frechet {ridgetorus}

Fréchet statistics on the torus

Description

Computes the Fréchet mean, variance, and standard deviation of a sample on the d-torus [-l, l)^d, d\geq 1, with -l \equiv l identified.

Usage

frechet_mean(x, l = pi, N = 500, draw_plot = FALSE)

frechet_ss(x, l = pi, N = 500, draw_plot = FALSE)

Arguments

x

sample of angles on [-l, l), a vector or a matrix.

l

half-period of the circular data. Can be a vector of length ncol(x) if x is a matrix. Defaults to pi.

N

size of the grid in [-l, l) for the exhaustive search of the mean. Defaults to 5e2.

draw_plot

draw a diagnostic plot showing the Fréchet loss function? Defaults to FALSE.

Value

• frechet_mean: a list with the marginal Fréchet means (mu), variances (var), and standard deviations (sd).

• frechet_ss: a list with the Fréchet variances (var) and the cumulative proportion of total variance explained (var_exp).

Examples

## Circular data

# Sample from a wrapped normal
x <- sdetorus::toPiInt(rnorm(n = 5e2, mean = 2, sd = 1))
frechet_mean(x = x)

# Sample from a bimodal distribution
x <- sdetorus::toPiInt(rnorm(n = 5e2, mean = c(1, -2), sd = c(0.5, 0.75)))
frechet_mean(x = x)

# Periodic data in [-2, 2)
x <- sdetorus::toInt(rnorm(n = 5e2, mean = c(-2, 1), sd = 2:1),
                     a = -2, b = 2)
frechet_mean(x = x, l = 2)

## Toroidal data

# Sample from a multivariate wrapped normal
n <- 50
S <- rbind(c(2.5, -0.2, 0.5),
           c(-0.2, 1.5, -0.5),
           c(0.5, -0.5, 0.75))
x <- sdetorus::toPiInt(mvtnorm::rmvnorm(n, mean = c(0, 1.5, -2), sigma = S))
(f <- frechet_mean(x = x))

# Total Fréchet variance is sum of marginal variances
sum(torus_dist(x, y = f$mu, squared = TRUE)) / n
sum(f$var)

# Cumulative proportion of variances
frechet_ss(x)