Local smoothing with circular predictors and linear or circular responses

Smooth a linear response over a circular predictor with circ_lin_loess(), or a circular response over a circular predictor with circ_circ_loess(). circ_loess() selects the response type with circ_y: FALSE (the default) chooses the linear response, and TRUE chooses the circular response.

Usage

circ_circ_loess(
  formula = NULL,
  data = NULL,
  angle = NULL,
  y = NULL,
  xseq = NULL,
  circ_space = NULL,
  span = 0.75,
  response_space = NULL,
  ...
)

circ_lin_loess(
  formula = NULL,
  data = NULL,
  angle = NULL,
  y = NULL,
  xseq = NULL,
  circ_space = NULL,
  span = 0.75,
  ...
)

circ_loess(
  formula = NULL,
  data = NULL,
  angle = NULL,
  y = NULL,
  xseq = NULL,
  circ_space = NULL,
  span = 0.75,
  circ_y = FALSE,
  response_space = NULL,
  ...
)

Arguments

formula

the formula, e.g., y ~ x

data

data to use

angle

a vector of angles (not used if a formula is provided)

y

dependent variable vector (not used if a formula is provided)

xseq

a grid to compute predictions on (optional, the default is to use 500 points spanning the circle)

circ_space

circular space to use (90, 180, 360, or 2*pi)

span

a span to adjust the degree of smoothing

response_space

Period of the circular response. Defaults to circ_space.

...

other arguments (ignored)

circ_y

Logical: use a circular response (TRUE) or linear response (FALSE, the default). The predictor is circular in both cases.

Value

A fitted object inheriting from circ_loess. The explicit functions additionally have class circ_lin_loess or circ_circ_loess. Components include:

• angle the angles in the data

• y the dependent variable vales in the data

• xseq the grid on which the loess values are estimated

• y_est the estimated loess values

• y_se uncertainty estimates (the linear fit retains the legacy calculation; the circular fit uses weighted SEM of locally unwrapped response residuals)

• w weights

• circ_space circular space used

• span span used

• response_space response period for the circular-response fit

• y_circ_sd weighted circular SD for the circular-response fit

Details

Weights for the regression are computed as \[w = (1-(d/d_{max})^3)^3\] where d is the angular difference between the point at which the estimate is computed and the angles in the data, and \(d_{max}\) is the maximum possible distance. If span is at least 1, all points are included and \(d_{max} = circ\_space \times span / 2\). Otherwise, the nearest proportion span of the observations is included and \(d_{max}\) is their maximal distance.

The predictor is circular in both functions. Only circ_circ_loess() uses circular response means. For a circular response, response_space may differ from circ_space; for example, an orientation predictor can have period 180 while a colour response has period 360. Predictions with no defined mean direction return NA. Prediction intervals are local approximations centred on the estimated direction; their endpoints are not wrapped.

See also

stats::loess()

Examples

angle <- c(-2, -1, 1, 2)
response <- c(179, 178, -178, -179)
circ_lin_loess(angle = angle, y = response, xseq = 0, circ_space = 360, span = 1)$y_est
#> [1] 0
circ_circ_loess(angle = angle, y = response, xseq = 0, circ_space = 360, span = 1)$y_est
#> [1] -180
library(ggplot2)
p <- ggplot(Pascucci_et_al_2019_data, aes(x = orientation, y = err)) +
  geom_point(alpha = 0.05) +
  labs(x = "Orientation, deg.", y = "Error, deg.")
p1 <- p + geom_smooth(method = "loess") + ggtitle("Standard LOESS")
p2 <- p + geom_smooth(method = "circ_lin_loess", method.args = list(circ_space = 180, span = 0.5)) +
  ggtitle("Circular LOESS, span = 0.5")
p3 <- p + geom_smooth(method = "circ_lin_loess", method.args = list(circ_space = 180, span = 0.2)) +
  ggtitle("Circular LOESS, span = 0.2")
(p1 + p2 + p3)
#> `geom_smooth()` using formula = 'y ~ x'
#> `geom_smooth()` using formula = 'y ~ x'
#> `geom_smooth()` using formula = 'y ~ x'