| Type: | Package |
| Title: | EM Algorithm for Sigmoid Normal Model |
| Version: | 1.0 |
| Date: | 2019-04-19 |
| Author: | Linsui Deng <denglinsui@gmail.com> |
| Maintainer: | Linsui Deng <denglinsui@gmail.com> |
| Description: | It provides a method based on EM algorithm to estimate the parameter of a mixture model, Sigmoid-Normal Model, where the samples come from several normal distributions (also call them subgroups) whose mean is determined by co-variable Z and coefficient alpha while the variance are homogeneous. Meanwhile, the subgroup each item belongs to is determined by co-variables X and coefficient eta through Sigmoid link function which is the extension of Logistic Link function. It uses bootstrap to estimate the standard error of parameters. When sample is indeed separable, removing estimation with abnormal sigma, the estimation of alpha is quite well. I used this method to explore the subgroup structure of HIV patients and it can be used in other domains where exists subgroup structure. |
| License: | GPL-2 | GPL-3 [expanded from: GPL (≥ 2)] |
| NeedsCompilation: | no |
| Packaged: | 2019-04-24 16:49:30 UTC; huihui |
| Repository: | CRAN |
| Date/Publication: | 2019-04-25 09:10:03 UTC |
EM Algorithm for Sigmoid Normal Model
Description
It provides a method based on EM algorithm to estimate the parameter of a mixture model, Sigmoid-Normal Model, where the samples come from several normal distributions (also call them subgroups) whose mean is determined by co-variable Z and coefficient alpha while the variance are homogeneous. Meanwhile, the subgroup each item belongs to is determined by co-variables X and coefficient eta through Sigmoid link function which is the extension of Logistic Link function. It uses bootstrap to estimate the standard error of parameters. When sample is indeed separable, removing estimation with abnormal sigma, the estimation of alpha is quite well. I used this method to explore the subgroup structure of HIV patients and it can be used in other domains where exists subgroup structure.
Details
The DESCRIPTION file:
| Package: | EMSNM |
| Type: | Package |
| Title: | EM Algorithm for Sigmoid Normal Model |
| Version: | 1.0 |
| Date: | 2019-04-19 |
| Author: | Linsui Deng <denglinsui@gmail.com> |
| Maintainer: | Linsui Deng <denglinsui@gmail.com> |
| Description: | It provides a method based on EM algorithm to estimate the parameter of a mixture model, Sigmoid-Normal Model, where the samples come from several normal distributions (also call them subgroups) whose mean is determined by co-variable Z and coefficient alpha while the variance are homogeneous. Meanwhile, the subgroup each item belongs to is determined by co-variables X and coefficient eta through Sigmoid link function which is the extension of Logistic Link function. It uses bootstrap to estimate the standard error of parameters. When sample is indeed separable, removing estimation with abnormal sigma, the estimation of alpha is quite well. I used this method to explore the subgroup structure of HIV patients and it can be used in other domains where exists subgroup structure. |
| License: | GPL(>=2) |
Index of help topics:
Ccompute Ccompute EMSNM-package EM Algorithm for Sigmoid Normal Model EM_parameter_sd Bootstrap Parameter Inference EM_result_sort Sort Parameter EMalgorithm Parameter Estimation EMbootstrap Bootstrap Method EMsimulation Simulation For Estimation Ggenerate Subgroup Determination Wgenerate Sigmoid Logistic Data Generation fnorm Density Value softmax Softmax Value standard Data Standardlization update_eta Updata Eta update_gamma Updata Alpha and Sigma weight_matrix Weighted Inner Product
The EMalgorithm is used to estimate the parameters, EMbootstrap is used to estimate the parameters with bootstrap method. In EMsimulation we can simulate the situation with given parameters, so parameter estimation can be verified.
