Skewness / kurtosis bergerak tertimbang eksponensial

15

Ada rumus on-line yang terkenal untuk menghitung rata-rata bergerak tertimbang secara eksponensial dan standar deviasi dari suatu proses (xn)n=0,1,2, . Untuk rata-rata,

μn=(1α)μn1+αxn

dan untuk varians

σn2=(1α)σn12+α(xnμn1)(xnμn)

dari mana Anda dapat menghitung simpangan baku.

Apakah ada rumus yang sama untuk perhitungan on-line dari momen ketiga dan keempat terpusat eksponensial? Intuisi saya adalah mereka harus mengambil formulir

M3,n=(1α)M3,n1+αf(xn,μn,μn1,Sn,Sn1)

dan

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

dari mana Anda dapat menghitung kemiringan dan kurtosis k n = M 4 , n / σ 4 n tetapi saya tidak dapat menemukan ekspresi bentuk-tertutup yang sederhana untuk fungsi-fungsi tersebut f dan g .γn=M3,n/σn3kn=M4,n/σn4fg


Sunting: Beberapa informasi lebih lanjut. Rumus pembaruan untuk varian bergerak adalah kasus khusus dari rumus untuk kovarians bergerak tertimbang eksponensial, yang dapat dihitung melalui

Cn(x,y)=(1α)Cn1(x,y)+α(xnx¯n)(yny¯n1)

di mana dan ˉ y n adalah cara bergerak eksponensial dari x dan y . Asimetri antara x dan y adalah ilusi, dan menghilang ketika Anda memperhatikan bahwa 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
sumber

Jawaban:

6

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

YX=xnYZ=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/μ23/2 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
sumber
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)σn12]

Updating formula for the kurtosis still open...

Chris Taylor
sumber
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