Package 'Rtractor'

Multifractal Spectrum via the Chhabra-Jensen Method

Description

Estimates the multifractal spectrum f(alpha) and generalised dimension spectrum D(q) of a strictly positive time series using the direct box-counting method of Chhabra & Jensen (1989). Clean-room C++ reimplementation from the published algorithm (the reference MATLAB implementation consulted, ChhabraJensen_Yuj_w0.m, co-authored by L. Franca, had no license header; see inst/COPYRIGHTS). The moments core was validated against a Python transliteration of that reference on synthetic test data (exact match to displayed precision).

Usage

chhabra_jensen(
  x,
  q_values = c(seq(-10, -0.1, by = 0.1), seq(1, 10, by = 0.1)),
  scales = NULL
)

Arguments

x	Numeric vector, strictly positive (treat it as a measure — e.g. apply a sigmoid transform first if your data can be negative). length(x) must be evenly divisible by 2^scale for every value in scales, i.e. dyadic lengths (powers of two) work best.
q_values	Numeric vector of multifractal orders. Per the original author's guidance, values strictly between 0 and 1 (exclusive) are numerically unstable for this method and best avoided – a warning is issued if any are supplied. Default skips that range: c(seq(-10, -0.1, by = 0.1), seq(1, 10, by = 0.1)).
scales	Integer vector of box-counting scale exponents; the box size at each scale is 2^scale. Default 1:floor(log2(length(x)) - 2), which keeps at least 4 points per box at the coarsest scale.

Value

A list with:

alpha	Singularity strength alpha(q).
falpha	Multifractal spectrum f(alpha(q)).
Dq	Generalised dimension spectrum D(q).
r_squared_alpha, r_squared_falpha, r_squared_Dq	R-squared of the linear fit underlying each of the above, per q – inspect these before trusting a given q value's estimate.
q	The q_values used.
mu_scale, Ma, Mf, Md	The underlying regression inputs, included for completeness.

References

Chhabra A, Jensen RV. Direct determination of the f(alpha) singularity spectrum. Phys Rev Lett 1989;62:1327-1330.

Franca LGS, Miranda JGV, Leite M, Sharma NK, Walker MC, Lemieux L, Wang Y. Fractal and multifractal properties of electrographic recordings of human brain activity: toward its use as a signal feature for machine learning in clinical applications. Front Physiol 2018;9:1767. Compares MF-DFA, MF-DMA, and Chhabra-Jensen on simulated and human intracranial EEG data and finds Chhabra-Jensen the most stable/reliable of the three – the basis for implementing this method (rather than MF-DFA) alongside mfdma() in Rtractor.

Examples

set.seed(1)
x <- abs(rnorm(1024)) + 0.01
res <- chhabra_jensen(x, scales = 1:6)
plot(res$alpha, res$falpha, type = "b")

---

Detrended Fluctuation Analysis (DFA)

Description

Estimates the scaling exponent alpha of a time series using Detrended Fluctuation Analysis (Peng et al. 1994). This is a direct C++ port of the reference dfa.c implementation distributed by PhysioNet (Mietus, Peng & Moody), validated to reproduce the original compiled binary's output exactly on synthetic test data — see inst/COPYRIGHTS.

Usage

dfa(
  x,
  order = 1,
  min_box = NULL,
  max_box = NULL,
  integrate = TRUE,
  sliding_window = FALSE
)

Arguments

x	Numeric vector. The time series to analyse.
order	Integer >= 1. Order of the polynomial detrending fit (1 = linear detrending, the DFA default; 2 = quadratic, etc.). Corresponds to nfit - 1 in the original dfa.c.
min_box, max_box	Integer or NULL. Smallest/largest box size to evaluate. Defaults (as in the original): min_box = 2 * (order + 1), max_box = length(x) / 4.
integrate	Logical. If TRUE (default), x is treated as an increment series and cumulatively summed before analysis (the usual DFA convention — e.g. pass RR-interval deviations, not a cumulative profile). Set FALSE if x is already an integrated/cumulative profile.
sliding_window	Logical. Use overlapping (sliding) windows instead of non-overlapping boxes. Default FALSE.

Value

A list with:

n	Box sizes evaluated (integer vector).
F	RMS fluctuation at each box size (numeric vector).
alpha	The DFA scaling exponent: the slope of log10(F) on log10(n).

