Package 'ClusTorus' reference manual

Title:	Prediction and Clustering on the Torus by Conformal Prediction
Description:	Provides various tools of for clustering multivariate angular data on the torus. The package provides angular adaptations of usual clustering methods such as the k-means clustering, pairwise angular distances, which can be used as an input for distance-based clustering algorithms, and implements clustering based on the conformal prediction framework. Options for the conformal scores include scores based on a kernel density estimate, multivariate von Mises mixtures, and naive k-means clusters. Moreover, the package provides some basic data handling tools for angular data.
Authors:	Sungkyu Jung [aut, cph], Seungki Hong [aut, cre], Kiho Park [ctb], Byungwon Kim [ctb]
Maintainer:	Seungki Hong <[email protected]>
License:	GPL-3
Version:	0.2.4
Built:	2025-02-07 04:28:02 UTC
Source:	https://github.com/sungkyujung/clustorus

Angular distance

Description

ang.dist computes element-wise angular distance between two angular values in $[0,2\pi)$ .

Usage

ang.dist(x, y)
ang.dist(x, y)

Arguments

x, y

angular data(both scalar or vector) whose elements are in $[0, 2\pi)$

Value

angular data (scalar or vector) whose elements are in $[0, 2\pi)$

References

Jung, S., Park, K., & Kim, B. (2021). Clustering on the torus by conformal prediction. The Annals of Applied Statistics, 15(4), 1583-1603.

Examples

x <- c(pi/3, 0)
y <- c(pi/4, pi/2)

ang.dist(x, y)
x <- c(pi/3, 0)
y <- c(pi/4, pi/2)

ang.dist(x, y)

Angular subtraction

Description

ang.minus computes element-wise angular subtraction defined as

$x - y := Arg(exp(i(x-y)))$

Usage

ang.minus(x, y)
ang.minus(x, y)

Arguments

x, y

angular data(scalar or vector) whose elements are in $[0, 2\pi)$

Value

returns a scalar or a vector whose elements are in $[-\pi, \pi)$ .

References

Jung, S., Park, K., & Kim, B. (2021). Clustering on the torus by conformal prediction. The Annals of Applied Statistics, 15(4), 1583-1603.

Examples

x <- c(pi/2, 0)
y <- c(pi, pi/3)

ang.minus(x, y)
x <- c(pi/2, 0)
y <- c(pi, pi/3)

ang.minus(x, y)

Pairwise L2 angular distance

Description

ang.pdist computes pairwise angular distances matrix.

Usage

ang.pdist(data)
ang.pdist(data)

Arguments

data

n x d angular data on $[0, 2\pi)^d$

Value

ang.pdist returns pairwise angular distances matrix with the class dist

References

Jung, S., Park, K., & Kim, B. (2021). Clustering on the torus by conformal prediction. The Annals of Applied Statistics, 15(4), 1583-1603.

Examples

data <- matrix(c(pi/3, pi/3, pi/2,
                 pi, pi/4, pi/2,
                 0, pi/3, pi/6),
               ncol = 3, byrow = TRUE)

ang.pdist(data)
data <- matrix(c(pi/3, pi/3, pi/2,
                 pi, pi/4, pi/2,
                 0, pi/3, pi/6),
               ncol = 3, byrow = TRUE)

ang.pdist(data)

Clustering on the torus by conformal prediction

Description

clus.torus returns clustering results of data on the torus based on inductive conformal prediction set

Usage

clus.torus(
  data,
  split.id = NULL,
  model = c("kmeans", "mixture"),
  mixturefitmethod = c("axis-aligned", "circular", "general"),
  kmeansfitmethod = c("general", "homogeneous-circular", "heterogeneous-circular",
    "ellipsoids"),
  J = NULL,
  level = NULL,
  option = NULL,
  verbose = TRUE,
  ...
)

## S3 method for class 'clus.torus'
plot(
  x,
  panel = 1,
  assignment = "outlier",
  data = NULL,
  ellipse = TRUE,
  type = NULL,
  overlay = FALSE,
  out = FALSE,
  ...
)
clus.torus(
  data,
  split.id = NULL,
  model = c("kmeans", "mixture"),
  mixturefitmethod = c("axis-aligned", "circular", "general"),
  kmeansfitmethod = c("general", "homogeneous-circular", "heterogeneous-circular",
    "ellipsoids"),
  J = NULL,
  level = NULL,
  option = NULL,
  verbose = TRUE,
  ...
)

## S3 method for class 'clus.torus'
plot(
  x,
  panel = 1,
  assignment = "outlier",
  data = NULL,
  ellipse = TRUE,
  type = NULL,
  overlay = FALSE,
  out = FALSE,
  ...
)

