Jiajuan Liang

This study investigates the performance of four parameter estimation methods for the Laplace distribution: Method of Moments (MM), Maximum Likelihood Estimation (MLE), Minimum Chi-Square Estimation using equiprobable cells (MCE-EQ), and Minimum Chi-Square Estimation using Representative Points (MCE-RP). Through comprehensive Monte Carlo simulations with sample sizes ranging from 50 to 400, we compare the root mean squared error (RMSE) of the location (μ) and scale (b) parameter estimates. Our results demonstrate that while MLE remains robust for location estimation, the MCE-RP method consistently outperforms other estimators—including MLE—for the scale parameter, particularly in small to moderate samples. The use of Representative Points, which provide an optimal discretization of the distribution, significantly enhances estimation precision. These findings are especially relevant for medical research, where accurate estimation of variability—such as in biomarker concentration levels or physiological response times—is critical for reliable sample size determination, risk assessment, and clinical decision-making. MCE-RP thus offers a superior, reliable estimator for the Laplace scale parameter, with direct implications for improving statistical inference in applied biomedical studies. Purpose: The purpose of this research is to empirically evaluate and compare the finite-sample performance of four estimation methods for the Laplace distribution’s parameters, with a focus on the novel application of Representative Points in minimum chi-square estimation. This work seeks to bridge the gap between theoretical estimation methods and practical applications, providing applied researchers with a more robust estimation tool when modeling data with Laplace characteristics, such as those commonly encountered in medical and biomedical studies. Methods: We conducted an extensive Monte Carlo simulation study to compare the four estimation methods: MM, MLE, MCE-EQ, and MCE-RP. For each method, we generated independent and identically distributed samples from a standard Laplace distribution (μ=0, b=1) with sample sizes n = 50, 100, 200, and 400. Each scenario was replicated 1,000 times. The performance of each estimator was assessed using the root mean squared error (RMSE) for both μ and b. The MCE-RP method utilized pre-computed Representative Points for the standard Laplace distribution, which were transformed according to preliminary MLE estimates to form an optimal cell structure for chi-square minimization. All nonlinear optimizations required for MCE-EQ and MCE-RP were implemented programmatically. Results: The simulation results indicate that MLE performs best for estimating the location parameter μ across all sample sizes. However, for the scale parameter b, the MCE-RP method consistently yields lower RMSE values compared to MLE, MM, and MCE-EQ. In many cases, particularly for smaller samples, the RMSE of MCE-RP is approximately half that of MLE for b. The advantage of MCE-RP is evident across varying numbers of Representative Points (m = 5, 10, 15, 20), with optimal performance often observed at m = 10 or 15. These findings confirm that MCE-RP provides a more precise and reliable estimator for the scale parameter, making it particularly advantageous in small-sample settings. Contribution: This paper contributes to the statistical methodology for the Laplace distribution by introducing and validating the use of Representative Points within a minimum chi-square estimation framework. The key contributions are: (1) demonstrating that MCE-RP significantly outperforms established methods for estimating the scale parameter; (2) providing empirical evidence that RP-based discretization enhances estimation efficiency, especially in finite samples; (3) offering practical guidance for applied researchers in fields such as medical statistics, where accurate scale estimation is crucial for variability assessment, power analysis, and reliable inference; and (4) laying a methodological foundation for extending the RP approach to other location-scale distributions.

