Thread Subject:

What does uncorrelatedness and independence imply?

Dear all,

I'm currently reading papers of statistical modelling. I encountered with the concepts of uncorrelatedness and independence. I understand the definitioins, but I am wondering what are the real effects they can make in statistical analysis?

For example, I have a dataset and I use certain technique (i.e. Principal Component Analysis) to separate the dataset into a set of uncorrelated vectors. On the other hand, I can separated the dataset into a set of independent vectors. What is the difference between these two sets of vectors? What property of uncorrelatedness and/or independence make such a difference?

Many thanks in advance. I do appreciate your kindly help.

Best wishes
Wenlong

"Wenlong" wrote in message <jrfoon$fq6$1@newscl01ah.mathworks.com>...
> Dear all,
>
> I'm currently reading papers of statistical modelling. I encountered with the concepts of uncorrelatedness and independence. I understand the definitioins, but I am wondering what are the real effects they can make in statistical analysis?
>
> For example, I have a dataset and I use certain technique (i.e. Principal Component Analysis) to separate the dataset into a set of uncorrelated vectors. On the other hand, I can separated the dataset into a set of independent vectors. What is the difference between these two sets of vectors? What property of uncorrelatedness and/or independence make such a difference?
>
> Many thanks in advance. I do appreciate your kindly help.
>
> Best wishes
> Wenlong

http://en.wikipedia.org/wiki/Uncorrelated
http://en.wikipedia.org/wiki/Correlation_and_dependence
http://en.wikipedia.org/wiki/Special:Search/intitle:correlation

Hopethis helps.

Greg

Hi, Greg

Thank you for your reply.

Actually, I am wondering why independent is loftier than uncorrelated? In other words, what is the advantage of independence than uncorrelatedness?

Thank you very much.

Wenlong

On 6/16/2012 10:49 AM, Wenlong wrote:
> Hi, Greg
>
> Thank you for your reply.
>
> Actually, I am wondering why independent is loftier than uncorrelated?
> In other words, what is the advantage of independence than
> uncorrelatedness?
>

Independence implies no correlation; the converse isn't necessarily so
in that correlation does not prove causation. (And, of course, there's
the pathological cases of many shapes in which the common measure of
linear correlation can be identically zero while there is an exact
relationship).

--

"Wenlong" wrote in message <jrfoon$fq6$1@newscl01ah.mathworks.com>...
> Dear all,
>
> I'm currently reading papers of statistical modelling. I encountered with the concepts of uncorrelatedness and independence. I understand the definitioins, but I am wondering what are the real effects they can make in statistical analysis?
>
> For example, I have a dataset and I use certain technique (i.e. Principal Component Analysis) to separate the dataset into a set of uncorrelated vectors.

Not true. What if you have d+1 vectors in a d-dimensional space?

>On the other hand, I can separated the dataset into a set of independent vectors.

Not true. How do you propose to do that?

>What is the difference between these two sets of vectors?
>What property of uncorrelatedness and/or independence make such a difference

Instead of thinking about a group of vectors, think about functions of
variables and the probability distributions of those variables.

1. Knowledge of x yields no statistical information about y
2. Knowledge of x yields some statistical information about y
3. The correlation coefficient of (x,y) is zero
4. The correlation coefficient of (x,y) is nonzero

1 ==> 3. However, 3 !==> 1.

Hope this helps.

Greg


vs

2. Knowledge of x gives