Blain Waan - 3 months ago 7

R Question

This question arose from one of my previous questions where the problem of generating two correlated series was resolved with some limitations. We were trying to produce two correlated series that followed exponential distributions with certain parameters. For example, a variable **tr** with mean **1** and another variable **t** with mean **2** with a correlation of **-0.5** were required satisfying the condition that **t>tr**). The following codes were tried in

`R`

`rho <- -0.5`

mu <- rep(0,2)

Sigma <- matrix(rho, nrow=2, ncol=2) + diag(2)*(1 - rho)

library(MASS)

compute.tr.t <- function(req.n, paccept) {

req.n <- round(req.n / paccept)

rawvars <- mvrnorm(req.n, mu=mu, Sigma=Sigma)

pvars <- pnorm(rawvars)

tr <- qexp(pvars[,1], 1/1)

t <- qexp(pvars[,2], 1/2)

keep <- which(t > tr)

return(data.frame(t=t[keep],tr=tr[keep]))

}

req.n <- n

paccept <- 1

res <- data.frame()

while (req.n > 0) {

new.res <- compute.tr.t(req.n, paccept)

res <- rbind(res, new.res)

req.n <- n - nrow(res)

paccept <- nrow(new.res) / n# updated paccept according to last step

}

The problem that occurred due to pruning data that do not satisfy the condition

- The means were not preserved.
- The correlation was not preserved.

See the output below. It is evident that because of imposing such a condition the locations shift.

`mean(res$tr)`

[1] 0.4660927

mean(res$t)

[1] 2.859441

print(cor(res$tr,res$t))

[1] -0.237159

Answer

**Updated answer** *every element* of t strictly greater than tr:

```
n <- 100
rho <- 0.5
mu <- rep(0,2)
Sigma <- matrix(rho, nrow=2, ncol=2) + diag(2)*(1 - rho)
library(MASS)
compute.tr.t <- function(req.n, paccept) {
req.n <- round(req.n / paccept)
rawvars <- mvrnorm(req.n, mu=mu, Sigma=Sigma)
pvars <- pnorm(rawvars)
tr <- qexp(pvars[,1], 1/1)
t <- qexp(pvars[,2], 1/2)
tr <- tr[(tr-mean(tr))^2 <.25 ] # can play with this value
t <- t[(t-mean(t))^2 <.25 ]
m <- min(length(t), length(tr))
t <- t[1:m]
tr <- tr[1:m]
return(data.frame(t=t,tr=tr))
}
req.n <- n
paccept <- 1
res <- data.frame()
while (req.n > 0) {
new.res <- compute.tr.t(req.n, paccept)
res <- rbind(res, new.res)
req.n <- n - nrow(res)
paccept <- nrow(new.res) / n
}
mean(res$t)
```

[1] 1.972218

```
mean(res$tr)
```

[1] 0.590776

```
table(res$t > res$tr) # should be all true, rarely you'll get 1 trivial false that you can kick out
```

`TRUE 132`

```
cor(res$t,res$tr) # suffered a little but not too bad, can probably improve
```

[1] .2527064

**Original answer** mean(t) > mean(tr) but not every element:

```
n <- 100
rho <- 0.5
mu <- rep(0,2)
Sigma <- matrix(rho, nrow=2, ncol=2) + diag(2)*(1 - rho)
library(MASS)
compute.tr.t <- function(req.n, paccept) {
req.n <- round(req.n / paccept)
rawvars <- mvrnorm(req.n, mu=mu, Sigma=Sigma)
pvars <- pnorm(rawvars)
tr <- qexp(pvars[,1], 1/1)
t <- qexp(pvars[,2], 1/2)
keep <- which(t > tr)
return(data.frame(t=t,tr=tr))
}
req.n <- n
paccept <- 1
res <- data.frame()
while (req.n > 0) {
new.res <- compute.tr.t(req.n, paccept)
res <- rbind(res, new.res)
req.n <- n - nrow(res)
paccept <- nrow(new.res) / n# updated paccept according to last step
}
mean(res$tr)
```

[1] 0.9399213

```
mean(res$t)
```

[1] 1.795431

```
print(cor(res$tr,res$t))
```

[1] 0.5075668

Since there's some randomness in this for good measure I ran it a 2nd time and got the following result:

```
mean(res$tr)
```

[1] 1.001255

```
mean(res$t)
```

[1] 1.922343

```
print(cor(res$tr,res$t))
```

[1] 0.6648311

After you run it once if you don't quite like the results a simple hack to meet any desired level of precision is:

```
while(
(cor(res$tr,res$t) > .55 | cor(res$tr,res$t) < .45)
){
n <- 100
rho <- 0.5
mu <- rep(0,2)
Sigma <- matrix(rho, nrow=2, ncol=2) + diag(2)*(1 - rho)
library(MASS)
compute.tr.t <- function(req.n, paccept) {
req.n <- round(req.n / paccept)
rawvars <- mvrnorm(req.n, mu=mu, Sigma=Sigma)
pvars <- pnorm(rawvars)
tr <- qexp(pvars[,1], 1/1)
t <- qexp(pvars[,2], 1/2)
keep <- which(t > tr)
return(data.frame(t=t,tr=tr))
}
req.n <- n
paccept <- 1
res <- data.frame()
while (req.n > 0) {
new.res <- compute.tr.t(req.n, paccept)
res <- rbind(res, new.res)
req.n <- n - nrow(res)
paccept <- nrow(new.res) / n# updated paccept according to last step
}
}
```