Create a Synthetic Data Stream — synthetic

This function creates a synthetic data stream with data points in roughly $[0, 1]^p$ by choosing points form k clusters following a sequence through these clusters. Each cluster has a density function following a d-dimensional normal distributions. In the test set outliers are introduced.

Usage

synthetic_stream(
  k = 10,
  d = 2,
  n_subseq = 100,
  p_transition = 0.5,
  p_swap = 0,
  n_train = 5000,
  n_test = 1000,
  p_outlier = 0.01,
  rangeVar = c(0, 0.005)
)

Arguments

k: number of clusters.
d: dimensionality of data set.
n_subseq: length of subsequence which will be repeat to create the data set.
p_transition: probability that the next position in the subsequence will belong to a different cluster.
p_swap: probability that two data points are swapped. This represents measurement errors (e.g., a data points arrive out of order) or that the data stream does not exactly follow the subsequence.
n_train: size of training set (without outliers).
n_test: size of test set (with outliers).
p_outlier: probability that a data point is replaced by an outlier (a randomly chosen point in $[0,1]^p$).
rangeVar: Used to create the random covariance matrices for the clusters. See genPositiveDefMat() in clusterGeneration for details.

Value

A list with the following elements:

test: test data.
train: training data.
sequence_test: sequence of the test data points through the clusters.
sequence_train: sequence of the training data points through the clusters.
swap_test: index where points are swapped.
swap_train: index where points are swapped.
outlier_position: logical vector for outliers in test data.
model: centers and covariance matrices for the clusters.

Details

The data generation process creates a data set consisting of k clusters in roughly $[0,1]^d$. The data points for each cluster are be drawn from a multivariate normal distribution given a random mean and a random variance/covariance matrix for each cluster. The temporal aspect is modeled by a fixed subsequence (of length n_subseq) through the k clusters. In each step in the subsequence we have a transition probability p_transition that the next data point is in the same cluster or in a randomly chosen other cluster, thus we can create slowly or fast changing data. For the complete sequence, the subsequence is repeated to create n_test/n_train data points. The data set is generated by drawing a data point from the cluster corresponding to each position in the sequence. Outliers are introduced by replacing data points in the data set with probability $p_outlier by randomly chosen data points in $[0,1]^d$. Finally, to introduce imperfection in the temporal sequence (e.g., because the data does not follow exactly a repeating sequence or because observations do not arrive in the correct order), we swap two consecutive observations with probability p_swap.

Examples

## create only test data (with outliers)
ds <- synthetic_stream(n_train = 0)

## plot test data
plot(ds$test, pch = ds$sequence_test, col = "gray")
text(ds$model$mu[, 1], ds$model$mu[, 2], 1:10)

## mark outliers
points(ds$test[ds$outlier_position, ],
  pch = 3, lwd = 2, col = "red")