This paper proposes using an Autoencoder (AE) prior to t-SNE or UMAP visualization for scRNA-seq data. Direct application of t-SNE/UMAP to the raw, sparse expression matrix often yields unstable, poorly separated clusters. To address this, the framework first employs an AE to learn a denoised, compact latent representation. Subsequent t-SNE or UMAP embedding of this latent space produces more robust visualizations with enhanced cluster consistency and structural separability. A real-data-based comparison shows that, when using the same AE-derived latent space, UMAP outperforms t-SNE. It achieves better cluster cohesion, stronger global structure preservation, greater robustness to initialization and data perturbation, and lower computational cost. Statistical validation via a projection F-test confirms that clusters in the AE latent space exhibit significant between-group mean differences, quantifying the observed visual improvement. The study concludes that AE-based representation learning creates an effective input space for nonlinear embedding, with the AE-UMAP pipeline emerging as a particularly stable and efficient choice for scRNA-seq exploratory analysis. Purpose: This study aims to investigate the effectiveness of AE based latent representations in enhancing nonlinear dimension reduction methods, namely t-SNE and UMAP, for single-cell gene expression data analysis. The performance of AE-based UMAP and AE-based t-SNE is systematically evaluated from multiple perspectives, including visualization quality, clustering consistency, structural preservation, and robustness. Methods: This paper constructs a two-step dimension reduction framework for single-cell gene expression data analysis. First, an AE is employed to compress high-dimensional, sparse, and noisy gene expression data into a low-dimensional latent representation. Subsequently, t-SNE and UMAP are applied to the learned AE latent space for nonlinear embedding and visualization. The performance of different methods is systematically evaluated under multiple experimental conditions using clustering consistency metrics, structure preservation measures, and a projected F-test. Results: Experimental results indicate that directly applying t-SNE or UMAP to the original expression data fails to stably recover meaningful clustering structures, whereas nonlinear dimension reduction performed on AE latent representations substantially improves visualization quality and clustering stability. Within the same latent space, t-SNE and UMAP exhibit comparable performance in terms of clustering accuracy; however, UMAP demonstrates superior performance with respect to cluster compactness, global structure preservation, stability across repeated experiments, and computational efficiency. Statistical testing further confirms the significance of between cluster differences in the AE latent space. Contribution: This study systematically reveals the critical role of AE latent representations in stabilizing nonlinear dimension reduction for single cell data and provides a quantitative comparison between t-SNE and UMAP within a unified latent space. The results demonstrate that UMAP applied to AE latent representations achieves superior performance in terms of visualization stability and computational efficiency, offering a more robust two step dimension reduction strategy for exploratory analysis of high dimensional single cell data.

In this paper, we apply a two-step dimension reduction method, PCA-t-SNE to a real gene expression dataset as case study. It turns out that the PCA-t-SNE can signigicantly improve the visualization and cluster separation of high-dimensional biological data. While t-SNE alone often fails to reveal clear cluster structures in complex datasets, our approach first applies Principal Component Analysis (PCA) to reduce noise and dimensionality, followed by t-SNE to condense the data into a two-dimensional space and then apply the k-means to clustering the two-dimensional data. We demonstrate that PCA-t-SNE produces more distinct and interpretable clusters compared to the standard t-SNE. Statistical validation via a projected F-test for MANOVA confirms that clusters derived from PCA-t-SNE exhibit significantly greater mean separation, with lower p-values, underscoring the enhanced discriminative power of the method. The proposed PCA-t-SNE plot proves particularly effective for nonlinear data where conventional t-SNE performs poorly, offering a robust visualization tool and supporting the utility of sequential dimension reduction in exploratory data analysis for biological and medical research. Purpose: This study aims to evaluate the effect from a combination of the classical PCA and the modern t-SNE technique for dimension reduction in clustering of high-dimensional gene expression data from the aspects of both visualization and MANOVA. Methods: This paper presents a combined approach to dimension reduction for high-dimensional gene expression data. The effect of the visual approach is re-enhanced by the classical MANOVA method for large sample sizes (n > p) and the newly developed MANOVA method for small sample sizes (n < p). Results: The proposed PCA t-SNE approach significantly improves the pure t-SNE approach for the selected gene expression dataset in the sense of clearer classification of the data from both visual observation and statistical significance tests. This provides a pre-processing of high-dimensional gene expression data before implementing the nonlinear dimension reduction, making the t-SNE approach more effective. Contribution: We carry out a successful application of the two-step dimension reduction method PCA-t-SNE to a real gene expression dataset as case study. The idea of the PCA-t-SNE approach to visualizing high-dimensional gene expression data, enhanced by the projection-type MANOVA tests, opens a new way to discrimination of complex high-dimensional with statistical significance in the case of high dimension with a small sample size (n < p). It enhances the clustering of those nonlinear-type of data where the pure t-SNE almost fails to discriminate the clusters, and provides insight into a two-step dimension reduction approach.

This paper introduces a statistical method for validating treatment effects in high-dimensional medical data with small sample sizes. The method compares multiple multivariate population means under multivariate normality, using spherical matrix distribution theory and principal component analysis (PCA) for dimension reduction. The resulting test statistic follows an exact F-distribution under the null hypothesis of equal means, even when the sample size is smaller than the data dimension. Unlike classical MANOVA, the approach does not require equal covariance matrices across groups, making it more robust for real-world biomedical data where variance-covariance homogeneity rarely holds. Monte Carlo simulations show the test achieves accurate type I error control and favorable power. Application to real medical datasets with high-dimensional biomarkers further demonstrates its practicality and interpretability. This work provides a rigorous and versatile advancement for high-dimensional inference in biomedical research and related fields.

