# Is epsilon error a standard known error or custom created by this paper?

I'm reading this computer vision paper, research paper link, about creating a model to estimate the real age and perceived age of the person in the image (or at least that's what I think it's about). The perceived age is decided by this method: each image is looked at by 10 independent individuals and they estimate the age of the person. The mean and standard deviation are taken from the age guesses of the 10 individuals.

The paper then goes on to use an epsilon error by this equation as part of the model evaluation for perceived age and state the following the evaluation employs fitting a normal distribution with the mean µ and standard deviation σ of the votes for each image. The paper also says that this epsilon error covers the aspect of the uncertainty of the ground truth age.

The results are then posted in a table showing different epsilon error values based on different models.

My questions are the following:

1. What is that equation and is it standardly used? It says

2. What is x in this equation? I thought it was some type of feature scaling of the image but that makes no sense because the results are all different in the table and completely out of context anyway. Is x supposed to be the predicted age?

What is that equation and is it standardly used?

I havn't seen exactly that before, so I guess it's not "standard", but it's not so unusual. The equation is "just" 1 - (normal distribution). If you think of the normal distribution as a measure of "how similar a sample is to the mean", then that equation turns the similarity (big is good) into a distance (small is good).

It looks related to "robust loss functions"; See below.

What is x in this equation? I thought it was some type of feature scaling of the image but that makes no sense because the results are all different in the table and completely out of context anyway. Is x supposed to be the predicted age?

Your second thought is right. $$x$$ is predicted age. They use that set up since they don't have a single correct answer for their data, but rather, many guesses/votes as what the correct age is.

From the paper:

The LAP challenge evaluation employs fitting a normal distribution with the mean μ and standard deviation σ of the votes for each image.

Say you have two images. For one, every guess you have for the age is the same. For another, you get a wide range of guesses for the age. The second image must be "harder" somehow. This equation gives you an error measure that tells you how the algorithm performed relative to the many people that guessed the true age.

### Robust losses

Check out this work

The "epsilon-error" equation from the paper you linked to will look sort-of like the bottom (orange) trace in Figure 1:

• Great, thanks! Much clearer now. Just 1 question about normal '1-(normal distribution)' part. Isn't (1/(std*sqrt(2*pie))) before the e part and if so what would justify removing it for this particular equation? – zipline86 Jun 17 '20 at 8:07
• Ah good question. That's the normalization term that makes sure the area under the curve is 1. This lowers the peak value of the bell curve when sigma is larger. That would mean that a perfect prediction ($x=$\mu\$) could have different errors due to sigma. Instead, the omit that normalization term, so that all perfect predictions have an error of zero. – bogovicj Jun 17 '20 at 11:27
• Great, thanks for the simple explanation! – zipline86 Jun 17 '20 at 11:48