anscombiser: Create Datasets with Identical Summary Statistics

Description

Anscombe's quartet (Anscombe, 1973) are a set of four two-variable datasets that have several common summary statistics but which have very different joint distributions. This becomes apparent when the data are plotted, which illustrates the importance of using graphical displays in Statistics. This package enables the creation of datasets that have identical marginal sample means and sample variances, sample correlation, least squares regression coefficients and coefficient of determination. The user supplies an initial dataset, which is shifted, scaled and rotated in order to achieve target summary statistics. The general shape of the initial dataset is retained. The target statistics can be supplied directly or calculated based on a user-supplied dataset.

Details

The main functions in anscombiser are

• anscombise, which modifies a user-supplied dataset so that it shares sample summary statistics with Anscombe's quartet.

• mimic, which modified a user-supplied dataset so that is shares sample summary statistics with another user-supplied dataset.

See vignette("intro-to-anscombiser", package = "anscombiser") for an overview of the package.

Author(s)

Maintainer: Paul J. Northrop p.northrop@ucl.ac.uk [copyright holder]

References

Anscombe, F. J. (1973). Graphs in Statistical Analysis. The American Statistician 27 (1): 17–21. doi:10.1080/00031305.1973.10478966

See Also

anscombise and mimic

Anscombe's Quartet Separated

Description

Provides Anscombe's Quartet as separate data frames.

Usage

anscombe1

anscombe2

anscombe3

anscombe4

Format

All datasets are objects of class data.frame with 11 rows and 2 columns.

Source

Anscombe's Quartet of 'Identical' Simple Linear Regressions: datasets::anscombe in the datasets package. The ith dataset is datasets::anscombe[, c(i, i + 4)].

References

Anscombe, Francis J. (1973). Graphs in statistical analysis. The American Statistician, 27, 17-21. doi:10.2307/2682899

Create new versions of Anscombe's quartet

Description

Modifies a dataset x so that it shares sample summary statistics with Anscombe's quartet.

Usage

anscombise(x, which = 1, idempotent = TRUE)

Arguments

Details

The input dataset x is modified by shifting, scaling and rotating it so that its sample mean and covariance matrix match those of the Anscombe quartet.

The rotation is based on the square root of the sample correlation matrix. If idempotent = FALSE then this square root is based on the Cholesky decomposition this matrix, using chol. If idempotent = TRUE the square root is based on the spectral decomposition of this matrix, using the output from eigen. This is a minimal rotation square root, which means that if the input data x already have the exactly/approximately the required summary statistics then the returned dataset is exactly/approximately the same as the target dataset.

Value

An object of class c("anscombe", "matrix", "array") with plot and print methods. This returned dataset has the following summary statistics in common with Anscombe's quartet.

• The sample means of each variable.

• The sample variances of each variable.

• The sample correlation matrix.

• The estimated regression coefficients from least squares linear regressions of each variable on each other variable.

The target and new summary statistics are returned as attributes old_stats and new_stats and the chosen Anscombe's quartet dataset as an attribute old_data.

See Also

mimic to modify a dataset to share sample summary statistics with another dataset.

datasets::anscombe for Anscombe's Quartet and anscombe for Anscombe's Quartet as 4 separate datasets.

input_datasets: input1 to input8 for some input datasest of the same size as those in Anscombe's quartet.

Examples

# Produce Anscombe-like datasets using input1 to input8

a1 <- anscombise(input1, idempotent = FALSE)
plot(a1)
a2 <- anscombise(input2)
plot(a2)
a3 <- anscombise(input3, idempotent = FALSE)
plot(a3)
a4 <- anscombise(input4, idempotent = FALSE)
plot(a4)
a5 <- anscombise(input5, idempotent = FALSE)
plot(a5)
a6 <- anscombise(input6)
plot(a6)
a7 <- anscombise(input7, idempotent = FALSE)
plot(a7)
a8 <- anscombise(input8, idempotent = FALSE)
plot(a8)

# Old faithful to new faithful
new_faithful <- anscombise(datasets::faithful, which = 4)
plot(new_faithful)
# Then check that the sample summary statistics are the same
plot(new_faithful, input = TRUE)

# Map of Italy
got_maps <- requireNamespace("maps", quietly = TRUE)
if (got_maps) {
  italy <- mapdata("Italy")
  new_italy <- anscombise(italy, which = 4)
  plot(new_italy)
}

Animation of several Anscombised datasets

Description

Create an animation to show datasets that share sample summary statistics with Anscombe's quartet.

