We generate `n`

observations from each of four
trivariate distributions such that the Euclidean distance
between each of the populations is a fixed constant,
`delta`

> 0.

1 |

`n` |
a vector (of length M = 5) of the sample sizes for each population |

`delta` |
the fixed distance between each population and the origin |

`seed` |
Seed for random number generation. (If NULL, does not set seed) |

To define the populations, let *x = (X_1, …,
X_p)'* be a multivariate uniformly distributed random
vector such that *X_j \sim U(a_j, b_j)* is an
independently distributed uniform random variable with
*a_j < b_j* for *j = 1, …, p*. Let
*Pi_m* denote the *m*th population *(m = 1,
…, 5)*. Then, we have the five populations:

*Π_1 = U(-1/2, 1/2) \times U(Δ - 1/2, Δ
+ 1/2) \times U(-1/2, 1/2) \times U(-1/2, 1/2),*

*Π_2 = U(Δ - 1/2, Δ + 1/2) \times
U(-1/2, 1/2) \times U(-1/2, 1/2) \times U(-1/2, 1/2),*

*Π_3 = U(-1/2, 1/2) \times U(-Δ - 1/2,
-Δ + 1/2) \times U(-1/2, 1/2) \times U(-1/2, 1/2),*

*Π_4 = U(-1/2, 1/2) \times U(-1/2, 1/2) \times
U(-Δ - 1/2, -Δ + 1/2) \times U(-1/2, 1/2),*

*Π_5 = U(-1/2, 1/2) \times U(-1/2, 1/2) \times
U(-1/2, 1/2) \times U(Δ - 1/2, Δ + 1/2).*

We generate *n_m* observations from population
*Π_m*.

For *Δ = 0* and *ρ_m = ρ*, *m = 1,
…, M*, the *M* populations are equal.

Notice that the support of each population is a unit
hypercube with 4 features. Moreover, for *Δ ≥
1*, the populations are mutually exclusive and entirely
separated.

named list containing:

- x:
A matrix whose rows are the observations generated and whose columns are the

`p`

features (variables)- y:
A vector denoting the population from which the observation in each row was generated.

1 2 3 4 5 6 7 8 9 10 11 | ```
data_generated <- sim_unif(seed = 42)
dim(data_generated$x)
table(data_generated$y)
data_generated2 <- sim_unif(n = 10 * seq_len(5), delta = 1.5)
table(data_generated2$y)
sample_means <- with(data_generated2,
tapply(seq_along(y), y, function(i) {
colMeans(x[i,])
}))
(sample_means <- do.call(rbind, sample_means))
``` |

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