SCOR: Spherically Constrained Optimization Routine

Description

A non convex optimization package that optimizes any function under the criterion, combination of variables are on the surface of a unit sphere, as described in the paper : Das et al. (2019) <arXiv:1909.04024> .

Author(s)

Maintainer: Debsurya De debsurya001@gmail.com

Authors:

• Priyam Das pdas@ncsu.edu

See Also

Useful links:

• https://github.com/synx21/SCOR/

Alzheimer's disease neuropsychometric marker dataset

Description

The dataset is a subset of the longitudinal cohort of Washington University (WU) Alzheimer's Disease Research Center (ADRC). In the AL dataset, measurements of 12 neuropsychological markers were collected on 108 independent individuals of age 75. The individuals were classified into 3 groups based on published clinical demential rating (CDR).

Usage

data(AL)

Format

A data frame with 108 observations on the following 12 variables.

Details

• ktemp. a numeric vector, measurements on the neuropsychometric test for “temporal factor”.

• kpar. a numeric vector, measurements on the neuropsychometric test for “parietal factor”.

• kfront. a numeric vector, measurements on the neuropsychometric test for “frontal factor”.

• zpsy005. a numeric vector, measurements on the neuropsychometric test for “digital span forward”.

• zpsy006. a numeric vector, measurements on the neuropsychometric test for “digital span backward”.

• zinfo. a numeric vector, measurements on the neuropsychometric test for “information”.

• zbentc. a numeric vector, measurements on the neuropsychometric test for “visual retention (10s)”.

• zbentd. a numeric vector, measurements on the neuropsychometric test for “visual retention (copy)”.

• zboston. a numeric vector,a numeric vector, measurements on the neuropsychometric test for “boston naming”.

• zmentcon. a numeric vector,measurements on the neuropsychometric test for “mental control”.

• zworflu. a numeric vector,measurements on the neuropsychometric test for “word fluency”.

• zassc. a numeric vector,measurements on the neuropsychometric test for “associate learning”.

Spherically Constrained Optimization

Description

SCOptim runs our optimization algorithm, efficient in estimating maximizing Hyper Volume Under Manifolds Estimators.

Usage

SCOptim(
  x0,
  func,
  rho = 2,
  phi = 0.001,
  max_iter = 50000,
  s_init = 2,
  tol_fun = 1e-06,
  tol_fun_2 = 1e-06,
  minimize = TRUE,
  time = 36000,
  print = FALSE,
  lambda = 0.001,
  parallel = FALSE
)

Arguments

Details

SCOptim is the modified version of RMPS, Recursive Modified Pattern Search. This is a blackbox algorithm efficient in optimizing non-differentiable functions. It works great in the shown cases of SHUM, EHUM and ULBA.

Value

The point where the value Of the Function is maximized under a sphere.

References

• Das, Priyam and De, Debsurya and Maiti, Raju and Chakraborty, Bibhas and Peterson, Christine B
  "Estimating the Optimal Linear Combination of Biomarkers using Spherically Constrained Optimization"
  (available at 'arXiv https://arxiv.org/abs/1909.04024).

Examples

f <- function(x)
return(x[2]^2 + x[3]^3 +x[4]^4)

SCOptim(rep(1,10), f)

SCOptim(c(2,4,6,2,1), f, minimize = FALSE, print = TRUE)
#Will Print the List and Find the Maximum

SCOptim(c(1,2,3,4), f, time = 10, lambda = 1e-2)
#Will perform no iterations after 10 secs, Sparsity Threshold is 0.01

Visualization Based On Youden Indices.

Description

A Box Plot Visualization Based On Youden Indices for less than equal to 3 categories.

Usage

YoupointsBoxPlot(beta, labels, x_mat, cat_names = NULL, grid_size = 100)

Arguments

Value

Box Plot Visualization Based On Youden Indices

Examples

beta <- c(-0.399,-0.155,-0.265,-0.184,
     -0.267,0.666,-0.187,0.273,0.0463,0.167,0.163,0.178)

YoupointsBoxPlot(beta, colnames(AL), AL, cat_names = c("Healthy", "MCI", "AD"))

Empirical Hyper Volume Under Manifolds

Description

An estimator of Hyper Volume Under Manifolds

Usage

estimate_EHUM(beta, labels, x_mat)

Arguments

Value

Empirical Hyper-volume Under Maniforlds Estimate

Examples

estimate_EHUM(rep(1, 12), colnames(AL), AL)


estimate_EHUM(1:10 , sample(c( rep("lab1", 10), rep("lab2", 10), rep("lab3", 10))),
matrix(rnorm(300), nrow = 10))

Smooth Approximations Of Empirical Hyper Volume Under Manifolds

Description

'SHUM' is a class of smoothed estimates of EHUM.

