# A method for extracting a condition to check whether a feature is related to an object

Let we have the object $$\bf S$$. This object has some properties such as length, temperature and other features. Assume that for the object $$\bf S$$ we selected $$n$$ features.For example the vector $${\bf v}=\{a_1,a_2\cdots ,a_n\}$$ is a list of features of $$\bf S$$ such that $$a_j \in \mathbb{R}$$ for $$1\leq j \leq n$$.

Suppose by performing $$m$$ experiments on the object $$\bf S$$ we get the following list of features: $${\bf v}^{(i)}=\{a_1^{(i)},a_2^{(i)}\cdots ,a_n^{(i)}\}, \quad 1\leq i \leq m. \tag{1}$$ such that in $$(1)$$ the list $${\bf L}_1=\{a_1^{(1)},a_1^{(2)}\cdots ,a_1^{(m)}\}$$ is obtained by performing $$m$$ experiments on the first feature of $$\bf S$$ such as the length of $$\bf S$$. Also the list $${\bf L}_2=\{a_2^{(1)},a_2^{(2)}\cdots ,a_2^{(m)}\}$$ is derived by performing $$m$$ experiments on the second feature of $$\bf S$$ such as the temperature of $$\bf S$$ and so on.

Assume we want to check that the vector $${\bf w}=\{b_1,b_2 \cdots ,b_n\}$$ is related to the object $$\bf S$$ which means $$b_1$$ is related to the length of $$\bf S$$, $$b_2$$ is related to the temperature of $$\bf S$$ and so on.

Question: Is there a method to get a condition based on the vectors $${\bf v}^{(i)}=\{a_1^{(i)},a_2^{(i)}\cdots ,a_n^{(i)}\}$$, $$1\leq i \leq m$$ such that by applying the condition we verify that whether $${\bf w}$$ is related to the object $$\bf S$$.

My try: I map the vectors $${\bf v}^{(i)}=\{a_1^{(i)},a_2^{(i)}\cdots ,a_n^{(i)}\}$$, $$1\leq i \leq m$$ into an interval $$\bf I$$. Then based on the condition $$\bf I$$, I verify claims as follows.

Let $$\mathrm{Var}_i$$ for $$1\leq i \leq m$$ be variances of $${\bf v}^{(i)}$$'s. Let $${\bf R}=\{\mathrm{Var}_1, \mathrm{Var}_2, \cdots \mathrm{Var}_m \}$$. Assume $$x=\min({\bf R})$$ and $$y=\max({\bf R})$$. Now suppose that $${\bf I}=(x,y)$$. Therfore, if $$\mathrm{Var}({\bf w}) \in {\bf I}$$, then we conclude that $$\bf w$$ is related to $$\bf S$$.

The disadvantage of this method is that a feature vector may be validated without belonging to $$\bf S$$.

This question is related to a practical problem and I would be grateful if you could suggest a solution

• If $w$ is not related to $S$, where would it have come from? It would be natural to estimate whether $w$ is more likely to have come from $S$ or from $T$, but the question provides no information about any alternative $T$. – Matt F. Oct 23 at 14:05