I've written a function to calculate the MLE estimates of a Generalised Pareto Distribution. When I use it with any data though I'm getting errors like this
1: In log(beta * ksi) : NaNs produced
2: In nlm(loglik, theta, stepmax = 5000, iterlim = 1000) :
NA/Inf replaced by maximum positive value
I was wondering if anyone could spot any mistakes with my code?
MLGPD<-function(data){
xi0 <- 1
beta0 <- 360
theta <- c(xi0, beta0)
excess <- data
assign("tmp", excess)
loglik <- function(theta){
ksi <- theta[1]
beta <- theta[2]
y <- ((tmp - 0.1)/beta)
f <- ((1/ksi)+1)*sum(log(1+y)) + length(tmp) * log(beta*ksi)
f
}
fit <- nlm(loglik, theta, stepmax = 5000, iterlim= 1000)
return(fit)
par.ests <- fit$x
return(par.ests)
}
#Checking our MLE algorithm works:
rgpd<-function(n,ksi, beta){
10000+beta*(((1-runif(n, min=0, max=1))^-ksi)-1)
}
rgpd1 <- rgpd(100, 1, 2.5)
MLGPD(rgpd1)
Thanks!
Related
I am implementing the Dirichlet Mixture Model using the EM algorithm in R, but am experiencing issues with the results. I generated two binomial distributions with fractions of (70%, 30%) and means of (0.05, 0.18), and trimmed 5% of the data set near 0. However, I am using a Beta distribution for clustering instead of a binomial distribution. Additionally, I am updating the mean and variance of the distributions rather than the alpha and beta parameters in order to impose constraints on the variance of each distribution.
I expected to obtain results similar to the ground truth settings, but instead I am getting pi values of (1, 0) and means of (0.09, 0.21). I am not sure if there are errors in my EM algorithm implementation or issues with parameter initialization.
I am including my R code for the data generation and DMM below. I would appreciate any help in identifying the cause of the problem and suggestions for how to resolve it.
library(dplyr)
library(data.table)
library(tidyverse)
set.seed(42)
#read count
cover <- 100
#Ground Truth Setting
subclone_f <- c(0.7, 0.3) # Ground Truth Setting - proportion
subclone_vaf <- c(0.05, 0.18) # Ground Truth Setting - mean
n_muts <- 45000
n_clone <-length(subclone_f)
#generating the virtual mutation notation: subclonal if 2, clonal if 1
mut_type <- sample.int(2, n_muts, prob = subclone_f, replace = TRUE)
mut_type
#generating negative binomial distribution(read count) for the given coverage
mut_reads <- rbinom(n_muts, cover, prob = subclone_vaf[mut_type]) %>% data.frame()
mut_reads
vaf <- (mut_reads/cover) %>% data.frame()
# Truncate the low count reads
n <- 0.95 * nrow(vaf) # cut-off setting
vaf_trim <- sapply(vaf, function(x) sort(x, decreasing = TRUE)[1:n])
colnames(vaf_trim) <- c("vaf")
hist(vaf_trim, breaks=seq(0,0.75,by=0.0001))
# Mixture Model
# Parameter Initialization (for 2 subclonality)
pi <- c(0.5, 0.5) # Mixture proportion weight: sums up to 1
alpha <- c(2,3)
beta <- c(20,5)
Mu[1] <- alpha[1] / (alpha[1] + beta[1])
Mu[2] <- alpha[2] / (alpha[2] + beta[2])
var[1] <- alpha[1]*beta[1] / ((alpha[1] + beta[1])^2 * (alpha[1] + beta[1] +1))
var[2] <- alpha[2]*beta[2] / ((alpha[2] + beta[2])^2 * (alpha[2] + beta[2] +1))
tau <-c(0.05, 0.05)
loglike[1] <- 0.5
loglike[2] <- 0.5
k <- 2
Nu <- 1/ (alpha + beta + 1) # control the variance: same across the distributions> Originally wanted to implement the same Nu for 2 distributions but I don't know how to do that.