Author(s)
Linsui Deng <denglinsui@gmail.com>
Maintainer: Linsui Deng <denglinsui@gmail.com>
Examples
#parameter initialization
etasize <- 2
classsize <- 2
alphasize <- 3
samplesize <- 100
expriments <- 30
etatest <- matrix(c(1,1,
0,0),etasize,classsize)
alphatest <- matrix(c(1,0,2,
4,3,5),alphasize,classsize)
sigmatest <- 0.5
#test of EMsimulation
EMsimulation_result <- EMsimulation(eta=etatest,alpha=alphatest,sigma=sigmatest,
samplesize=samplesize,expriments=expriments,
compact_flag=TRUE,C0=5,C1=0.5,C2=5)
index <- which(EMsimulation_result$sigma<0.8)
EMsimulation_result_sort <- EM_result_sort(EMsimulation_result$alpha[index,,],
EMsimulation_result$eta[index,,])
EM_parameter <- EM_parameter_sd(EMsimulation_result_sort$alpha,
EMsimulation_result_sort$eta,
EMsimulation_result$sigma[index])
#test of EMbootstrap
samplesize <- 1000
X <- matrix(c(matrix(1,samplesize),
rnorm(samplesize*(etasize-1))+1),samplesize,etasize)
Z <- matrix(c(matrix(1,samplesize),rbinom(prob=1/2,size=1,n=samplesize),
rnorm(samplesize*(alphasize-2))+1),samplesize,alphasize)
Wtest <- Wgenerate(alpha=alphatest,eta=etatest,sigma=sigmatest,X=X,Z=Z)
boots_samplesize <- 100
boots_expriments <- 30
samplesize <- dim(Wtest$X)[1]
EMbootstrap_theta <- EMbootstrap(Wtest$X,Wtest$Y,Wtest$Z,samplesize,
boots_samplesize,boots_expriments,
classsize=2,compact_flag=TRUE,C0=5,C1=0.2,C2=5)
index <- which(EMbootstrap_theta$sigma<0.8)
EMsimulation_result_sort <- EM_result_sort(EMbootstrap_theta$alpha[index,,],
EMbootstrap_theta$eta[index,,])
EM_parameter <- EM_parameter_sd(EMsimulation_result_sort$alpha,
EMsimulation_result_sort$eta,
EMbootstrap_theta$sigma[index])
Ccompute
Description
Compute the probability of Y in given parameters alphat, sigmat, etat and variables X, Z by the Bayesian Formula under the assumption of Sigmoid-Normal Model.
Usage
Ccompute(alphat, sigmat, etat, X, Z, Y)
Arguments
alphat |
the coeffients of the mean of each subgroup |
sigmat |
the variance of Y |
etat |
the coeffients determining subgroup |
X |
the covariables of the mean of each subgroup |
Z |
the covaraibles determining subgroup |
Y |
the respond variable |
Value
the probability of Y in given parameters alphat, sigmat, etat and variables X, Z under the assumption of Sigmoid-Normal Model.
Author(s)
Linsui Deng
Examples
#some variables
samplesize <- 1000
classsize <- 6
etasize <- 3
alphasize <- 2
Xtest <- data.frame(matrix(rnorm(samplesize*etasize),samplesize,etasize))
Ztest <- matrix(rnorm(samplesize*alphasize),samplesize,alphasize)
etatest <- matrix(seq(1.15,1,length=etasize*classsize),etasize,classsize)
alphatest <- matrix(seq(1.15,1,length=alphasize*classsize),alphasize,classsize)
Wtest <- Wgenerate(alpha=alphatest,eta=etatest,X=Xtest,Z=Ztest)
#test of Ccompute
sigmatest <- 1
Ctest <-
Ccompute(alphat=alphatest,sigmat=sigmatest,
etat=etatest,X=Wtest$X,Z=Wtest$Z,Y=Wtest$Y)
Bootstrap Parameter Inference
Description
Estimating the parameters and their stand error through the sorted parameters estimated by bootstrap method.