References

Peng C-K, Buldyrev SV, Havlin S, Simons M, Stanley HE, Goldberger AL. Mosaic organization of DNA nucleotides. Phys Rev E 1994;49:1685-1689.

Examples

set.seed(1)
dfa(rnorm(2000))

---

Higuchi Fractal Dimension

Description

Estimates the fractal dimension of a time series using Higuchi's (1988) curve-length algorithm. A clean-room C++ reimplementation, validated against a MATLAB reference implementation on synthetic test data — see inst/COPYRIGHTS.

Usage

higuchi_fd(x, k_max = 10L)

Arguments

x	Numeric vector. The time series to analyse.
k_max	Integer (>= 2). Maximum sub-series interval k. Choose based on where the log-log curve plateaus for your signal (see Doyle et al. for guidance in postural-sway / physiological contexts).

Value

A list with:

k	The k values evaluated, 1:k_max.
L	Average curve length at each k.
hfd	The Higuchi Fractal Dimension: negative slope of log(L) on log(1/k).

References

Higuchi T. Approach to an irregular time series on the basis of the fractal theory. Physica D 1988;31(2):277-283.

Examples

set.seed(1)
higuchi_fd(rnorm(1000), k_max = 10)

---

Hjorth Mobility and Complexity

Description

Computes the Hjorth parameters (Hjorth 1970): mobility, a proxy for mean frequency derived from the variance ratio of the first difference to the signal itself; and complexity, a proxy for bandwidth/irregularity derived from the second difference. Ported from Lucas Franca's own mrpheus package (AASM staging feature pipeline). Uses Bessel-corrected (ddof = 1) variance throughout, deliberately matching R's var() convention – this was mrpheus's original design choice, and differs slightly from the antropy Python library's population-variance (ddof = 0) convention. The two converge as length(x) grows but differ meaningfully for short series; see inst/COPYRIGHTS.

Usage

hjorth_parameters(x)

Arguments

x	Numeric vector. The time series to analyse.

Value

A named list with mobility and complexity.

References

Hjorth B. EEG analysis based on time domain properties. Electroencephalogr Clin Neurophysiol 1970;29(3):306-310.

Examples

set.seed(1)
hjorth_parameters(rnorm(1000))

---

Multifractal Detrending Moving Average (MFDMA)

Description

Estimates the multifractal scaling properties of a time series using the detrending moving average algorithm (Gu & Zhou 2010). Clean-room C++ reimplementation from the published algorithm (the reference MATLAB implementation consulted, MFDMA_1D.m, had no license header; see inst/COPYRIGHTS). The segment-fluctuation core was validated against a Python transliteration of that reference on synthetic test data (exact match to displayed precision).

Usage

mfdma(
  x,
  n_min = 10L,
  n_max = NULL,
  n_scales = 30L,
  theta = 0,
  q = seq(-4, 4, by = 0.1)
)

Arguments

x	Numeric vector. The time series to analyse.
n_min, n_max	Integer. Lower/upper bound of the segment size n. Following the reference implementation's guidance: n_min around 10; n_max around 10% of length(x). Defaults: n_min = 10, n_max = round(length(x) / 10).
n_scales	Integer. Number of segment sizes to evaluate (log-spaced between n_min and n_max). Default 30.
theta	Numeric in ⁠[0, 1]⁠. Position of the moving-average window: 0 (default, recommended) = backward MFDMA, 0.5 = centered, 1 = forward.
q	Numeric vector. Multifractal orders to evaluate. Default seq(-4, 4, by = 0.1).

Value

A list with:

n	Segment sizes evaluated.
Fq	Matrix of the q-th order fluctuation function (segment size x q).
tau	Multifractal scaling exponent tau(q).
alpha	Singularity strength alpha(q) (trimmed at both ends by the local-slope smoothing window; shorter than q).
f	Multifractal spectrum f(alpha).
q	The q values corresponding to alpha/f (trimmed to match).

References

Gu GF, Zhou WX. Detrending moving average algorithm for multifractals. Phys Rev E 2010;82:011136.

