compute_elastic_mean function

Compute a elastic mean for a collection of curves

Computes a Fréchet mean for the curves stored in data_curves) with respect to the elastic distance. Constructor function for class elastic_mean.

compute_elastic_mean(
  data_curves,
  knots = seq(0, 1, len = 5),
  type = c("smooth", "polygon"),
  closed = FALSE,
  eps = 0.01,
  pen_factor = 100,
  max_iter = 50
)

Arguments

• data_curves: list of data.frames with observed points in each row. Each variable is one coordinate direction. If there is a variable t, it is treated as the time parametrization, not as an additional coordinate.
• knots: set of knots for the mean spline curve
• type: if "smooth" linear srv-splines are used which results in a differentiable mean curve if "polygon" the mean will be piecewise linear.
• closed: TRUE if the curves should be treated as closed.
• eps: the algorithm stops if L2 norm of coefficients changes less
• pen_factor: penalty factor forcing the mean to be closed
• max_iter: maximal number of iterations

Returns

an object of class elastic_mean, which is a list

with entries - type: "smooth" if mean was modeled using linear srv-splines or "polygon" if constant srv-splines are used

• coefs: spline coeffiecients

• knots: spline knots

• data_curves: list of data.frames with observed points in each row. First variable t gives the initial parametrization, second variable t_optim

  the optimal parametrization when the curve is aligned to the mean.

• closed: TRUE if the mean is supposed to be a closed curve.

Examples

curve <- function(t){
  rbind(t*cos(13*t), t*sin(13*t))
}
set.seed(18)
data_curves <- lapply(1:4, function(i){
  m <- sample(10:15, 1)
  delta <- abs(rnorm(m, mean = 1, sd = 0.05))
  t <- cumsum(delta)/sum(delta)
  data.frame(t(curve(t)) + 0.07*t*matrix(cumsum(rnorm(2*length(delta))),
             ncol = 2))
})

#compute elastic means
knots <- seq(0,1, length = 11)
smooth_elastic_mean <- compute_elastic_mean(data_curves, knots = knots)
plot(smooth_elastic_mean)

knots <- seq(0,1, length = 15)
polygon_elastic_mean <- compute_elastic_mean(data_curves, knots = knots, type = "poly")
lines(get_evals(polygon_elastic_mean), col = "blue", lwd = 2)

#compute closed smooth mean, takes a little longer

knots <- seq(0,1, length = 11)
closed_elastic_mean <- compute_elastic_mean(data_curves, knots = knots, closed = TRUE)
plot(closed_elastic_mean)