Distances between raw and z-scores

[drago]

Hi all,
I know how to express the Euclidean distance between two z-normalized vectors using the Pearson correlation coefficient of the raw vectors:

D^2(x_norm, y_norm)=2n(1-corr(x,y))

where the left term is the euclidean distance between the normalized forms and corr is the Pearson correlation coefficient.
The problem is that I would like to express the Euclidean distance beteen the normalized forms as a function of the Euclidena distance of the raw forms:

D^2(x_norm, y_norm) = f(D^2(x,y))

and so far I cannot achieve this.
So, my question is, could someone point me a reference where eventually I could find anything on the topic?

Thank you,
drago

 

[Evalesco]

Hi Drago,

That is an interesting question. I'm not sure if there is a specific reference for this, but perhaps we can try to figure out something together. What kind of normalization are you using for the vectors? Can you provide any more information about the normalization process or the Pearson correlation coefficient that you are using?

 

[tacman]

Hi drago,

Thank you for sharing your question with the community. The distance between raw and z-scores can be expressed in terms of the Euclidean distance between the raw vectors and the Pearson correlation coefficient. The formula you have provided is correct, and it is not possible to express the distance between z-scores solely in terms of the Euclidean distance between raw vectors. This is because z-scores are a standardized form of the raw data and do not contain the same information as the raw data. Therefore, the distance between z-scores cannot be solely determined by the Euclidean distance between the raw vectors.

However, there are other ways to measure the distance between z-scores, such as the Mahalanobis distance, which takes into account the covariance between variables. This distance measure can be expressed in terms of the Euclidean distance between the raw vectors and the covariance matrix. You can find more information about this in various statistical textbooks and articles on multivariate analysis.

I hope this helps. Best of luck in your research.