Usage

anscombise_gif(
  x,
  which = 1,
  idempotent = TRUE,
  theme_name = "classic",
  ease = "cubic-in-out",
  transition_length = 3,
  state_length = 1,
  wrap = TRUE
)

Arguments

Details

For this function to work the packages ggplot2 and gganimate must be installed.

Value

An object of class c("gganim", "gg", "ggplot") with an additional attribute new_data that is a data frame with 3 variables, x, y and dataset containing the datasets output from anscombise.

The returned object may be displayed using by typing its name, e.g., anim or saved as a GIF file using anim_save, e.g., gganimate::anim_save("anscombe.gif", anim).

See Also

anscombise modifies a dataset so that it shares sample summary statistics with Anscombe's quartet.

input_datasets: input1 to input8 for some input datasets of the same size as those in Anscombe's quartet.

Examples

# Animate some Anscombe-like datasets produced using input1 to input8
x <- list(input1, input2, input3, input4, input5, input6, input7, input8)
idem <- c(FALSE, TRUE, FALSE, FALSE, FALSE, TRUE, FALSE, FALSE)
anim <- anscombise_gif(x, idempotent = idem)

Internal anscombiser functions

Description

Internal anscombiser functions

Usage

is_wholenumber(x, tol = .Machine$double.eps^0.5)

is_pos_def(x, tol = 1e-06)

minimal_rotation(x)

make_stats(x, stats, idempotent = FALSE)

Details

These functions are not intended to be called by the user.

Calculate Anscombe's summary statistics

Description

Calculates a particular set of summary statistics for a dataset.

Usage

get_stats(x)

Arguments

Value

A named list of summary statistics containing

• n The sample size.

• means The sample means of each variable.

• variances The sample means of each variable.

• correlation The sample correlation matrix.

• intercepts,slopes,rsquared Matrices whose (i,j)th entries are the estimated regression coefficients in a regression of x[, i] on x[, j] and the resulting coefficient of determination R^2.

Examples

get_stats(anscombe[, c(1, 5)])

Input datasets for use by anscombise()

Description

Provides input datasets from which anscombe will produce transformed datasets that behave like Anscombe's quartet of datasets, that is, with the same traditional statistical properties but different general behaviours. Use plot(input1), for example, to see the behaviours of the datasets.

Usage

input1

input2

input3

input4

input5

input6

input7

input8

Format

All datasets are objects of class matrix (inherits from array) with 11 rows and 2 columns.

Source

None. Created for use in 'anscombiser'.

References

Anscombe, Francis J. (1973). Graphs in statistical analysis. The American Statistician, 27, 17-21. doi:10.2307/2682899

Extract longitude and latitude values

Description

Extracts longitude and latitude values for a particular region from the world map supplied by the maps package.

Usage

mapdata(region = ".", map = "world", exact = FALSE, ...)

Arguments

Value

A dataframe with two columns: long and lat for longitude and latitude.

Examples

See the examples in mimic.

Modify a dataset to mimic another dataset

Description

Modifies a dataset x so that it shares sample summary statistics with a target dataset x2.

Usage

mimic(x, x2, idempotent = TRUE, ...)

Arguments

Details

The input dataset x is modified by shifting, scaling and rotating it so that its sample mean and covariance matrix match those of the target dataset x2.

The rotation is based on the square root of the sample correlation matrix. If idempotent = FALSE then this square root is based on the Cholesky decomposition this matrix, using chol. If idempotent = TRUE the square root is based on the spectral decomposition of this matrix, using the output from eigen. This is a minimal rotation square root, which means that if the input data x already have the exactly/approximately the required summary statistics then the returned dataset is exactly/approximately the same as the target dataset x2.

Value

An object of class c("anscombe", "matrix", "array") with plot and print methods. This returned dataset has the following summary statistics in common with x2.

• The sample means of each variable.

• The sample variances of each variable.

• The sample correlation matrix.

• The estimated regression coefficients from least squares linear regressions of each variable on each other variable.

