Package 'clugenr'

Description

Typically, the angle between two vectors v1 and v2 can be obtained with:

acos((v1 %*% v2) / (norm(v1, "2") * norm(v2, "2")))

However, this approach is numerically unstable. The version provided here is numerically stable and based on the Angle Between Vectors Julia package by Jeffrey Sarnoff (MIT license), implementing an algorithm provided by Prof. W. Kahan in these notes (see page 15).

Usage

angle_btw(v1, v2)

Arguments

Value

Angle between v1 and v2 in radians.

Examples

angle_btw(c(1.0, 1.0, 1.0, 1.0), c(1.0, 0.0, 0.0, 0.0)) * 180 / pi

Description

Determine the angles between the average cluster direction and the cluster-supporting lines. These angles are obtained from a wrapped normal distribution (\(\mu=0\), \(\sigma=\) angle_disp ) with support in the interval \(\left[-\pi/2,\pi/2\right]\). Note this is different from the standard wrapped normal distribution, the support of which is given by the interval \(\left[-\pi,\pi\right]\).

Usage

angle_deltas(num_clusters, angle_disp)

Arguments

Value

Angles between the average cluster direction and the cluster-supporting lines, given in radians in the interval \(\left[-\pi/2,\pi/2\right]\)

Note

This function is stochastic. For reproducibility set a PRNG seed with set.seed.

Examples

set.seed(123)
arad <- angle_deltas(4, pi / 8) # Angle dispersion of 22.5 degrees
arad                            # What angles deltas did we get?
arad * 180 / pi                 #  Show angle deltas in degrees

Description

Determine cluster centers using the uniform distribution, taking into account the number of clusters (num_clusters) and the average cluster separation (clu_sep).

More specifically, let \(c=\)num_clusters, \(\mathbf{s}=\)clu_sep, \(\mathbf{o}=\)clu_offset, \(n=\)length(clu_sep) (i.e., number of dimensions). Cluster centers are obtained according to the following equation:

where \(\mathbf{C}\) is the \(c \times n\) matrix of cluster centers, \(\mathbf{U}\) is an \(c \times n\) matrix of random values drawn from the uniform distribution between -0.5 and 0.5, and \(\mathbf{1}\) is an \(c \times 1\) vector with all entries equal to 1.

Usage

clucenters(num_clusters, clu_sep, clu_offset)

Arguments

Value

A \(c \times n\) matrix containing the cluster centers.

Note

This function is stochastic. For reproducibility set a PRNG seed with set.seed.

Examples

set.seed(321)
clucenters(3, c(30, 10), c(-50,50))

Description

This is the main function of clugenr, and possibly the only function most users will need.

Usage

clugen(
  num_dims,
  num_clusters,
  num_points,
  direction,
  angle_disp,
  cluster_sep,
  llength,
  llength_disp,
  lateral_disp,
  allow_empty = FALSE,
  cluster_offset = NA,
  proj_dist_fn = "norm",
  point_dist_fn = "n-1",
  clusizes_fn = clusizes,
  clucenters_fn = clucenters,
  llengths_fn = llengths,
  angle_deltas_fn = angle_deltas,
  seed = NA
)

Arguments

Details

If a custom function was given in the clusizes_fn parameter, it is possible that num_points may have a different value than what was specified in the num_points parameter.

The terms "average" and "dispersion" refer to measures of central tendency and statistical dispersion, respectively. Their exact meaning depends on the optional arguments.

Value

A named list with the following elements:

• points: A num_points x num_dims matrix with the generated points for all clusters.

• clusters: A num_points factor vector indicating which cluster each point in points belongs to.

• projections: A num_points x num_dims matrix with the point projections on the cluster-supporting lines.

• sizes: A num_clusters x 1 vector with the number of points in each cluster.

• centers: A num_clusters x num_dims matrix with the coordinates of the cluster centers.

• directions: A num_clusters x num_dims matrix with the final direction of each cluster-supporting line.

• angles: A num_clusters x 1 vector with the angles between the cluster-supporting lines and the main direction.

• lengths: A num_clusters x 1 vector with the lengths of the cluster-supporting lines.

Note

This function is stochastic. For reproducibility set a PRNG seed with set.seed.