Arguments

`data`	n x d matrix of toroidal data on $[0, 2\pi)^d$ or $[-\pi, \pi)^d$ . Default is `NULL`.
`split.id`	a n-dimensional vector consisting of values 1 (estimation) and 2(evaluation)
`model`	A string. One of "mixture" and "kmeans" which determines the model or estimation methods. If "mixture", the model is based on the von Mises mixture, fitted with an EM algorithm. It supports the von Mises mixture and its variants based conformity scores. If "kmeans", the model is also based on the von Mises mixture, but the parameter estimation is implemented with the elliptical k-means algorithm. It supports the log-max-mixture based conformity score only. If the dimension of data space is greater than 2, only "kmeans" is supported. Default is `model = "kmeans"`.
`mixturefitmethod`	A string. One of "circular", "axis-aligned", and "general" which determines the constraint of the EM fitting. Default is "axis-aligned". This argument only works for `model = "mixture"`.
`kmeansfitmethod`	A string. One of "general", ellipsoids", "heterogeneous-circular" or "homogeneous-circular". If "general", the elliptical k-means algorithm with no constraint is used. If "ellipsoids", only the one iteration of the algorithm is used. If"heterogeneous-circular", the same as above, but with the constraint that ellipsoids must be spheres. If "homogeneous-circular", the same as above but the radii of the spheres are identical. Default is "general". This argument only works for `model = "kmeans"`.
`J`	the number of components for mixture model fitting. If `J` is a vector, then `hyperparam.torus` is used to choose optimal `J`. If `J == NULL`, then `J = 4:30` is used.
`level`	a scalar in $[0,1]$ . The level of the conformal prediction set used for clustering. If `level == NULL`, then `hyperparam.alpha` is used to choose optimal `level`
`option`	A string. One of "elbow", "risk", "AIC", or "BIC", which determines the criterion for the model selection. "risk" is based on the negative log-likelihood, "AIC" for the Akaike Information Criterion, and "BIC" for the Bayesian Information Criterion. "elbow" is based on minimizing the criterion used in Jung et. al.(2021). This argument is only used if `J` is a vector or `NULL`.
`verbose`	boolean index, which indicates whether display additional details as to what the algorithm is doing or how many loops are done. Default is `TRUE`.
`...`	Further arguments that will be passed to `icp.torus` and `hyperparam.torus`
`x`	`clus.torus` object
`panel`	One of 1 or 2 which determines the type of plot. If `panel = 1`, `x$cluster.obj` will be plotted, if `panel = 2`, `x$icp.torus` will be plotted. If `panel = 3`, `x$hyperparam.select` will be plotted. Default is `panel = 1`.
`assignment`	A string. One of "outlier", "log.density", "posterior", "mahalanobis". Default is "outlier".
`ellipse`	A boolean index which determines whether plotting ellipse-intersections. Default is `TRUE`. Only available for `panel = 2`.
`type`	A string. One of "mix", "max" or "e". This argument is only available if `icp.torus` object is fitted with `model = "mixture"`. Default is `NULL`. If `type != NULL`, argument `ellipse` automatically becomes `FALSE`. If "mix", it plots based on von Mises mixture. If "max", it plots based on von Mises max-mixture. If "e", it plots based on ellipse-approximation.
`overlay`	A boolean index which determines whether plotting ellipse-intersections on clustering plots. Default is `FALSE`. Only available for `panel = 1`.
`out`	An option for returning the ggplot object. Default is `FALSE`.

Details

clus.torus is a user-friendly all-in-one function which implements following procedures automatically: 1. compute conformity scores for given model and fitting method, 2. choose optimal model and level based on prespecified criterion, and 3. make clusters based on the chosen model and level. Procedure 1-3 can be independently done with icp.torus, hyperparam.torus, hyperparam.J, hyperparam.alpha and cluster.assign.torus. If you want to see more detail for each procedure, please see icp.torus, hyperparam.J, hyperparam.alpha hyperparam.torus, cluster.assign.torus.

Value

clus.torus returns a clus.torus object, which consists of following 3 different S3 objects;

cluster.obj: cluster.obj object; clustering assignment results for several methods. For detail, see cluster.assign.torus.
icp.torus: icp.torus object; containing model parameters and conformity scores. For detail, see icp.torus.
hyperparam.select: hyperparam.torus object (if J = NULL or a sequence of numbers, and level = NULL or a sequence of numbers), hyperparam.J object (if level is a scalar), or hyperparam.alpha object (if J is a scalar); contains information for the optimally chosen model (number of components J) and level (alpha) based on prespecified criterion. For detail, see hyperparam.torus, hyperparam.J, and hyperparam.alpha.

References

Jung, S., Park, K., & Kim, B. (2021). Clustering on the torus by conformal prediction. The Annals of Applied Statistics, 15(4), 1583-1603.

Mardia, K. V., Kent, J. T., Zhang, Z., Taylor, C. C., & Hamelryck, T. (2012). Mixtures of concentrated multivariate sine distributions with applications to bioinformatics. Journal of Applied Statistics, 39(11), 2475-2492.

Shin, J., Rinaldo, A., & Wasserman, L. (2019). Predictive clustering. arXiv preprint arXiv:1903.08125.

Examples


data <- toydata2[, 1:2]
n <- nrow(data)
clus.torus(data = data, model = "kmeans", kmeansfitmethod = "general", J = 5:30, option = "risk")

data <- toydata2[, 1:2]
n <- nrow(data)
clus.torus(data = data, model = "kmeans", kmeansfitmethod = "general", J = 5:30, option = "risk")

Clustering by connected components of ellipsoids

Description

cluster.assign.torus returns clustering assignment for data given icp.torus objects, which can be constructed with icp.torus.

plot.clus.torus plots clustering results, which is given by cluster.obj object, with some options.

Usage

cluster.assign.torus(icp.object, data = NULL, level = NULL)

## S3 method for class 'cluster.obj'
plot(
  x,
  assignment = c("outlier", "log.density", "posterior", "mahalanobis"),
  overlay = FALSE,
  out = FALSE,
  ...
)
cluster.assign.torus(icp.object, data = NULL, level = NULL)

## S3 method for class 'cluster.obj'
plot(
  x,
  assignment = c("outlier", "log.density", "posterior", "mahalanobis"),
  overlay = FALSE,
  out = FALSE,
  ...
)

Arguments

`icp.object`	an object must be an `icp.torus` object, which contains all values to compute the conformity score constructed with `icp.torus`, or a `hyperparam.torus` object which is generated by `hyperparam.torus`.
`data`	n x d matrix of toroidal data on $[0, 2\pi)^d$ or $[-\pi, \pi)^d$ . If `data = NULL`, then data within the `icp.object` is used.
`level`	a scalar in $[0,1]$ . If argument `icp.object` is an `icp.torus` object, the default value for `level` is 0.1. If argument `icp.object` is a `hyperparam.torus` object and `level = NULL`, then `level` is set as the optimal level `hyperparam.torus$alphahat`.
`x`	`cluster.obj` object
`assignment`	A string. One of "outlier", "log.density", "posterior", "mahalanobis". Default is "outlier".
`overlay`	A boolean index which determines whether plotting ellipse-intersections on clustering plots. Default is `FALSE`.
`out`	An option for returning the ggplot object. Default is `FALSE`.
`...`	additional parameter for ggplot2::ggplot()

Value

clustering assignment for data, given icp.torus objects

cluster.id.by.log.density: cluster assignment result based on approximate log-density.
cluster.id.by.posterior: cluster assignment result based on the posterior probability.
cluster.id.outlier: cluster assignment result which regards data not included in conformal prediction set as outliers.
cluster.id.by.Mah.dist: cluster assignment result based on Mahalanobis distance.
level: used level which determines the size of clusters(conformal prediction set).
data: input data which are assigned to each cluster.
icp.torus: icp.torus object which is used for cluster assignment.

