(a) Generate 1000 samples where each consists of 50 independent exponential random variables with
mean 1. Estimate the mean of each sample. Draw a histogram of the means.
(b) Perform a KS test on each sample against the null hypothesis that they are from an exponential
random variable with a mean that matches the mean of the data set. Draw a histogram of the
1000 values of D.
i did part a with this code
set.seed(0)
simdata = rexp(50000, 1)
matrixdata = matrix(simdata,nrow=50,ncol=1000)
means.exp = apply(matrixdata,2,mean)
means.exp
hist(means.exp)
but im stuck on part (b)
You can use lapply on the column indices:
# KS test on every column
# H0: pexp(rate = 1/mean(column))
lst.ks <- lapply(1:ncol(matrixdata), function(i)
ks.test(matrixdata[, i], "pexp", 1.0/means.exp[i]))
Or directly without having to rely on means.exp:
lst.ks <- lapply(1:ncol(matrixdata), function(i)
ks.test(matrixdata[, i], "pexp", 1.0/mean(matrixdata[, i])))
Here 1.0/means.exp[i] corresponds to the rate of the exponential distribution.
PS. Using means.exp = colMeans(matrixdata) is faster than apply(matrixdata, 2, mean), see e.g. here for a relevant SO post.
To extract the test statistic and store it in a vector simply sapply over the KS test results:
# Extract test statistic as vector
Dstat <- sapply(lst.ks, function(x) x$statistic);
# (gg)plot Dstat
ggplot(data.frame(D = Dstat), aes(D)) + geom_histogram(bins = 30);
Related
Using the mvrnorm() from the MASS package, now we can simulate realizations of multivariate normal distributions. This function works as follows:
library(MASS)
MASS::mvrnorm(
n = 10, # Number of realizations,
mu = c(1, 5), # Parameter vector mu,
Sigma = my_cov_matrix(1, 3, 0.2) # Parameter matrix Sigma
)
What does this output mean? Why are there two columns with ten random variables each?
The task is as follows:
Now, I created a function my_mvrnorm(n, mu_1, mu_2, sigma_1, sigma_2, rho), which simulates realizations of the corresponding multivariate normal distribution depending on mu and the matrix n and stores them in a tibble with the column names X and Y. In addition, this tibble is to contain a third column rho, in which all entries are filled with rho.
This should look like the following then:
But I couldn't write a function yet, because I don't quite understand what the values in table X and Y should be. Can someone help me?
Attempt:
my_mvrnorm <- function(n, mu_1, mu_2, sigma_1, sigma_2, rho){
mu = c(mu_1, mu_2)
sigma = my_cov_matrix(sigma_1, sigma_2, rho)
tb <- tibble(
X = ,
Y = ,
rho = rep(rho, n)
)
return(tb)
}
The n = 10 specification says do 10 samples. The mu = c(1, 5) specification says do two means. So, you get a 10 X 2 matrix as the result. If you check, the first column has a mean close to 2, and the second a mean close to 5. Is my_cov_matrix defined somewhere else?
For a paper I'm writing I have subsetted a larger dataset into 3 groups, because I thought the strength of correlations between 2 variables in those groups would differ (they did). I want to see if subsetting my data into random groupings would also significantly affect the strength of correlations (i.e., whether what I'm seeing is just an effect of subsetting, or if those groupings are actually significant).
To this end, I am trying to generate n new data frames by randomly sampling 150 rows from an existing dataset, and then want to calculate correlation coefficients for two variables in those n new data frames, saving the correlation coefficient and significance in a new file.
But, HOW?
I can do it manually, e.g., with dplyr, something like
newdata <- sample_n(Random_sample_data, 150)
output <- cor.test(newdata$x, newdata$y, method="kendall")
I'd obviously like to not type this out 1000 or 100000 times, and have been trying things with loops and lapply (see below) but they've not worked (undoubtedly due to something really obvious that I'm missing!).
Here I have tried to assign each row to a different group, with 10 groups in total, and then to do correlations between x and y by those groups:
Random_sample_data<-select(Range_corrected, x, y)
cat <- sample(1:10, 1229, replace=TRUE)
Random_sample_cats<-cbind(Random_sample_data,cat)
correlation <- function(c) {
c <- cor.test(x,y, method="kendall")
return(c)
}
b<- daply(Random_sample_cats, .(cat), correlation)
Error message:
Error in cor.test(x, y, method = "kendall") :
object 'x' not found
Once you have the code for what you want to do once, you can put it in replicate to do it n times. Here's a reproducible example on built-in data
result = replicate(n = 10, expr = {
newdata <- sample_n(mtcars, 10)
output <- cor.test(newdata$wt, newdata$qsec, method="kendall")
})
replicate will save the result of the last line of what you did (output <- ...) for each replication. It will attempt to simplify the result, in this case cor.test returns a list of length 8, so replicate will simplify the results to a matrix with 8 rows and 10 columns (1 column per replication).
