Alternating Conditional Expectation (ACE) Algorithm for robust regression analysis of simulated dataset in the presence of homoscedastic and heteroscedastic noise

Regression analysis, aimed at efficiently predicting outcomes by examining the relationships between variables, is acknowledged as a crucial research area in the field of chemometrics. The relationship among variables and the nature of noise are two important components of information in regression modeling. When these parts are known, the regression is carried out easily. But what about the unknown data? This means data without any prior information about the relationships between variables and patterns of noise. The aim of this study is to answer this challenging question. Actually, classical regression procedures may encounter problems in this situation because these methods are based solely on constant and normally distributed residuals [1, 2]. Therefore, an alternative solution must be provided. This solution utilizes robust methods. In this work, the performance of several regression methods including simple least squares (SLS) as a classical regression technique, weighted least squares (WLS) method, and Alternating Conditional Expectation (ACE) as a robust method were evaluated for univariate regression analysis of a simulated dataset with 100 data points. These data points were mathematically simulated according to a nonlinear equation: y = 2.1 + 0.4 x2. The study was conducted in four steps. In the first step, the data produced without noise were analyzed using regression approaches. Then, homoscedastic (constant) noise with values of 0.1 and 10 were added to raw data, and these noisy datasets were utilized as inputs for regression models in second and third steps, respectively. In the last step, heteroscedastic (nonconstant) noise was added to raw data for further analysis. The statistical parameters such as R2 of 1.000, R2adj of 1.000, sum of squares of residuals (SSE) of 3.65E- 05, the variance inflation factor (VIF) of 2714149.979, and the Bayes Information Criterion (BIC) of - 1473.126508 were the results of ACE for heteroscedastic data. The results demonstrate that ACE has the most efficient performance compared to other methods. This superiority stems from transformation-based nature of this approach. ACE suggests the best transformation function with the highest correlation between response and descriptor variables. Variable transformation makes the error variance stable and normalizes its distribution. Therefore, highly satisfying outputs are obtained without needing to consider the relationship among the variables and the noise pattern [3, 4].