Examples

set.seed(1)
x <- rnorm(4000)
res <- mfdma(x, n_min = 10, n_max = 400, n_scales = 20)
plot(res$alpha, res$f, type = "b")

---

Multiscale Entropy (MSE)

Description

Estimates the complexity of a time series across a range of temporal scales (Costa, Goldberger & Peng 2002): the series is coarse-grained (non-overlapping block-averaged) at each scale factor, and sample_entropy() is computed on each coarse-grained series, using a tolerance held fixed relative to the original (not the coarse-grained) series' standard deviation – this is the standard MSE convention and essential for entropy values to be comparable across scales. Direct C++ port of the coarse-graining + sample-entropy core in PhysioNet's reference mse.c (Costa), validated to reproduce the compiled reference binary's output exactly (to its own displayed precision) on synthetic test data. See inst/COPYRIGHTS.

Usage

multiscale_entropy(x, scale_max = 20L, m = 2L, r = 0.15)

Arguments

x	Numeric vector. The time series to analyse.
scale_max	Integer >= 1. Largest scale factor to evaluate; MSE is computed at every integer scale from 1 to scale_max. Default 20 (the standard MSE convention).
m	Integer >= 1. Template length, passed to sample_entropy(). Default 2 (the standard MSE convention).
r	Numeric > 0. Tolerance as a fraction of the original series' standard deviation, passed to sample_entropy(). Default 0.15 (the standard MSE convention).

Value

A list with:

scale	The scale factors evaluated, 1:scale_max.
mse	Sample entropy at each scale (may contain NA at large scales, where the coarse-grained series becomes too short to estimate reliably).
m, r	The parameters used, echoed back for reference.

References

Costa M, Goldberger AL, Peng CK. Multiscale entropy analysis of complex physiologic time series. Phys Rev Lett 2002;89:068102.

Examples

set.seed(1)
res <- multiscale_entropy(rnorm(2000), scale_max = 10)
plot(res$scale, res$mse, type = "b")

---

Number of Zero Crossings

Description

Counts sign changes in a time series – a simple time-domain proxy for dominant frequency / signal roughness, commonly reported alongside Hjorth parameters and fractal dimension in EEG complexity work. Ported from Lucas Franca's own mrpheus package (AASM staging feature pipeline), itself validated for exact parity against the antropy Python library; re-validated here directly against antropy 0.2.2 on synthetic test data (exact match). See inst/COPYRIGHTS.

Usage

num_zerocross(x)

Arguments

x	Numeric vector. The time series to analyse.

Value

A length-1 integer: the number of zero crossings.

Examples

set.seed(1)
num_zerocross(rnorm(1000))

---

Permutation Entropy

Description

Estimates the complexity of a time series using permutation entropy (Bandt & Pompe 2002): the Shannon entropy of the distribution of ordinal patterns ("motifs") of length order found in the series. Ported from Lucas Franca's own mrpheus package (part of its AASM sleep-staging feature pipeline), itself validated for exact parity against the antropy Python library; re-validated here directly against antropy 0.2.2 on synthetic test data (exact match to displayed precision). See inst/COPYRIGHTS.

Usage

perm_entropy(x, order = 3L, delay = 1L, normalize = TRUE)

Arguments

x	Numeric vector. The time series to analyse.
order	Integer >= 2. Length of the ordinal pattern (embedding dimension). Default 3.
delay	Integer >= 1. Time delay between pattern elements. Default 1.
normalize	Logical. If TRUE (default), divide by ⁠log(order!)⁠ so the result falls in ⁠[0, 1]⁠. If FALSE, return the raw Shannon entropy in nats (natural log). Note this differs from the antropy Python library, which computes the raw value in bits (log base 2); the normalized value is base-independent and matches antropy exactly, but the raw values are not directly comparable between the two. See inst/COPYRIGHTS.

Value

A length-1 numeric: the permutation entropy.

References

Bandt C, Pompe B. Permutation entropy: a natural complexity measure for time series. Phys Rev Lett 2002;88:174102.

Examples

set.seed(1)
perm_entropy(rnorm(1000))

---

Petrosian Fractal Dimension

Description

Estimates fractal dimension from the rate of sign changes in a time series' first difference (a fast proxy for signal irregularity). Ported from Lucas Franca's own mrpheus package (AASM staging feature pipeline), itself validated for exact parity against the antropy Python library; re-validated here directly against antropy 0.2.2 on synthetic test data (exact match). See inst/COPYRIGHTS.

Usage

petrosian_fd(x)

Arguments

x	Numeric vector. The time series to analyse.