References

Jung, S., Park, K., & Kim, B. (2021). Clustering on the torus by conformal prediction. The Annals of Applied Statistics, 15(4), 1583-1603.

Gilitschenski, I., & Hanebeck, U. D. (2012, July). A robust computational test for overlap of two arbitrary-dimensional ellipsoids in fault-detection of kalman filters. In 2012 15th International Conference on Information Fusion (pp. 396-401). IEEE.

Examples

data <- toydata1[, 1:2]
icp.torus <- icp.torus(data, model = "kmeans",
                       kmeansfitmethod = "general",
                       J = 4, concentration = 25)
level <- 0.1

cluster.assign.torus(icp.torus, level = level)
data <- toydata1[, 1:2]
icp.torus <- icp.torus(data, model = "kmeans",
                       kmeansfitmethod = "general",
                       J = 4, concentration = 25)
level <- 0.1

cluster.assign.torus(icp.torus, level = level)

Conformal prediction set indices with kernel density estimation

Description

cp.torus.kde computes conformal prediction set indices (TRUE if in the set) using kernel density estimation as conformity score.

Usage

cp.torus.kde(data, eval.point = grid.torus(), level = 0.1, concentration = 25)

## S3 method for class 'cp.torus.kde'
plot(x, level.id = 1, ...)
cp.torus.kde(data, eval.point = grid.torus(), level = 0.1, concentration = 25)

## S3 method for class 'cp.torus.kde'
plot(x, level.id = 1, ...)

Arguments

`data`	n x d matrix of toroidal data on $[0, 2\pi)^d$
`eval.point`	N x N numeric matrix on $[0, 2\pi)^d$ . Default input is `NULL`, which represents the fine grid points on $[0, 2\pi)^d$ .
`level`	either a scalar or a vector, or even `NULL`. Default value is 0.1.
`concentration`	positive number which has the role of $\kappa$ of von Mises distribution. Default value is 25.
`x`	`cp.torus.kde` object
`level.id`	an integer among `1:length(cp.torus$level)`.
`...`	additional parameter for ggplot2::ggplot()

Value

If level is NULL, then return kde at eval.point and at data points.

If level is a vector, return the above and prediction set indices for each value of level.

References

Jung, S., Park, K., & Kim, B. (2021). Clustering on the torus by conformal prediction. The Annals of Applied Statistics, 15(4), 1583-1603.

Di Marzio, M., Panzera, A., & Taylor, C. C. (2011). Kernel density estimation on the torus. Journal of Statistical Planning and Inference, 141(6), 2156-2173.

Examples

data <- ILE[1:200, 1:2]
cp.torus.kde(data, eval.point = grid.torus(),
             level = 0.05, concentration = 25)
data <- ILE[1:200, 1:2]
cp.torus.kde(data, eval.point = grid.torus(),
             level = 0.05, concentration = 25)

6VXX: Structure of the SARS-CoV-2 spike glycoprotein(closed state)

Description

The torsion angle dataset of SARS-CoV-2 spike glycopreotein.

Usage

data_6VXX
data_6VXX

Format

data_6VXX consists of following informations:

phi: main chain torsion angle for atoms C,N,CA,C.
psi: main chain torsion angle for atoms N,CA,C,N.
omega: main chain torsion angle for atoms CA,C,N,CA.
alpha: virtual torsion angle between consecutive C-alpha atoms.
chi1: side chain torsion angle for atoms N,CA,CB,*G.
chi2: side chain torsion angle for atoms CA,CB,*G,*D.
chi3: side chain torsion angle for atoms CB,*G,*D,*E.
chi4: side chain torsion angle for atoms *G,*D,*E,*Z.
chi5: side chain torsion angle for atoms *D,*E,*Z, NH1.
coords: numeric matrix of ‘justified’ coordinates.
tbl: a numeric matrix of psi, phi and chi torsion angles.

Source

This data can be downloaded in https://www.rcsb.org/structure/6VXX, or with using R package bio3d. Precisely, we use the code: bio3d::torsion.pdb(bio3d::read.pdb("6vxx"))

References

Walls, A. C., Park, Y. J., Tortorici, M. A., Wall, A., McGuire, A. T., & Veesler, D. (2020). Structure, function, and antigenicity of the SARS-CoV-2 spike glycoprotein. Cell, 181(2), 281-292. Retrived from https://www.wwpdb.org/pdb?id=pdb_00006vxx

K-Means Clustering to K-Spheres Clustering on Torus

Description

ellip.kmeans.torus prepares the parameters for conformity scores which are derived by k-means clustering on torus.

Usage

ellip.kmeans.torus(
  data,
  centers = 10,
  type = c("homogeneous-circular", "heterogeneous-circular", "ellipsoids", "general"),
  init = c("kmeans", "hierarchical"),
  d = NULL,
  additional.condition = TRUE,
  THRESHOLD = 1e-10,
  maxiter = 200,
  verbose = TRUE,
  ...
)
ellip.kmeans.torus(
  data,
  centers = 10,
  type = c("homogeneous-circular", "heterogeneous-circular", "ellipsoids", "general"),
  init = c("kmeans", "hierarchical"),
  d = NULL,
  additional.condition = TRUE,
  THRESHOLD = 1e-10,
  maxiter = 200,
  verbose = TRUE,
  ...
)

Arguments

`data`	data n x d matrix of toroidal data on $[0, 2\pi)^d$
`centers`	either the number of clusters or a set of initial cluster centers. If a number, a random set of row in x is chosen as the initial centers.
`type`	character which must be "homogeneous-circular", "heterogeneous-circular", or "general". If "homogeneous-circular", the radii of k-spheres are identical. If "heterogeneous-circular", the radii of k-spheres may be different. If "ellipsoids", cluster with k-ellipsoids without optimized parameters. If, "general", clustering with k-ellipsoids. The parameters to construct the ellipses are optimized with elliptical k-means algorithm, which is modified for toroidal space. See references for the detail. Default is "homogeneous-circular".
`init`	determine the initial parameter for option "general". Must be "kmeans" or "hierarchical". If "kmeans", the initial parameters are obtained with extrinsic kmeans method. If "hierarchical", the initial parameters are obtained with hierarchical clustering method. Default is "hierarchical".
`d`	pairwise distance matrix(`dist` object) for `init = "hierarchical"`, which used in hierarchical clustering. If `init = "hierarchical"` and `d = NULL`, `d` will be automatically filled with `ang.pdist(data)`.
`additional.condition`	boolean index. If `TRUE`, a singular matrix will be altered to the scalar identity.
`THRESHOLD`	number of threshold for difference between updating and updated parameters. Default is 1e-10.
`maxiter`	the maximal number of iteration. Default is 200.
`verbose`	boolean index, which indicates whether display additional details as to what the algorithm is doing or how many loops are done. Default is `TRUE`.
`...`	Further arguments for argument `init`. If `init = "kmeans"`, these are for `kmeans`. If `init = "hierarchical"`, there are for `hclust`.

