Question on a kind of metric

[JMF]

I don't suppose this is a particularly recreational problem, but
somebody might be interested in it. It actually comes from my work.

I'm calculating a kind of weighted average of a whole bunch of values.
In plain English, it's "The sum of various percentages of the values
divided by the sum total."

That is, w1v1 + w2v2 + ... + wnvn divided by v1 + v2 + ... vn where
the w's are percentages -- that is, values between 0 and 1.

Since it's "the sum of percentages of the values divided by the sum
total", it's clearly less than 1. Now here's the part I'm wondering about:

Suppose you divide them up into groups. That is, suppose, say, you have
30 values, and you compute this formula for the first ten values. And
for the second ten. And for the third ten.

What is the relationship of the results for the three groups to the
overall result?

Is the overall result the average of the three formulas? Or something else?

As far as I can figure, there's no particular relation. The only way to
get to the overall result is to do the whole thing.

(If you're curious about where this comes from, it's simple: this is a
kind of metric for electronic components. I'm wondering what happens
when you do the metric on its subcomponents -- in this example, I
postulated three such subcomponents -- and then try to combine those
three metrics to get the overall metric for the overall component.)

So, my first question is: does the overall result have any particular
relationship to the results you get when you divide the values up into
groups and do those groups on their own?

As I said, it seems to me like the answer is NO. But I might be wrong.

Second question: suppose you constrained the calculation to be less than
some amount -- for example, 0.6 -- for all groups. Then is the overall
figure guaranteed to be less than 0.6?

In my opinion, the answer to that second question is YES. But I might
be wrong there, too.

Anybody know for sure?

Thanks.

[Leon Aigret]

On Sat, 16 Mar 2013 16:34:33 +0100, JMF <jo...@favaro.net> wrote:

>I don't suppose this is a particularly recreational problem, but
>somebody might be interested in it. It actually comes from my work.
>
>I'm calculating a kind of weighted average of a whole bunch of values.
>In plain English, it's "The sum of various percentages of the values
>divided by the sum total."
>
>That is, w1v1 + w2v2 + ... + wnvn divided by v1 + v2 + ... vn where
>the w's are percentages -- that is, values between 0 and 1.
>
>Since it's "the sum of percentages of the values divided by the sum
>total", it's clearly less than 1. Now here's the part I'm wondering about:
>
>Suppose you divide them up into groups. That is, suppose, say, you have
>30 values, and you compute this formula for the first ten values. And
>for the second ten. And for the third ten.
>
>What is the relationship of the results for the three groups to the
>overall result?

Do you mean something like (w1v1 + ... + w30v30) / (v1 + ... v30) =

(w1v1 + ... + w10v10) / (v1 + ... + v10) *
* (v1 + ... + v10) / (v1 + ... + v30) +

+ (w11v11 + ... + w20v20) / (v11 + ... + v20) *
* (v11 + ... + v20) / (v1 + ... + v30) +

+ (w21v21 + ... + w30v30) / (v21 + ... + v30) *
* (v21 + ... + v30) / (v1 + ... + v30) ?

Leon

[JMF]

Leon,

First of all many thanks for your interest.

That's very interesting - I can see now what you mean - there is a
relationship indeed. Now, this is the specific relationship I was wondering

about:

Suppose you do the three calculations like you describe above, obtaining
three separate ratios, like this:


ratio1 = (w1v1 + ... + w10v10) / (v1 + ... + v10)

ratio2 = (w11v11 + ... + w20v20) / (v11 + ... + v20)

ratio3 = (w21v21 + ... + w30v30) / (v21 + ... + v30)

Now, suppose you wanted to work from this point onward ONLY with those
three ratios.

Is there any way to relate them to the "overall ratio"

ratioOverall = (v1 + ... + v30) / (v1 + ... + v30)

That is, is there ANY formula that directly relates those four values?

ratioOverall = SomeFunctionOf(ratio1, ratio2, ratio3)?

Your formula shows that they're related, but "needs" the sum of v1+ .. +v30.

That's a step forward, and it's a lot more than I knew before (!) but I
was wondering whether any more steps forward are possible, where you

don't have to calculate any of the overall sums but can get that overall
ratio only from the previously calculated three ratios.

As I said earlier, I thought nothing at all was possible. You showed me
that I can get partway there. My feeling is that it's probably not

possible to get all the way there, working only with those three ratios.
What do you think?