Value

A length-1 numeric: the Petrosian fractal dimension.

References

Petrosian A. Kolmogorov complexity of finite sequences and recognition of different preictal EEG patterns. Proceedings of the Eighth IEEE Symposium on Computer-Based Medical Systems 1995:212-217.

Examples

set.seed(1)
petrosian_fd(rnorm(1000))

---

Simulate a Multifractal Time Series Using the p-Model

Description

Generates a multiplicative binomial cascade ("p-model") time series (Meneveau & Sreenivasan 1987), optionally fractionally integrated in Fourier space to a prescribed power-spectrum slope (Davis et al. 1997) to produce a continuous, nonstationary series superposed on the multifractal cascade.

Usage

pmodel(n_values = 256L, p = 0.375, slope = NULL, seed = 42L)

Arguments

n_values	Integer. Desired length of the output series. Internally rounded up to the next power of two for the cascade construction, then truncated back to n_values. Default 256.
p	Numeric in ⁠(0, 1)⁠. The p-model parameter: values near 0 or 1 produce a very peaked (strongly multifractal) series; values near 0.5 produce a calm series (p = 0.5 gives an exactly constant unit vector – no multifractality at all). Default 0.375.
slope	Numeric or NULL. If supplied, the p-model series is fractionally integrated in Fourier space to have this power-spectrum slope, producing a continuous, nonstationary series (slopes between -1 and -3 give stationary increments; steeper than -3 is differentiable) – see Davis et al. (1997) for the regimes. If NULL (default), no fractional integration is applied.
seed	Integer or NULL. Seeds the random sign draws at every cascade level. The original MATLAB implementation reseeds to a fixed seed (42) at every level unconditionally, making the cascade fully deterministic by design (each level's random signs are literally a prefix of the next level's, since the RNG stream restarts from the same point every time). This port preserves that same reseed-every-level structure, with seed (default 42, matching the original) controlling it. Pass seed = NULL to draw fresh randomness at every level instead (no reseeding) – unlike the original, which has no way to disable this.

Details

Clean-room R reimplementation of pmodel.m, written by the late Victor Venema (no license header present in the source file; see inst/COPYRIGHTS). The underlying random sign draws use R's own RNG rather than attempting to replicate MATLAB's specific generator bit-for-bit (not feasible across languages, and not meaningful for a synthetic-data generator characterised by its cascade rule rather than particular RNG bytes) – see the seed argument below for how the original's reseed-every-level behaviour is preserved. Validated against two RNG-independent structural properties that hold for any random sign sequence: at p = 0.5 the cascade is provably an exact constant vector of 1s, and total mass is conserved exactly at every cascade level (sum(y) doubles per level) for any p – see tests/testthat/test-pmodel.R.

Value

If slope is NULL: a numeric vector of length n_values, the p-model series itself. If slope is supplied: a list with

x	The fractionally integrated series.
y	The underlying p-model series before integration.

References

Meneveau C, Sreenivasan KR. Simple multifractal cascade model for fully developed turbulence. Phys Rev Lett 1987;59:1424-1427.

Davis A, Marshak A, Cahalan R, Wiscombe W. The Landsat scale break in stratocumulus as a three-dimensional radiative transfer effect: implications for cloud remote sensing. J Atmos Sci 1997;54(2):241-260.

Venema V, Bachner S, Rust HW, Simmer C. Statistical characteristics of surrogate data based on geophysical measurements. Nonlin Processes Geophys 2006;13:449-466. (Requested citation for this code by its original author, Victor Venema.)

Examples

y <- pmodel(1024, p = 0.3)
range(y)

# p = 0.5 always gives a constant series, regardless of seed:
all(pmodel(64, p = 0.5) == 1)

---

Recurrence Microstates Entropy (maximum-entropy threshold search)

Description

Estimates the complexity of a (windowed) time series using the entropy of recurrence microstates (Corso et al. 2018), searching over the vicinity threshold epsilon to find the value that maximises the Shannon entropy of the microstate distribution — a parameter-free alternative to fixing epsilon by hand, related to the optimal-threshold method of Prado et al. (2018).