Usage

estimate_SHUM(beta, labels, x_mat, p = 0)

Arguments

Value

Smooth approximation of the empirical Hyper-volume Under Manifolds Estimate

References

• Maiti, Raju and Li, Jialiang and Das, Priyam and Feng, Lei and Hausenloy, Derek and Chakraborty, Bibhas
  "A distribution-free smoothed combination method of biomarkers to improve diagnostic accuracy in multi-category classification"
  (available at 'arXiv https://arxiv.org/abs/1904.10046).

Examples

estimate_SHUM(rep(1, 12), colnames(AL), AL)
estimate_SHUM(rep(1, 12), colnames(AL), AL, p = 1)


estimate_SHUM(1:10 , sample(c( rep("lab1", 10), rep("lab2", 10), rep("lab3", 10))),
matrix(rnorm(300), nrow = 10))

Upper And Lower Bound Approach

Description

'ULBA' is an another approach to Hyper Volume Under Manifold Problem

Usage

estimate_ULBA(beta, labels, x_mat)

Arguments

Value

Upper and Lower Bound Approach on empirical Hyper-volume Under Manifolds Estimate

Examples

estimate_ULBA(rep(1, 12), colnames(AL), AL)


estimate_ULBA(1:10 , sample(c( rep("lab1", 10), rep("lab2", 10), rep("lab3", 10))),
matrix(rnorm(300), nrow = 10))

Optimizing Different Estimators Of Hyper Volume Under Manifold

Description

As we know 'SCOptim' is efficient in estimating maximizing Hyper Volume Under Manifolds Estimators, we made some pre-functions that optimizes specific Problems of EHUM,SHUM and ULBA.

Usage

optimized_EHUM(
  beta_start,
  labels,
  x_mat,
  rho = 2,
  phi = 0.001,
  max_iter = 50000,
  s_init = 2,
  tol_fun = 1e-06,
  tol_fun_2 = 1e-06,
  minimize = FALSE,
  time = 36000,
  print = FALSE,
  lambda = 0.001,
  parallel = TRUE
)

optimized_SHUM(
  beta_start,
  labels,
  x_mat,
  p = 0,
  rho = 2,
  phi = 0.001,
  max_iter = 50000,
  s_init = 2,
  tol_fun = 1e-06,
  tol_fun_2 = 1e-06,
  minimize = FALSE,
  time = 36000,
  print = FALSE,
  lambda = 0.001,
  parallel = TRUE
)

optimized_ULBA(
  beta_start,
  labels,
  x_mat,
  rho = 2,
  phi = 0.001,
  max_iter = 50000,
  s_init = 2,
  tol_fun = 1e-06,
  tol_fun_2 = 1e-06,
  minimize = FALSE,
  time = 36000,
  print = FALSE,
  lambda = 0.001,
  parallel = TRUE
)

Arguments

Details

Optimization of EHUM, SHUM and ULBA using SCOptim.

Value

Optimum Values Of HUM Estimates

Examples

R <- optimized_SHUM(rep(1, 12), colnames(AL), AL, parallel = FALSE)
estimate_SHUM(R, colnames(AL), AL)
# This run will take about 10 mins on average based on computational capacity of the system
# Optimum value of HUM estimate noticed for this case : 0.8440681


R <- optimized_EHUM(rep(1, 12), colnames(AL), AL, parallel = FALSE)
estimate_EHUM(R, colnames(AL), AL)
# Optimum value of HUM estimate noticed for this case : 0.8403805

R <- optimized_ULBA(rep(1, 12), colnames(AL), AL, parallel = FALSE)
estimate_ULBA(R, colnames(AL), AL)
# Optimum value of HUM estimate noticed for this case : 0.9201903

Finding Youden Indices

Description

A function to find Youden Indices and Cutpoints for number of categories less than equal to 3.

Usage

youden_points(beta, labels, x_mat, grid_size = 100)

Arguments

Value

Youden Indices and Cut Points

Examples

beta <- c(-0.399,-0.155,-0.265,-0.184,
     -0.267,0.666,-0.187,0.273,0.0463,0.167,0.163,0.178)

youden_points(beta, colnames(AL), AL)