Package 'SCOR' reference manual

Title:	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.
Authors:	Debsurya De, Priyam Das
Maintainer:	Debsurya De <[email protected]>
License:	MIT + file LICENSE
Version:	1.0.0.2
Built:	2025-02-20 05:14:43 UTC
Source:	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 publsihed clinical demential rating (CDR).

Usage

data(AL)
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”.

Empirical Hyper Volume Under Manifolds

Description

An estimator of Hyper Volume Under Manifolds

Usage

estimate_EHUM(beta, labels, x_mat)
estimate_EHUM(beta, labels, x_mat)

Arguments

`beta`	The parameter we measure EHUM based on.
`labels`	The labels of the Columns of the data matrix.
`x_mat`	The Data Matrix

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))

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

Arguments

`beta`	The parameter we measure SHUM based on.
`labels`	The labels of the Columns of the data matrix.
`x_mat`	The Data Matrix
`p`	p decides whether to use $s_n(x)$ or $\phi_n(x)$ . p = 1 stands for $\phi_n(x)$ and p = 0 stands for $s_n(x)$

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 http://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))

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)
estimate_ULBA(beta, labels, x_mat)

Arguments

`beta`	The parameter we measure ULBA based on.
`labels`	The labels of the Columns of the data matrix.
`x_mat`	The Data Matrix

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))


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 'SCOR' 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 = 10^(-3),
  max_iter = 50000, s_init = 2, tol_fun = 10^(-6),
  tol_fun_2 = 10^(-6), minimize = FALSE, time = 6e+05,
  print = FALSE, lambda = 10^(-3), parallel = TRUE)

optimized_SHUM(beta_start, labels, x_mat, p = 0, rho = 2,
  phi = 10^(-3), max_iter = 50000, s_init = 2, tol_fun = 10^(-6),
  tol_fun_2 = 10^(-6), minimize = FALSE, time = 6e+05,
  print = FALSE, lambda = 10^(-3), parallel = TRUE)

optimized_ULBA(beta_start, labels, x_mat, rho = 2, phi = 10^(-3),
  max_iter = 50000, s_init = 2, tol_fun = 10^(-6),
  tol_fun_2 = 10^(-6), minimize = FALSE, time = 6e+05,
  print = FALSE, lambda = 10^(-3), parallel = TRUE)
optimized_EHUM(beta_start, labels, x_mat, rho = 2, phi = 10^(-3),
  max_iter = 50000, s_init = 2, tol_fun = 10^(-6),
  tol_fun_2 = 10^(-6), minimize = FALSE, time = 6e+05,
  print = FALSE, lambda = 10^(-3), parallel = TRUE)

optimized_SHUM(beta_start, labels, x_mat, p = 0, rho = 2,
  phi = 10^(-3), max_iter = 50000, s_init = 2, tol_fun = 10^(-6),
  tol_fun_2 = 10^(-6), minimize = FALSE, time = 6e+05,
  print = FALSE, lambda = 10^(-3), parallel = TRUE)

optimized_ULBA(beta_start, labels, x_mat, rho = 2, phi = 10^(-3),
  max_iter = 50000, s_init = 2, tol_fun = 10^(-6),
  tol_fun_2 = 10^(-6), minimize = FALSE, time = 6e+05,
  print = FALSE, lambda = 10^(-3), parallel = TRUE)

Arguments

`beta_start`	The initial guess for optimum $\beta$ by user
`labels`	Sample Sizes vector of that has number of elements in each category. It works like the labels of data matrix.
`x_mat`	The Data Matrix
`rho`	Step Decay Rate with default value 2
`phi`	Lower Bound Of Global Step Size. Default value is $10^{-6}$
`max_iter`	Max Number Of Iterations In each Run. Default Value is 50,000.
`s_init`	Initial Global Step Size. Default Value is 2.
`tol_fun`	Termination Tolerance on the function value. Default Value is $10^{-6}$
`tol_fun_2`	Termination Tolerance on the difference of solutions in two consecutive runs. Default Value is $10^{-6}$
`minimize`	Binary Command to set SCOR on minimization or maximization. FALSE is for minimization which is set default.
`time`	Time Alloted for execution of SCOR
`print`	Binary Command to print optimized value of objective function after each interation. FALSE is set fault
`lambda`	Sparsity Threshold. Default value is $10^{-3}$
`parallel`	Binary Command to ask SCOR to perform parallel computing. Default is set at TRUE.
`p`	This parameter exists for the case of optimized_SHUM only.p decides whether to use $s_n(x)$ or $\phi_n(x)$ . p = 1 stands for $\phi_n(x)$ and p = 0 stands for $s_n(x)$

Details

Optimization of EHUM, SHUM and ULBA using SCOR.

