Synthetic Data Generation and Random Projection Methods for Assessing Multivariate Similarity: A Low-Dimensional Methodological Study

Data modeling in biomedical research often operates in the small-sample regime, where the number of observations is small relative to the data dimensionality; the detrimental effects of limited sample sizes are well documented in cancer studies. Synthetic data offers a potential solution to data shortfalls provided that the data generated is an adequate facsimile of the underlying distribution; the adequacy of such synthetic data remains an open-ended problem. In this work, we evaluate a synthetic generator proposed previously. We (1) develop a nonparametric method for assessing multivariate similarity based on the Cramer-Wold theorem and random projection testing, (2) investigate when the absence of bivariate correlation approximates independence in a non-normal setting, and (3) evaluate artifacts induced by data compression. The presentation is primarily methodological; low-dimensional data were used so each stage of the generation process could be analyzed explicitly. The generator applies a series of transformations to the observed data to accommodate the small-sample size, resulting in an uncoupled representation, where uncorrelated marginal distributions are modeled with optimized univariate kernel density estimation. Random projection outcomes were compared with a two-sample test and modeled as Bernoulli trials. Replicate outcomes were aggregated within each projection direction and pooled across directions, yielding a scalable standardized normal test statistic, under a working binomial approximation. The two-sample test significance level was decoupled from that governing the final normal inference. The projection framework also evaluated the full multivariate covariance structure. The generator produced high-fidelity multivariate synthetic data when bivariate correlation approximated independence in the non-normal setting. In highly compressed data, residual modes were best modeled as normally distributed regardless of their intrinsic distributional form. Ongoing work includes applying these methods to higher-dimensional, diverse data.