# Calculating the trend of an angle when it crosses 360 -> 0

I have a variable that measures an angle describing the relative position of two objects (i.e., can range from 0-359), and I would like to quantify how this has changed through time.

For example, here we have the relative position of the two items changing by 1 degree per year:

year <- seq(1981, 2020)
angle <- c(seq(341, 359), seq(0, 20)) However, taking the slope here is meaningless because of the "crossover" that occurs in the year 2000. I have a number of different samples, and some have this issue and some don't. I don't know a priori which samples will have this issue, nor when the crossover occurs, so I can't just apply some sort of offset (i.e., add 360 to the last 20 years).

Is there an accepted way to calculate angular trends, accounting for the fact that 0 = 360?

Think of the angle $$y$$ at any time $$t$$ as the accumulation of small changes in the angle. Symbolically, when $$f(t)$$ is the rate of change of the angle at time $$t$$ and $$t_0$$ is the beginning of the observations,

$$y(t) = y(t_0) + \int_{t_0}^t f(t)\,\mathrm{d}t.$$

Your problem is that $$y(t)$$ has been recorded modulo $$360$$ degrees -- perhaps with some error $$\epsilon(t).$$ That is, you have observed only the values

$$y^{*}(t) = y(t) + \epsilon(t) \mod 360.$$

You can, however, reconstruct $$y(t) + \epsilon(t)$$ provided you have sufficiently frequent observations. For successive times $$t \lt s,$$ notice

$$y^{*}(s) - y^{*}(t) = y(s) - y(t) + \epsilon(s) - \epsilon(t) \mod 360 = \int_t^s f(t)\,\mathrm{d}t + \delta$$

where $$\delta$$ equals the contribution of the errors $$\epsilon(s)-\epsilon(t)$$ plus, perhaps, some integral multiple of $$360$$ whenever there has been an angular break between $$y^{*}(t)$$ and $$y^{*}(s).$$ Now, provided the size of the total error $$|\epsilon(s)-\epsilon(t)|$$ is less than $$180$$ degrees and provided the angle didn't go around more than once, we can figure out whether a break occurred: if $$|\epsilon(s)-\epsilon(t)| \gt 180,$$ add or subtract $$360$$ degrees from $$\delta$$ to place it into the interval from $$-180$$ to $$+180$$ degrees.

Although we can't observe these errors directly, if we are sampling frequently enough to make the increments $$y(t_i) - y(t_{i-1})$$ fairly small, we simply apply this adjustment to the observed differences. Thus,

Whenever $$|y^{*}(s)-y^{*}(t)| \gt 180,$$ add or subtract $$360$$ degrees from $$\delta$$ to place it into the interval from $$-180$$ to $$+180$$ degrees.

Equivalently, compute the differences modulo $$180$$ but express them in the range from $$-180$$ to $$+180$$ degrees rather than (as is conventional) the range from $$0$$ to $$360.$$

Let's call the adjusted value $$\delta^{*}(t,s),$$ so that

$$y^{*}(s) - y^{*}(t) = \int_t^s f(t)\,\mathrm{d}t + \delta(t,s)^{*}.$$

This is equality, not equality modulo $$360.$$ We may now remove the effect of recording the angles modulo $$360$$ by summing these adjusted differences. When the observations are made at times $$t_0 \lt t_1\lt \cdots \lt t_n,$$ we have

\begin{aligned} y^{*}(t_i) &= y^{*}(t_0) + \left[y^{*}(t_1) - y^{*}(t_0)\right] + \cdots + \left[y^{*}(t_i) - y^{*}(t_{i-1})\right] \\ &=y(t_0) + \int_{t_0}^{t_i} f(t)\,\mathrm{d}t + \delta(t_0,t_1)^{*} + \delta(t_1,t_2)^{*} + \cdots + \delta(t_{i-1},t_i)^{*} \\ &= y(t_i) + \left[\epsilon(t_i) - \epsilon(t_0)\right]. \end{aligned}

The problem with computation modulo $$360$$ is gone: you may now use any procedure you like to model the response $$y^{*}(t).$$

Here is an illustration with a fairly difficult dataset. The data were generated according to the model $$y(t) = 30t \mod 360$$ and observed annually from 1980 through 2020 with iid Normally distributed error of standard deviation $$60$$ degrees (a large amount). The trend is barely discernible in the raw data, but the angle adjustment algorithm has visibly aligned them. We may fit a least squares model to the adjusted data, for instance, producing this result: The expanded vertical scale for the raw data shows details of the fit and their deviations from it. Incidentally, in this example the estimate of the slope is $$28.0 \pm 0.74$$ degrees, not remarkably different from the true value of $$30$$ degrees (the p-value for this comparison is $$1.1\%$$).

I will end by remarking that when the standard deviation of the errors $$\epsilon(t)$$ is large (greater than $$180/2/\sqrt{2} \approx 64$$ degrees, roughly), sometimes the angular adjustment will be incorrect. This will show up in the model residuals as a sudden change by a value around 360 degrees. Thus, a routine analysis of the model residuals can detect such problems, enabling you to modify the adjusted values for a better fit. The details of this will depend on your model and the fitting procedure.

This R code created the figures. At "adjust the angles" it shows how the angle adjustment can be computed efficiently.

#
# Specify the data-generation process.
#
year <- 1980:2020 # Dates to use
beta <- 30        # Annual rate of change
sigma <- 60       # Error S.D.
#
# Generate the data.
#
set.seed(17)
angle <- (year * beta + rnorm(length(year), 0, sigma)) %% 360
X <- data.frame(year, angle)
#
#
X$total angle <- with(X, { d <- (diff(angle) + 180) %% 360 - 180 cumsum(c(angle, d)) }) # # Fit a model to the adjusted angles. # fit <- lm(total angle ~ year, X) # # Analyze the fit. # b <- coefficients(fit) y.hat <- predict(fit) #--Compute dates the fit must wrap around from 360 to 0: y.breaks <- seq(floor(min(y.hat) / 360)*360, max(y.hat), by=360) year.breaks <- (y.breaks - b) / b #--Make the plots: u <- ceiling(max(X$total angle)/360)
par(mfcol=c(1,2))

#--The fits:
plot(X$$year, X$$angle, pch=19, ylim=c(0, 360), yaxp=c(0, 360, 4),
col="gray", ylab="Angle (degrees)", xlab="Year",
main="Raw Data and Fit")
for (x in year.breaks)
abline(c(-x * b, b), col="Red", lwd=2)

plot(X$$year, X$$total angle, ylim=c(0,u*360),  yaxp=c(0, u*360, u),
xlab="Year", ylab="Total angle",
plot(X$$year, X$$angle, ylim=c(0,u*360),  yaxp=c(0, u*360, u),
plot(X$$year, X$$total angle, ylim=c(0,u*360),