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Sometimes we need to generate correlated data for exhibition purposes, technical assessments, testing etc. We have provided a walk-through example of how to generate correlated data in Python using the `scikit-learn` library. In R, as far as I know, there is not any library that allows us to generate correlated data. For that reason, we will work with the simulated data from the Multivariate Normal Distribution. I would suggest having a look at the variance-covariance matrix and the relationship between correlation and covariance.

## Generate Correlated Data

We will generate 1000 observations from the Multivariate Normal Distribution of 3 Gaussians as follows:

• V1~N(10,1), V2~N(5,1), V3~N(2,1)
• The correlation of V1 vs V2 is around -0.8, the correlation of V1 vs V2 is around -0.7 and the correlation of V2 vs V3 is around 0.9
```library(MASS)
library(tidyverse)
library(GGally)

set.seed(5)

# create the variance covariance matrix
sigma<-rbind(c(1,-0.8,-0.7), c(-0.8,1, 0.9), c(-0.7,0.9,1))

# create the mean vector
mu<-c(10, 5, 2)

# generate the multivariate normal distribution
df<-as.data.frame(mvrnorm(n=1000, mu=mu, Sigma=sigma))

ggpairs(df)
```

As we can see, we generated the correlated data with the expected outcome in terms of mean, variance, and correlation.

## Generate Categorical Correlated Data

In the case where we want to generate categorical data, we work in two steps. First, we generate the continuous correlated data as we did above, and then we transform it to categorical by creating bins.

### Binary Variables

Let’s see how we can create a Binary variable taking values 0 and 1:

```df<-df%>%mutate(MyBinary = ifelse(V1>median(V1), 1 ,0))
```

### Binary Variables with Noise

In the above example, we set the value “1” when the V1 variable is greater than the median and “0” otherwise. Let’s say that we want to create more noise data, i.e. not fully correlated. Let’s say that we want to apply the following rule:

• If the variable is greater than the median, then assign 1 with a 75% probability
• If the variable is less than the median, then assign 1 with a 25% probability
```df<-df%>%mutate(MyNoisyBinary = ifelse(V1>median(V1), sample(c(0,1),n(), replace = TRUE, p=c(0.25, 0.75)) ,
sample(c(0,1),n(), replace = TRUE, p=c(0.75, 0.25))))
```

### Categorical Variables

Similarly, we can generate categorical variables of many levels. Let’s say that we want to create the `age group` from V1 with the following rules:

• If it is less than the Q1 then Age Group 1
• If it is less than the Q2 then Age Group 2
• if it is less than the Q3 then Age Group 3
• else, Age Group 4
```df<-df%>%mutate(AgeGroup= case_when(V1<quantile(V1,0.25)~"Group 1",
V1<quantile(V1,0.5)~"Group 2",
V1<quantile(V1,0.75)~"Group 3",
TRUE~"Group 4"))
```

### Categorical Variables with Noise

Similarly, we can add some noise to our generated categorical variables as follows:

```df<-df%>%mutate(MyNoisyCat= case_when(V1<quantile(V1,0.25)~sample(c("G1","G2","G3","G4"),n(), replace = TRUE, p=c(0.70, 0.1, 0.1, 0.1)),
V1<quantile(V1,0.5)~sample(c("G1","G2","G3","G4"),n(), replace = TRUE, p=c(0.1, 0.7, 0.1, 0.1)),
V1<quantile(V1,0.75)~sample(c("G1","G2","G3","G4"),n(), replace = TRUE, p=c(0.1, 0.1, 0.7, 0.1)),
TRUE~sample(c("G1","G2","G3","G4"),n(), replace = TRUE, p=c(0.1, 0.1, 0.1, 0.7))))

ggpairs(df)

```