The target and new summary statistics are returned as attributes old_stats and new_stats. If x2 is supplied then it is returned as a attribute old_data.

See Also

anscombise modifies a dataset so that it shares sample summary statistics with Anscombe's quartet.

Examples

### 2D examples

# The UK and a dinosaur
got_maps <- requireNamespace("maps", quietly = TRUE)
got_datasauRus <- requireNamespace("datasauRus", quietly = TRUE)
if (got_maps && got_datasauRus) {
  library(maps)
  library(datasauRus)
  dino <- datasaurus_dozen_wide[, c("dino_x", "dino_y")]
  UK <- mapdata("UK")
  new_UK <- mimic(UK, dino)
  plot(new_UK)
}

# Trump and a dinosaur
if (got_datasauRus) {
  library(datasauRus)
  dino <- datasaurus_dozen_wide[, c("dino_x", "dino_y")]
  new_dino <- mimic(dino, trump)
  plot(new_dino)
}

## Examples of passing summary statistics

# The default is zero mean, unit variance and no correlation
new_faithful <- mimic(faithful)
plot(new_faithful)

# Change the correlation
mat <- matrix(c(1, -0.9, -0.9, 1), 2, 2)
new_faithful <- mimic(faithful, correlation = mat)
plot(new_faithful)

### A 3D example

new_randu <- mimic(datasets::randu, datasets::trees)
# The samples summary statistics are equal
get_stats(new_randu)
get_stats(datasets::trees)

Animation of several mimicking datasets

Description

Create an animation to show datasets that mimic a target dataset x2.

Usage

mimic_gif(
  x,
  x2,
  idempotent = TRUE,
  theme_name = "classic",
  ease = "cubic-in-out",
  transition_length = 3,
  state_length = 1,
  wrap = TRUE
)

Arguments

Details

For this function to work the packages ggplot2 and gganimate must be installed.

Value

An object of class c("gganim", "gg", "ggplot") with an additional attribute new_data that is a data frame with 3 variables, x, y and dataset containing the datasets output from mimc.

The returned object may be displayed using by typing its name, e.g., anim or saved as a GIF file using anim_save, e.g., gganimate::anim_save("anscombe.gif", anim).

See Also

mimic to modify a dataset to share sample summary statistics with another dataset.

input_datasets: input1 to input8 for some input datasets of the same size as those in Anscombe's quartet.

Examples

# Create 8 datasets that mimic Anscombe's first dataset
x <- list(input1, input2, input3, input4, input5, input6, input7, input8)
anim <- mimic_gif(x, anscombe1)

Plot method for objects of class "anscombe"

Description

plot method for objects inheriting from class "anscombe".

Usage

## S3 method for class 'anscombe'
plot(x, input = FALSE, stats = TRUE, digits = 3, legend_args = list(), ...)

Arguments

Details

This function is only applicable in 2 dimensions, that is, when length(attr(x, "new_stats")$means) = 2.

Value

Nothing is returned.

Examples

See the examples in anscombise and mimic.

See Also

anscombise and mimic.

Print method for objects of class "anscombe"

Description

print method for class "anscombe".

Usage

## S3 method for class 'anscombe'
print(x, ...)

Arguments

Details

Just extracts the new dataset from x and prints it using print.

Value

The argument x, invisibly.

See Also

anscombise and mimic

Create a list of summary statistics

Description

Creates a list of summary statistics to pass to mimic.

Usage

set_stats(d = 2, means = 0, variances = 1, correlation = diag(2))

Arguments

Details

The vectors means and variances are recycled using rep_len to have length d.

Value

A list containing the following components.

• means a d-vector of sample means.

• variances a d-vector sample variances.

• correlation a d by d correlation matrix.

Examples

# Uncorrelated with zero means and unit variances
set_stats()
# Sample correlation = 0.9
set_stats(correlation = matrix(c(1, 0.9, 0.9, 1), 2, 2))

Donald Trump

Description

A dataset that provides an image of Donald Trump's face.

Usage

trump

Format

A matrix with 4885 rows and 2 columns: x and y.

Source

This image was created by Accentaur from the Noun Project. https://thenounproject.com/term/donald-trump/727774/

Examples

plot(trump)