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$x_i$ takes values of angles, so it is a circular variable. Is there some method to perform cubic splines or some interpolation on the values of $y_i$, taking into account the circularity of $x_i$? …

(Modeled in ggplot using a gam specifying a "cc" circular cubic smoother). … I would like to model the (nonlinear) change in each predictor as a function of time using something like a two-dimensional spline. …

Here is a Generalized Additive Model (GAM) fit using a circular spline with three knots (that is, comparable in complexity to a sine wave). … I used the R package mgcv to create this fit, employing its built-in circular splines (type "cc"). …

As my other answer is already long, I'll use the actual data linked in a comment to illustrate circular splines for handling hourly data and further analysis of the time by treatment interaction. … Several packages help here: library(tidyr) # formatting data library(nnet) # multinomial fit library(pbs) # for circular spline library(aod) # for Wald test library(emmeans) library(ggplot2) Data …

.; the only difference is in the way the model is parameterized sinusoidal variation (as in the linked question): 2 parameters, 3 var-cov parameters circular splines - adjustable complexity However, … If something more complex is going on (which you might detect when looking at residual plots), you could use a quadratic model or a regression spline. …

be to use the built-in spline basis functions with periodic boundary conditions ! … bs="cp" gives a cyclic version of a P-spline …

A worked example can be found here: circular periodic time series or fit with periodic splines …

The model, fit as a continuous function of time with a circular spline, will be evaluated at each hourly value. That's how the data were collected. …

There is nothing per se wrong or flawed about representing wind direction, a circular variable, via a cyclic cubic regression spline. … If you have data covering the interval 20-270 degrees, the spline will be constrained to make 20 == 270, thus introducing a bias. …

After I explained them about model building, model checking, spline bases, circular splines, penalty versus no penalty, confounding and a bit more, they understood that even a 2-day course by a trained …

Here is a piecewise constant spline with two knots. It is determined by its three levels or, equivalently, by a "baseline" level and two jumps. … For instance, I recently had to develop circular quadratic splines for a regression that involved an angular independent variable (an orientation in the plane modulo $180$ degrees). …

One method that might be effective is to create a smoothing spline model with a radial kernel that takes the coordinates of the pixel relative to the center and the thicknesses of the films as arguments … I'm assuming the images are essentially circular around some origin that you can calculate image-wise. …

(If the locations go fully around the circumference of the circle you could use circular statistics to deal with the wrap-around problem at an angle of 0/360 degrees.) …

You should model that predictor flexibly, for example as a cubic spline, to capture any non-linearity in the relationship. … You can quickly get into circular reasoning, so make sure that your modeling doesn't fall into that trap. …

So using stop time in the regression is essentially using the outcome to help predict the outcome, which is circular reasoning. …