Vrftsjtryk

I am doing a course on machine learning. In it I am asked to convert a matrix's columns to z-scores. It says:

Subtract the mean value of each feature from the dataset (columns of
X). After subtracting the mean, additionally scale (divide) the
feature values by their respective “standard deviations.”

This is not hard for me and I have written a little python function to this goal, but I was wondering what the vectorized version of this function would look like. So I set up some equations that I could then express in python, but I have done some unconventional things to achieve this and I would like to know what the conventional way would be.

Here is what I've got:

$$
z: R^{mtimes n}rightarrow R^{mtimes n}\
z( X) =left( X-1_{m} mu ( x)^{T}right) otimes 1_{m} sigma ( x)^{T}\
$$

Insecurities:

• The circle operator in this equation is the hadamard product. Does this even have an official operator? Should I describe what is meant with that operator? Or is there some way around it?


• I've added the function's mapping on top as is usual in certain programming languages like Haskell. I find this to be very handy in terms of doing linear algebra where you have to keep track of the dimensions of vectors and matrices. Is this common in expressing math as well?

Here are mu and sigma:

$$
mu :R^{mtimes n} rightarrow R^{n}\
mu ( X) =frac{1}{m} x^{T} 1_{m}\
$$

Same insecurities here.

$$
sigma : R^{mtimes n}rightarrow R^{n}\
sigma ( X) =frac{1}{m}sqrt{left(sumlimits ^{n}_{j=0}left( X-1_{m} mu ( x)^{T}right)^{2}_{ij}right)_{j}}\
$$

Insecurities:

• I have to square each cell of the resulting matrix and sum the results in to a vector. I am using a fictional i index. How should I describe what i is?


• I later have to square each element of the vector. Here I use j, but outside the summation. Again: how should I describe this?

Your insights are much appreciated as I would like to learn the right/conventional way of doing linear algebra and don't have access to a teacher. I am curious to know how would your solution would look like.

I find your question very interesting. Hopefully we can chat about it in greater detail.

Is it the case that you feel comfortable programming a solution to this problem, but you are not confident is writing a mathematical exposition of the solution?

Or is it that you want to refactor your code to make it more in line with "standard" mathematical steps?

Cheers