![]() ![]() It is also necessarily linear in each variable separately, which can also be seen geometrically. permutations and combinations, the various ways in which objects from a set may be selected, generally without replacement, to form subsets. Such a function is necessarily alternating. ![]() v_n calculates the signed volume of the parallelpiped given by the vectors v_1. The determinant of a matrix with columns v_1. noun complete change in character or condition 'the permutations. From a geometric persepective, that is how alternating functions come into play. If you swap two vectors that reverse the orientation of the parellelpiped, so you should get the negative of the previous answer. In R^n it is useful to have a similar function that is the signed volume of the parallelpiped spanned by n vectors. Also, we may see that that correlation between actual features importances and calculated depends on the model’s score: higher the score lower the correlation (Figure 10 Spearman. There is no difference between importance calculated using SHAP of built-in gain. If you swap x and y you get the negative of your previous answer. Permutation importance suffers the most from highly correlated features. It cares about the direction of the line from x to y and gives you positive or negative based on that direction. It really gives you a bit more than length because is a signed notion of length. On the real line function of two variables (x,y) given by x-y gives you a notion of length. There is a geometric side, which gives some motivation for his answer, because it isn't clear offhand why multilinear alternating functions should be important. I think Paul's answer gets the algebraic nub of the issue. There is a close connection between the space of alternating $k$-linear functions and the $k$-order wedge product of a space, so I could have very similarly developed the determinant based on the wedge product, but alternating $k$-linear functions are easier conceptually. In particular that $\det(MN) = \det(M)\det(N)$. Certain properties of determinants that are difficult to prove from the Liebnitz formula are almost trivial from this definition. This is only one of many possible definitions of the determinant.Ī more "immediately meaningful" definition could be, for example, to define the determinant as the unique function on $\mathbb R^f \in A^n(V)$$Īll the properties of determinants, including the permutation formula can be developed from this.
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