Interesuje mnie, czy „korelacja” trzech zmiennych jest czymś, a jeśli tak, co by to było?
Współczynnik korelacji momentu Pearsona
Teraz pytanie dotyczące 3 zmiennych: Is
byle co?
W R wydaje się to czymś możliwym do interpretacji:
> a <- rnorm(100); b <- rnorm(100); c <- rnorm(100)
> mean((a-mean(a)) * (b-mean(b)) * (c-mean(c))) / (sd(a) * sd(b) * sd(c))
[1] -0.3476942
Zwykle patrzymy na korelację między 2 zmiennymi przy ustalonej wartości trzeciej zmiennej. Czy ktoś może to wyjaśnić?
correlation
pearson-r
PascalVKooten
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Odpowiedzi:
To jest rzeczywiście coś. Aby się dowiedzieć, musimy sprawdzić, co wiemy o samej korelacji.
Macierz korelacji z wartościami wektora losowej zmiennejX=(X1,X2,…,Xp) jest wariancją kowariancji matrycy, albo po prostu „wariancji” znormalizowanej wersji X . Oznacza to, że każdy Xi jest zastępowany przez jego najnowszą, przeskalowaną wersję.
KowariancjaXi i Xj jest oczekiwanie od iloczynu ich skupionych wersjach. Oznacza to, że pisząc X′i=Xi−E[Xi] i X′j=Xj−E[Xj] , mamy
WariancjaX , którą napiszę Var(X) , nie jest pojedynczą liczbą. Jest to tablica wartości
Sposób myślenia o kowariancji zamierzonego uogólnienia polega na uznaniu jej za tensor . Oznacza to, że jest to cały zbiór wielkościvij , indeksowanych przez i oraz j od 1 do p , których wartości zmieniają się w szczególnie prosty przewidywalny sposób, gdy X ulega transformacji liniowej. W szczególności niech Y=(Y1,Y2,…,Yq) będzie kolejną zmienną losową o wartości wektorowej zdefiniowaną przez
The constantsaji (i and j are indexes--j is not a power) form a q×p array A=(aji) , j=1,…,p and i=1,…,q . The linearity of expectation implies
In matrix notation,
All the components ofVar(X) actually are univariate variances, due to the Polarization Identity
This tells us that if you understand variances of univariate random variables, you already understand covariances of bivariate variables: they are "just" linear combinations of variances.
The expression in the question is perfectly analogous: the variablesXi have been standardized as in (1) . We can understand what it represents by considering what it means for any variable, standardized or not. We would replaced each Xi by its centered version, as in (2) , and form quantities having three indexes,
These are the central (multivariate) moments of degree3 . As in (4) , they form a tensor: when Y=AX , then
The indexes in this triple sum range over all combinations of integers from1 through p .
The analog of the Polarization Identity is
On the right hand side,μ3 refers to the (univariate) central third moment: the expected value of the cube of the centered variable. When the variables are standardized, this moment is usually called the skewness. Accordingly, we may think of μ3(X) as being the multivariate skewness of X . It is a tensor of rank three (that is, with three indices) whose values are linear combinations of the skewnesses of various sums and differences of the Xi . If we were to seek interpretations, then, we would think of these components as measuring in p dimensions whatever the skewness is measuring in one dimension. In many cases,
The first moments measure the location of a distribution;
The second moments (the variance-covariance matrix) measure its spread;
The standardized second moments (the correlations) indicate how the spread varies inp -dimensional space; and
The standardized third and fourth moments are taken to measure the shape of a distribution relative to its spread.
To elaborate on what a multidimensional "shape" might mean, observed that we can understand PCA as a mechanism to reduce any multivariate distribution to a standard version located at the origin and equal spreads in all directions. After PCA is performed, then,μ3 would provide the simplest indicators of the multidimensional shape of the distribution. These ideas apply equally well to data as to random variables, because data can always be analyzed in terms of their empirical distribution.
Reference
Alan Stuart & J. Keith Ord, Kendall's Advanced Theory of Statistics Fifth Edition, Volume 1: Distribution Theory; Chapter 3, Moments and Cumulants. Oxford University Press (1987).
Appendix: Proof of the Polarization Identity
Letx1,…,xn be algebraic variables. There are 2n ways to add and subtract all n of them. When we raise each of these sums-and-differences to the nth power, pick a suitable sign for each of those results, and add them up, we will get a multiple of x1x2⋯xn .
More formally, letS={1,−1}n be the set of all n -tuples of ±1 , so that any element s∈S is a vector s=(s1,s2,…,sn) whose coefficients are all ±1 . The claim is
Indeed, the Multinomial Theorem states that the coefficient of the monomialxi11xi22⋯xinn (where the ij are nonnegative integers summing to n ) in the expansion of any term on the right hand side is
In the sum(1) , the coefficients involving xi11 appear in pairs where one of each pair involves the case s1=1 , with coefficient proportional to s1 times si11 , equal to 1 , and the other of each pair involves the case s1=−1 , with coefficient proportional to −1 times (−1)i1 , equal to (−1)i1+1 . They cancel in the sum whenever i1+1 is odd. The same argument applies to i2,…,in . Consequently, the only monomials that occur with nonzero coefficients must have odd powers of all the xi . The only such monomial is x1x2⋯xn . It appears with coefficient (n1,1,…,1)=n! in all 2n terms of the sum. Consequently its coefficient is 2nn! , QED.
We need take only half of each pair associated withx1 : that is, we can restrict the right hand side of (1) to the terms with s1=1 and halve the coefficient on the left hand side to 2n−1n! . That gives precisely the two versions of the Polarization Identity quoted in this answer for the cases n=2 and n=3 : 22−12!=4 and 23−13!=24 .
Of course the Polarization Identity for algebraic variables immediately implies it for random variables: let eachxi be a random variable Xi . Take expectations of both sides. The result follows by linearity of expectation.
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Hmmm. If we run...
it does seem to center on 0 (I haven't done a real simulation), but as @ttnphns alludes, running this (all variables the same)
also seems to center on 0, which certainly makes me wonder what use this could be.
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sd
or variance is a function of squaring, as is covariance. But with 3 variables, cubing occurs in the numerator while denominator remains based on originally squared termsIf You need to calculate "correlation" between three or more variables, you could not use Pearson, as in this case it will be different for different order of variables have a look here. If you are interesting in linear dependency, or how well they are fitted by 3D line, you may use PCA, obtain explained variance for first PC, permute your data and find probability, that this value may be to to random reasons. I've discuss something similar here (see Technical details below).
Matlab code
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