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I have an set of items, each with an age (measured in days). For each age I wish to calculate a level of importance (weight) that will be used in a following function.

The younger the age the larger the weight (more recent items are more influential and carry more weight).

I don't wish for the age/weight relationship to be linear. The weight should drop off more steeply as the age increases.

What formula would generate a weight value when passed in an age value?

How can I control the 'drop-off' rate of weight as the age increases (the curve of the graph)?

Simple language and explanations rather than lots of formula would be appreciated by this novice :-)

UPDATE: Using Tanner's assumptions to more specifically ask the question:

The weight drops off more steeply as the age increases, the weight will eventually hit zero. The age value that will produce a weight of zero will be called A.

When the age is zero I know what the weight will be; I call this weight W.

d is the initial drop-off rate: at the beginning, if the age increases by a small value (say 1), the weight will decrease proportionally by d.

I'm not allowed to post images yet but you can view a graph mockup of what I'm attempting at https://img.skitch.com/20120307-bxu8c6t2crubq6c59kprwgcdrc.png

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Does the weight degrease by the same amount each day? –  user16124 Mar 8 '12 at 2:29
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2 Answers

If I understand correctly, since the weight drops off more steeply as the age increases, the weight must eventually hit zero. I'm going to assume that you know which age value will produce a weight of zero; call this age value $A$.

Also, allow me to assume that when the age is zero, you know what you want the weight to be; call this weight $W$. Finally, let's say that $d$ is the initial drop-off rate: at the beginning, if the age increases by a small value (say $1$), the weight will decrease proportionally (say by $d$).

In fancy mathematical terms, we want $f(A) = 0$, $f(0) = W$, and $f'(0) = -d$. (The derivative, $f'$, is the function's rate of increase.) A quadratic equation will have the properties you're after. Using Wolfram Alpha to help us out, we come up with this formula for $f$:

$$f(x) = \frac{A d - W}{A^2} x^2 - d x + W.$$

We call on Wolfram Alpha again to test this formula out, and it seems to do the trick.

(If you like, I can go into a bit more detail about just how I came up with the formula.)

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Hi Tanner, it seems like you're on the right track and your assumptions are correct. I lack the expertise to know if the formula provided meets my needs. I've updated the question with more information that might help you determine this. Thanks! –  nutcracker Mar 7 '12 at 5:53
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There are many possibilities. One would be $weight=\exp(age/t_0)$, where you choose $t_0$ to be how fast things age. In time $t_0$ the weight drops by a factor $e\approx 2.718$

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