4 3.7 1.5 0.2 s e t o s a
12 4.8 3.4 1.6 0.2 s e t o s a
13 4.8 3.0 1.4 0.1 s e t o s a
...
148 6.5 3.0 5.2 2.0 v i r g i n i c a
149 6.2 3.4 5.4 2.3 v i r g i n i c a
150 5.9 3.0 5.1 1.8 v i r g i n i c a
6 / 27

7. Graco del conjunto de datos
x
p a i r s
p a i r s $S p e c i e s )Sepal.Length
2.02.53.03.54.0
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0.51.01.52.02.5
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Sepal.Width
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l l
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ll
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lll
l
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ll
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ll
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ll
l l
l
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ll
l
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l
l
l
l
ll
l
l
l
l
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l
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l
l
l
l
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l
ll
l
l
l
l
l
l
l
l
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ll
l
l
l
l
l
lll
l
l
l
l
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ll
l
l
l
l
l
l
l
l l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
ll
l
l
ll
ll
l
l
l
l
l
l
l
lll
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
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l
l
l
l
l
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l
l
l
l
l
l
l
l
l
l
l
l
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l l
l
l
l
l
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l
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lll
l
l
l
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l
l
l
l
l
l
l
l
l
l
l
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l
l
l
l
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l
l
l
l
l
l
l
l
l
l
l
l
l
ll
l
l
l
l
l
l
l
ll
l
l
l
l
l
l
l
l l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l l
l
l
ll
l l
l
l
l
l
l
l
l
ll l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l l
l
l
l
l
l
l
l
l
lll
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
ll
l
l
l
l
l
l
l
Petal.Length
1
2
3
4
5
6
7
ll
l
l
l
l
l
l
l
ll
l
l
l
l
l
l
l
l
l
l
l
l
l
l
ll
l
l
ll
ll
l
l
l
l
l
l
l
lll
l
l
l
l
l
l
l
l
l
l
l
l
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l
l
l
l
l
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l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
ll
l
l
l
l
l
l
l
l
lll
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
ll
l
l
l
l
l
l
l
4.55.56.57.5
0.5
1.0
1.5
2.0
2.5
lllll
l
l
ll
l
ll
ll
l
ll
l ll
l
l
l
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ll
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llll
l
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l l
ll
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ll ll
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l l
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lll
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l l
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l
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ll
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l l
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lll
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llll l
l
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lll
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1234567
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Petal.Width Sepal.Length
2.02.53.03.54.0
l
l
l
l
l
l
l
l
l
l
l
ll
l
l
l
l
l
l
l
l
l
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ll
l
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ll
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ll
l
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l
l
l
l
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l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
ll
l
l
l
l
0.51.01.52.02.5
4.5
5.5
6.5
7.5
l
l
l
l
l
l
l
l
l
l
l
ll
l
l
l
l
l
l
l
l
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ll
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ll
l
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ll
l
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ll
l
l
l
l
l
l
l
l
l
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l
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ll
l
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l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
ll
l
l
l
l
2.0
2.5
3.0
3.5
4.0
l
l
l
l
l
l
ll
l
l
l
l
ll
l
l
l
l
ll
l
l
l
l
l
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l
l
l
l
l
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l
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l
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ll
l
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ll
l
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ll
l
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l
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ll
ll
l
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l l
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l l
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l
l
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l
l
l
l
l l
l
l
l
l
l
l
l
l
l
ll
l
l
l
l
l
lll
l
l
l
l
l
l
l
l
Sepal.Width
l
l
l
l
l
l
ll
l
l
l
l
ll
l
l
l
l
ll
l
l
l
l
l
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l
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ll
l l
l
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ll
l
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ll
l
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l l
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l
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lll
l
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lll
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lll
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l l
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lll
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l l
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l l
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ll
l l
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ll l
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Petal.Length
1
2
3
4
5
6
7
ll
l
l
l
l
l
l
l
ll
l
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l
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lll
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lll
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ll
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l
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4.55.56.57.5
0.5
1.0
1.5
2.0
2.5
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Petal.Width
7 / 27

8. Estadsticas de resumen
Analogamente a la xys
2
para el caso unidimensional. Podemos
denir sus
x=
1
n
n
X
i=1
xi;
S=
1
n1
n
X
i=1
(xix)(xix)
>
:
En efecto,x= (x1; : : : ;xp)
>
yS= (srs), donde
xj=
1
n
n
X
i=1
xij;
srs=
1
n1
n
X
i=1
(xirxr)(xisxs);
parar; s= 1; : : : ; p.
8 / 27

