TEORIA DE LA ACTIVIDAD RUTINARIA ADPATADA A LOS DCSV

First we assume the first eigenvalue is much larger than the others. Suppose the

✏2-

condition holds, the following proposition states that under the general setting described at

the beginning of this section, the first sample eigenvectoruˆ

1

converges to its population coun-

terpartu

1

(consistency) or tends to be perpendicular tou

1

(strong consistency) according to

the magnitude of the first eigenvalue

1

, while all the other sample eigenvectors are strongly

inconsistent regardless of the magnitude

1

.

Proposition 1.

For a fixed

n, let

⌃

=

U⇤U

T

,

d

=

n

+ 1, n

+ 2, . . . ,

be a sequence of

covariance matrices. LetXbe ad⇥ndata matrix from ad-variate distribution with mean 0

and covariance matrix⌃. Let_Sˆ

_{= ˆU⇤}ˆ_Uˆ

T

_{be the sample covariance matrix estimated from}

X

for eachd. Let↵1

>0. Assume the following

1. The components ofZ

=⇤

1/2

U

T

X

have uniformly bounded fourth moments and are

3. The✏2-condition holds andPd

_i=2 i

=O(d).

If↵1

>1, then the first sample eigenvector is consistent and the others are strongly inconsis-

tent in the sense that

Angle(ˆu

1,

u

1

)

prob

!0asd! 1,

and

Angle(ˆui,ui)

prob

_!

⇡

2

asd! 1,

for alli= 2, . . . , n.

If0<↵1

<1, then all sample eigenvectors are strongly inconsistent, i.e.,

Angle(ˆui,ui)

prob

_!

⇡

2

asd! 1,

for alli= 1, . . . , n.

When

↵1

>

1, we can say that

1

is much larger than the others, which means that

the signal (dominates the background noise) is captured by the first PC. This results the

consistency ofuˆ

1

and strong inconsistency ofuiˆ

,

for alli= 2, . . . , n. when0<↵1

<1, we

can say that all eigenvalues are of the same order of magnitude and satisfy the✏-condition,

which means that there is only background noise. This results that all samples eigenvectors

are strongly inconsistent.

The Proposition 1 can be generalized to the case that multiple eigenvalues are much larger

than the others. This leads to two different types of result.

First is the case that the firstpeigenvalues are consistent. Consider a covariance structure

with multiple spikes, that is,peigenvalues,p >1, which are much larger than the background

a distinct order of magnitude, for example,

1,d

=

O(d

3

),

2,d

=

O(d

2

)and the sum of the

rest is order ofd.

Proposition 2.

For a fixedn, let⌃,X

andS

be as before. Let↵1

>· · ·>↵

p

>1for some

p < n. Suppose the following conditions hold

1. The same as assumption 1 of Proposition 1.

2.

i

/d

↵i

!c

i

for somec

i

>0,

for alli= 1, . . . , p.

3. The✏

p+1-condition holds and

Pd

i=p+1 i

=O(d).

Then the firstpsample eigenvectors are consistent and the others are strongly inconsistent in

the sense that

Angle(ˆui,ui)

prob

!0asd! 1for alli= 1, . . . , p,

Angle(ˆui,ui)

prob

!

⇡

2

asd! 1for alli=p+ 1, . . . , n.

When↵1

>

· · ·

>

↵

p

>

1, we can say that the firstpeigenvalues are much larger than

the others. Then the signal is captured by the first

p

principal components and the signal

level decreases from the1

st

to thep

th

principal component. This results the consistency of

ˆ

u

1, . . . ,

upˆ

. For the rest of the principal components, because their eigenvalues are much

smaller and satisfy the✏

k+1

-condition, only the noise is captured by these higher order prin-

cipal components. This results the inconsistency ofuˆ

p+1

, . . . ,uˆ

n

.

of the same order of magnitude, that is, lim

d!1 1,d

/

p,d

=c >1for some constantc. Then

the firstpsample eigenvectors are neither consistent nor strongly inconsistent. However, all

of those random directions converge to the subspace spanned by the firstppopulation eigen-

vectors. Essentially, when eigenvalues are of the same order of magnitude, the eigenvectors

cannot be separated but a subspace consistent with the proper subspace.

Proposition 3.

For a fixedn, let⌃,X

andS

be as before. Let↵1

>1andp < n. Suppose

the following conditions hold

1. The same as assumption 1 of Proposition 1.

2.

i

/d

↵1

!c

i

for somec

i

>0,

for alli= 1, . . . , p.

3. The✏

p+1-condition holds and

Pd

i=p+1 i

=O(d).

Then the first

psample eigenvectors are subspace consistent with the subspace spanned by

the firstppopulation eigenvectors, and the others are strongly inconsistent in the sense that

Angle(ˆui,span{u

1, . . . ,

up})

prob

!0asd

! 1,

for alli= 1, . . . , p,

Angle(ˆu

i

,u

i

)

prob

!

⇡

2

asd! 1,

for alli=p+ 1, . . . , n.

Propositions 1, 2 and 3 are combined and generalized in the main theorem as follows

Theorem 5.

For a fixed

n, let

⌃

d

,

X

and

S

be as before. Let

↵1, . . . ,↵

p

be such that

↵1

>

_{· · ·}

>

↵

p

>

1

for some

p < n. Let

k1, . . . , k

p

be nonnegative integers such that

Pp

that

J

l

=

(

1 +

l 1

X

j=0

k

j

,2 +

l 1

X

j=0

k

j

, . . . , k

l

+

l 1

X

j=0

k

j

)

,

for alll= 1, . . . , p+ 1.

Suppose the following conditions hold

1. The components ofZ

=⇤

1/2

U

T

X

have uniformly bounded fourth moments and are

⇢-mixing for some permutation.

2.

i

/d

↵l

!c

i

for somec

i

>0,

for alli2J

l

,

8l= 1, . . . , p.

3. The✏

+1-condition, i.e., Pd i=+1 2i

(

Pd i=+1 i

)

2

!0holds and

P

i2Jp+1 i

=O(d).

Then the sample eigenvectors whose labels are in the group

J

l

,

for all

l

= 1, . . . , p, are

subspace-consistent with the space spanned by the population eigenvectors whose labels are

inJ

l

and the others are strongly inconsistent in the sense that

Angle(ˆu

i

,span{u

j

:j

2J

l}

)

prob

!0asd_{! 1},

for alli₂J

l

,

for alll= 1, . . . , p,

(5.2)

Angle(ˆui,ui)

prob

_!

⇡

2

asd! 1,

for alli=+ 1, . . . , n.

Remark 1.

In the special case that the cardinality ofJ

l

,k

l

, equals 1, then(5.2)implies that

ˆ

ui

is consistent withuifor alli2J

l

.

Remark 2.

There is a case where the strongly inconsistent eigenvectors whose labels are

in

J

p+1

can be considered to be subspace-consistent. Let

be the very large subspace

spanned by the population eigenvectors whose labels are in

J

p+1

for each

d, i.e.,

=

span{uj

:j

2J

p+1

}=span{u

+1, . . . ,

ud}. Then

Angle(ˆui,

)

prob

_!0asd_{! 1},

for alli₂J

p+1

.

In document Política criminal y delitos contra la seguridad vial (página 94-98)