Value

Optimum Values Of HUM Estimates

Examples


optimized_SHUM(rep(1,12),colnames(AL),AL)
# Optimum value of HUM estimate noticed for this case : 0.8440681

optimized_EHUM(rep(1,12),colnames(AL),AL)
# Optimum value of HUM estimate noticed for this case : 0.8403805

optimized_ULBA(rep(1,12),colnames(AL),AL)
# Optimum value of HUM estimate noticed for this case : 0.9201903

optimized_SHUM(rep(1,12),colnames(AL),AL)
# Optimum value of HUM estimate noticed for this case : 0.8440681

optimized_EHUM(rep(1,12),colnames(AL),AL)
# Optimum value of HUM estimate noticed for this case : 0.8403805

optimized_ULBA(rep(1,12),colnames(AL),AL)
# Optimum value of HUM estimate noticed for this case : 0.9201903

Spherically Constrained Optimization Routine

Description

'SCOR' is our optimization algorithm, efficient in estimating maximizing Hyper Volume Under Manifolds Estimators.

Usage

SCOR(x0, func, rho = 2, phi = 10^(-3), max_iter = 50000,
  s_init = 2, tol_fun = 10^(-6), tol_fun_2 = 10^(-6),
  minimize = TRUE, time = 6e+05, print = FALSE, lambda = 10^(-3),
  parallel = FALSE)
SCOR(x0, func, rho = 2, phi = 10^(-3), max_iter = 50000,
  s_init = 2, tol_fun = 10^(-6), tol_fun_2 = 10^(-6),
  minimize = TRUE, time = 6e+05, print = FALSE, lambda = 10^(-3),
  parallel = FALSE)

Arguments

`x0`	The initial guess by user
`func`	The function to be optimized
`rho`	Step Decay Rate with default value 2
`phi`	Lower Bound Of Global Step Size. Default value is $10^{-6}$
`max_iter`	Max Number Of Iterations In each Run. Default Value is 50,000.
`s_init`	Initial Global Step Size. Default Value is 2.
`tol_fun`	Termination Tolerance on the function value. Default Value is $10^{-6}$
`tol_fun_2`	Termination Tolerance on the difference of solutions in two consecutive runs. Default Value is $10^{-6}$
`minimize`	Binary Command to set SCOR on minimization or maximization. TRUE is for minimization which is set default.
`time`	Time Alloted for execution of SCOR
`print`	Binary Command to print optimized value of objective function after each interation. FALSE is set fault
`lambda`	Sparsity Threshold. Default value is $10^{-3}$
`parallel`	Binary Command to ask SCOR to perform parallel computing. Default is set at FALSE.

Details

SCOR is the modified version of RMPS, Recursive Modified Pattern Search. This is a blackbox algorithm efficienct 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 http://arxiv.org/abs/1909.04024).

Examples

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

SCOR(rep(1,10),f)

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

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

SCOR(c(2,6,2,7,8),f, parallel = TRUE)
#Will do Parallel Computing

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

SCOR(rep(1,10),f)

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

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

SCOR(c(2,6,2,7,8),f, parallel = TRUE)
#Will do Parallel Computing

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

Arguments

`beta`	The parameter we do HUM based on
`labels`	The labels of the Columns of the data matrix.
`x_mat`	The Data Matrix
`grid_size`	The size of increment in the grid we check cutpoints against. Default value is 100.

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)


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)

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

Arguments

`beta`	The parameter we do HUM based on
`labels`	The labels of the Columns of the data matrix
`x_mat`	The Data Matrix
`cat_names`	The vector of strings containing category names.
`grid_size`	The size of increment in the grid we check cutpoints against. Default value is 100.

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"))
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"))

Package 'SCOR'

Help Index

Alzheimer's disease neuropsychometric marker dataset

Description

Usage

Format

Details

Empirical Hyper Volume Under Manifolds

Description

Usage

Arguments

Value

Examples

Smooth Approximations Of Empirical Hyper Volume Under Manifolds

Description

Usage

Arguments

Value

References

Examples

Upper And Lower Bound Approach

Description

Usage

Arguments

Value

Examples

Optimizing Different Estimators Of Hyper Volume Under Manifold

Description

Usage

Arguments

Details

Value

Examples

Spherically Constrained Optimization Routine

Description

Usage

Arguments

Details

Value

References

Examples

Finding Youden Indices

Description

Usage

Arguments

Value

Examples

Visualization Based On Youden Indices.

Description

Usage

Arguments

Value

Examples