9. Estadsticas de resumen
Observacion:
Algunas propiedades del, surgen de escribir
formas compactas que dependen de la matriz de datosX= (xij)2R
np
. En efecto,
x=
1
n
n
X
i=1
xi=
1
n
X
>
1n:
Sea
Q=
n
X
i=1
(xix)(xix)
>
=
n
X
i=1
(xix)x
>
i

n
X
i=1
(xix)x
>
=
n
X
i=1
xix
>
i
x
n
X
i=1
x
>
i
=
n
X
i=1
xix
>
i
nxx
>
;
pues
P
n
i=1
(xix)x
>
=0.
2
2
No se hara distincion sobre el
9 / 27

10. Estadsticas de resumen
Notando
n
X
i=1
xix
>
i
=X
>
X;
sigue que
S=
1
n1
Q=
1
n1
n
X
>
Xn

1
n
X
>
1n

1
n
X
>
1n

>
o
=
1
n1

X
>
X
1
n
X
>
1n1
>
n
X

=
1
n1
X
>
CX
conC=In
1
n
1n1
>
nla. Es sencillo mostrar que
C
>
=C; C
2
=C;
es decirCes matriz de proyeccion. Esto permite mostrar el siguiente resultado.
10 / 27

11. Estadsticas de resumen
Resultado 1:
La matriz de covarianzaS, es semidenida positiva.
Demostracion:
Seaa2R
p
, vector no nulo. Tenemos que,
a
>
Sa=
1
n1
a
>
X
>
CXa=
1
n1
a
>
X
>
C
2
Xa
=
1
n1
u
>
u0; u=CXa;
es decir,Ses matriz semidenida positiva.
3
3
Ssera denida positiva sinp+ 1.
11 / 27

12. Estadsticas de resumen
La pvariables de interes, es dada por:
R= (rij);
donde
rjk=
P
n
i=1
(xijxj)(xikxk)
pP
n
i=1
(xijxj)
2
P
n
i=1
(xikxk)
2
=
sjk
p
sjjskk
;
paraj; k= 1; : : : ; p, conS= (sjk).
Sea,D= diag(s11; s22; : : : ; spp). As, podemos escribir
R=D
1=2
SD
1=2
; S=D
1=2
RD
1=2
:
12 / 27

13. Estadsticas de resumen: Datos Iris
> x
#
> xbar
> S
>
#
> xbar
S e p a l . L e n g t h S e p a l . W i d t h P e t a l . L e n g t h P e t a l . W i d t h
5 . 8 4 3 3 3 . 0 5 7 3 3 . 7 5 8 0 1 . 1 9 9 3
> S
S e p a l . L e n g t h S e p a l . W i d t h P e t a l . L e n g t h P e t a l . W i d t h
S e p a l . L e n g t h 0 . 6 8 5 7 -0.0424 1.27 0 . 5 1 6
S e p a l . W i d t h -0.0424 0 . 1 9 0 0 -0.33 -0.122
P e t a l . L e n g t h 1 . 2 7 4 3 -0.3297 3.12 1 . 2 9 6
P e t a l . W i d t h 0 . 5 1 6 3 -0.1216 1.30 0 . 5 8 1
#
> xbar
>
> z
13 / 27

