Generate a synthetic data set with an ordered non-numeric outcome, together with conditional probabilities and covariates' marginal effects.
generate_ordered_data(n)A list storing a data frame with the observed data, a matrix of true conditional probabilities, and a matrix of true marginal effects at the mean of the covariates.
First, a latent outcome is generated as follows:
$$Y_i^* = g ( X_i ) + \epsilon_i$$
with:
$$g ( X_i ) = X_i^T \beta$$
$$X_i := (X_{i, 1}, X_{i, 2}, X_{i, 3}, X_{i, 4}, X_{i, 5}, X_{i, 6})$$
$$X_{i, 1}, X_{i, 3}, X_{i, 5} \sim \mathcal{N} \left( 0, 1 \right)$$
$$X_{i, 2}, X_{i, 4}, X_{i, 6} \sim \textit{Bernoulli} \left( 0, 1 \right)$$
$$\beta = \left( 1, 1, 1/2, 1/2, 0, 0 \right)$$
$$\epsilon_i \sim logistic (0, 1)$$
Second, the observed outcomes are obtained by discretizing the latent outcome into three classes using uniformly spaced threshold parameters.
Third, the conditional probabilities and the covariates' marginal effects at the mean are generated using standard textbook formulas. Marginal effects are approximated using a sample of 1,000,000 observations.
Di Francesco, R. (2023). Ordered Correlation Forest. arXiv preprint arXiv:2309.08755.
## Generate synthetic data.
set.seed(1986)
data <- generate_ordered_data(1000)
head(data$true_probs)
#> P(Y=1) P(Y=2) P(Y=3)
#> [1,] 0.1780543 0.3900898 0.4318559
#> [2,] 0.4252436 0.3927161 0.1820403
#> [3,] 0.3558784 0.4145215 0.2296001
#> [4,] 0.3128062 0.4215497 0.2656441
#> [5,] 0.3413966 0.4175279 0.2410755
#> [6,] 0.4785603 0.3693182 0.1521215
data$me_at_mean
#> P'(Y=1) P'(Y=2) P'(Y=3)
#> x1 -0.2051655 -0.0003337965 0.2054993
#> x2 -0.1947837 -0.0166079075 0.2113916
#> x3 -0.1025828 -0.0001668983 0.1027497
#> x4 -0.1001678 -0.0044354927 0.1046033
#> x5 0.0000000 0.0000000000 0.0000000
#> x6 0.0000000 0.0000000000 0.0000000
sample <- data$sample
Y <- sample$Y
X <- sample[, -1]
## Fit ocf.
forests <- ocf(Y, X)