Usage

recurrence_microstate_entropy(
  x,
  block = 3L,
  n_samples = 18000L,
  eps_min = 1e-09,
  eps_max = 0.499999999,
  frac = 10L,
  frac2 = 10L,
  seed = NULL
)

Arguments

x	Numeric vector. The (already windowed) time series segment to analyse. In the original study, x was min-max normalised to ⁠[0, 1]⁠ per window before calling this function — do the same if you want comparable eps_max values across windows/subjects.
block	Integer. Side length of the square recurrence microstate block; the microstate space has 2^(block^2) possible patterns. Default 3 (matching the reference implementation).
n_samples	Integer. Number of random point-pairs sampled to estimate the recurrence microstate distribution. Default 18000.
eps_min, eps_max	Numeric. Search range for the vicinity threshold. Defaults match the reference implementation.
frac, frac2	Integer. Number of steps in the coarse search pass and number of refinement iterations, respectively. Defaults match the reference implementation.
seed	Optional integer. Seeds the random pair sampling for reproducibility. The original Julia script reseeds from system entropy on every run (⁠Random.seed!()⁠ with no argument) and is therefore not reproducible run-to-run by design; pass a seed here if you need reproducible results, or leave NULL to match the original's fresh-randomness-every-call behaviour.

Details

Ported from circadia-bio/maxEntropy (Entropy.jl, MIT licensed), the script used to produce the entropy estimates in Ferre et al. (2024, PLOS ONE). The deterministic core loop was validated against a Python transliteration of the original on synthetic test data (exact match on S_max and eps_max given identical sampled index pairs). Only the reusable Max_Entropy() routine was ported — the original script's main() (file I/O over named subject directories) is study-specific scaffolding, not part of the general-purpose method.

Value

A list with:

microstate_probs	Probability of each of the 2^(block^2) microstates at the entropy-maximising threshold.
entropy_max	The maximum Shannon entropy found (S_max).
eps_max	The vicinity threshold at which entropy_max was achieved.

References

Corso G, Prado TL, dos Santos Lima GZ, Kurths J, Lopes SR. Quantifying entropy using recurrence matrix microstates. Chaos 2018;28(8):083108.

Prado TL, dos Santos Lima GZ, Lobao-Soares B, do Nascimento GC, Corso G, Fontenele-Araujo J, Kurths J, Lopes SR. Optimizing the detection of nonstationary signals by using recurrence analysis. Chaos 2018;28(8):085703.

Ferre IBS, Corso G, dos Santos Lima GZ, Lopes SR, Leocadio-Miguel MA, Franca LGS, de Lima Prado T, Araujo JF. Cycling reduces the entropy of neuronal activity in the human adult cortex. PLOS ONE 2024;19(10):e0298703.

Examples

set.seed(1)
x <- (sin(seq(0, 20 * pi, length.out = 300)) + 1) / 2
recurrence_microstate_entropy(x, seed = 1)

---

Retrieve an Rtractor palette

Description

Returns a named character vector of hex colour codes for the requested palette. Suitable for direct use in ggplot2::scale_fill_manual() or ggplot2::scale_colour_manual().

Usage

rtractor_palette(palette = "main", n = NULL, reverse = FALSE)

Arguments

palette	Name of the palette. One of "main", "core", "diverging", "seq_blue", "seq_sage", "seq_coral". Defaults to "main".
n	Number of colours to return. If NULL (default) all colours in the palette are returned. If n is smaller than the palette length the first n colours are returned; if larger an error is thrown.
reverse	Logical. Reverse the order of the palette? Default FALSE.

Value

A named (or unnamed for gradient palettes) character vector of hex colour codes.

Examples

rtractor_palette()
rtractor_palette("core")
rtractor_palette("seq_blue", n = 3)
rtractor_palette("diverging", reverse = TRUE)

---

List all available Rtractor palettes

Description

Prints the names and sizes of all palettes defined in the package.

Usage

rtractor_palettes()

Value

A named integer vector of palette lengths, invisibly.

Examples

rtractor_palettes()

---

Sample Entropy (SampEn)

Description

Estimates the complexity of a time series using sample entropy (Richman & Moorman 2000): the negative log ratio of template matches of length m + 1 to template matches of length m, using a fixed Chebyshev-distance tolerance. Direct C++ port of the counting core in PhysioNet's reference mse.c (Costa), validated to reproduce the compiled reference binary's output exactly (to its own displayed precision) on synthetic test data. See inst/COPYRIGHTS.