n_cluster <- nrow(data.frame(pi))
logdbeta <- function(x, alpha, beta) {
sum(sapply(x, function(x) {dbeta(x, alpha, beta, log = TRUE)}))
}
estBetaParams <- function(mu, var) {
alpha <- ((1 - mu) / var - (1 / mu)) * mu ^ 2
beta <- alpha * (1 / mu - 1)
return(params = list(alpha = alpha, beta = beta))
}
# Loop for the EM algorithm
while(abs(loglike[k]-loglike[k-1]) >= 0.00001) {
# E step
total <- (pi[1]*dbeta(vaf_trim, alpha[1], beta[1])) + (pi[2]*dbeta(vaf_trim, alpha[2], beta[2]))
tau1 <- pi[1]*(dbeta(vaf_trim, alpha[1], beta[1]))/ total
tau2 <- pi[2]*(dbeta(vaf_trim, alpha[2], beta[2]))/ total
# M step
pi[1] <- sum(tau1)/length(vaf_trim) # Update Pi(weight)
pi[2] <- sum(tau2)/length(vaf_trim)
Mu[1] <- sum(tau1*vaf_trim)/sum(tau1) # Update Mu
Mu[2] <- sum(tau2*vaf_trim)/sum(tau2)
#Nu <- alpha + beta
Nu <- 1/ (alpha + beta + 1)
# Our main aim was to share the same coefficient for all dist
var[1] <- Mu[1] * (1-Mu[1]) * Nu[1] # Update Variance
var[2] <- Mu[2] * (1-Mu[2]) * Nu[2]
#Update in terms of alpha and beta
estBetaParams(Mu[1], var[1])
estBetaParams(Mu[2], var[2])
# Maximize the loglikelihood
loglike[k+1]<-sum(tau1*(log(pi[1])+logdbeta(vaf_trim,Mu[1],var[1])))+sum(tau2*(log(pi[2])+logdbeta(vaf_trim,Mu[2],var[2])))
k<-k+1
}
# Print estimates
EM <- data.table(param = c("pi", "Mean"), pi = pi, Mean = Mu)
knitr::kable(EM)
I am trying to figure out how to sample from a custom density in rJAGS but am running into issues. having searched the site, I saw that there is a zeroes (or ones) trick that can be employed based on BUGS code but am having a hard time with its implementation in rJAGS. I think I am doing it correctly but keep getting the following error:
Error in jags.model(model1.spec, data = list(x = x, N = N), n.chains = 4, :
Error in node dpois(lambda)
Length mismatch in Node::setValue
Here is my rJAGS code for reproducibility:
library(rjags)
set.seed(4)
N = 100
x = rexp(N, 3)
L = quantile(x, prob = 1) # Censoring point
censor = ifelse(x <= L, 1, 0) # Censoring indicator
x[censor == 1] <- L
model1.string <-"
model {
for (i in 1:N){
x[i] ~ dpois(lambda)
lambda <- -N*log(1-exp(-(1/mu)))
}
mu ~ dlnorm(mup, taup)
mup <- log(.0001)
taup <- 1/49
R <- 1 - exp(-(1/mu) * .0001)
}
"
model1.spec<-textConnection(model1.string)
jags <- jags.model(model1.spec,
data = list('x' = x,
'N' = N),
n.chains=4,
n.adapt=100)
Here, my negative log likelihood of the density I am interested in is -N*log(1-exp(-(1/mu))). Is there an obvious mistake in the code?
Using the zeros trick, the variable on the left-hand side of the dpois() relationship has to be an N-length vector of zeros. The variable x should show up in the likelihood somewhere. Here is an example using the normal distribution.
