Consider the scenario in which I am measuring certain $f(a,x)$, which i want to be the target value for some related input $g(a,x)$. In other words, I am trying to map

$$g(a,x)\Rightarrow f(a,x)$$

I know that

$$f(a,x) = a x^{-3/4}$$

With $x \in [1,20]$ and $a=[1,2,3]$. I now perform "measurements" for $x=[1,2,...,20]$ and $a=[1,2,3]$.The histogram of my target values looks awful:

enter image description here

And that means that the neural network will favor smaller output values. I believe there are two solutions to make the distribution more "even" so that the neural network treats each target value equally:

1. Transform the output variable

This is troublesome, because for example the mapping $f'(a,x)=f(a,x)^{-3/4}$ still leads to a skewed distribution because of the varying constant $a$:

enter image description here

2. Perform measurements for certain $x_i$ such that the output is more evenly distributed. Constraint: I must always measure for each $a$.

I still want to sample the whole range, so I could do


$$x_i'= x_i^{-4/3}$$

But this only gives "plateaus" of uniformity, one for each unique $a$. You can see this by increasing the number of points to sample:

enter image description here

So my question: what is the best way to generate/transform the data for my $f(x)$ with $a=[1,2,3]$ so that the target values are distributed in such a way that it does not favour certain values?

PS: $f(a,x)$ is something that I have to measure in the real world, I cannot do more than 20 measurements.

  • $\begingroup$ Can you explain how we should read those diagrams? What is on the x and y axes? This would improve the readability of your post. Also, can you explain why you want the distribution of your output/target values to be more uniform? Are you trying to build a balanced dataset: is this your purpose? $\endgroup$ – nbro Nov 5 '20 at 11:35

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