Suppose there is a dataset $X = [x_1, \cdots, x_n]$ where each element $x_i$ has a set of features (weights) $A_i = [a^1_i, a^2_i, \cdots , a^k_i]$. We then can have a multiple weighted average according to each feature like following: $\bar{x}^1=\frac{\sum_i a^1_ix_i}{\sum_i a^1_i}$ and $\bar{x}^2=\frac{\sum_i a^2_ix_i}{\sum_i a^2_i}$; $\cdots$ $\bar{x}^k=\frac{\sum_i a^k_ix_i}{\sum_i a^k_i}$
Is there a standard method of combining all the $k$ weighted average and what is the pros/cons of such approach? Some ways I can think of are
- [1] Averaging all the weighted average: $\bar{x} = \frac{1}{k}\sum_j\bar{x^j}$ (We can then add a secondary set of weights for each of the $k$ features to this sum.)
- [2] duplicate the set for $k$ times, taking the weighted sum of each set. taking the sum of weighted sum and divided by the sum of all weights of all sets.
I want to create an average metric that takes into account all the $k$ features but the issue is the features are not of the same magnitude or unit so it seems like approach [2] would suffer if one feature is an order of magnitude bigger than the rest. Is there any way to deal with this/combining feature beside normalizing them?