set.seed(519)
N <- 100
x <- rnorm(100, mean=3)
z <- rep(0, N)
C <- 10
pi <- pi
model1.string <-"
model {
for (i in 1:N){
lambda[i] <- pow(2*pi*sig2, -0.5) * exp(-.5*pow(x[i]-mu, 2)/sig2)
loglam[i] <- log(lambda[i]) + C
z[i] ~ dpois(loglam[i])
}
mu ~ dnorm(0,.1)
tau ~ dgamma(1,.1)
sig2 <- pow(tau, -1)
sumLL <- sum(log(lambda[]))
}
"
model1.spec<-textConnection(model1.string)
set.seed(519)
jags <- jags.model(model1.spec,
data = list('x' = x,
'z' = z,
'N' = N,
'C' = C,
'pi' = pi),
inits = function()list(tau = 1, mu = 3),
n.chains=4,
n.adapt=100)
samps1 <- coda.samples(jags, c("mu", "sig2"), n.iter=1000)
summary(samps1)
Iterations = 101:1100
Thinning interval = 1
Number of chains = 4
Sample size per chain = 1000
1. Empirical mean and standard deviation for each variable,
plus standard error of the mean:
Mean SD Naive SE Time-series SE
mu 4.493 2.1566 0.034100 0.1821
sig2 1.490 0.5635 0.008909 0.1144
2. Quantiles for each variable:
2.5% 25% 50% 75% 97.5%
mu 0.6709 3.541 5.218 5.993 7.197
sig2 0.7909 0.999 1.357 1.850 2.779
I am trying to implement a linear regression in R from scratch without using any packages or libraries using the following data:
UCI Machine Learning Repository, Bike-Sharing-Dataset
The linear regression was easy enough, here is the code:
data <- read.csv("Bike-Sharing-Dataset/hour.csv")
# Select the useable features
data1 <- data[, c("season", "mnth", "hr", "holiday", "weekday", "workingday", "weathersit", "temp", "atemp", "hum", "windspeed", "cnt")]
# Split the data
trainingObs<-sample(nrow(data1),0.70*nrow(data1),replace=FALSE)
# Create the training dataset
trainingDS<-data1[trainingObs,]
# Create the test dataset
testDS<-data1[-trainingObs,]
x0 <- rep(1, nrow(trainingDS)) # column of 1's
x1 <- trainingDS[, c("season", "mnth", "hr", "holiday", "weekday", "workingday", "weathersit", "temp", "atemp", "hum", "windspeed")]
# create the x- matrix of explanatory variables
x <- as.matrix(cbind(x0,x1))
# create the y-matrix of dependent variables
y <- as.matrix(trainingDS$cnt)
m <- nrow(y)
solve(t(x)%*%x)%*%t(x)%*%y
The next step is to implement the batch update gradient descent and here is where I am running into problems. I dont know where the errors are coming from or how to fix them, but the code works. The problem is that the values being produced are radically different from the results of the regression and I am unsure of why.
The two versions of the batch update gradient descent that I have implemented are as follows (the results of both algorithms differ from one another and from the results of the regression):
# Gradient descent 1
gradientDesc <- function(x, y, learn_rate, conv_threshold, n, max_iter) {
plot(x, y, col = "blue", pch = 20)
m <- runif(1, 0, 1)
c <- runif(1, 0, 1)
yhat <- m * x + c
MSE <- sum((y - yhat) ^ 2) / n
converged = F
iterations = 0
while(converged == F) {
## Implement the gradient descent algorithm
m_new <- m - learn_rate * ((1 / n) * (sum((yhat - y) * x)))
c_new <- c - learn_rate * ((1 / n) * (sum(yhat - y)))
m <- m_new
c <- c_new
yhat <- m * x + c
MSE_new <- sum((y - yhat) ^ 2) / n
if(MSE - MSE_new <= conv_threshold) {
abline(c, m)
converged = T
return(paste("Optimal intercept:", c, "Optimal slope:", m))
}
iterations = iterations + 1
if(iterations > max_iter) {
abline(c, m)
converged = T
return(paste("Optimal intercept:", c, "Optimal slope:", m))
}
}
return(paste("MSE=", MSE))
}
AND:
grad <- function(x, y, theta) { # note that for readability, I redefined theta as a column vector
gradient <- 1/m* t(x) %*% (x %*% theta - y)
return(gradient)
}
grad.descent <- function(x, maxit, alpha){
theta <- matrix(rep(0, length=ncol(x)), ncol = 1)
for (i in 1:maxit) {
theta <- theta - alpha * grad(x, y, theta)
}
return(theta)
}
If someone could explain why these two functions are producing different results I would greatly appreciate it. I also want to make sure that I am in fact implementing the gradient descent correctly.
Lastly, how can I plot the results of the descent with varying learning rates and superimpose this data over the results of the regression itself?
EDIT
Here are the results of running the two algorithms with alpha = .005 and 10,000 iterations:
1)
> gradientDesc(trainingDS, y, 0.005, 0.001, 32, 10000)
TEXT_SHOW_BACKTRACE environmental variable.
