Background

I want to compare some predicted data ($x_i$) to experimental data ($y_i$), shown below. Due to underlying symmetries of my scenario, a pair $(x_i,y_i)$ is equivalent to $(-x_i,-y_i)$ and I chose to flip both signs of all points with $x_i<0$ to make all the $x_i$ non-negative (this should have little bearing on my question)¹.

example data

What I want

Now, I want to evaluate the overall quality of my predictions, where the following make for a good prediction:

  • Small absolute predictions correspond to small absolute experimental values (the sign is not that important)
  • Big absolute predictions correspond to big absolute experimental values of the same sign (the absolute value does not need to match well).

Roughly speaking, I consider points in the red-shaded area above to be good.

I am looking for a statistics that quantifies this and is ideally intuitively understandable by a general scientific audience. I would use this statistics to communicate the magnitude of the effect and also to compare my data to an appropriate permutation null model.

What I have so far

My best choice so far is a correlation coefficient with the centre/mean being anchored on zero, i.e.:

$$ \hat{r}(x,y) = \frac{\sum\limits_i x_i y_i}{||x||·||y||} = \frac{\sum\limits_i x_i y_i}{\sqrt{\sum\limits_i x_i^2} \sqrt{\sum\limits_i y_i^2}} .$$

Without the anchoring on zero, I would also positively evaluate a case like the above, but shifted down by 0.2, which is obviously not good predictions.

To arrive at this, I substitute $\bar{x}=0$ and $\bar{y}=0$ in: $$ r(x,y) = \frac{\sum\limits_i (x_i-\bar{x}) (y_i-\bar{y})}{\sqrt{\sum\limits_i (x_i-\bar{x})^2} \sqrt{\sum\limits_i (y_i-\bar{y})^2}} .$$

Questions

  • Is there a name or literature on $\hat{r}$ as defined above?
  • Are there any alternative, established statistics fulfilling my requirements?

Lengthy footnote

¹ In case you want to know, the predicted observable quantifies a trend in yield in a microbial microcosm in dependence on the inoculum ratio within a pair of two microbial species. A positive sign indicates a trend in one direction; a negative sign in the other. Any data point $(-x_i,-y_i)$ becomes $(x_i,y_i)$ if I flip the order of the species within the pair. As this order is completely arbitrary, I may (and should) choose it such that the plot becomes compact and doesn’t contain a random meaningless distinction between data points (namely the sign of $x_i$). I might as well have chosen to keep the sign for $y_i$ non-negative. Since I need to flip both signs simultaneously, I can only choose one, however.

As a simple analogy, consider evaluating predictions of different football match outcomes by goal difference (ignoring any home-team advantage). A prediction–result pair (5:3,4:1) is equivalent to (3:5,1:4). To avoid “noise” for your analysis from the arbitrary order of teams, you should reorder the participants in each match in a way that collapses this symmetry. For example, you could reorder them such that the first team is never predicted to lose.