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I would like to describe a hypothesis, formally. So I have defined $p = {t_1,t_2,t_3, \cdots, t_n}$

and I would like to say

"A pattern $p_i$ is more important than $p_j$, if $t \in p_i$ is more important than $t \in p_j$", I know that it is a bit vague to describe in this manner, so what will be a better option, the ideas I have in mind.

  • To define an importance function or weight function saying $f(p) \propto f(t) $
  • To describe purely in words.

I kind of want to avoid 2nd option as I want to find a way and learn how to formally define relationship between variables.

Thank you.

Regards, Andy

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This is vague indeed. Is $p$ an (unordered) set or an (ordered) n-tupel? –  Hauke Strasdat Dec 20 '11 at 9:08
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Assuming it is an n-tuple, do you mean: "A pattern $p$ is more important than $q$, if and only if $p_i$ is more important than $q_i$ for all $i=1,...,n$?" ($p_i$ is the $i$th element of $p$...) –  Hauke Strasdat Dec 20 '11 at 9:12
    
It is an unordered tuple, I found some papers that describe this if it is an ordered tuple, e.g. Wu et al. "Deploying approaches for pattern refinement in text mining" –  cherhan Dec 20 '11 at 12:37
    
Okay, in this case, do you mean: "A pattern $p$ is more important than $q$, if and only if for all $a\in p$ and all $b \in q$ it holds that $a$ is more important than $b$"? I am just trying to understand what you mean. I can give a more formal, complete answer, once I do ;-) –  Hauke Strasdat Dec 20 '11 at 13:29
    
Thanks Hauke, I think I know the problem. It should be $\sum_{t \in p}$, if the sum is greater, means it is more important, does that make sense to you now? –  cherhan Dec 20 '11 at 13:52
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1 Answer

up vote 1 down vote accepted

Let $P$ and $Q$ be unordered $n$-tuples of real numbers. More formally, $P\subset \mathbb{R}$ and $Q\subset \mathbb{R}$ with $|M|=|N|=n$.

Let $f$ be a function which assigns an importance weight to each element in $P$ and $Q$. Formally, $f$ is a function $f:P\cup Q\rightarrow \mathbb{R}^+$.

Definition $$P \text{ is more important than } Q :\Leftrightarrow \sum_{t\in P}f(t) \ge \sum_{u \in Q}f(u)$$

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Thanks a lot :) –  cherhan Dec 20 '11 at 14:26
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