Industrial control systems are widely used in fields such as power, chemical, and petrochemical industries, playing a crucial role in ensuring production and livelihoods of people. However, the complex hierarchical structure and numerous controllable nodes of industrial control systems pose serious security challenges. In recent years, attacks on industrial control systems have been occurring frequently, causing significant damage to the safe operation of industrial production and even resulting in severe security incidents. For scenarios of false data injection attacks on industrial control systems in chemical field, the limitations of principal components analysis (PCA) anomaly detection for non-Gaussian data are analyzed. To address this issue, independent component analysis (ICA) method is introduced as a complementary method to extract independent components from non-Gaussian data. By combining the advantages of PCA and ICA, a joint detection method that integrates both has been proposed. In the proposed method, the $T^{2}$ statistic based on PCA, the $T^{8}$ statistic for the main part and the SPE statistic for the residual part based on ICA are combined to reconstruct a new unique joint detection indicator, the Bayesian inference criterion (BIC) detection indicator, by using the Bayesian probability. Finally, the Tennessee-Eastman (TE) process model is taken as the research object, and a simulation verification with injecting attacks with sine wave false data was conducted to validate the effectiveness of the proposed method. The result indicates that the proposed method has a good detection capability for the signal changes caused by the attacks.