Value

returns a list, containing all values which determines the shape and location of spheres.

References

Jung, S., Park, K., & Kim, B. (2021). Clustering on the torus by conformal prediction. The Annals of Applied Statistics, 15(4), 1583-1603.

Shin, J., Rinaldo, A., & Wasserman, L. (2019). Predictive clustering. arXiv preprint arXiv:1903.08125.

Examples

data <- ILE[1:200, 1:2]

ellip.kmeans.torus(data, centers = 3, type = "general", init = "hierarchical")
data <- ILE[1:200, 1:2]

ellip.kmeans.torus(data, centers = 3, type = "general", init = "hierarchical")

Fitting mixtures of bivariate von Mises distribution

Description

EMsinvMmix returns fitted parameters of J-mixture of bivariate sine von Mises distributions.

Usage

EMsinvMmix(
  data,
  J = 4,
  parammat = EMsinvMmix.init(data, J),
  THRESHOLD = 1e-10,
  maxiter = 100,
  type = c("circular", "axis-aligned", "general"),
  kmax = 500,
  verbose = TRUE
)
EMsinvMmix(
  data,
  J = 4,
  parammat = EMsinvMmix.init(data, J),
  THRESHOLD = 1e-10,
  maxiter = 100,
  type = c("circular", "axis-aligned", "general"),
  kmax = 500,
  verbose = TRUE
)

Arguments

`data`	n x 2 matrix of toroidal data on $[0, 2\pi)^2$
`J`	number of components of mixture density
`parammat`	6 x J parameter data with the following components: `parammat[1, ]` : the weights for each von Mises sine density `parammat[n + 1, ]` : $\kappa_n$ for each von Mises sine density for n = 1, 2, 3 `parammat[m + 4, ]` : $\mu_m$ for each von Mises sine density for m = 1, 2
`THRESHOLD`	number of threshold for difference between updating and updated parameters.
`maxiter`	the maximal number of iteration.
`type`	a string one of "circular", "axis-aligned", "general", and "Bayesian" which determines the fitting method.
`kmax`	the maximal number of kappa. If estimated kappa is larger than `kmax`, then put kappa as `kmax`.
`verbose`	boolean index, which indicates whether display additional details as to what the algorithm is doing or how many loops are done.

Details

This algorithm is based on ECME algorithm. That is, constructed with E - step and M - step and M - step maximizes the parameters with given type.

If type == "circular", then the mixture density is just a product of two independent von Mises.

If type == "axis-aligned", then the mixture density is the special case of type == "circular": only need to take care of the common concentration parameter.

Iftype == "general", then the fitting the mixture density is more complicated than before, check the detail of the reference article.

Value

returns approximated parameters for bivariate normal distribution with list:

list$Sigmainv[j] : approximated covariance matrix for j-th bivariate normal distribution, approximation of the j-th von Mises.

list$c[j] : approximated $|2\pi\Sigma|^{-1}$ for j-th bivariate normal distribution, approximation of the j-th von Mises.

References

Jung, S., Park, K., & Kim, B. (2021). Clustering on the torus by conformal prediction. The Annals of Applied Statistics, 15(4), 1583-1603.

Examples


data <- ILE[1:200, 1:2]

EMsinvMmix(data, J = 3,
           THRESHOLD = 1e-10, maxiter = 200,
           type = "general", kmax = 500, verbose = FALSE)

data <- ILE[1:200, 1:2]

EMsinvMmix(data, J = 3,
           THRESHOLD = 1e-10, maxiter = 200,
           type = "general", kmax = 500, verbose = FALSE)

Grid on torus

Description

grid.torus returns an equally-spaced grid on torus.

Usage

grid.torus(d = 2, grid.size = 100)
grid.torus(d = 2, grid.size = 100)

Arguments

`d`	number for dimension. Default is 2.
`grid.size`	number of grid for each axis. Default value is 100.

Value

returns (grid.size) x (grid.size) numeric matrix which indicates the grid points on torus.

Examples

grid.torus(d = 2, grid.size = 100)
grid.torus(d = 2, grid.size = 100)

Selecting optimal level based on the runs of the number of clusters

Description

hyperparam.alpha evaluates the numbers of clusters for various levels, and select the optimal level based on the runs of the cluster numbers.

Usage

hyperparam.alpha(icp.torus, alphavec = NULL, alpha.lim = 0.15)

## S3 method for class 'hyperparam.alpha'
plot(x, ...)
hyperparam.alpha(icp.torus, alphavec = NULL, alpha.lim = 0.15)

## S3 method for class 'hyperparam.alpha'
plot(x, ...)

Arguments

`icp.torus`	an object containing all values to compute the conformity score, which will be constructed with `icp.torus.score`.
`alphavec`	either a scalar or a vector, or even `NULL` for the levels. Default value is `NULL`. If `NULL`, then `alphavec` is automatically generated as a sequence from 0 to `alpha.lim`.
`alpha.lim`	a positive number lower than 1, which is the upper bound of Default is 0.15.
`x`	`hyperparam.alpha` object
`...`	additional parameter for ggplot2::ggplot()

Value

returns a hyperparam.alpha object which contains a data.frame for the numbers of clusters corresponding to the levels and the optimal level.

Examples


data <- toydata2[, 1:2]
n <- nrow(data)
split.id <- rep(2, n)
split.id[sample(n, floor(n/2))] <- 1
icp.torus <- icp.torus(data, split.id = split.id, model = "kmeans",
                       kmeansfitmethod = "ge", init = "h",
                       J = 25, verbose = TRUE)
hyperparam.alpha(icp.torus)

data <- toydata2[, 1:2]
n <- nrow(data)
split.id <- rep(2, n)
split.id[sample(n, floor(n/2))] <- 1
icp.torus <- icp.torus(data, split.id = split.id, model = "kmeans",
                       kmeansfitmethod = "ge", init = "h",
                       J = 25, verbose = TRUE)
hyperparam.alpha(icp.torus)

Selecting optimal number of mixture components based on various criteria

Description

hyperparam.J evaluates criterion for each icp.torus objects, and select the optimal number of mixture components based on the evaluated criterion.

