Converting a circular outcome variable to a linear one

Question

I have a problem where I want to measure deviations from zero degrees. This outcome variable is a circular measure, since a deviation of -180 degrees is equivalent to 180 degrees.

However, I don't want to complicate my model (using a linear mixed effects model) by using circular statistics, so I was wondering if I can use the absolute deviation expressed as a percentage of 180?

So for instance, -180 and 180 degrees would both give 100% deviation, while 90 and -90 would both give 50% deviation. Is this a legitimate fix? Namely, would their be any caveats I need to be aware of by 'linear-izing' my circular outcome measure?

**Edit: To give more context into my problem. I am looking to predict peaks in a real-time signal. I make a prediction and see how far off I was to the closest peak (0 degrees corresponds to peaks, while 180 degrees corresponds to antipeaks). I'm interested in computing how 'accurate' my different prediction strategies are, so I was considering just looking at deviations from zero degrees. I'm not sure whether this is a completely circular problem in the first place, since an outcome measure of is bounded between 180 and -180 degrees.

Answer 1

I don't want to complicate my model (using a linear mixed effects model) by using circular statistics, so I was wondering if I can use the absolute deviation expressed as a percentage of 180?

...Is this a legitimate fix?

There is not sufficient information in order to tell whether this is legitimate or not.

The problem with circular systems is that they wrap around themselves. For instance it you make three quarter turns to the left then you end up one quarter turn to the right.

So a large random step/movement/change/deviation/effect (whatever you want to call it) might end up as being measured/observed as a small step and in the opposite direction. What you observe as a single quarter step might in reality be three quarter steps.

If you treat the circle as a linear variable then you will not be taking this into account and you will wrongly interpret the values.

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If the nature of your data is such that you do not get this effect of revolving/wrappingaround the circle. That is, if your changes are small enough that you will not see values, or only negligible few values, that make a deviation of more than a half circle, then you can use a linearized variable.

You speak about 'using the absolute value only' and you want to ignore the direction of change. It is unclear why you (need to) do this. Depending on the task and data that you have you might choose to do this. It is not wrong in principle and it does occur. However, in order to be able to say whether it is good for your case, the details about the case need to be know.

Answer 2

You cannot validly linearize a circular measure which spans 360°, assuming the circularity of that measure is valid.

Any transformation which "linearizes" a circular measure must necessarily privilege some value as being maximally linearly distant from some other value by virtue of lying on the other side of whatever point the transformation uses as its point of either "unwinding" or "flattening" the circle. This maximal linear distance will be a fiction created entirely as an artifact of the transformation, and will not exist in the original circular measure. The same holds true in a continuous circular measure of degrees, radians, etc. In fact, by carefully choosing the privileged point in your transformation function, you could probably fabricate any relationship you wanted between outcome and predictor, by ensuring that certain linearized values become either the largest, smallest, or middlemost.

Modular measurements—whether discrete or continuous—have important characteristics which have no representation in linear forms. This is why treating the modular numbers adorning the face of a clock make no sense as truly (linear) natural numbers. For a simple example, as we actually read the 12 hour clock, $1 - 12 = 1$, $3 - 9 = 9 - 3 = 6$, etc. But linearizing the clock's hours into integers would mean that $1 - 12 = -11$, and that $3 - 9 \ne 9 - 3$.

Answer 3

Since you are interested in deviation from 0 (and not the direction), it would be appropriate to use $|\theta|$ as your variable.

You've defined the problem in a way such that $-90$ and $+90$ (and similarly, $-2$ and $+2$) are the same outcome so one can take the absolute value and replace the circular problem with a linear scale going from $0$ to $180$.

Your solution is equivalent to what I describe but rescaling (dividing by $1.8$) to go from $0$ to $100$ instead.

Circular statistics are vital when we care about position on a circle and need to account for the ends of the line wrapping around. In your problem instead of connecting the ends of the line together we are folding the line in half (not just matching $-180$ to $180$, but matching every $-\theta$ to $\theta$).