14. Estadsticas de resumen: Datos Iris
#
> z
> z
$cov
S e p a l . L e n g t h S e p a l . W i d t h P e t a l . L e n g t h P e t a l . W i d t h
S e p a l . L e n g t h 0 . 6 8 5 6 9 3 5 - 0 . 0 4 2 4 3 4 0 1 . 2 7 4 3 1 5 4 0 . 5 1 6 2 7 0 7
S e p a l . W i d t h - 0 . 0 4 2 4 3 4 0 0 . 1 8 9 9 7 9 4 - 0 . 3 2 9 6 5 6 4 - 0 . 1 2 1 6 3 9 4
P e t a l . L e n g t h 1 . 2 7 4 3 1 5 4 - 0 . 3 2 9 6 5 6 4 3 . 1 1 6 2 7 7 9 1 . 2 9 5 6 0 9 4
P e t a l . W i d t h 0 . 5 1 6 2 7 0 7 - 0 . 1 2 1 6 3 9 4 1 . 2 9 5 6 0 9 4 0 . 5 8 1 0 0 6 3
$c e n t e r
S e p a l . L e n g t h S e p a l . W i d t h P e t a l . L e n g t h P e t a l . W i d t h
5 . 8 4 3 3 3 3 3 . 0 5 7 3 3 3 3 . 7 5 8 0 0 0 1 . 1 9 9 3 3 3
$n . obs
[1] 150
$cor
S e p a l . L e n g t h S e p a l . W i d t h P e t a l . L e n g t h P e t a l . W i d t h
S e p a l . L e n g t h 1 . 0 0 0 0 0 0 0 - 0 . 1 1 7 5 6 9 8 0 . 8 7 1 7 5 3 8 0 . 8 1 7 9 4 1 1
S e p a l . W i d t h - 0 . 1 1 7 5 6 9 8 1 . 0 0 0 0 0 0 0 - 0 . 4 2 8 4 4 0 1 - 0 . 3 6 6 1 2 5 9
P e t a l . L e n g t h 0 . 8 7 1 7 5 3 8 - 0 . 4 2 8 4 4 0 1 1 . 0 0 0 0 0 0 0 0 . 9 6 2 8 6 5 4
P e t a l . W i d t h 0 . 8 1 7 9 4 1 1 - 0 . 3 6 6 1 2 5 9 0 . 9 6 2 8 6 5 4 1 . 0 0 0 0 0 0 0
14 / 27

15. Estadsticas de resumen
Transformaciones lineales
Considere:
y
i
=Axi+b; i= 1; : : : ; n;
dondeA2R
qp
yb2R
p
. Entonces,
y=
1
n
n
X
i=1
y
i
=A
1
n
n
X
i=1
xi+b=Ax+b;
mientras que
y
i
y=Axi+bAxb=A(xix):
De este modo,
SY=
1
n1
n
X
i=1
(y
i
y)(y
i
y)
>
=
1
n1
n
X
i=1
A(xix)(xix)
>
A
>
=
1
n1
A
n
X
i=1
(xix)(xix)
>
A
>
=ASXA
>
:
15 / 27

16. Estadsticas de resumen
Transformacion de Mahalanobis
En particular, para la transformacion,
zi=S
1=2
(xix); i= 1; : : : ; n;
dondeS=S
1=2
S
1=2
conS
1=2
un S, sigue que
z=0; y SZ=Ip:
Observacion:
En la practica podemos considerar los siguientes metodos para obtenerS
1=2
:
Idescomposicion.
Idescomposicion.
4
4
Este procedimiento no es recomendado.
16 / 27

17. Estadsticas de resumen
SupongaS>0y considere la
S=UU
>
;
dondeUes matriz ortogonal y= diag(1; : : : ; p)con1 p>0son los
valores propios deS. De este modo podemos considerarS
1=2
=U
1=2
U
>
;
5
con

1=2
= diag(
1=2
1
; : : : ;
1=2
p). Esto lleva a la transformacion
zi=U
1=2
U
>
(xix); i= 1; : : : ; n: (1)
Usando la S=GG
>
, conGmatriz triangular
inferior. En este caso,S
1
= (GG
>
)
1
=G
>
G
1
y hacemos
zi=G
1
(xix); i= 1; : : : ; n: (2)
5
aun otra alternativa es considerarS
1=2
=U
1=2
de ah queS
1=2
=
1=2
U
>
.
17 / 27

18. Estadsticas de resumen
Observacion:
Si consideramosS=UU
>
y hacemos
y
i
=U
>
(xix); i= 1; : : : ; n: (3)
Entonces, es facil notar que
y=0;
mientras que
SY=U
>
SU=U
>
UU
>
U=:
Ademas,
trSY= tr=
p
X
j=1
j;
jSYj=jj=
p
Y
j=1
j:
La transformacion en.
18 / 27

19. Estadsticas de resumen
Denicion 1 (Distancia de Mahalanobis):
Considere una muestra denobservacionesx1; : : : ;xn. De este modo, la
Mahalanobis
Di=f(xix)
>
S
1
(xix)g
1=2
; i= 1; : : : ; n;
como la distancia de lai-esima observacion hacia el xponde-
rada por la matriz de covarianza.
Observacion:
Note que las distanciasDipueden ser calculadas de forma bastante eciente usando
(2), como:
Di=f(xix)
>
S
1
(xix)g
1=2
= (z
>
i
zi)
1=2
;
es mas,zies obtenido como solucion del sistemaGzi=xixparai= 1; : : : ; n.
19 / 27