This paper introduces a novel methodology for multiple mean comparison of clusters identified in gene expression data through the t-distributed Stochastic Neighbor Embedding (t-SNE) plot, which is a powerful dimensionality re- duction technique for visualizing high-dimensional gene expression data. Our approach integrates the t-SNE visualization with rigorous statistical testing to validate the differences between identified clusters, bridging the gap between exploratory and confirmatory data analysis. We applied our methodology to two real-world gene expression datasets for which the t-SNE plots provided clear separation of clusters corresponding to different expression levels. Our findings underscore the value of combining the t-SNE visualization with multiple mean comparison in gene expression analysis. This integrated approach enhances the interpretability of complex data and provides a robust statistical framework for validating observed patterns. While the classical MANOVA method can be applied to the same multiple mean comparison, it requires a larger total sample size than the data dimension and mostly relies on an asymptotic null distribution. The proposed approach in this paper has broad applicability in the case of high dimension with small sample sizes and an exact null distribution of the test statistic. Objective: Propose a two-step approach to analysis of gene expression data. Gene expression data usually possess a complicated nonlinear structure that cannot be visualized under simple linear dimension reduction like the principal component analysis (PCA) method. We propose to employ the existing t-SNE approach to dimension reduction first so that clusters among data can be clearly visualized and then multiple mean comparison methods can be further employed to carry out statistical inference. We propose the PCA-type projected exact F-test for multiple mean comparison among the clusters. It is superior to the classical MANOVA method in the case of high dimension and relatively large number of clusters. Results: Based on a simple Monte Carlo study on a comparison between the projected F-test and the classical MANOVA Wilks’ Lambda-test and an illustration of two real datasets, we show that the projected F-test has better empirical power performance than the classical Wilks’ Lambda-test. After applying the t-SNE plot to real gene expression data, one can visualize the clear cluster structure. The projected F-test further enhances the interpretability of the t-SNE plot, validating the significant differences among the visualized clusters. Conclusion: Our findings suggest that the combination of the t-SNE visualization and multiple mean comparison through the PCA-projected exact F-test is a valuable tool for gene expression analysis. It not only enhances the interpretability of high-dimensional data but also provides a rigorous statistical framework for validating the observed patterns.

This paper introduces a novel chisquare test for spherical symmetry, utilizing statistical representative points. The proposed representative-point-based chisquare statistic is shown, through a Monte Carlo study, to considerably improve the power performance compared to the traditional equiprobable chisquare test in many high-dimensional cases. While the test requires relatively large sample sizes to approximate the chisquare distribution, obtaining critical values from existing chisquare tables is simpler compared to many existing tests for spherical symmetry. A real-data application demonstrates the robustness of the proposed method against different choices of representative points. This paper argues that the use of representative points provides a new perspective in high-dimensional goodness-of-fit testing, offering an alternative approach to evaluating spherical symmetry in such contexts. By leveraging the flexibility of choosing the number of representative points, this method ensures more reliable detection of departures from spherical symmetry, especially in high-dimensional datasets. Overall, this research highlights the practical advantages of the proposed approach in statistical analysis, emphasizing its potential as a powerful tool in goodness-of-fit tests within the realm of high-dimensional data.

Testing multivariate normality in high-dimensional data analysis has been a long-lasting topic in the area of goodness of fit. Numerous methods for this purpose can be found in the literature. Reviews on different methods given by influential researchers show that new methods keep emerging in the literature from different perspectives. The theory of statistical representative points provides a new perspective to construct tests for multivariate normality. To avoid the difficulty and huge computational load in finding the statistical representative points from a high-dimensional probability distribution, we develop an approach to constructing a test for high-dimensional normal distribution based on the representative points of the simple univariate beta distribution. The representative-points-based approach is extended to the the case that the sample size may be smaller than the dimension. A Monte Carlo study shows that the new test is able to control type I error rates fairly well for both large and small sample sizes when faced with a high dimension. The power of the new test against some non-normal distributions is generally or substantially improved for a set of selected alternative distributions. A real-data example is given for a simple application illustration.