Fourier Series Smoothing

geom_fourier() and stat_fourier() fit a truncated Fourier (discrete Fourier transform, DFT) series to the supplied x/y observations and render the reconstructed smooth curve. The data are first aggregated at duplicate x positions, interpolated to a uniform grid, optionally de-trended, transformed via stats::fft(), and then reconstructed from the requested number of harmonics.

Usage

geom_fourier(
  mapping = NULL,
  data = NULL,
  stat = "fourier",
  position = "identity",
  ...,
  n_harmonics = NULL,
  detrend = NULL,
  arrow = NULL,
  arrow.fill = NULL,
  lineend = "butt",
  linejoin = "round",
  linemitre = 10,
  na.rm = FALSE,
  show.legend = NA,
  inherit.aes = TRUE
)

stat_fourier(
  mapping = NULL,
  data = NULL,
  geom = "line",
  position = "identity",
  ...,
  n_harmonics = NULL,
  detrend = NULL,
  na.rm = FALSE,
  show.legend = NA,
  inherit.aes = TRUE
)

Arguments

mapping

Set of aesthetic mappings created by aes(). If specified and inherit.aes = TRUE (the default), it is combined with the default mapping at the top level of the plot. You must supply mapping if there is no plot mapping.

data

The data to be displayed in this layer. There are three options:

If NULL, the default, the data is inherited from the plot data as specified in the call to ggplot().

A data.frame, or other object, will override the plot data. All objects will be fortified to produce a data frame. See fortify() for which variables will be created.

A function will be called with a single argument, the plot data. The return value must be a data.frame, and will be used as the layer data. A function can be created from a formula (e.g. ~ head(.x, 10)).

position

A position adjustment to use on the data for this layer. This can be used in various ways, including to prevent overplotting and improving the display. The position argument accepts the following:

• The result of calling a position function, such as position_jitter(). This method allows for passing extra arguments to the position.

• A string naming the position adjustment. To give the position as a string, strip the function name of the position_ prefix. For example, to use position_jitter(), give the position as "jitter".

• For more information and other ways to specify the position, see the layer position documentation.

...

Other arguments passed on to layer()'s params argument. These arguments broadly fall into one of 4 categories below. Notably, further arguments to the position argument, or aesthetics that are required can not be passed through .... Unknown arguments that are not part of the 4 categories below are ignored.

• Static aesthetics that are not mapped to a scale, but are at a fixed value and apply to the layer as a whole. For example, colour = "red" or linewidth = 3. The geom's documentation has an Aesthetics section that lists the available options. The 'required' aesthetics cannot be passed on to the params. Please note that while passing unmapped aesthetics as vectors is technically possible, the order and required length is not guaranteed to be parallel to the input data.

• When constructing a layer using a stat_*() function, the ... argument can be used to pass on parameters to the geom part of the layer. An example of this is stat_density(geom = "area", outline.type = "both"). The geom's documentation lists which parameters it can accept.

• Inversely, when constructing a layer using a geom_*() function, the ... argument can be used to pass on parameters to the stat part of the layer. An example of this is geom_area(stat = "density", adjust = 0.5). The stat's documentation lists which parameters it can accept.

• The key_glyph argument of layer() may also be passed on through .... This can be one of the functions described as key glyphs, to change the display of the layer in the legend.

n_harmonics

Integer or NULL. Number of Fourier harmonics to retain. NULL (default) uses all harmonics up to the Nyquist limit, giving an interpolating fit. Smaller values produce smoother curves.

detrend

Character string or NULL. De-trending method applied before the FFT; one of "lm", "loess", or NULL (default). See the Detrending section for details.

arrow

Arrow specification, as created by grid::arrow().

arrow.fill

fill colour to use for the arrow head (if closed). NULL means use colour aesthetic.

lineend

Line end style (round, butt, square).

linejoin

Line join style (round, mitre, bevel).

linemitre

Line mitre limit (number greater than 1).

na.rm

If FALSE, the default, missing values are removed with a warning. If TRUE, missing values are silently removed.

show.legend

logical. Should this layer be included in the legends? NA, the default, includes if any aesthetics are mapped. FALSE never includes, and TRUE always includes. It can also be a named logical vector to finely select the aesthetics to display. To include legend keys for all levels, even when no data exists, use TRUE. If NA, all levels are shown in legend, but unobserved levels are omitted.

inherit.aes

If FALSE, overrides the default aesthetics, rather than combining with them. This is most useful for helper functions that define both data and aesthetics and shouldn't inherit behaviour from the default plot specification, e.g. annotation_borders().

geom, stat

Override the default connection between geom_fourier() and stat_fourier().