Usage

sample_entropy(x, m = 2L, r = 0.15)

Arguments

x	Numeric vector. The time series to analyse.
m	Integer >= 1. Template length. Default 2 (the standard MSE convention, per Costa et al.).
r	Numeric > 0. Tolerance, as a fraction of x's own standard deviation (i.e. the actual Chebyshev-distance tolerance used is sd(x) * r). Default 0.15 (the standard MSE convention).

Details

This is also the building block multiscale_entropy() applies at each coarse-grained scale – see R/multiscale.R.

Value

A length-1 numeric: the sample entropy. If there are too few valid template pairs to compare (length(x) - m < 2), returns NA. If there are zero matches at either template length (undefined ratio), returns the conventional fallback -log(1 / (N*(N-1))) used by the reference implementation, where N = length(x) - m.

References

Richman JS, Moorman JR. Physiological time-series analysis using approximate entropy and sample entropy. Am J Physiol Heart Circ Physiol 2000;278(6):H2039-H2049.

Examples

set.seed(1)
sample_entropy(rnorm(1000))

---

Rtractor discrete colour scale

Description

Applies an Rtractor qualitative palette to the colour aesthetic. Defaults to "core" (excludes the dark ink colour, which is reserved for text/axis annotation rather than data encoding).

Usage

scale_colour_rtractor(palette = "core", reverse = FALSE, ...)

scale_color_rtractor(palette = "core", reverse = FALSE, ...)

Arguments

palette	Palette name. Default "core".
reverse	Logical. Reverse the palette? Default FALSE.
...	Additional arguments passed to ggplot2::discrete_scale().

Examples

library(ggplot2)
ggplot(mtcars, aes(wt, mpg, colour = factor(cyl))) +
  geom_point(size = 3) +
  scale_colour_rtractor() +
  theme_rtractor()

---

Rtractor continuous colour scale

Description

Interpolates across an Rtractor palette for continuous colour data. The "diverging" palette is recommended for signed quantities (e.g. Lyapunov exponent sign, spectrum asymmetry); the ⁠seq_*⁠ palettes for unipolar data (e.g. entropy magnitude).

Usage

scale_colour_rtractor_c(palette = "seq_blue", reverse = FALSE, ...)

scale_color_rtractor_c(palette = "seq_blue", reverse = FALSE, ...)

Arguments

palette	Palette name. Default "seq_blue".
reverse	Logical. Reverse the palette? Default FALSE.
...	Additional arguments passed to ggplot2::scale_colour_gradientn().

---

Rtractor discrete fill scale

Description

Applies an Rtractor qualitative palette to the fill aesthetic.

Usage

scale_fill_rtractor(palette = "core", reverse = FALSE, ...)

Arguments

palette	Palette name. Default "core".
reverse	Logical. Reverse the palette? Default FALSE.
...	Additional arguments passed to ggplot2::discrete_scale().

Examples

library(ggplot2)
ggplot(mpg, aes(class, fill = drv)) +
  geom_bar() +
  scale_fill_rtractor() +
  theme_rtractor()

---

Rtractor continuous fill scale

Description

Interpolates across an Rtractor palette for continuous fill data.

Usage

scale_fill_rtractor_c(palette = "seq_blue", reverse = FALSE, ...)

Arguments

palette	Palette name. Default "seq_blue".
reverse	Logical. Reverse the palette? Default FALSE.
...	Additional arguments passed to ggplot2::scale_colour_gradientn().

---

Rtractor ggplot2 theme

Description

A clean ggplot2 theme built on ggplot2::theme_minimal(), visually distinct from theme_circadia() (softer palette, same structural conventions) so that Rtractor figures are recognisable at a glance.

Usage

theme_rtractor(
  base_size = 14,
  base_family = "",
  grid = "none",
  legend_position = "right"
)

Arguments

base_size	Base font size in points. Default 14.
base_family	Base font family. Default "" (ggplot2 default).
grid	Which grid lines to show. One of "none" (default), "y" (horizontal only), "xy" (both), "x" (vertical only).
legend_position	Position of the legend passed to ggplot2::theme(). Default "right".

Value

A ggplot2::theme() object.

Examples

library(ggplot2)
ggplot(mtcars, aes(wt, mpg, colour = factor(cyl))) +
  geom_point(size = 2) +
  scale_colour_rtractor() +
  theme_rtractor()

---