Usage

hyperparam.J(icp.torus.objects, option = c("risk", "AIC", "BIC"))

## S3 method for class 'hyperparam.J'
plot(x, ...)
hyperparam.J(icp.torus.objects, option = c("risk", "AIC", "BIC"))

## S3 method for class 'hyperparam.J'
plot(x, ...)

Arguments

`icp.torus.objects`	a list whose elements are icp.torus objects, generated by `icp.torus`.
`option`	a string one of "risk", "AIC", or "BIC", which determines the criterion for the model selection. "risk" is based on the negative log-likelihood, "AIC" for the Akaike Information Criterion, and "BIC" for the Bayesian Information Criterion.
`x`	`hyperparam.J` object
`...`	additional parameter for ggplot2::ggplot()

Value

returns a hyperparam.J object which contains a data.frame for the evaluated criterion corresponding to each number of components, the optimal number of components, and the corresponding icp.torus object.

References

Jung, S., Park, K., & Kim, B. (2021). Clustering on the torus by conformal prediction. The Annals of Applied Statistics, 15(4), 1583-1603.

Akaike, H. (1974). A new look at the statistical model identification. IEEE transactions on automatic control, 19(6), 716-723.

Schwarz, G. (1978). Estimating the dimension of a model. The annals of statistics, 461-464.

Examples


data <- toydata1[,1:2]
n <- nrow(data)
split.id <- rep(2,n)
split.id[ sample(n,floor(n/2)) ] <- 1

Jvec = 4:20
icp.torus.objects <- icp.torus(data, split.id = split.id, model = "kmeans", J = Jvec)

hyperparam.J(icp.torus.objects, option = "AIC")
data <- toydata1[,1:2]
n <- nrow(data)
split.id <- rep(2,n)
split.id[ sample(n,floor(n/2)) ] <- 1

Jvec = 4:20
icp.torus.objects <- icp.torus(data, split.id = split.id, model = "kmeans", J = Jvec)

hyperparam.J(icp.torus.objects, option = "AIC")

Selecting optimal hyperparameters for the conformal prediction set

Description

hyperparam.torus selects optimal hyperparameters for constructing the conformal prediction set, based on the type of postulated model and the criterion.

Usage

hyperparam.torus(
  icp.torus.objects,
  option = NULL,
  alphavec = NULL,
  alpha.lim = NULL,
  eval.point = NULL
)

## S3 method for class 'hyperparam.torus'
plot(x, color = "auto", ...)
hyperparam.torus(
  icp.torus.objects,
  option = NULL,
  alphavec = NULL,
  alpha.lim = NULL,
  eval.point = NULL
)

## S3 method for class 'hyperparam.torus'
plot(x, color = "auto", ...)

Arguments

`icp.torus.objects`	list whose elements are icp.torus objects, generated by `icp.torus`
`option`	A string. One of "elbow", "risk", "AIC", or "BIC", which determines the criterion for the model selection. "risk" is based on the negative log-likelihood, "AIC" for the Akaike Information Criterion, and "BIC" for the Bayesian Information Criterion. "elbow" is based on minimizing the criterion used in Jung et. al.(2021). Default is `option = "elbow"` for 2-dimensional cases and `option = "risk"` for d(>2)-dimensional cases.
`alphavec`	either a scalar or a vector, or even `NULL` for the levels. Default value is `NULL`. If `NULL`, then `alphavec` is automatically generated as a sequence from 0 to `alpha.lim`.
`alpha.lim`	a positive number lower than 1. Default value is `NULL`. If `NULL`, then `alpha.vec` is is 0.5 for `option = "elbow"`, and 0.15 for options c("risk", "AIC", or "BIC").
`eval.point`	N x N numeric matrix on $[0, 2\pi)^2$ . Default input is `grid.torus`.
`x`	`hyperparam.torus` object
`color`	A string for plotting `hyperparam.torus` object, whose criterion option is `option = "elbow"`. One of "auto", "sequential", or "qualitative". If `color = "auto"`, color assignment will be done automatically based on the number of J or concentration. If `color = "sequential"`, color assignment will be done by regarding each J or concentration as quantitative variable. If `color = "qualitative"`, color assignment will be done by regarding each J or concentration as qualitative variable. Default is `color = "auto"`.
`...`	additional parameter for ggplot2::ggplot()

Value

returns a list object which contains data.frame objects for the evaluated criterion corresponding to each hyperparameter, selected hyperparameters based on the designated criterion, and an icp.torus object based the selected hyperparameters.

References

Jung, S., Park, K., & Kim, B. (2021). Clustering on the torus by conformal prediction. The Annals of Applied Statistics, 15(4), 1583-1603.

Akaike, H. (1974). A new look at the statistical model identification. IEEE transactions on automatic control, 19(6), 716-723.

Schwarz, G. (1978). Estimating the dimension of a model. The annals of statistics, 461-464.

Examples


data <- toydata2[, 1:2]
n <- nrow(data)
split.id <- rep(2, n)
split.id[sample(n, floor(n/2))] <- 1
Jvec <- 3:35
icp.torus.objects <- icp.torus(data, split.id = split.id, model = "kmeans",
                                      kmeansfitmethod = "ge", init = "h",
                                      J = Jvec, verbose = TRUE)
hyperparam.torus(icp.torus.objects, option = "risk")

data <- toydata2[, 1:2]
n <- nrow(data)
split.id <- rep(2, n)
split.id[sample(n, floor(n/2))] <- 1
Jvec <- 3:35
icp.torus.objects <- icp.torus(data, split.id = split.id, model = "kmeans",
                                      kmeansfitmethod = "ge", init = "h",
                                      J = Jvec, verbose = TRUE)
hyperparam.torus(icp.torus.objects, option = "risk")

Conformity score for inductive prediction sets

Description

icp.torus prepares all values for computing the conformity score for specified methods.

plot.icp.torus plots icp.torus object with some options.

