Create a Smoothed Vector Field Layer

geom_stream_smooth() creates a ggplot2 layer that visualizes a smooth vector field based on raw vector data. The function fits a multivariate linear model (by default, using the formula cbind(fx, fy) ~ x * y) to predict the vector displacements at any given location. It also handles different input formats by converting polar coordinates or endpoint data to vector displacements.

Usage

geom_stream_smooth(
  mapping = NULL,
  data = NULL,
  stat = StatStreamField,
  position = "identity",
  ...,
  na.rm = FALSE,
  show.legend = NA,
  inherit.aes = TRUE,
  n = 11,
  xlim = NULL,
  ylim = NULL,
  normalize = TRUE,
  center = FALSE,
  type = "vector",
  formula = cbind(fx, fy) ~ x * y,
  eval_points = NULL,
  lineend = "butt",
  linejoin = "round",
  linemitre = 10,
  arrow = grid::arrow(angle = 20, length = unit(0.015, "npc"), type = "closed")
)

stat_stream_smooth(
  mapping = NULL,
  data = NULL,
  geom = GeomStream,
  position = "identity",
  ...,
  na.rm = FALSE,
  show.legend = NA,
  inherit.aes = TRUE,
  n = 11,
  xlim = NULL,
  ylim = NULL,
  normalize = TRUE,
  center = FALSE,
  type = "vector",
  formula = cbind(fx, fy) ~ x * y,
  eval_points = NULL,
  lineend = "butt",
  linejoin = "round",
  linemitre = 10,
  arrow = grid::arrow(angle = 20, length = unit(0.015, "npc"), type = "closed")
)

Arguments

mapping

A set of aesthetic mappings created by ggplot2::aes(). Required: Must include x and y; vector displacements are defined by fx and fy.

data

A data frame containing the raw vector data.

stat

The statistical transformation to use on the data. Defaults to "vector_smooth".

position

Position adjustment, either as a string or the result of a position adjustment function.

...

Additional arguments passed to the layer. If a fixed parameter color is not provided, then color = "blue" is used.

na.rm

Logical. If FALSE (the default), missing values are removed with a warning.

show.legend

Logical. Should this layer be included in the legends?

inherit.aes

Logical. If FALSE, overrides the default aesthetics rather than combining with them.

n

An integer vector specifying the grid resolution for smoothing.

xlim

Numeric vector of length 2 specifying the domain limits in the \(x\)-direction. Defaults to \(c(-1, 1)\).

ylim

Numeric vector of length 2 specifying the domain limits in the \(y\)-direction. Defaults to \(c(-1, 1)\).

normalize

Logical. If TRUE, the vector endpoints are scaled to unit length before being scaled by L (default: TRUE).

center

Logical. If TRUE, the vector is recentered so that the original (x, y) becomes the midpoint (default is TRUE for geom_vector() and FALSE for geom_vector2()).

type

Character. Either "stream" (default) or "vector". "stream" computes a full streamline by integrating in both directions (if center = TRUE), while "vector" computes a single vector.

formula

A formula specifying the multivariate linear model used for smoothing. Defaults to cbind(fx, fy) ~ x * y.

eval_points

A data frame of evaluation points, or NULL. When provided, it specifies the grid where the smoothing model is evaluated; if NULL, a grid is generated based on n.

lineend

Line end style (round, butt, square).

linejoin

Line join style (round, mitre, bevel).

linemitre

Line mitre limit (number greater than 1).

arrow

An optional grid::arrow() specification to add arrowheads to the vectors (default: grid::arrow(angle = 25, length = unit(0.025, "npc"), type = "closed")).

geom

The geometric object used to render the streamline (only used in stat_stream(); defaults to GeomStream).

Value

A ggplot2 layer that can be added to a ggplot object to display a smoothed vector field.

norm

Computed as the Euclidean norm of the displacement, \(\sqrt{fx^2 + fy^2}\), this variable is used to normalize and scale the vector lengths.

t

The integration time or evaluation time at each computed point along the smoothed field (when applicable).

d

The incremental distance between consecutive computed points.

l

The cumulative arc length along the smoothed vector field, calculated as the cumulative sum of d.

