The main purpose of an application performance monitoring/management (APM) software is to ensure the highest availability, efficiency and security of applications. An APM software accomplishes the main goals through automation, measurements, analysis and diagnostics. Gartner specifies the three crucial capabilities of APM softwares. The first is an end-user experience monitoring for revealing the interactions of users with application and infrastructure components. The second is application discovery, diagnostics and tracing. The third key component is machine learning (ML) and artificial intelligence (AI) powered data analytics for predictions, anomaly detection, event correlations and root cause analysis. Time series metrics, logs and traces are the three pillars of observability and the valuable source of information for IT operations. Accurate, scalable and robust time series forecasting and anomaly detection are the requested capabilities of the analytics. Approaches based on neural networks (NN) and deep learning gain an increasing popularity due to their flexibility and ability to tackle complex nonlinear problems. However, some of the disadvantages of NN-based models for distributed cloud applications mitigate expectations and require specific approaches. We demonstrate how NN-models, pretrained on a global time series database, can be applied to customer specific data using transfer learning. In general, NN-models adequately operate only on stationary time series. Application to nonstationary time series requires multilayer data processing including hypothesis testing for data categorization, category specific transformations into stationary data, forecasting and backward transformations. We present the mathematical background of this approach and discuss experimental results based on implementation for Wavefront by VMware (an APM software) while monitoring real customer cloud environments.