Usage

icp.torus(
  data,
  split.id = NULL,
  model = c("kmeans", "kde", "mixture"),
  mixturefitmethod = c("axis-aligned", "circular", "general"),
  kmeansfitmethod = c("general", "homogeneous-circular", "heterogeneous-circular",
    "ellipsoids"),
  init = c("hierarchical", "kmeans"),
  d = NULL,
  additional.condition = TRUE,
  J = 4,
  concentration = 25,
  kmax = 500,
  THRESHOLD = 1e-10,
  maxiter = 200,
  verbose = TRUE,
  ...
)

## S3 method for class 'icp.torus'
logLik(object, ...)

## S3 method for class 'icp.torus'
predict(object, newdata, ...)

## S3 method for class 'icp.torus'
plot(
  x,
  data = NULL,
  level = 0.1,
  ellipse = TRUE,
  out = FALSE,
  type = NULL,
  ...
)
icp.torus(
  data,
  split.id = NULL,
  model = c("kmeans", "kde", "mixture"),
  mixturefitmethod = c("axis-aligned", "circular", "general"),
  kmeansfitmethod = c("general", "homogeneous-circular", "heterogeneous-circular",
    "ellipsoids"),
  init = c("hierarchical", "kmeans"),
  d = NULL,
  additional.condition = TRUE,
  J = 4,
  concentration = 25,
  kmax = 500,
  THRESHOLD = 1e-10,
  maxiter = 200,
  verbose = TRUE,
  ...
)

## S3 method for class 'icp.torus'
logLik(object, ...)

## S3 method for class 'icp.torus'
predict(object, newdata, ...)

## S3 method for class 'icp.torus'
plot(
  x,
  data = NULL,
  level = 0.1,
  ellipse = TRUE,
  out = FALSE,
  type = NULL,
  ...
)

Arguments

`data`	n x d matrix of toroidal data on $[0, 2\pi)^d$ or $[-\pi, \pi)^d$ . Default is `NULL`.
`split.id`	a n-dimensional vector consisting of values 1 (estimation) and 2(evaluation)
`model`	A string. One of "kde", "mixture", and "kmeans" which determines the model or estimation methods. If "kde", the model is based on the kernel density estimates. It supports the kde-based conformity score only. If "mixture", the model is based on the von Mises mixture, fitted with an EM algorithm. It supports the von Mises mixture and its variants based conformity scores. If "kmeans", the model is also based on the von Mises mixture, but the parameter estimation is implemented with the elliptical k-means algorithm illustrated in Appendix. It supports the log-max-mixture based conformity score only. If the dimension of data space is greater than 2, only "kmeans" is supported. Default is `model = "kmeans"`.
`mixturefitmethod`	A string. One of "circular", "axis-aligned", and "general" which determines the constraint of the EM fitting. Default is "axis-aligned". This argument only works for `model = "mixture"`.
`kmeansfitmethod`	A string. One of "general", ellipsoids", "heterogeneous-circular" or "homogeneous-circular". If "general", the elliptical k-means algorithm with no constraint is used. If "ellipsoids", only the one iteration of the algorithm is used. If"heterogeneous-circular", the same as above, but with the constraint that ellipsoids must be spheres. If "homogeneous-circular", the same as above but the radii of the spheres are identical. Default is "general". This argument only works for `model = "kmeans"`.
`init`	Methods for choosing initial values of "kmeans" fitting. Must be "hierarchical" or "kmeans". If "hierarchical", the initial parameters are obtained with hierarchical clustering method. If "kmeans", the initial parameters are obtained with extrinsic k-means method. Additional arguments for k-means clustering and hierarchical clustering can be designated via argument `...`. If no options are designated, `nstart=1` for `init="kmeans"` and `method="complete"` for `init="hierarchical"` are used. Default is "hierarchical".
`d`	pairwise distance matrix(`dist` object) for `init = "hierarchical"`, which used in hierarchical clustering. If `init = "hierarchical"` and `d = NULL`, `d` will be automatically filled with `ang.pdist(data)`.
`additional.condition`	boolean index. If `TRUE`, a singular matrix will be altered to the scaled identity.
`J`	A scalar or numeric vector for the number(s) of components for `model = c("mixture", "kmeans")`. Default is `J = 4`.
`concentration`	A scalar or numeric vector for the concentration parameter(s) for `model = "kde"`. Default is `concentration = 25`.
`kmax`	the maximal number of kappa. If estimated kappa is larger than `kmax`, then put kappa as `kmax`.
`THRESHOLD`	number for difference between updating and updated parameters. Default is 1e-10.
`maxiter`	the maximal number of iteration. Default is 200.
`verbose`	boolean index, which indicates whether display additional details as to what the algorithm is doing or how many loops are done. Moreover, if `additional.condition` is `TRUE`, the warning message will be reported.
`...`	additional parameters. For plotting icp.torus, these parameters are for ggplot2::ggplot().
`object`	`icp.torus` object
`newdata`	n x d matrix of toroidal data on $[0, 2\pi)^d$ . Dimension d must be the same as data used for `icp.torus` object.
`x`	`icp.torus` object
`level`	either a numeric scalar or a vector in $[0,1]$ . Default value is 0.1.
`ellipse`	A boolean index which determines whether plotting ellipses from mixture models. Default is `TRUE`. (This option is used only when the `icp.torus` object `x` is fitted by model `kmeans` or `mixture`.)
`out`	An option for returning the ggplot object. Default is `FALSE`.
`type`	A string. One of "mix", "max" or "e". This argument is only available if `icp.torus` object is fitted with `model = "mixture"`. Default is `NULL`. If `type != NULL`, argument `ellipse` automatically becomes `FALSE`. If "mix", it plots based on von Mises mixture. If "max", it plots based on von Mises max-mixture. If "e", it plots based on ellipse-approximation.

Value

icp.torus returns an icp.torus object, containing all values to compute the conformity score (if J or concentration is a single value). if J or concentration is a vector containing multiple values, then icp.torus returns a list of icp.torus objects

References

Jung, S., Park, K., & Kim, B. (2021). Clustering on the torus by conformal prediction. The Annals of Applied Statistics, 15(4), 1583-1603.