Usage
EM_parameter_sd(EMsimulation_sort_alpha, EMsimulation_sort_eta, EMsimulation_sort_sigma)
Arguments
EMsimulation_sort_alpha |
sorted alpha estimated by bootstrap method. |
EMsimulation_sort_eta |
sorted eta estimated by bootstrap method. |
EMsimulation_sort_sigma |
sorted sigma estimated by bootstrap method. |
Value
sigma |
the estimation of sigma |
sigma_sd |
the estimation of standard error of sigma |
alpha |
the estimation of alpha |
alpha_sd |
the estimation of standard error of alpha |
eta |
the estimation of eta |
eta_sd |
the estimation of standard error of eta |
Author(s)
Linsui Deng
Examples
#parameter initialization
etasize <- 2
classsize <- 2
alphasize <- 3
samplesize <- 100
expriments <- 30
etatest <- matrix(c(1,1,
0,0),etasize,classsize)
alphatest <- matrix(c(1,0,2,
4,3,5),alphasize,classsize)
sigmatest <- 0.5
EMsimulation_result <- EMsimulation(eta=etatest,alpha=alphatest,sigma=sigmatest,
samplesize=samplesize,expriments=expriments,
compact_flag=TRUE,C0=5,C1=0.5,C2=5)
index <- which(EMsimulation_result$sigma<0.8)
EMsimulation_result_sort <- EM_result_sort(EMsimulation_result$alpha[index,,],
EMsimulation_result$eta[index,,])
#test of EM_parameter_sd
EM_parameter <- EM_parameter_sd(EMsimulation_result_sort$alpha,
EMsimulation_result_sort$eta,
EMsimulation_result$sigma[index])
Sort Parameter
Description
Since the number of subgroup types is always beyond 1, the order of subgroup may be different, we can sort them in this function.
Usage
EM_result_sort(EMsimulation_result_alpha, EMsimulation_result_eta)
Arguments
EMsimulation_result_alpha |
alpha estimated by bootstrap method. |
EMsimulation_result_eta |
eta estimated by bootstrap method. |
Value
alpha |
sorted alpha estimated by bootstrap method. |
eta |
sorted eta estimated by bootstrap method. |
Author(s)
Linsui Deng
Examples
#parameter initialization
etasize <- 2
classsize <- 2
alphasize <- 3
samplesize <- 100
expriments <- 30
etatest <- matrix(c(1,1,
0,0),etasize,classsize)
alphatest <- matrix(c(1,0,2,
4,3,5),alphasize,classsize)
sigmatest <- 0.5
#test of EMsimulation
EMsimulation_result <- EMsimulation(eta=etatest,alpha=alphatest,sigma=sigmatest,
samplesize=samplesize,expriments=expriments,
compact_flag=TRUE,C0=5,C1=0.5,C2=5)
#test of EM_result_sort
EMsimulation_result_sort <- EM_result_sort(EMsimulation_result$alpha,
EMsimulation_result$eta)
Parameter Estimation
Description
Estimate paramters value by EM algorithm under the assumption of Sigmoid-Normal Model.
Usage
EMalgorithm(X, Y, Z, etat, alphat, sigmat, classsize = 2, learning_rate = 0.1,
regular_parameter_eta = 0.001, max_iteration = 10000,
max_iteration_eta = 10000, compact_flag = FALSE, C0 = 5, C1 = 2, C2 = 9)
Arguments
X |
the covariables of the mean of each subgroup |
Y |
the respond variable |
Z |
the covaraibles determining subgroup |
etat |
the coeffients determining subgroup |
alphat |
the coeffients of the mean of each subgroup |
sigmat |
the variance of Y |
classsize |
the number of subgroup types in your model assumption |
learning_rate |
learning rate of updating eta |
regular_parameter_eta |
regular value of updating eta by gradiant descending methond. |
max_iteration |
maximum steps of interation to avoid unlimited looping. |
max_iteration_eta |
maximal steps of eta interation to avoid unlimited looping. |
compact_flag |
if the value of eta is limited in a compact set, set it TRUE |
C0 |
the maximum of intercept of eta. |
C1 |
the minimum of the norm of slope of eta |
C2 |
the maximum of the norm of slope of eta |
Value
alpha |
alpha estimation |
eta |
eta estimation |
sigma |
sigma estimation |
Author(s)
Linsui Deng
Examples
#data generation
samplesize <- 1000
classsize <- 2
etasize <- 3
alphasize <- 3
set.seed(1)
Xtest <- data.frame(matrix(rnorm(samplesize*etasize),samplesize,etasize))
etatest <- matrix(c(1,2,-1,
0,0,0),etasize,classsize)
Ztest <- matrix(rnorm(samplesize*alphasize),samplesize,alphasize)
alphatest <- matrix(c(1,0,2,
5,0,7),alphasize,classsize)
sigmatest <- 5
Wtest <- Wgenerate(alpha=alphatest,eta=etatest,X=Xtest,Z=Ztest,sigma=sigmatest)
eta_initial <- matrix(c(rnorm(3),0,0,0),etasize,classsize)
alpha_initial<- matrix(rnorm(alphasize*classsize)*3,alphasize,classsize)
sigma_initial <- 1
EMtheta <- EMalgorithm(X=Wtest$X,Z=Wtest$Z,Y=Wtest$Y,classsize=2,
etat=eta_initial,alphat=alpha_initial,sigmat=sigma_initial,
learning_rate=0.01,regular_parameter_eta=0.001,
max_iteration=1000,max_iteration_eta=10000,
compact_flag = TRUE, C0 = 5, C1 = 2, C2 = 9)
Bootstrap Method
Description
Estimate the value of parameters several times by bootstrap method where the parameters are estimated by EM algorithm. In this way, we can observe the distribution of parameter.
