Here, we want to predict a linear dependent variable from circular independent variables. There are several ways to approach this. The main thing to check is whether the relation between your dependent variable (let's say $Y$) and the circular predictor (say $\theta$) has a sinusoidal shape. This is often the case, but not necessarily. Below is an example of data of this shape.
th <- rnorm(100, 1, 4) %% (2*pi)
err <- rnorm(100, mean = 0, sd = 0.8)
icp <- 10
bc <- 2
bs <- 3
y <- icp + bc * cos(th) + bs * sin(th) + err
plot(th, y)
If the data does have this shape, roughly, a good simple model for the data is then given by splitting the circular predictor $\theta$ up in a sine and a cosine component, and running a regular linear regression on these two components, in this case by:
lm(y ~ cos(th) + sin(th))
>Call:
>lm(formula = y ~ cos(th) + sin(th))
>
>Coefficients:
>(Intercept) cos(th) sin(th)
> 10.12 2.04 2.95
Of course, this can be done for multiple predictors as well. A good introduction on this may be found in Pewsey, Neuhauser & Ruxton (2013), Circular Statistics in R.
As mentioned before, we may add terms as in a Fourier regression, but this can only be recommended if the relationship structurally exhibits very different forms, because higher-order Fourier regression introduces, IIRC, a large number of difficult to interpret parameters.
See the documentation:
help(lm.circular)
"If type=="c-l" or lm.circular.cl is called directly, this function
implements the homoscedastic version of the maximum likelihood
regression model proposed by Fisher and Lee (1992). The model assumes
that a circular response variable theta has a von Mises distribution
with concentration parameter kappa, and mean direction related to a
vector of linear predictor variables according to the relationship: mu
+ 2*atan(beta'*x), where mu and beta are unknown parameters, beta being a vector of regression coefficients. The function uses Green's
(1984) iteratively reweighted least squares algorithm to perform the
maximum likelihood estimation of kappa, mu, and beta. Standard errors
of the estimates of kappa, mu, and beta are estimated via large-sample
asymptotic variances using the information matrix. An estimated
circular standard error of the estimate of mu is then obtained
according to Fisher and Lewis (1983, Example 1)."
Thus you should compare with a different model
> nls(y~a+2*atan(b*x),start=c(a=0.06337,b=0.022344),data=list(x=x,y=y))
Nonlinear regression model
model: y ~ a + 2 * atan(b * x)
data: list(x = x, y = y)
a b
0.07112 0.02231
residual sum-of-squares: 12.36
Number of iterations to convergence: 12
Achieved convergence tolerance: 5.838e-06
this 'nls' function does not use the same underlying distributions for the residual terms but does provide similar coefficients.
Clearly you made your posted problem very simplified in order to make it easier to be understood.
Could you add your real case? (to spice up the question)
Best Answer
You may use the circular package. It's probably best to convert your data to radians by multiplying by $\pi/180$, or using
xcirc <- rad(x).The type, unit, zero, modulo flags are mostly there for plotting and storing information about a dataset. For your analysis, they don't matter.
Here is some example analysis, adapted from
example(lm.circular)