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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

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  • $\begingroup$ 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 $T$. $\endgroup$ – Matt F. Oct 23 at 14:05

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