Usage
EMbootstrap(X, Y, Z, samplesize, boots_samplesize, boots_expriments, classsize = 2,
learning_rate = 0.1, regular_parameter_eta = 0.001, max_iteration = 10000,
max_iteration_eta = 10000, compact_flag = FALSE, C0 = 5, C1 = 2, C2 = 9)
Arguments
X |
the co-variables of the mean of each subgroup |
Y |
the respond variable |
Z |
the co-variables determining subgroup |
samplesize |
the size of this sample. |
boots_samplesize |
the size of chosen sample in one step of bootstrap |
boots_expriments |
the steps of bootstrap |
classsize |
the number of subgroup types in your model assumption |
learning_rate |
learning rate of updating eta |
regular_parameter_eta |
regular value of updating eta by gradient descending method. |
max_iteration |
maximum steps of iteration to avoid unlimited looping. |
max_iteration_eta |
maximal steps of eta iteration to avoid unlimited looping. |
compact_flag |
if the value of eta is limited in a compact set, set it TRUE |
C0 |
the maximum of intercept of eta. |
C1 |
the minimum of the norm of slope of eta |
C2 |
the maximum of the norm of slope of eta |
Details
Actually, the method can be extended to other parameter estimation where the standard error of parameter can't be calculated in a simple way.
Value
alpha |
alpha estimated by bootstrap method. |
eta |
eta estimated by bootstrap method. |
sigma |
sigma estimated by bootstrap method. |
Author(s)
Linsui Deng
Examples
#parameter initialization
etasize <- 2
classsize <- 2
alphasize <- 3
samplesize <- 1000
etatest <- matrix(c(1,1,
0,0),etasize,classsize)
alphatest <- matrix(c(1,0,2,
4,3,5),alphasize,classsize)
sigmatest <- 0.5
#test of EMbootstrap
X <- matrix(c(matrix(1,samplesize),
rnorm(samplesize*(etasize-1))+1),samplesize,etasize)
Z <- matrix(c(matrix(1,samplesize),rbinom(prob=1/2,size=1,n=samplesize),
rnorm(samplesize*(alphasize-2))+1),samplesize,alphasize)
Wtest <- Wgenerate(alpha=alphatest,eta=etatest,sigma=sigmatest,X=X,Z=Z)
boots_samplesize <- 100
boots_expriments <- 30
samplesize <- dim(Wtest$X)[1]
EMbootstrap_theta <- EMbootstrap(Wtest$X,Wtest$Y,Wtest$Z,samplesize,
boots_samplesize,boots_expriments,
classsize=2,compact_flag=TRUE,C0=5,C1=0.2,C2=5)
index <- which(EMbootstrap_theta$sigma<0.8)
EMsimulation_result_sort <- EM_result_sort(EMbootstrap_theta$alpha[index,,],
EMbootstrap_theta$eta[index,,])
EM_parameter <- EM_parameter_sd(EMsimulation_result_sort$alpha,
EMsimulation_result_sort$eta,
EMbootstrap_theta$sigma[index])
Simulation For Estimation
Description
It simulates the experiments with given alpha, eta and sigma to verify the EMalgorithm
Usage
EMsimulation(eta, alpha, sigma, samplesize, expriments,
compact_flag = FALSE, C0 = 5, C1 = 2, C2 = 9)
Arguments
eta |
the true value of eta |
alpha |
the true value of alpha |
sigma |
the true value of sigma |
samplesize |
the size of sample |
expriments |
the times of experiments |
compact_flag |
if the value of eta is limited in a compact set, set it TRUE |
C0 |
the maximum of intercept of eta. |
C1 |
the minimum of the norm of slope of eta |
C2 |
the maximum of the norm of slope of eta |
Value
alpha |
alpha estimated in simulation. |
eta |
eta estimated in simulation. |
sigma |
sigma estimated in simulation. |
Author(s)
Linsui Deng
Examples
#parameter initialization
etasize <- 2
classsize <- 2
alphasize <- 3
samplesize <- 100
expriments <- 30
etatest <- matrix(c(1,1,
0,0),etasize,classsize)
alphatest <- matrix(c(1,0,2,
4,3,5),alphasize,classsize)
sigmatest <- 0.5
#test of EMsimulation
EMsimulation_result <- EMsimulation(eta=etatest,alpha=alphatest,sigma=sigmatest,
samplesize=samplesize,expriments=expriments,
compact_flag=TRUE,C0=5,C1=0.5,C2=5)
Subgroup Determination
Description
In data genaration, determining the subgroup of each item belonging to through random number and Sigmoid Link function.