Examples

# 2D example
x <- clugen(2, 5, 1000, c(1, 3), 0.5, c(10, 10), 8, 1.5, 2)
graphics::plot(x$points, col = x$clusters, xlab = "x", ylab = "y", asp = 1)
# 3D example
x <- clugen(3, 5, 1000, c(2, 3, 4), 0.5, c(15, 13, 14), 7, 1, 2)

Description

Merges the fields (specified in fields) of two or more data sets (passed as lists). The fields to be merged need to have the same number of columns. The corresponding merged field will contain the rows of the fields to be merged, and will have a common "supertype".

Usage

clumerge(..., fields = c("points", "clusters"), clusters_field = "clusters")

Arguments

Details

The clusters_field parameter specifies a field containing integers that identify the cluster to which the respective points belongs to. If clusters_field is specified (by default it's specified as "clusters"), cluster assignments in individual datasets will be updated in the merged dataset so that clusters are considered separate. This parameter can be set to NA, in which case no field will be considered as a special cluster assignments field.

This function can be used to merge data sets generated with the clugen function, by default merging the points and clusters fields in those data sets. It also works with arbitrary data by specifying alternative fields in the fields parameter. It can be used, for example, to merge third-party data with clugen-generated data.

Value

A list whose fields consist of the merged fields in the original data sets.

Examples

a <- clugen(2, 5, 100, c(1, 3), 0.5, c(10, 10), 8, 1.5, 2)
b <- clugen(2, 3, 250, c(-1, 3), 0.5, c(13, 14), 7, 1, 2)
ab <- clumerge(a, b)

Description

Each point is placed around its projection using the normal distribution (\(\mu=0\), \(\sigma=\) lat_disp ).

Usage

clupoints_n(projs, lat_disp, line_len, clu_dir, clu_ctr)

Arguments

Details

This function's main intended use is by the main clugen function, generating the final points when the point_dist_fn parameter is set to "n".

Value

Generated points (\(p \times n\) matrix).

Note

This function is stochastic. For reproducibility set a PRNG seed with set.seed.

Examples

set.seed(123)
ctr <- c(0, 0)
dir <- c(1, 0)
pdist <- c(-0.5, -0.2, 0.1, 0.3)
proj <- points_on_line(ctr, dir, pdist)
clupoints_n(proj, 0.01, NA, dir, NA)

Description

Each point is placed on a hyperplane orthogonal to that line and centered at the point's projection, using the normal distribution (\(\mu=0\), \(\sigma=\) lat_disp ).

Usage

clupoints_n_1(projs, lat_disp, line_len, clu_dir, clu_ctr)

Arguments

Details

This function's main intended use is by the main clugen function, generating the final points when the point_dist_fn parameter is set to "n-1".

Value

Generated points (\(p \times n\) matrix).

Note

This function is stochastic. For reproducibility set a PRNG seed with set.seed.

Examples

set.seed(123)
ctr <- c(0, 0)
dir <- c(1, 0)
pdist <- c(-0.5, -0.2, 0.1, 0.3)
proj <- points_on_line(ctr, dir, pdist)
clupoints_n_1(proj, 0.1, NA, dir, NA)

Description

Generate points from their \(n\)-dimensional projections on a cluster-supporting line, placing each point on a hyperplane orthogonal to that line and centered at the point's projection. The function specified in dist_fn is used to perform the actual placement.

Usage

clupoints_n_1_template(projs, lat_disp, clu_dir, dist_fn)

Arguments

Details

This function is used internally by clupoints_n_1 and may be useful for constructing user-defined final point placement strategies for the point_dist_fn parameter of the main clugen function.

Value

Generated points (\(p \times n\) matrix).

Note

This function is stochastic. For reproducibility set a PRNG seed with set.seed.

Examples

set.seed(123)
ctr <- c(0, 0)
dir <- c(1, 0)
pdist <- c(-0.5, -0.2, 0.1, 0.3)
proj <- points_on_line(ctr, dir, pdist)
clupoints_n_1_template(proj, 0, dir, function(p, l) stats::runif(p))

Description

Cluster sizes are determined using the normal distribution (\(\mu=\) num_points \(/\) num_clusters, \(\sigma=\mu/3\)), and then assuring that the final cluster sizes add up to num_points via the fix_num_points function.

Usage

clusizes(num_clusters, num_points, allow_empty)

Arguments

Value

Number of points in each cluster (vector of length num_clusters).

Note

This function is stochastic. For reproducibility set a PRNG seed with set.seed.

Examples

set.seed(123)
sizes <- clusizes(4, 1000, TRUE)
sizes
sum(sizes)

Description

Certifies that, given enough points, no clusters are left empty. This is done by removing a point from the largest cluster and adding it to an empty cluster while there are empty clusters. If the total number of points is smaller than the number of clusters (or if the allow_empty parameter is set to TRUE), this function does nothing.