Di Marzio, M., Panzera, A., & Taylor, C. C. (2011). Kernel density estimation on the torus. Journal of Statistical Planning and Inference, 141(6), 2156-2173.

Shin, J., Rinaldo, A., & Wasserman, L. (2019). Predictive clustering. arXiv preprint arXiv:1903.08125.

Examples


data <- toydata1[, 1:2]

icp.torus <- icp.torus(data, model = "kmeans",
                       kmeansfitmethod = "general",
                       J = 4, concentration = 25)

data <- toydata1[, 1:2]

icp.torus <- icp.torus(data, model = "kmeans",
                       kmeansfitmethod = "general",
                       J = 4, concentration = 25)

Inductive prediction sets for each level

Description

icp.torus.eval evaluates whether each pre-specified evaluation point is contained in the inductive conformal prediction sets for each given level.

Usage

icp.torus.eval(icp.torus, level = 0.1, eval.point = grid.torus())
icp.torus.eval(icp.torus, level = 0.1, eval.point = grid.torus())

Arguments

`icp.torus`	an object containing all values to compute the conformity score, which will be constructed with `icp.torus`.
`level`	either a scalar or a vector, or even `NULL`. Default value is 0.1.
`eval.point`	N x N numeric matrix on $[0, 2\pi)^2$ . Default input is `grid.torus`.

Value

returns a cp object with the boolean values which indicate whether each evaluation point is contained in the inductive conformal prediction sets for each given level.

References

Jung, S., Park, K., & Kim, B. (2021). Clustering on the torus by conformal prediction. The Annals of Applied Statistics, 15(4), 1583-1603.

Examples


data <- toydata1[, 1:2]

icp.torus <- icp.torus(data, model = "kmeans",
                       mixturefitmethod = "general",
                       J = 4, concentration = 25)

icp.torus.eval(icp.torus, level = c(0.1, 0.08), eval.point = grid.torus())

data <- toydata1[, 1:2]

icp.torus <- icp.torus(data, model = "kmeans",
                       mixturefitmethod = "general",
                       J = 4, concentration = 25)

icp.torus.eval(icp.torus, level = c(0.1, 0.08), eval.point = grid.torus())

ILE: Structure of the Isoleucine

Description

An isomer of leucine, essential branched-chain aliphatic amino acid found in many proteins.

Usage

ILE
ILE

Format

This list contains the following components:

phi: main chain torsion angle for atoms C,N,CA,C.
psi: main chain torsion angle for atoms N,CA,C,N.
chi1: side chain torsion angle for atoms N,CA,CB,*G.
chi2: side chain torsion angle for atoms CA,CB,*G,*D.

Details

ILE data is generated with collection of different pdb files. To select adequate protein data, we use PISCES server. (the method is introduced in articles of references.) To select high-quality protein data, we use several benchmarks: resolution : 1.6A(angstrom) or better, R-factor : 0.22 or better, Sequence percentage identity: <= 25 Then, we select ILE only angular data for each protein data. To see the detail code, visit https://github.com/sungkyujung/ClusTorus

Source

This data is extracted from PISCES server http://dunbrack.fccc.edu/pisces/

References

Data description is from https://www.rcsb.org/ligand/ILE.

The data extracting method is from Harder, T., Boomsma, W., Paluszewski, M., Frellsen, J., Johansson, K. E., & Hamelryck, T. (2010). Beyond rotamers: a generative, probabilistic model of side chains in proteins. BMC bioinformatics, 11(1), 1-13.

Kernel density estimation using circular von Mises distribution

Description

kde.torus returns a kde using independent multivariate von mises kernel.

Usage

kde.torus(data, eval.point = NULL, concentration = 25)
kde.torus(data, eval.point = NULL, concentration = 25)

Arguments

`data`	n x d matrix of toroidal data on $[0, 2\pi)^d$
`eval.point`	N x N numeric matrix on $[0, 2\pi)^d$ . Default input is `NULL`, which represents the fine grid points on $[0, 2\pi)^d$ .
`concentration`	positive number which has the role of $\kappa$ of von Mises distribution. Default value is 25.

Value

kde.torus returns N-dimensional vector of kdes evaluated at eval.point

References

Jung, S., Park, K., & Kim, B. (2021). Clustering on the torus by conformal prediction. The Annals of Applied Statistics, 15(4), 1583-1603.

Di Marzio, M., Panzera, A., & Taylor, C. C. (2011). Kernel density estimation on the torus. Journal of Statistical Planning and Inference, 141(6), 2156-2173.

Examples

data <- ILE[1:200, 1:2]

kde.torus(data)
data <- ILE[1:200, 1:2]

kde.torus(data)

K-Means Clustering on Torus

Description

kmeans.torus implements extrinsic k-means clustering on toroidal space.

Usage

kmeans.torus(data, centers = 10, ...)

## S3 method for class 'kmeans.torus'
predict(object, newdata, ...)
kmeans.torus(data, centers = 10, ...)

## S3 method for class 'kmeans.torus'
predict(object, newdata, ...)

Arguments

`data`	n x d matrix of toroidal data on $[0, 2\pi)^d$
`centers`	either the number of clusters or a set of initial cluster centers. If a number, a random set of row in x is chosen as the initial centers.
`...`	additional parameter
`object`	`kmeans.torus` object
`newdata`	n x d matrix of toroidal data on $[0, 2\pi)^d$ . Dimension d must be the same as data used for `kmeans.torus` object.

Details

In Euclidean space, we know that the total sum of squares is equal to the summation of the within cluster sum of squares and the between cluster centers sum of squares. However, toroidal space does not satisfy the property; the equality does not hold. Thus, you need to be careful to use the sum of squares.

Extrinsic k-means algorithm uses the ambient space for $[0, 2\pi)^d$ . Each datum is transformed to a vector in 2d-dimensional Euclidean space, whose elements are sine and cosine values of the datum, then a usual k-means algorithm is applied to transformed data.

Value

returns a kmeans object, which contains

extrinsic.results: extrinsic k-means clustering results using ordinary kmeans algorithm.
centers: A matrix of cluster centers.
membership: A vector of integers indicating the cluster to which each point is allocated.
size: The number of points in each cluster.
withinss: Vector of within-cluster sum of squares, one component per cluster.
totss: The total sum of squares, based on angular distance.
betweenss: The between-cluster sum of squares.