Thanks,

John

[JMF]

> Is there any way to relate them to the "overall ratio"
>
> ratioOverall = (v1 + ... + v30) / (v1 + ... + v30)

Well, sorry about that ... of course I meant in the numerator (w1v1 +
... + w30v30) ...

[Ray Koopman]

On Mar 16, 8:34 am, JMF <j...@favaro.net> wrote:
> I don't suppose this is a particularly recreational problem, but
> somebody might be interested in it. It actually comes from my work.
>
> I'm calculating a kind of weighted average of a whole bunch of values.
> In plain English, it's "The sum of various percentages of the values
> divided by the sum total."
>
> That is, w1v1 + w2v2 + ... + wnvn divided by v1 + v2 + ... vn where
> the w's are percentages -- that is, values between 0 and 1.
>
> Since it's "the sum of percentages of the values divided by the sum
> total", it's clearly less than 1. Now here's the part I'm wondering about:

You seem to have reversed the roles of "weights" and "values".

(w1*v1 + w2*v2 + ... + wn*vn) / (v1 + v2 + ... + vn)

is a weighted average of the w's, with the v's as weights.

If you want a weighted average of the v's, with the w's as weights
then you should divide by the sum of the w's, not the sum of the v's.

Which do you want?

[JMF]

Sorry for the imprecision. I think I probably simply used the wrong
terminology calling it a weighted average (because it kind of looked
like that to me).

I want that exact formula I presented. But you're right, it's a mistake
of mine to call it a weighted average when it really isn't.

Thanks,

John

[Leon Aigret]

On Sun, 17 Mar 2013 13:19:31 +0100, JMF <jo...@favaro.net> wrote:

>Suppose you do the three calculations like you describe above, obtaining
>three separate ratios, like this:
>
>
>ratio1 = (w1v1 + ... + w10v10) / (v1 + ... + v10)
>
>ratio2 = (w11v11 + ... + w20v20) / (v11 + ... + v20)
>
>ratio3 = (w21v21 + ... + w30v30) / (v21 + ... + v30)
>
>Now, suppose you wanted to work from this point onward ONLY with those
>three ratios.
>
>Is there any way to relate them to the "overall ratio"
>
>ratioOverall = (v1 + ... + v30) / (v1 + ... + v30)
>
>That is, is there ANY formula that directly relates those four values?
>
>ratioOverall = SomeFunctionOf(ratio1, ratio2, ratio3)?

With just the three ratios there simply is not enough information
left. If for example v1, ... , v10 were 10 times as big, ratio1-3
would all remain exactly the same, but the value of ratioOverall
should move closer to ratio1.

The minimum extra effort probably consists of linking each ratio with
the sum of all v's that were used for its computation (e.g.
vsum1 = v1 + ... + v10). This results in

ratioOverall = (ratio1 vsum1 + ... + ratio3 vsum3) / vsumOverall,

with vsumOverall = vsum1 + vsum2 + vsum3.

Leon

[JMF]

Thanks, Leon, I think you're exactly right. There simply isn't enough
information in the three ratios to deduce the fourth. In fact, as you
point out, any number of combinations of the underlying values could
produce that fourth ratio.

In case you're curious, this is related to the problem I mentioned
earlier: you have an electronic product to make, which consists of three
subcomponents. This is a kind of reliability metric that needs to be
calculated for the overall product. You give out the three subcomponents
to three different suppliers.

What instructions do you give to the suppliers regarding the "partial
metric" on their own component? You have to say something, because
otherwise they might produce values that make the overall metric not
stay within its required bounds.

The only way I see to do it is to constrain them to make their three
ratios no bigger than the required overall ratio. Then (I think) the
overall ratio is guaranteed to be within the required bounds. Example:
the overall ratio is required by the safety or reliability standard to
be no more than 0.6. Then you say to your three suppliers: "Your own
partial metric on your values cannot be more than 0.6."

The potential disadvantage of this is that you may unnecessarily
constrain them. Because (as Leon points out), one could theoretically
overload one of the components with high values, underload one of the
others with lower values, but when you put it all together the whole
thing still works. But that places the burden on you of KNOWING that you
can overload one and underload the other -- and that's not always so
easy to decide.

Anyway, many thanks for the analysis.

John