Usage
Ggenerate(eta, X, seed = 0)
Arguments
eta |
the coeffients determining subgroup |
X |
the covariables determining subgroup |
seed |
random seed |
Value
the classes items belonging to, it's a vector. If X1 belongs to class 3, then the 1st row 3rd colmn is 1 and the rest of 1st row are 0.
Author(s)
Linsui Deng
Examples
#some variables
samplesize <- 1000
classsize <- 6
etasize <- 3
alphasize <- 2
#test of Ggenerate
Xtest <- data.frame(matrix(rnorm(samplesize*etasize),samplesize,etasize))
etatest <- matrix(seq(1.15,1,length=etasize*classsize),etasize,classsize)
Gtest1 <- Ggenerate(etatest,Xtest)
Gtest2 <- Ggenerate(etatest,Xtest,1)
Sigmoid Logistic Data Generation
Description
Generate data satisties Sigmoid Logistic Model to check EMalgorithm.
Usage
Wgenerate(alpha, sigma = 1, eta, samplesize = 0, X, Z, seed1 = 0, seed2 = 0)
Arguments
alpha |
the coeffients of the mean of each subgroup |
sigma |
the variance of Y |
eta |
the coeffients determining subgroup |
samplesize |
the size of sample you wanna generate |
X |
the covariables of the mean of each subgroup |
Z |
the covaraibles determining subgroup |
seed1 |
random seed of generating Y |
seed2 |
random seed of generating G |
Value
X |
the covariables of the mean of each subgroup |
Z |
the covaraibles determining subgroup |
Y |
the generated respond variable |
G |
the classes items belonging to |
Author(s)
Linsui Deng
Examples
#some variables
samplesize <- 1000
classsize <- 6
etasize <- 3
alphasize <- 2
#test of Wgenerate
Xtest <- data.frame(matrix(rnorm(samplesize*etasize),samplesize,etasize))
Ztest <- matrix(rnorm(samplesize*alphasize),samplesize,alphasize)
etatest <- matrix(seq(1.15,1,length=etasize*classsize),etasize,classsize)
alphatest <- matrix(seq(1.15,1,length=alphasize*classsize),alphasize,classsize)
Wtest <- Wgenerate(alpha=alphatest,eta=etatest,X=Xtest,Z=Ztest)
Density Value
Description
Calculate the density value of respond value Y under each mean and homogeneous variance.
Usage
fnorm(Y, mu, sigma)
Arguments
Y |
the respond variable |
mu |
different mean of each subgroup |
sigma |
standard error |
Value
the density value of Y under different mu and common sigma.
Author(s)
Linsui Deng
Examples
fnormtest <- fnorm(matrix(1:6,3,2),matrix(seq(1,3,length=6),3,2),1)
Softmax Value
Description
Calculate the Softmax Value of each subgroup to represent the probability of items belonging to specific class.
Usage
softmax(eta, X)
Arguments
eta |
the coeffients determining subgroup |
X |
the covariables determining subgroup |
Value
Softmax Value of each subgroup
Author(s)
Linsui Deng
Examples
#some variables
samplesize <- 1000
classsize <- 6
etasize <- 3
alphasize <- 2
#test of softmax
Xtest <- data.frame(matrix(rnorm(samplesize*etasize),samplesize,etasize))
etatest <- matrix(seq(1.15,1,length=etasize*classsize),etasize,classsize)
softmax_value <- softmax(etatest,Xtest)
Data Standardlization
Description
Standardlize the data to elimilate the effect of scale
Usage
standard(X)
Arguments
X |
the original data |
Value
standarlized data
Author(s)
Linsui Deng
Examples
#data generata
Y <- rnorm(100)*2+5
Y_sta <- standard(Y)
Updata Eta
Description
Updata eta in step t+1 with given data and coeffients estimated in step t.