References

Jung, S., Park, K., & Kim, B. (2021). Clustering on the torus by conformal prediction. The Annals of Applied Statistics, 15(4), 1583-1603.

Gao, Y., Wang, S., Deng, M., & Xu, J. (2018). RaptorX-Angle: real-value prediction of protein backbone dihedral angles through a hybrid method of clustering and deep learning. BMC bioinformatics, 19(4), 73-84.

Examples

data <- ILE[1:200, 1:2]

kmeans.torus(data, centers = 2,
             iter.max = 100, nstart = 1)
data <- ILE[1:200, 1:2]

kmeans.torus(data, centers = 2,
             iter.max = 100, nstart = 1)

Transform the angular data to be on principal interval

Description

on.torus transforms d-dimensional angular data to be on $[0, 2\pi)^d$ .

Usage

on.torus(x)
on.torus(x)

Arguments

`x`	d-dimensional angular data(vector or matrix) whose unit is the radian.

Value

d-dimensional radian-unit angular data on $[0, 2\pi)^d$ .

Examples

data <- SARS_CoV_2 * pi / 180

on.torus(data)
data <- SARS_CoV_2 * pi / 180

on.torus(data)

SARS-CoV-2: chain B of Structure of the SARS-CoV-2 spike glycoprotein(closed state)

Description

The torsion angle dataset of the chain B of SARS-CoV-2 spike glycopreotein. This data is originally from first two main torsion angles of data_6VXX.

Usage

SARS_CoV_2
SARS_CoV_2

Format

This data.frame contains the following columns:

phi: main chain torsion angle for atoms C,N,CA,C.
psi: main chain torsion angle for atoms N,CA,C,N.

Details

This data is obtained with following codes:

Source

This data can be downloaded in https://www.rcsb.org/structure/6VXX, or with using R package bio3d. To see the precise extracting code, visit https://github.com/sungkyujung/ClusTorus/tree/master/data-raw

References

Toroidal subtraction

Description

tor.minus computes angular subtraction bewtween n x d toroidal data and a d dimensional vector.

Usage

tor.minus(data, mu)
tor.minus(data, mu)

Arguments

`data`	n x d matrix of toroidal data
`mu`	a d-dimensinal vector

Value

angular subtraction bewtween n x d toroidal data and a d dimensional vector.

References

Jung, S., Park, K., & Kim, B. (2021). Clustering on the torus by conformal prediction. The Annals of Applied Statistics, 15(4), 1583-1603.

Examples

data <- ILE[1:200, 1:2]
Mu1 <- c(4.5, 3)
tor.minus(data, Mu1)
data <- ILE[1:200, 1:2]
Mu1 <- c(4.5, 3)
tor.minus(data, Mu1)

toydata1: Labelled Data for 5 Clusters

Description

Artificially generated data on the 2 dimensional torus

Usage

toydata1
toydata1

Format

This data.frame contains the following components:

phi: column for the first angle
psi: column for the second angle
label: column for the clustering membership

Details

toydata1 is an artificial data generated from a mixture of 5 clusters, where three clusters are sampled from bivariate normal distributions and the other two are each sampled from the uniform distribution on a rectangle.

References

Jung, S., Park, K., & Kim, B. (2021). Clustering on the torus by conformal prediction. The Annals of Applied Statistics, 15(4), 1583-1603.

toydata2: Labelled Data for 3 Clusters

Description

Artificially generated data on the 2 dimensional torus

Usage

toydata2
toydata2

Format

This data.frame contains the following components:

phi: column for the first angle
psi: column for the second angle
label: column for the clustering membership

Details

toydata2 is an artificial data generated from a mixture of 3 clusters, where the first cluster is sampled from a spherical normal distribution, the second cluster is from the uniform distribution on a large “L”-shaped region, and the third cluster of size 50 is sampled from the uniform distribution on the entire 2-dimensional torus.

References

Jung, S., Park, K., & Kim, B. (2021). Clustering on the torus by conformal prediction. The Annals of Applied Statistics, 15(4), 1583-1603.

Weighted extrinsic mean direction and mean resultant length

Description

wtd.stat.ang computes weighted extrinsic mean direction and mean resultant length.

Usage

wtd.stat.ang(data, w)
wtd.stat.ang(data, w)

Arguments

`data`	angular data whose elements are in $[0, 2\pi)$
`w`	numeric vector whose each element is non-negative and `sum(w) == 1`. Moreover, the length of `w` is the same with `nrow(data)`.

Value

list which is consisting of the following components:

Mean weighted extrinsic mean direction

R mean resultant length

References

Jung, S., Park, K., & Kim, B. (2021). Clustering on the torus by conformal prediction. The Annals of Applied Statistics, 15(4), 1583-1603.

Examples

data <- matrix(c(pi/3, pi/3, pi/2,
                 pi, pi/4, pi/2,
                 0, pi/3, pi/6),
               ncol = 3, byrow = TRUE)
w <- c(0.3, 0.3, 0.4)
wtd.stat.ang(data, w)
data <- matrix(c(pi/3, pi/3, pi/2,
                 pi, pi/4, pi/2,
                 0, pi/3, pi/6),
               ncol = 3, byrow = TRUE)
w <- c(0.3, 0.3, 0.4)
wtd.stat.ang(data, w)

Package 'ClusTorus'

Help Index

Angular distance

Description

Usage

Arguments

Value

References

Examples

Angular subtraction

Description

Usage

Arguments

Value

References

Examples

Pairwise L2 angular distance

Description

Usage

Arguments

Value

References

See Also

Examples

Clustering on the torus by conformal prediction

Description

Usage

Arguments

Details

Value

References

See Also

Examples

Clustering by connected components of ellipsoids

Description

Usage

Arguments

Value

References

See Also

Examples

Conformal prediction set indices with kernel density estimation

Description

Usage

Arguments

Value

References

See Also

Examples

6VXX: Structure of the SARS-CoV-2 spike glycoprotein(closed state)

Description

Usage

Format

Source

References

See Also

K-Means Clustering to K-Spheres Clustering on Torus

Description

Usage

Arguments

Value

References

See Also

Examples

Fitting mixtures of bivariate von Mises distribution

Description

Usage

Arguments

Details

Value

References

Examples

Grid on torus

Description

Usage

Arguments

Value

Examples

Selecting optimal level based on the runs of the number of clusters

Description