simdata_friedman(n = rep(15, K), p = 10, experiment = 1,
seed = NULL)NULL, does not set seed)Here, we provide a brief description of each of the six experimental configurations. For more information, see Friedman (1989). We use Friedman's original setup except we fix the number of observations rather than randomly choosing the number of class observations.
Define $I_p$ as the $p$-dimensional identity matrix.
Experiment #1 -- Equal, Spherical Covariance Matrices
Each of the three classes are generated from a population with covariance matrix, $I_p$. The population mean of the first class is the origin. The means of the other two classes are taken to be 3.0 in two orthogonal directions.
Experiment #2 -- Unequal, Spherical Covariance Matrices
Let each of the classes have covariance matrix $k *I_p$, where $k$ is the class number $(1 \le k \le 3)$. Similar to experiment #1, the population mean of the first class is the origin; the means for classes 2 and 3 are shifted in orthogonal directions, class 2 by a distance of 3.0 and class 3 by a distance of 4.0.
Experiment #3 -- Equal, Highly Ellipsoidal Covariance Matrices
The covariance matrices of all three classes are equal and highly ellipsoidal. The location differences between the classes are concentrated in the low-variance subspace. The $j$th eigenvalue $(j = 1, \ldots, p)$ of the common covariance matrices is $$e_j = [9 (j - 1) / (p - 1) + 1]^2,$$ so that the ratio of the largest to smallest eigenvalues is 100.
The population mean of the first class is the origin. The mean vectors for the second and third classes are in terms of the population eigenvalues. The mean of the $j$th feature for class 2 is $$\mu_{2j} = 2.5 \sqrt{e_j / p} \frac{p - j}{p/2 - 1},$$ where $e_j$ is the $j$th eigenvalue given above. The mean of the $j$th feature for class 3 is $$\mu_{3j} = (-1)^j \mu_{2j}.$$
Experiment #4 -- Equal, Highly Ellipsoidal Covariance Matrices
Similar to Experiment #3, the covariance matrices of all three classes are equal and highly ellipsoidal. However, in this experiment the location differences between the classes are concentrated in the high-variance subspace. The $j$th eigenvalue $(j = 1, \ldots, p)$ of the common covariance matrices is $$[9 * (j - 1) / (p - 1) + 1]^2,$$ so the ratio of the largest to smallest eigenvalues is 100.
The population mean of the first class is the origin. The mean vectors for the second and third classes are in terms of the population eigenvalues. The mean of the $j$th feature for class 2 is $$\mu_{2j} = 2.5 \sqrt{e_j / p} \frac{j - 1}{p/2 - 1},$$ where $e_j$ is the $j$th eigenvalue given above. The mean of the $j$th feature for class 3 is $$\mu_{3j} = (-1)^j \mu_{2j}.$$
Experiment #5 -- Unequal, Highly Ellipsoidal Covariance Matrices
In this experiment, the class covariance matrices are highly ellipsoidal and very unequal. The eigenvalues for the first class are given by $$e_{1j} = [9 (j - 1) / (p - 1) + 1]^2,$$ so that the ratio of the largest to smallest eigenvalues is 100. The eigenvalues for the second class are $$e_{2j} = [9 (p - j) / (p - 1) + 1]^2.$$ The eigenvalues for class 3 are given by $$e_{3j} = {9 [j - (p - 1) / 2] / (p - 1) }^2.$$
For the first two classes, the ratio of the largest to the smallest eigenvalues is 100, but their high and low variance subspaces are complementary of each other. This ratio for the third class is $(p + 1)^2$. The third class has low variance in the subspace of intermediate variance for the first two classes, and high variance where they have their complementary high/low variances.
Each class' mean vector is the origin so that the class distributions differ only in their covariance matrices.
Experiment #6 -- Unequal, Highly Ellipsoidal Covariance Matrices
This experiment uses the same covariance structures described for Experiment #5. The population means, however, are different. The mean vector of the first class is the origin. The mean of the $j$th feature for class 2 is $$\mu_{2j} = 14 / \sqrt{p},$$ and class 3's mean vector is defined such that $$\mu_{3j} = (-1)^j \mu_{2j}.$$
# Generates 10 observations from three multivariate normal populations having
# the covariance structure given in Friedman's (1989) fifth experiment.
sample_sizes <- c(10, 20, 30)
p <- 20
data <- simdata_friedman(n = sample_sizes, p = p, experiment = 5, seed = 42)
dim(data$x)
table(data$y)
# Generates 15 observations from each of three multivariate normal
# populations having the covariance structure given in Friedman's (1989)
# sixth experiment.
sample_sizes <- c(15, 15, 15)
p <- 20
set.seed(42)
data2 <- simdata_friedman(n = sample_sizes, p = p, experiment = 6)
dim(data2$x)
table(data2$y)Run the code above in your browser using DataLab