[1] "Optimal intercept: 2183458.95872599 Optimal slope: 62417773.0184353"
2)
> print(grad.descent(x, 10000, .005))
[,1]
x0 8.3681113
season 19.8399837
mnth -0.3515479
hr 8.0269388
holiday -16.2429750
weekday 1.9615369
workingday 7.6063719
weathersit -12.0611266
temp 157.5315413
atemp 138.8019732
hum -162.7948299
windspeed 31.5442471
To give you an example of how to write functions like this in a slightly better way, consider the following:
gradientDesc <- function(x, y, learn_rate, conv_threshold, max_iter) {
n <- nrow(x)
m <- runif(ncol(x), 0, 1) # m is a vector of dimension ncol(x), 1
yhat <- x %*% m # since x already contains a constant, no need to add another one
MSE <- sum((y - yhat) ^ 2) / n
converged = F
iterations = 0
while(converged == F) {
m <- m - learn_rate * ( 1/n * t(x) %*% (yhat - y))
yhat <- x %*% m
MSE_new <- sum((y - yhat) ^ 2) / n
if( abs(MSE - MSE_new) <= conv_threshold) {
converged = T
}
iterations = iterations + 1
MSE <- MSE_new
if(iterations >= max_iter) break
}
return(list(converged = converged,
num_iterations = iterations,
MSE = MSE_new,
coefs = m) )
}
For comparison:
ols <- solve(t(x)%*%x)%*%t(x)%*%y
Now,
out <- gradientDesc(x,y, 0.005, 1e-7, 200000)
data.frame(ols, out$coefs)
ols out.coefs
x0 33.0663095 35.2995589
season 18.5603565 18.5779534
mnth -0.1441603 -0.1458521
hr 7.4374031 7.4420685
holiday -21.0608520 -21.3284449
weekday 1.5115838 1.4813259
workingday 5.9953383 5.9643950
weathersit -0.2990723 -0.4073493
temp 100.0719903 147.1157262
atemp 226.9828394 174.0260534
hum -225.7411524 -225.2686640
windspeed 12.3671942 9.5792498
Here, x refers to your x as defined in your first code chunk. Note the similarity between the coefficients. However, also note that
out$converged
[1] FALSE
so that you could increase the accuracy by increasing the number of iterations or by playing around with the step size. It might also help to scale your variables first.
I'm trying to estimate a GARCH (1,1) model using maximum likelihood with simulated data. This is what I got:
library(fGarch)
set.seed(1)
garch11<-garchSpec(model = list())
x<-garchSim(garch11, n = 1000)
y <- t(x)
r <- y[1, ]
### Calculate Residuals
CalcResiduals <- function(theta, r)
{
n <- length(r)
omega<-theta[1]
alpha11<-theta[2]
beta11<-theta[3]
sigma.sqs <- vector(length = n)
sigma.sqs[1] <- 0.02
for (i in 1:(n-1)){
sigma.sqs[i+1] <- omega + alpha11*(r[i]^2) + beta11*sigma.sqs[i]
}
return(list(et=r, ht=sigma.sqs))
}
###Calculate the log-likelihood
GarchLogl <- function(theta, r){
res <- CalcResiduals(theta,r)
sigma.sqs <- res$ht
r <- res$et
return(-sum(dnorm(r[-1], mean = 0, sd = sqrt(sigma.sqs[-1]), log = TRUE)))
}
fit2 <- nlm(GarchLogl, # function call
p = rep(1,3), # initial values = 1 for all parameters
hessian = FALSE, # also return the hessian matrix
r = r , # data to be used
iterlim = 500) # maximum iteration
Unfortunately I get the following error message and no results
There were 50 or more warnings (use warnings() to see the first 50)
1: In sqrt(sigma.sqs[-1]) : NaNs produced
2: In nlm(GarchLogl, p = rep(1, 3), hessian = FALSE, data <- r, ... :
NA/Inf durch größte positive Zahl ersetzt
Do you have any idea whats wrong with my code? Thanks a lot!
I need help, I am doing t test using monte carlo metod but the results shows
Error in t.test(x, alternative = "greater", mu = 500) : unused
arguments (alternative = "greater", mu = 500).
Below is my code
#Compute an empirical probability of type I using Monte Carlo Method
# m is number of replicates
n <- 20
alpha <- .05
mu0 <-500
sigma <- 100
m <- 10000
p <- numeric(m)
for (j in 1:m){
x <- rnorm(n, mu0, sigma)
ttest <- t.test(x, alternative="greater", mu=500)
p[j] <-ttest$p.value
}
p.hat <- mean(p < alpha)
se.hat <-sqrt(p.hat*(1-p.hat)/m)
print(c(p.hat, se.hat))