Creating a function based on data

[JFonseka]

Homework Statement

Now this isn't a homework question, it's something me and some others are looking into, and someone posted this function, and I'm not sure how he worked it out:

Creating a function for Elvis' album sales based on the RIAA certifications of his albums that have been certified so far

69 albms - over 500,000 sales certified
39 albums - over 1,000,000
19 albums - over 2,000,000
11 albums - over 3,000,000
4 albums - over 4,000,000 and 5,000,000
2 albums - over 6,000,000
1 albums - over 9,000,000

Let exclude the 4 albums over 4 million, which could be estimated on their own and disturb the linear repartition of figures.

I set X = 500000*x

Then Elvis statistics are almost perfectly equal to the function 1/(1,23*x + 0,278).

Let x = 1 if you want to find albums that sold over 500,000
Let x = 2 if you want to find albums that sold over 1 million
etc.

So what he's done is used the figures from the albums certified to create an equation. So therefore if you substitute 1 for x in the function, which is actually equal to 500,000 you will get all the albums that certified for over 500,000 sales, it's not exact, but it's a close enough function, how did he work that function out?

Homework Equations

None

The Attempt at a Solution

I have no idea!

 

[hotvette]

Fitting a function to a set of data points is an age old problem that is often as much art as it is science. Unless there is some theoretical basis that can be used to hint at the functional relationship, about the only option is intelligent guessing. Once you've zeroed in on a function, you can use least squares analysis to determine the unknown parameters if there are more data points than unknown parameters

In this case, it isn't hard to tell that there is some sort of an inverse relationship between y and x (as x gets bigger, y gets smaller). Thus a logical first guess might be y = a/x + b. After a few trial and error attempts, it isn't far fetched to eventually try y = 1/(ax + b). Also, scaling x by 500,000 beforehand makes sense because the values are so large.

 

[sean trom]

a function can be created using a calculator that presumably uses least square regression techniques.

from using your data (and excluding the 4 milion and 5 million data points) an exponential function of a = 60.4(0.78^s) where a = number of albums and s = sales*500000. This equation has a correlation co-efficient of -0.98 whereas using linear regression, only -0.79 is achieved.

this line of best fit suits most of your data points and the one that is the most far off is the first data point