20. Estadsticas de resumen
Usando
gij= (xix)
>
S
1
(xjx); i; j= 1; : : : ; n:
Mardia (1970)
6
denio medidas de, dadas por
b1p=
1
n
2
n
X
i=1
n
X
j=1
g
3
ij
; b 2p=
1
n
n
X
i=1
g
2
ii
;
respectivamente.
Bajo normalidad, debemos tener:
b1p= 0; b 2p=p(p+ 2):
Observacion:
Las estadsticasb1pyb2pson invariantes bajo transformaciones afn:
y
i
=Axi+b:
6
Biometrika57, 519-530.
20 / 27

21. Distancia de Mahalanobis: Datos Iris
#
> nobs
> p
>
#
> D2
> s k e w n e s s ( x )
[1] 2 . 6 9 7 2 2
> k u r t o s i s ( x )
[1] 2 3 . 7 3 9 6 6
attr
[1] - 0 . 2 6 0 3 4 2 1
> p
[1] 24
#
>
+ xlab = " C u a n t i l e s teor cos " ,
+ ylab = " D i s t a n c i a s o r d e n a d a s " )
>
21 / 27

22. Distancia de Mahalanobis: Datos Irisl
l
lll
l
l
lllll
llllllllll
lllllllllllllllllllllllllllllllllllllllllllllllllllll
llll
llllllllllllllllll
l
l
ll
l
l
llllllll
ll
l
lll
ll
ll
l
l
l
l
l
ll
lll
ll
ll
l
l
ll
l
l
ll
l
ll
l
l l
l
0 5 10 15
0
2
4
6
8
10
12
Cuantiles teorícos
Distancias ordenadas l
l
l
ll
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
ll
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
−2 −1 0 1 2
−3
−2
−1
0
1
2
3
Cuantiles teóricos
Distancias transformadas
22 / 27

23. Un algoritmo online para calcularxyS(Clarke, 1971)
7
Considere
xn=
1
n
n
X
i=1
xi:
De este modo,
xn=
1
n
n1
X
i=1
xi+xn

=
1
n

(n1)xn1+xn

=
1
n

nxn1xn1+xn

=xn1+
n
n
; (4)
conn=xnxn1.
Ecuacion en
7
Applied Statistics20, 206-209.
23 / 27

24. Un algoritmo online para calcularxyS(Clarke, 1971)
Sea
Q
n
=
n
X
i=1
(xixn)(xixn)
>
:
Es facil notar que:
Q
n
=Q
n1
+

1
1
n

n
>
n: (5)
conn=xnxn1.
Ecuaciones
24 / 27

25. Calculo de la varianza muestral. Algoritmo online (1-paso)
8
Algoritmo AS 41:Promedio y matriz de covarianza muestral.
Entrada:Matriz de datosX
>
= (x1; : : : ;xn).
Salida :Promedio y matriz de covarianza,xyS.
1begin
2 M x1
3 Q 0
4 fori= 2tondo
5 xiM
6 M M+
1
i

7 Q Q+

1
1
i


>
8 end
9 x M
10 S
1
n1
Q
11end
8
Algoritmo implementado en la funcion fastmatrix.
25 / 27

26. Propiedades basicas de los momentos muestrales
Suponga quex1; : : : ;xnson una muestra aleatoria, tal que E(xi) =y
Cov(xi) =. De este modo,
E(x) =
1
n
n
X
i=1
E(xi) =;
mientras que
Cov(x) =
1
n
2
n
X
i=1
Cov(xi) =
1
n
:
Seay
i
=xi, asy=xy E(y
i
) =0, Cov(y
i
) =, parai= 1; : : : ; n.
Ademas
n
X
i=1
(xix)(xix)
>
=
n
X
i=1
(y
i
y)(y
i
y)
>
=
n
X
i=1
y
i
y
>
i
nyy
>
26 / 27

27. Propiedades basicas de los momentos muestrales
Ahora
E(Q) =
n
X
i=1
E(y
i
y
>
i
)nE(yy
>
)
=
n
X
i=1
Cov(y
i
)nCov(y)
=nn

1
n


= (n1):
Es decir,xySson y, respectivamente.
Observacion:
Evidentemente
b
=
1
n
Q=

n1
n

S;
es un .
27 / 27