## 2 Creation and Analysis of Synthetic Data Sets  
### Creation of Synthetic Data  
Recall that the intraclass correlation coefficient (ICC) is a measure of the correlation expected for repeated measurements of the same feature on different occasions. Unlike the Pearson r correlation coefficient, which is typically applied as an interclass measure of relative agreement (i.e., two series can be correlated even if they differ substantially in level and spread), the ICC is an intraclass measure of absolute agreement [7]. Measures from the same set of subjects at two different times are intraclass measurements (same metric and variance). ICC ranges from 0.0 to 1.0 with the latter corresponding to perfect temporal persistence. Our goal is to create synthetic features with a specified target ICC (denoted \(ICC_{Target}\)). Let \(X_{ijs}\) denote the measurement of feature \(j\) (\(j=1,\ldots,K\)) on session (occasion) \(s\) (\(s=1,\ldots,S\)) for individual \(i\) (\(i=1,\ldots,N\)). Although the ICC can be calculated based on many sessions, in our experience, biometric assessment is typically performed comparing only two points in time. Therefore, henceforth we will set \(S=2\). We generate normally distributed features such that the theoretical intraclass correlation of repeated measurements of the same feature on the same subject is \(ICC_{Target}\) while the theoretical correlation of measurements of different features on the same individual and the theoretical correlation of measurements from different individuals are zero. In practice when data are simulated there are small variations in the empirical ICCs and there are small intercorrelations between features (and individuals) due to chance.  
The R code to create our datasets is included in [8]. Here we describe the method briefly. A more formal presentation is spelled out in Algorithm 1. The starting point is to populate the full set of session one measurements \(X_{ij1}\) with random draws from a standard normal distribution (mean zero and variance one). Then the measurements for the second session are set equal to the value of the given feature from the first session, \(X_{ij2}=X_{ij1}\) for (\(i=1,\ldots,N\), \(j=1,\ldots,K\)). At this point both sessions have the same data and each feature has ICC equal to 1.0 (perfect persistence). We obtain the desired ICC by adding a draw from a normal distribution with \(\text{mean}=0\) and \(\text{variance}=(1-ICC_{Target})/ICC_{Target}\) to each of the measurements. At the end we apply a z-score transform to each feature (with all sessions concatenated together) so that they all have mean 0 and standard deviation one. It can be shown that the resulting measurements have the desired ICC (up to simulation noise).  
Using this method, we can create features which are normally distributed, that have specified ICCs, with as many subjects and sessions as we desire. These features all have mean = 0 and SD = 1. These features are generally independent, but there are some small intercorrelations between features due to chance. To illustrate the approach, we generated data for 10,000 subjects, 1000 features and 2 occasions with \(ICC_{Target}=0.7\). Figure 2A shows a histogram of the resulting empirical ICCs. Figure 2B shows a histogram of the resulting inter-feature correlations.