Assimetria em movimento exponencial ponderada / curtose

15

Existem fórmulas on-line conhecidas para calcular médias móveis exponencialmente ponderadas e desvios padrão de um processo (xn)n=0,1,2, . Para a média,

μn=(1α)μn1+αxn

e pela variação

σ2n=(1α)σ2n1+α(xnμn1)(xnμn)

a partir do qual você pode calcular o desvio padrão.

Existem fórmulas semelhantes para o cálculo on-line de momentos terceira e quarta centrais ponderados exponenciais? Minha intuição é que eles devem assumir a forma

M3,n=(1α)M3,n1+αf(xn,μn,μn1,Sn,Sn1)

e

M4,n=(1α)M4,n1+αf(xn,μn,μn1,Sn,Sn1,M3,n,M3,n1)

from which you could compute the skewness γn=M3,n/σ3n and the kurtosis kn=M4,n/σ4n but I've not been able to find simple, closed-form expression for the functions f and g.


Edit: Some more information. The updating formula for moving variance is a special case of the formula for the exponential weighted moving covariance, which can be computed via

Cn(x,y)=(1α)Cn1(x,y)+α(xnx¯n)(yny¯n1)

onde e ˉ y n são os meios de deslocamento exponencial de x e y . A assimetria entre x e y é ilusória, e desaparece quando se notar que y - ˉ y n = ( 1 - α ) ( y - ˉ y n - 1 ) .x¯ny¯nxyxyyy¯n=(1α)(yy¯n1)

Formulas like this can be computed by writing the central moment as an expectation En(), where weights in the expectation are understood to be exponential, and using the fact that for any function f(x) we have

En(f(x))=αf(xn)+(1α)En1(f(x))

It's easy to derive the updating formulas for the mean and variance using this relation, but it's proving to be more tricky for the third and fourth central moments.

Chris Taylor
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Respostas:

6

The formulas are straightforward but they are not as simple as intimated in the question.

Let Y be the previous EWMA and let X=xn, which is presumed independent of Y. By definition, the new weighted average is Z=αX+(1α)Y for a constant value α. For notational convenience, set β=1α. Let F denote the CDF of a random variable and ϕ denote its moment generating function, so that

ϕX(t)=EF[exp(tX)]=Rexp(tx)dFX(x).

With Kendall and Stuart, let μk(Z) denote the non-central moment of order k for the random variable Z; that is, μk(Z)=E[Zk]. The skewness and kurtosis are expressible in terms of the μk for k=1,2,3,4; for example, the skewness is defined as μ3/μ3/22 where

μ3=μ33μ2μ1+2μ13 and μ2=μ2μ12

are the third and second central moments, respectively.

By standard elementary results,

1+μ1(Z)t+12!μ2(Z)t2+13!μ3(Z)t3+14!μ4(Z)t4+O(t5)=ϕZ(t)=ϕαX(t)ϕβY(t)=ϕX(αt)ϕY(βt)=(1+μ1(X)αt+12!μ2(X)α2t2+)(1+μ1(Y)βt+12!μ2(Y)β2t2+).

To obtain the desired non-central moments, multiply the latter power series through fourth order in t and equate the result term-by-term with the terms in ϕZ(t).

whuber
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I am having some formula visualization problem, possibly whenever a ' is used, with both IE and Firefox, would you please care checking? Thanks!
Quartz
1
@Quartz Thanks for the heads up. This used to display properly, so evidently there has been some change in the processing of the TEX markup. I found a workaround by enclosing all single quotes within braces. (This change has probably broken a few dozen posts on this site.)
whuber
0

I think that the following updating formula works for the third moment, although I'd be glad to have someone check it:

M3,n=(1α)M3,n1+α[xn(xnμn)(xn2μn)xnμn1(μn12μn) μn1(μnμn1)23(xnμn)σ2n1]

Updating formula for the kurtosis still open...

Chris Taylor
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Why the ... in the above formula?
Chris
Line continuation.
Chris Taylor
Did your equation prove to be correct? I asked a similar question in R. stats.stackexchange.com/q/234460/70282
Chris
Did you account for the division by N in the third moment? Skewness is the ratio of the 3rd moment and the standard deviation^3 like so: Skew = m3 / sqrt(variance)^3 The third moment is defined as: m3 = sum( (x-mean)^3 )/n
Chris