Usage

fix_empty(clu_num_points, allow_empty = FALSE)

Arguments

Details

This function is used internally by clusizes and might be useful for custom cluster sizing implementations given as the clusizes_fn parameter of the main clugen function.

Value

Number of points in each cluster, after being fixed by this function (vector of size \(c\)).

Examples

clusters <- c(3, 4, 5, 0, 0)    # A vector with some empty elements
clusters <- fix_empty(clusters) # Apply this function
clusters                        # Check that there's no more empty elements

Description

Certifies that the values in the clu_num_points array, i.e. the number of points in each cluster, add up to num_points. If this is not the case, the clu_num_points array is modified in-place, incrementing the value corresponding to the smallest cluster while sum(clu_num_points) < num_points, or decrementing the value corresponding to the largest cluster while sum(clu_num_points) > num_points.

Usage

fix_num_points(clu_num_points, num_points)

Arguments

Details

This function is used internally by clusizes and might be useful for custom cluster sizing implementations given as the clusizes_fn parameter of the main clugen function.

Value

Number of points in each cluster, after being fixed by this function.

Examples

clusters <- c(1, 6, 3)                   # 10 total points
clusters <- fix_num_points(clusters, 12) # But we want 12 total points
clusters                                 # Check that we now have 12 points

Description

Line lengths are determined using the folded normal distribution (\(\mu=\) llength, \(\sigma=\) llength_disp ).

Usage

llengths(num_clusters, llength, llength_disp)

Arguments

Value

Lengths of cluster-supporting lines (vector of size num_clusters).

Note

This function is stochastic. For reproducibility set a PRNG seed with set.seed.

Examples

set.seed(123)
llengths(4, 20, 3.5)

Description

Determine coordinates of points on a line with center and direction, based on the distances from the center given in dist_center.

This works by using the vector formulation of the line equation assuming direction is a \(n\)-dimensional unit vector. In other words, considering \(\mathbf{d}=\) as.matrix(direction) (\(n \times 1\) vector), \(\mathbf{c}=\) as.matrix(center) (\(n \times 1\) vector), and \(\mathbf{w}=\) as.matrix(dist_center) (\(p \times 1\) vector), the coordinates of points on the line are given by:

where \(\mathbf{P}\) is the \(p \times n\) matrix of point coordinates on the line, and \(\mathbf{1}\) is a \(p \times 1\) vector with all entries equal to 1.

Usage

points_on_line(center, direction, dist_center)

Arguments

Value

Coordinates of points on the specified line (\(p \times n\) matrix).

Examples

points_on_line(c(5, 5), c(1, 0), seq(-4, 4, length.out=5)) # 2D, 5 points

points_on_line(c(-2, 0, 0, 2), c(0, 0, -1, 0), c(10, -10)) # 4D, 2 points

Description

Get a random unit vector orthogonal to u.

Usage

rand_ortho_vector(u)

Arguments

Value

A random unit vector orthogonal to u.

Note

This function is stochastic. For reproducibility set a PRNG seed with set.seed.

Examples

r <- stats::runif(3)      # Get a random 3D vector
r <- r / norm(r, "2")     # Normalize it
o <- rand_ortho_vector(r) # Get a random unit vector orthogonal to r
r %*% o            # Check that r and o are orthogonal (result should be ~0)

Description

Get a random unit vector with num_dims components.

Usage

rand_unit_vector(num_dims)

Arguments

Value

A random unit vector with num_dims components.

Note

This function is stochastic. For reproducibility set a PRNG seed with set.seed.

Examples

r <- rand_unit_vector(4)
norm(r, "2")

Description

Get a random unit vector which is at angle radians of vector u. Note that u is expected to be a unit vector itself.

Usage

rand_vector_at_angle(u, angle)

Arguments

Value

Random unit vector with \(n\) components which is at angle radians with vector u.

Note

This function is stochastic. For reproducibility set a PRNG seed with set.seed.

Examples

u <- c(1.0, 0, 0.5, -0.5)            # Define a 4D vector
u <- u / norm(u, "2")                # Normalize the vector
v <- rand_vector_at_angle(u, pi / 4) # Get a vector at 45 degrees
arad <- acos((u %*% v) / norm(u,"2") * norm(v, "2")) # Get angle in radians
arad * 180 / pi # Convert to degrees, should be close to 45 degrees