Generating "Random" Datasets with Statistical Patterns

Question

Does anyone know if there any packages in common statistical computing software (e.g. R) that have the ability to simulate realistic random data with statistical patterns?

It's quite straightforward to simulate random data that does not contain any statistical patterns, for example:

var_1 = rnorm(100,10,1)
var_2 = rnorm(100,5,5)
var_3 = rnorm(100,1,1)

id = 1:100

response <- c(0,1)

response_var <- as.factor(sample(response, 100, replace=TRUE, prob=c(0.5, 0.5)))

my_data = data.frame(id, var_1, var_2, var_3, response)

       id     var_1    var_2     var_3 response
1  1 10.008459 4.698752 0.6666546        0
2  2 10.471192 6.710553 2.6666892        1
3  3  9.901345 3.899702 0.6533916        0
4  4 10.343638 5.633607 0.5578606        1
5  5  8.560387 0.662563 0.8842000        0
6  6 10.055957 1.522140 0.8124197        1

But are there any ways to simulate this kind of data so that are there "statistical patterns"? For example, response class "0" is more associated with larger values of var_1 and smaller values of var_2 and var_3? Or any general methods to simulate clustered data containing groups of statistically similar individuals?

Of course, if you spend enough time, you can figure out how to do this manually by simulating multiple datasets and combining them together - but are there any statistical packages that allow you to do something like this for datasets containing many variables?

Thanks!

Note: As an example, I included an example of "crescent shaped" data being simulated with random noise (using R) and a random forest model being used to predict this data - but the data itself is still quite simplistic and doesn't contain the type of statistical patterns/clusters that I want:

#load library
library(RSSL)
library(ggplot2)
library(mlr)

#generate first data
d <- generateCrescentMoon(1000,2,1)
d$c = ifelse(d$Class == "+", "1","0")
d$Class = NULL

#generate second data
c <- sample(0:1, 500, TRUE)

X1 <- runif(500, min=-5, max=0)
X2 <- runif(500, min=-10, max=10)

a = data.frame(X1,X2,c)
a$c = as.factor(a$c)

g = rbind(a,d)

ggplot(g, aes(x=X1, y=X2, color=c, shape=c)) +  geom_point()


#mlr

aa = makeClassifTask(data = g, target = "c")

#specify and train machine learning algorithms
learners = list(
    makeLearner("classif.svm", kernel = "linear"),
    makeLearner("classif.svm", kernel = "polynomial"),
    makeLearner("classif.svm", kernel = "radial"),
    "classif.rpart",
    "classif.randomForest",
    "classif.knn"
)

plotLearnerPrediction(learner = learners[[5]], task = aa)
 plotLearnerPrediction(learner = learners[[4]], task = aa)

Additional References:

• https://rviews.rstudio.com/2020/09/09/fake-data-with-r/
• https://cran.r-project.org/web/packages/MixSim/index.html

Answer 1

Since "statistical patterns" is an infinitely broad category, this is an overly broad question. In terms of the available software packages, this depends a great deal on what statistical model you wish to simulate from. Here are a few standard ones you might be interested in, and how to implement them in R.

---

Continuous data

• Multivariate Gaussian: To generate data from this distribution (with some specified correlation matrix) you can use the rmvnorm function in the mvtnorm package.

• Regression models: If you have already generated the explanatory variables for a regression model (e.g., with the multivariate Gaussian distribution) you can then generate the response variable in the model directly using the model equation using a randomly generated error term. This allows you to generate data from Gaussian regression models, logistic regression models, other GLMs, etc.

• Time-series models: To generate data from the stationary Gaussian ARMA model (with some specified parameters) you can use the rGARMA function in the ts.extend package (see also O'Neill 2021).

---

Discrete data

• Simple-random-sampling: To generate data using simple-random-sampling from a specified population of values you can use the sample function in the base package. This function can accomodate simple-random-sampling with or without replacement.

• Balls-in-bins model: To generate data from the extended balls-in-bins model you can use the sample.ballbin function in the occupancy package.