h***t
发帖数: 2540 | 1 Let's say you have 3 random variables X, Y, Z with corr(X,Y)=0.70, corr(X,Z)
=0.80. What is the minimum value for corr(Y,Z)？
Thanks | z*******g
发帖数: 18 | 2 method 1. correlation between two variable can be expressed as the angle
between them, so corr(X,Y) = cos(x1)=0.7, corr(x,Z)=cos(x2)=0.8,
so the minimum value of corr(Y,Z) = cos(x1+x2);
method 2. using the property that the covariance matrix is positive defined.
Z)
【在 h***t 的大作中提到】
: Let's say you have 3 random variables X, Y, Z with corr(X,Y)=0.70, corr(X,Z)
: =0.80. What is the minimum value for corr(Y,Z)？
: Thanks
| l******n
发帖数: 9344 | 3 cos(x1+x2) is the max, not min.
no minimum, -1 is the lower bound and can't be achieved
defined.
【在 z*******g 的大作中提到】
: method 1. correlation between two variable can be expressed as the angle
: between them, so corr(X,Y) = cos(x1)=0.7, corr(x,Z)=cos(x2)=0.8,
: so the minimum value of corr(Y,Z) = cos(x1+x2);
: method 2. using the property that the covariance matrix is positive defined.
:
: Z)
| h***t
发帖数: 2540 | 4 Indeed I also suspect -1 is the lower bound, but can you prove -1 is the
strict lower bound?
Thanks
【在 l******n 的大作中提到】
: cos(x1+x2) is the max, not min.
: no minimum, -1 is the lower bound and can't be achieved
:
: defined.
| k*****y
发帖数: 744 | 5 Assume var(X) = var(Y) = var(Z) = 1.
Write
Y = 0.7*X + sqrt(0.51)*y
Z = 0.8*X + 0.6*z
where cov(X,y) = cov(X,z) = 0, and var(y) = var(z) = 1.
Then cov(Y,Z) = 0.56 + sqrt(0.51)*0.6 cov(y, z). Therefore
max = 0.56 + sqrt(0.51)*0.6
min = 0.56 - sqrt(0.51)*0.6
Z)
【在 h***t 的大作中提到】
: Let's say you have 3 random variables X, Y, Z with corr(X,Y)=0.70, corr(X,Z)
: =0.80. What is the minimum value for corr(Y,Z)？
: Thanks
| h***t
发帖数: 2540 | 6 this is incorrect, the min value definitely can be negative.
【在 k*****y 的大作中提到】
: Assume var(X) = var(Y) = var(Z) = 1.
: Write
: Y = 0.7*X + sqrt(0.51)*y
: Z = 0.8*X + 0.6*z
: where cov(X,y) = cov(X,z) = 0, and var(y) = var(z) = 1.
: Then cov(Y,Z) = 0.56 + sqrt(0.51)*0.6 cov(y, z). Therefore
: max = 0.56 + sqrt(0.51)*0.6
: min = 0.56 - sqrt(0.51)*0.6
:
: Z)
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