Finding a point on a curve given a series of points

Question

I have a series of values (x & y) which I can put into Excel and create a curve, as shown below.

Using this graph, I can see what the value of y might be for a given value of x (and vise-versa)

e.g.

• for an x value of 815 - y is approx. 750
• for a y value of 400 - x is approx 812.5

Is there a simple algorithm that will take a series of points and return a value for x or y given y or x respectively?

I've been reading lots on calculating various curves; but nothing seems to be working towards my goal, and it all gets very complex very quickly. The answer may be that I need to plow through that maths, but just incase there is a simple answer I'm asking here!

The accuracy isn't super important, so long as it's as good as me reading the excel graph...

Answer 1

You might want to look into regressions. Simply put, applying a regression to a set of x/y values enables you to get a function representing the relationship between x and y. With this function, it is easy to get y for any x.

There are many possible models for regression depending on the shape of your data. Given your data, it looks like a linear regression might be a good fit. In this case, you will get the straight line that best represents your data. Once you have this line as a equation y = ax +b, you can get from x to y or from y to x easily. a and b are computed automatically (lookup least square error to learn more about how).

Regressions and particularly linear regressions are very common statistical tools, so any language/program you use should have a tool for that.

Answer 2

A simple method can be through linear interpolation between points (which is what you do when you "eyeball" a solution when looking at the plot.

You can perform linear interpolaiton in a simple matter using the following method:

If you want to know a y value based on x, do the following:

1. take the two (x,y) points that are the closest points from the right or left to the wated point in the x axis (in the example you got, the point x=815 is in the [814,817] section so taking the points

   1.1. (x1,y1) = (814, 600)

   1.2. (x2,y2) = (817, 1050)

2. calculate the slope of the section using the following formula:

slope = (y2 - y1) / (x2 - x1)

In our case, this gives:

slope = (1050 - 600) / (817 - 814) = 150

3. comute the y value of the wanted x using the following:

y = (x-x1) * slope + y1

again, in our example this gives:

y = (815-814)*150 + 600 = 750

If you want to know the 'x' value, based on y, do the following:

1. take the two (x,y) points that are the closest points from the right or left to the wated point in the y axis (in the example you got, the point y=400 is in the [301,500] section so taking the points

   1.1. (x1,y1) = (812, 301)

   1.2. (x2,y2) = (813, 500)

2. compute the slope, using the same formula:

slope = (y2 - y1) / (x2 - x1)

In our case, this will be:

slope = (500-301) / (813-812) = 199

3. compute the wanted x value by:

x = x1 + (y - y1) / slope

and again, in our case it will be:

x = 812 + (400-301) / 199 = 812.4974874

Answer 3

In addition to Tomer Geva answer. Excel(and other similar programs) have builtin function for linear interpolation called FORECAST that works exactly as you want. And if you need to get x by y, just swap x range and y range.