Help interpreting this equation

Hi,

This is my first post here and I hope that I am in the correct subforum (Tongueout).

So, I have this equation:$\displaystyle r_{Gm}(y)=\left \{ Y_{j}|D_{Gm}(Y_{j})\leq D_{Gm}(y),j=1,...,m \right \}/m$

Also there is a # before the curly bracket, I don't know why the equation writer won't allow me to put it.

Now to the question. Disregard what the variables are, I just want to know how this equation works. How can I interpret it? If I have the variables how do I get rGm?

Thank you for your help.

Re: Help interpreting this equation

Here's what I see in that.

For a given $\displaystyle y$ $\displaystyle r_{Gm}(y)$ is the cardinality of the set $\displaystyle \left\{ Y_j \vert D_{Gm}(Y_j)\leq D_{Gm}(y), \,\,j=1,\ldots , m \right\}$ divided by $\displaystyle m$.

Since you have m values $\displaystyle \left\{Y_1,Y_2,\ldots , Y_m \right\}$ you count how many of them are such that for a given y $\displaystyle D_{Gm}(Y_i)\leq D_{Gm}(y)$ holds. Obviously that number is somewhere between 0 and m. Take that number and divide it by m and you get the $\displaystyle r_{Gm}(y)$ which then obviously is always a number between 0 and 1.

$\displaystyle r_{Gm}(y)$ can be thought of as a percentage provided you multiply it by 100.

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Re: Help interpreting this equation

Thank you Mathoman.

I understand in principle want you have explained me but with the data I have I'm not getting the right values. (Or maybe I did not understand as well as I think.) Maybe you can use an example?( I'm attaching a table where the column r(X) should be the result of the equation)

Attachment 23617

Re: Help interpreting this equation

Ok I'll make up an example, trivial one.

Let $\displaystyle {\cal Y}=\left\{-10,1,2,10,30,20\right\}$. Hence $\displaystyle m=6$

Let $\displaystyle D_{Gm}(x)=x^2-2$.

For a given $\displaystyle y$, say $\displaystyle y=1$, $\displaystyle D_{Gm}(y)=D_{Gm}(1)=1^2-2=-1$.

Then $\displaystyle r_{Gm}(1)=\frac{\left\vert \left\{Y_j \vert Y_j^2-2 \leq -1,\,\, j=1,\ldots, 6 \right\} \right\vert}{6}$.

Evaluate $\displaystyle D_{Gm}$ for every element in $\displaystyle {\cal Y}$ - apply $\displaystyle D_{Gm}$ elementwise to $\displaystyle {\cal Y}$ - $\displaystyle D_{Gm}({\cal Y})=\left\{98,-1,2,98,898,398 \right\}$.

Now count how many values are $\displaystyle \leq -1$: 1.

$\displaystyle r_{Gm}(1)=\frac{\left\vert \left\{Y_j \vert Y_j^2-2 \leq -1,\,\, j=1,\ldots, 6 \right\} \right\vert}{6}=\frac{1}{6}$.

If y=11, then $\displaystyle D_{Gm}(y)=D_{Gm}(11)=11^2-2=119$.

In $\displaystyle { \cal Y}$ there are 4 values for which function $\displaystyle D_{Gm}$ evaluates to a value $\displaystyle \leq 119$ so:

$\displaystyle r_{Gm}=\frac{4}{6}=\frac{2}{3}$.

That's my way of looking at it, hope it helps.

Re: Help interpreting this equation

Thank you for the example. So I will try to follow your example with mine(table in my previous post):

The values for $\displaystyle D_{Gm}$ are on the second column D(X), and we have a set of 80 values so m=80.

If I calculate the $\displaystyle r_{Gm}$ of the first value (0.0028) according to your previous example I should check which values are equal or less than 0.0028. Checking the table I have (if I counted correctly) 36 values. So doing $\displaystyle r_{GM}=\frac{36}{80}$ we get 0.45 and not the 0.082 which is the given result.

Maybe you can see where I'm making the mistake?

Re: Help interpreting this equation

My guess is that you are counting the wrong values.

The values of $\displaystyle \left\{\ Y_1, Y_2,\ldots,Y_m \right\} $ are some other values not given in that table. That table contains values X, D(X), r(X). Unless I'm mistaking, the first row with X=1, D(X)=0.0028 and r(X)=0.082 should be interpreted as follows:

For $\displaystyle X=1$, based on the definition of the function $\displaystyle D$ (which is not explicitly given), $\displaystyle D(X)=D(1)=0.0028$.

Then $\displaystyle r(X)=r(1)=\frac{\left\vert \left\{ Y_j \vert D(Y_j)\leq 0.0028, j=1,\ldots,m \right\} \right\vert}{m}=0.082$.

Table doesn't tell you how many Y's there are, nor what are their respective values, nor what are the values of D(Y). You just get the final result.

Re: Help interpreting this equation

The values on the column X are 80 random values of a distribution(these values are not written on the table, just the number from 1 to 80 in order to represent them). With each of this values and another function I get the values in the column D(X). For this case in the equation we can regard X=Y, so the equation of the beginning can be written as $\displaystyle r_{Gm}(x)=\left \{ X_{j}|D_{Gm}(X_{j})\leq D_{Gm}(x),j=1,...,m \right \}/m$

Quote:

Originally Posted by

**MathoMan** For $\displaystyle X=1$, based on the definition of the function $\displaystyle D$ (which is not explicitly given), $\displaystyle D(X)=D(1)=0.0028$.

Then $\displaystyle r(X)=r(1)=\frac{\left\vert \left\{ Y_j \vert D(Y_j)\leq 0.0028, j=1,\ldots,m \right\} \right\vert}{m}=0.082$.

I agree with your interpretation, but then which values are this $\displaystyle Y_j \vert D(Y_j)$ ? The random values?

Re: Help interpreting this equation

Maybe the values are repeated, but only counted once. Really don't know anything else that could help. Maybe someone else has a better idea. Its 2am here, and I'm off to bed. Nighty night! :)

Re: Help interpreting this equation

I'm not sure about this, but I think you have two distinct sets of numbers. One of them is the one represented by your first column. I'm not sure what the other set is, but it's probably the one you're asking about ( the $\displaystyle Y_j \vert D(Y_j)$). Now, from your interpretation, you may have tried using the same set of numbers for both, which is clearly not the case, because at least one of the values on the third column should be 1 (i.e. the $\displaystyle X_i$ that generates the biggest $\displaystyle D(X_i)$ should give $\displaystyle r(X_i)=1$.

Another thing to note is that most likely $\displaystyle m \neq 80$, because for most values on the 3rd column, when multiplied by 80, does not give you an integer (which it should be because that represents the cardinality of a set).

So, here's something to look into, you said that the X's you got are from a distribution. I'm guessing that the distribution has more than 80 values, and that the number of values it has should be your m, and that those values are the one's that represent the $\displaystyle Y_j \vert D(Y_j)$ that you're asking about.

Hope that's it :D