Aesthetics

geom_stream_smooth() supports the following aesthetics (required aesthetics are in bold):

• x: The x-coordinate of the vector's starting point.

• y: The y-coordinate of the vector's starting point.

• fx: The displacement along the x-axis.

• fy: The displacement along the y-axis.

• color: The fixed color for the vector. Defaults to "blue".

• linewidth: The thickness of the vector line.

• linetype: The type of the vector line (e.g., solid or dashed).

• alpha: The transparency level of the vector.

• arrow: Specifies arrowheads for the vectors.

Details

Data Conversion: If xend/yend are missing or all NA, the function computes them. It first checks for vector displacements (fx and fy); if present, it computes \(xend = x + fx,\quad yend = y + fy.\) Otherwise, it checks for polar coordinates (angle and distance) and computes \(xend = x + distance \times \cos(angle \times 180/\pi),\quad yend = y + distance \times \sin(angle \times 180/\pi).\) An error is thrown if neither set is available.

Smoothing: The multivariate linear model is fitted using the provided formula and data. This model is then used to predict vector displacements at any specified grid point, generating a smooth approximation of the vector field.

Prediction Intervals: Two types of prediction intervals can be displayed:

• Ellipse: Depicts the joint uncertainty (covariance) in the predicted fx and fy.

• Wedge: Indicates the range of possible vector directions and magnitudes.

Examples

if (FALSE) { # \dontrun{
# Define a true vector field function
f <- function(u) {
  x <- u[1]; y <- u[2]
  c(x^2 - y^2, x^2 + y^2 - 2)
}

# Alternative example function
f <- function(u) c(-u[2], u[1])

# Visualize the vector field
ggplot() + geom_stream_field(fun = f, xlim = c(-2, 2), ylim = c(-2, 2))

# Generate design points
n <- 20
df <- data.frame(x = runif(n, -2, 2), y = runif(n, -2, 2))

# Sample function values at design points
fdf <- as.data.frame(t(apply(df, 1, f)))
colnames(fdf) <- c("fx", "fy")
df <- cbind(df, fdf)

# Visualize raw vector field data
ggplot(df) + geom_vector(aes(x, y, fx = fx, fy = fy))

# Add smoothed layer using default model
ggplot(df) +
  geom_vector(aes(x, y, fx = fx, fy = fy)) +
  geom_stream_smooth(formula = cbind(fx, fy) ~ x * y)

# Use a more complex polynomial model
ggplot(df) +
  geom_vector(aes(x, y, fx = fx, fy = fy)) +
  geom_stream_smooth(formula = cbind(fx, fy) ~ poly(x, 2) * poly(y, 2), data = df)

# Fit a linear model and use it for prediction
fhat <- function(u) {
  model <- lm(cbind(fx, fy) ~ x * y, data = df)
  predict(model, newdata = data.frame(x = u[1], y = u[2])) |> as.numeric()
}

# Visualize estimated field with the raw vector field
ggplot(df) +
  geom_stream_field(fun = fhat, normalize = FALSE, color = "#3366FF") +
  geom_vector(aes(x, y, fx = fx, fy = fy))

# Generate a hexagonal grid
hex_lattice <- grid_hex(xlim = c(-5, 5), ylim = c(-5, 5), d = 1)

# Use the hexagonal grid in geom_stream_field
ggplot(data = df) +
  geom_vector(aes(x, y, fx = fx, fy = fy), color = "black", normalize = FALSE) +
  geom_stream_smooth(eval_points = hex_lattice)

# user specified point

eval_pts <- data.frame(x = c(0, 1), y = c(2, -1))

ggplot(data = df) +
  geom_vector(aes(x, y, fx = fx, fy = fy), color = "black", normalize = FALSE) +
  geom_stream_smooth(eval_points = eval_pts)
} # }