# Computing distance and standarization of features

Intro: suppose we have $$n$$ observations with $$m$$ features, represented by a $$n\times m$$-matrix $$X$$, and two specific points $$x,y\in\mathbb{R}^m$$, and we are interested in distances between $$X$$ and $$x,y$$, respectively. Since the features may be of different orders of magnitude, we standardize every feature of $$X$$, then calculate respective z-scores for coordinates of $$x$$, $$y$$ in order to be scale-independent (or to measure separation with respect to standard deviations of each feature). Then we may apply some particular distance function (e.g. euclidean, Mahalanobis) to compute and compare $$d_X(X,x)$$ and $$d_X(X,y)$$.

(Here $$d_X$$ means that the distance is calculated after the standarization with respect to $$X$$.)

Say we face the opposite task, i.e. given two series of observations ($$n\times m$$-matrices) $$X$$, $$Y$$ and a specific point $$x\in\mathbb{R}^m$$, decide which one of $$X,Y$$ is closer to $$x$$, in the scale-independent sense sketched above. Two strategies seem to be valid:

1. standardize $$X$$, calculate z-score of $$x$$, then calculate distance $$d_X(X,x)$$; standardize separately $$Y$$, again calculate z-score of $$x$$, then calculate distance $$d_Y(Y,x)$$; in the end compare $$d_X(X,x)$$ and $$d_Y(Y,x)$$

2. first concatenate $$X$$ and $$Y$$ to a full $$2n\times m$$-matrix $$Z$$ of all observations, standardize $$Z$$, calculate z-score of $$x$$, compare $$d_Z(X,x)$$ and $$d_Z(Y,x)$$.

Question: which strategy is appropriate? My guess is that, assuming the reasoning from Intro, the first one seems to be more natural.