1 Dimensional Representation of a Gaussian Distribution

[dacruick]

Hi,

I currently have a Gaussian distribution (Normalized Frequency on the y-axis and a value we can just call x on the x-axis).

So for the sake of simplicity, let's say that I ignore any values below 0 and any values above 1 on the x-axis. Then what I will do is take 10 equal segments (0.1 units in length) of the remaining gaussian distribution. Then this is where I get confused. What I want to have is a 1 dimensional representation of this distribution. The y-axis will be represented by the distances between two x-values. So for example, if the slope between 2 points is a large positive number, the distance between the two points will be decreasing because the frequency is increasing. If the slope is 0, then the distance between the points will be constant, because there is a constant frequency.

Does this make any sense?

I feel like this idea can be easily done and the solution is dangling right in front of me but I just can't seem to get it.

Thank you in advance for any help.

dacruick

 

[Mute]

It sounds like basically what you want to do is plot the values y = f(x) = \exp(-x^2/2)/\sqrt{2\pi} on a number line. Pick 10 (or however many you want) equally spaced values of x, calculate f(x), and then plot the values of f(x) on a number line.

 

[dacruick]

well that's the thing, I don't want the y values on the number line. I want the x values on the number line. Those x-values cannot be equally spaced, I want the space between the x values to correspond to the y value.

I want the space between the x values to reflect the distribution (frequency). But this will have to be a relative distribution. That I believe is my issue. Since I want to be assigning 10 points in each interval, The spacing cannot be dependent on the height because near the average x value there will be higher frequencies. So it might have to be slope dependent??

 

[Mute]

Ok, in that case you pretty much do the same thing, just in reverse. Pick ten equally spaced points along the y axis, draw lines from points on the y-axis to the curve y = f(x), then straight down to the x-axis. The points on the x-axis are non-uniformly spaced (for general f(x)) and the distance between them tells you about how f(x) changes.

I've attached a sketch to show you what I mean.

To get exactly what you want, perhaps you want to use f'(x) instead of f(x), but the basic idea is the same. The main issue you might encounter is a function which would have a multivalued inverse (such as a gaussian). For that you'll have to choose a section that's well defined to do this for.

It seems like this still isn't quite what you're looking for, but hopefully it helps give you an idea of what you could do.

EDIT: Actually, what you might want to do this for is the integral of f(x). That way if your function has a region of constant slope, the integral will be linear there, and this method will give you evenly spaced points on the x-axis for that region.

 

[dacruick]

this is going to be harder than I thought haha. I don't really see a solution right now.