Value

A ggplot2::layer() object that can be added to a ggplot2::ggplot().

Period convention

The DFT treats the input as one period of an infinitely repeating signal. The correct period for \(N\) uniformly-spaced samples with spacing \(\Delta x\) is \(P = N \cdot \Delta x\), not \(x_{max} - x_{min}\). Using the latter (a closed interval) implicitly maps the last sample to \(t = 1\), which coincides with \(t = 0\) of the next period, causing a boundary discontinuity and Gibbs-phenomenon ringing whenever the first and last y values differ. This implementation uses the half-open period.

Detrending

Before the FFT is applied the data can be de-trended so that slow, non-periodic trends do not dominate the low-frequency coefficients:

NULL (default)

No de-trending; the raw signal is transformed.

"lm"

Subtract a global ordinary-least-squares linear fit.

"loess"

Subtract a LOESS smooth. Falls back to "lm" with a message if the group is too small for LOESS (fewer than 4 observations).

The trend is added back before the final curve is returned, so the output is always on the original y-scale.

Nyquist limit

The maximum number of harmonics recoverable from \(N\) observations is \(\lfloor N/2 \rfloor\). Requesting more triggers a message and the limit is used instead.

Irregular spacing

The input data is linearly interpolated onto a uniform grid before the FFT. If the original x-spacing is highly irregular (e.g. monthly time series data), the interpolation may introduce artefacts in sparse regions. A message is emitted when the coefficient of variation of the x-spacing exceeds 0.5.

See also

stats::fft() for the underlying Fast Fourier Transform, lm() and loess() for the optional detrending fits, geom_catenary() and geom_chaikin() for other curve-fitting geoms.

Aesthetics

geom_fourier() understands the following aesthetics. Required aesthetics are displayed in bold and defaults are displayed for optional aesthetics:

•	x
•	y
•	alpha	→ NA
•	colour	→ via theme()
•	group	→ inferred
•	linetype	→ via theme()
•	linewidth	→ via theme()

Learn more about setting these aesthetics in vignette("ggplot2-specs").

Examples

library(ggplot2)

n <- 50
df1 <- data.frame(
  x = seq(0, 1, length.out = n),
  y = sin(seq(0, 2 * pi, length.out = n)) + rnorm(n, sd = 0.2)
)

# Basic usage – Interpolating fit (all harmonics)
p <- ggplot(df1, aes(x, y)) +
  geom_point(alpha = 0.5)
p + geom_fourier()


# Use 1 harmonic only
p + geom_fourier(n_harmonics = 1)


# De-trending a linearly drifting signal
set.seed(2)
x <- seq(0, 4 * pi, length.out = n)
df2 <- data.frame(
  x = x,
  y = sin(x) + x * 0.3 + rnorm(n, sd = 0.15)
)

ggplot(df2, aes(x, y))  +
geom_point(alpha = 0.35) +
  geom_fourier(aes(colour = "detrend = NULL"), n_harmonics = 3) +
  geom_fourier(aes(colour = "detrend = \"lm\""), n_harmonics = 3,
               detrend = "lm")


# Multiple groups
set.seed(3)
x <- seq(0, 2 * pi, length.out = n/2)
df3 <- rbind(
  data.frame(x = x,
             y = sin(x) + rnorm(n/2, sd = 0.2),
             grp = "sine"),
  data.frame(x = x,
             y = cos(x) + rnorm(n/2, sd = 0.2),
             grp = "cosine")
)

ggplot(df3, aes(x, y, colour = grp)) +
  geom_point(alpha = 0.5) +
  geom_fourier()


# when the data is not uniformly-spaced, the Fourier
# curve will not pass through every data point
df4 <- data.frame(
  x = c(1:10, 19:20),
  y = sin(seq_len(12))
)

ggplot(df4, aes(x, y)) +
  geom_fourier()
#> Warning: Highly irregular x-spacing detected (coefficient of variation = 1.4). The
#> uniform-grid interpolation may introduce artefacts.