Usage
update_eta(fun, alphat, sigmat, etat, X, Y, Z, learning_rate_eta = 0.001,
regular_parameter_eta = 0.001, max_iteration_eta = 10000)
Arguments
fun |
the function updata eta |
alphat |
the estimated coeffients of the mean of each subgroup in step t |
sigmat |
the estimated standard error of Y in step t |
etat |
the estimated coeffients determining subgroup in step t |
X |
the covariables of the mean of each subgroup |
Z |
the covaraibles determining subgroup |
Y |
the respond variable |
learning_rate_eta |
learning rate of updating eta |
regular_parameter_eta |
regular value of updating eta by gradiant descending methond. |
max_iteration_eta |
maximal steps of eta interation to avoid unlimited looping. |
Value
alpha |
alpha estimated in step t. |
eta |
eta estimated in step t+1. |
sigma |
sigma estimated in step t. |
Author(s)
Linsui Deng
Examples
#some variables
samplesize <- 1000
classsize <- 6
etasize <- 3
alphasize <- 2
Xtest <- data.frame(matrix(rnorm(samplesize*etasize),samplesize,etasize))
Ztest <- matrix(rnorm(samplesize*alphasize),samplesize,alphasize)
etatest <- matrix(seq(1.15,1,length=etasize*classsize),etasize,classsize)
alphatest <- matrix(seq(1.15,1,length=alphasize*classsize),alphasize,classsize)
sigmatest <- 0.1
Wtest <- Wgenerate(alpha=alphatest,eta=etatest,X=Xtest,Z=Ztest)
#test of update_eta
thetaupdate_eta <- update_eta(fun=eta_gradient_fun,alphat=alphatest,sigmat=sigmatest,
etat=etatest,X=Wtest$X,Z=Wtest$Z,Y=Wtest$Y,
learning_rate=0.1,regular_parameter=0.001,max_iteration=10000)
Updata Alpha and Sigma
Description
Updata alpha and sigma in t+1 step with given data and coeffients estimated in step t.
Usage
update_gamma(alphat, sigmat, etat, X, Z, Y)
Arguments
alphat |
the estimated coeffients of the mean of each subgroup in step t |
sigmat |
the estimated standard error of Y in step t |
etat |
the estimated coeffients determining subgroup in step t |
X |
the covariables of the mean of each subgroup |
Z |
the covaraibles determining subgroup |
Y |
the respond variable |
Value
alpha |
alpha estimated in step t+1. |
eta |
eta estimated in step t. |
sigma |
sigma estimated in step t+1. |
Author(s)
Linsui Deng
Examples
#some variables
samplesize <- 1000
classsize <- 6
etasize <- 3
alphasize <- 2
Xtest <- data.frame(matrix(rnorm(samplesize*etasize),samplesize,etasize))
Ztest <- matrix(rnorm(samplesize*alphasize),samplesize,alphasize)
etatest <- matrix(seq(1.15,1,length=etasize*classsize),etasize,classsize)
alphatest <- matrix(seq(1.15,1,length=alphasize*classsize),alphasize,classsize)
sigmatest <- 0.1
Wtest <- Wgenerate(alpha=alphatest,eta=etatest,X=Xtest,Z=Ztest)
#test of updata_gamma
thetaupdate_gamma <- update_gamma(alphat=alphatest,sigmat=sigmatest,
etat=etatest,X=Wtest$X,Z=Wtest$Z,Y=Wtest$Y)
Weighted Inner Product
Description
Calculate the weighted inner product of A and B with the weight W. This result is useful in Logistic Regression.
Usage
weight_matrix(A, W, B)
Arguments
A |
the left matrix |
W |
the weight |
B |
the right matrix |
Value
the wighted inner product, noticing it can be vector when A or B is 2 dimension.
Author(s)
Linsui Deng
Examples
#data generation
A <- matrix(rnorm(200),100,2)
W <- rnorm(100)
B <- matrix(rnorm(100),100,1)
weighted <- weight_matrix(A,W,B)