You may want to clean up the results a little bit so that, e.g., you only save the p-value. Here, we store only the p-value, so the result is a vector with one p-value per replication, not a matrix:
result = replicate(n = 10, expr = {
newdata <- sample_n(mtcars, 10)
cor.test(newdata$wt, newdata$qsec, method="kendall")$p.value
})
I am trying to inject anomalies into a dataset, essentially changing certain values, based on a condition. I have a dataset, there are 10 subsets. The condition is that anomalies would be 2.8-3 times the standard deviation of each segment away from the mean of that subset. For that, I am dividing the dataset into 10 equal parts, then calculating the mean and standard deviation of each subset, and changing certain values by putting them 3 standard deviations of that subset away from the mean of that subset. The code looks like the following:
set.seed(1)
x <- rnorm(sample(1:35000, 32000, replace=F),0,1) #create dataset
y <- cumsum(x) #cumulative sum of dataset
j=1
for(i in c(1:10)){
seg = y[j:j+3000] #name each subset seg
m = mean(seg) #mean of subset
print(m)
s = sd(seg) # standard deviation of subset
print(s)
o_data = sample(j:j+3000,10) #draw random numbers from j to j + 3000
print(o_data)
y[o_data] = m + runif(10, min=2.8, max=3) * s #values = mean + 2.8-3 * sd
print(y[o_data])
j = j + 3000 # increment j
print(j)
}
The error I get is that standard deviation is NA, so I am not able to set the values.
What other approach is there by which I can accomplish the task? I have the inject anomalies which are 2.8-3 standard deviations away from the rolling mean essentially.
You have a simple error in your code. when you wrote
seg = y[j:j+3000] I believe that you meant seg = y[j:(j+3000)]
Similarly o_data = sample(j:j+3000,10) should be o_data = sample(j:(j+3000),10)
I am trying to fit a for Loop in R in order to run correlations for multiple subsets in a data frame and then store the results in a vector.
What I have in this loop is a data frame with 2 columns, x and y, and 30 rows of different continuous measurement values in each column. The process should be repeated 100 times. The data can be invented.
What I need, is to compute the Spearman's rho for the first five rows (between x and y) and then for increasing subsets (e.g., the sixth first rows, the sevenths first rows etc.). Then, I'd need to store the rho results in a vector that I can further use.
What I had in mind (but does not work):
sortvector <- 1:(30)
for (i in 1:100)
{
sortvector <- sample(sortvector, replace = F)
xtemp <- x[sortvector]
rho <- cor.test(xtemp,y, method="spearman")$estimate
}
The problem is that the code gives me one value of rho for the whole dataframe, but I need it for increments of subsets.
How can I get rho for subsets of increasing values in a for-loop? And how can i store the coefficients in a vector that i can use afterwards?
Any help would be much appreciated, thanks.
Cheers
The easiest approach is to convertfor loop into sapply function, which returns a vector of rho's as a result of your bootstrapping:
sortvector <- 1:(30)
x <- rnorm(30)
y <- rnorm(30)
rho <- sapply(1:100, function(i) {
sortvector <- sample(sortvector, replace = F)
xtemp <- x[sortvector]
cor.test(xtemp, y, method = "spearman")$estimate
})
head(rho)
Output:
rho rho rho rho rho rho
0.014460512 -0.239599555 0.003337041 -0.126585095 0.007341491 0.264516129
I wish to calculate the distance between two 3-dimensional posterior distributions. The draws are stored at two 30,000x3 matrices.
So far I have been successful in calculating Total Variation distance between two 2-dimensional posteriors (two 30,000x2 matrices) by splitting the grid into bins. However, I am having trouble calculating the divergence between posteriors with more parameters. Some examples of related distance measures can be found here.
NOTE: I do not wish to calculate the distance between the marginals (column-wise entries), rather than obtain an overall value after comparing the joint distributions in R.
I would really appreciate it if somebody could point out what I am missing here.
EDIT 1: Some example code for calculating Total variation distance between posterior samples stored in two matrices has been added below:
EDIT 2: This is a R question.
set.seed(123)
comparison.2D <- matrix(rnorm(40000*2,0,1),ncol=2)
ground.truth.2D <- matrix(rnorm(40000*2,0,2),ncol=2)
# Function to calculate TVD between matrices with 2 columns:
Total.Variation.Distance.2D<-function(true,
comparison,
burnin,
window.size){
# Bandwidth for theta.1.
my_bw_x<-window.size
# Bandwidth for theta.2.
my_bw_y<-window.size
range_x<-range(c(true[-c(1:burnin),1],comparison[-c(1:burnin),1]))
range_y<-range(c(true[-c(1:burnin),2],comparison[-c(1:burnin),2]))
xx <- seq(range_x[1],range_x[2],by=my_bw_x)
yy <- seq(range_y[1],range_y[2],by=my_bw_y)
true.pointidxs <- matrix( c( findInterval(true[-c(1:burnin),1], xx),
findInterval(true[-c(1:burnin),2], yy) ), ncol=2)
comparison.pointidxs <- matrix( c( findInterval(comparison[-c(1:burnin),1], xx),
findInterval(comparison[-c(1:burnin),2], yy) ), ncol=2)
# Count the frequencies in the corresponding cells:
square.mat.dims <- max(length(xx),nrow=length(yy))
frequencies.true <- frequencies.comparison <- matrix(0, ncol=square.mat.dims, nrow=square.mat.dims)
for (i in 1:dim(true.pointidxs)[1]){
frequencies.true[true.pointidxs[i,1], true.pointidxs[i,2]] <- frequencies.true[true.pointidxs[i,1],
true.pointidxs[i,2]] + 1
frequencies.comparison[comparison.pointidxs[i,1], comparison.pointidxs[i,2]] <- frequencies.comparison[comparison.pointidxs[i,1],
comparison.pointidxs[i,2]] + 1
}# End for
# Normalize frequencies matrix:
frequencies.true <- frequencies.true/dim(true.pointidxs)[1]
frequencies.comparison <- frequencies.comparison/dim(comparison.pointidxs)[1]
TVD <-0.5*sum(abs(frequencies.comparison-frequencies.true))
return(TVD)
}# End function
TVD.2D <- Total.Variation.Distance.2D(true=ground.truth.2D, comparison=comparison.2D,burnin=10000,window.size=0.05)