Is there a signed (ie anti-symmetric) version of SMAPE?

Question

The symmetric mean absolute percent error (SMAPE) is a symmetrized version of percent error with the formula:

$$\frac{200\%}{n}\sum_i\frac{|x_i - y_i|}{|x_i| + |y_i|}$$

SMAPE is symmetric: interchanging the $x$'s and $y$'s doesn't change the value. However, there are applications where it would be nice if an interchange of $x$ and $y$ changed the sign of the difference measure without changing the magnitude (ie if the measure was anti-symmetric).

For example, say that the value of an asset falls from $1 \rightarrow .95$, then recovers $.95 \rightarrow 1$. If you take the SMAPE of the price change events you get $5.13\%$. Instead I'd like to find an approach that would give an an answer of $0\%$ in this case.

Edit:

5 minutes after I wrote this question I thought of a simple answer (posted below). It would still be nice to know if there's an anti-symmetric percent difference measure that has actually been used before in the literature.

You are asking for a multiple of the "Relative Percent Difference." See https://stats.stackexchange.com/questions/86708 for a generalization. — whuber, May 30 '19 at 19:45

tel · Answer 1 · 2019-05-30T19:25:51.880

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I suppose one obvious way of getting an anti-SMAPE would be to just use the plain, non-absolute, difference in the numerator:

$$\textrm{ASMAPE} = \frac{200\%}{n}\sum_i\frac{x_i - y_i}{|x_i| + |y_i|}$$

edited May 30 '19 at 19:25

answered May 30 '19 at 19:16

tel

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Since sign cancels absolute value, both can be omitted. – Tim May 30 '19 at 19:21
Whoops. Thanks for pointing that out – tel May 30 '19 at 19:23
You can also drop 200% since it doesn't make much sense anymore--it would be percentage of what? – Tim May 30 '19 at 19:45
@Tim the factor $200\% = 2\times 100\%$ doubles the ratio and converts that into a percent. It does make sense. – whuber May 30 '19 at 20:01
The factor of $2$ comes from the fact that putting it in the denominator yields the average of the predictions and the actuals: $$ \frac{|x_i|+|y_i|}{2}. $$ – Stephan Kolassa Jan 05 '21 at 17:27

Is there a signed (ie anti-symmetric) version of SMAPE?

Edit:

1 Answers1