## 3 Methodology  
### Feature-Based TL: TCA and JDA  
When operating conditions change, machines tend to produce data with different distributions, making traditional machine learning models, trained under previous conditions, inaccurate under new operating conditions [6]. Feature-based TL can correct such cross-condition distribution discrepancy by finding optimal mapping functions to map the source and target domain data into a common feature space, where their distribution discrepancy can be reduced [10; 19]. A graphical description of feature-based TL is shown in Figure 3[19; 20]. Data from the source and target domains with different distributions are transformed into features in the feature space, where a distance metric could be applied to measure the discrepancy between the source and target domain features. This  
Figure 2: (**a**) Temperature profile of PACK for operating condition A. (**b**) Temperature profile of PACK for operating condition B. (**c**) Temperature profile of PACK for operating condition C. (**d**) Temperature profile of PACK for operating condition D.  
information then propagates back to the feature mapping stage to allow optimisation of the mapping functions with the aim of minimising the discrepancy between the source and target domain features in the feature space. Finally, the distributions of the source and target domain features are sufficiently aligned to allow a domain-invariant classifier to act accurately on both domains. There are two types of distribution of source and target domain data, which feature-based TL can consider--marginal and conditional distributions [21]. The mathematical expression for the distributions will be introduced later in this section but the marginal distribution refers to the distribution of data regardless of the associated labels, whereas the conditional distribution is approximated and calculated as the distribution of data with a common label association. Among the three major categories of TL approach in fault diagnosis, instance-based, feature-based, and parameter-based TL, the feature-based TL is determined to be the optimal approach for the problem of interest. The instance-based approach, which reweighs the source domain instances and uses them as auxiliary datasets for target domain problems, is not ideal, because identical conditional distribution is assumed for source and target domain [22]. The parameter-based TL, which reuses hyperparameters in pretrained models from source domain in target domain, is not chosen, since this approach may fail at significant distribution discrepancy [22]. The specific feature-based TL methods used in this work are transfer component analysis (TCA) and joint distribution alignment (JDA). Both algorithms are commonly studied feature-based TL algorithms, with successful application to bearings and gears, and other niche applications including spacecraft attitude systems, ball screws, and reciprocating compressors [19; 23; 24; 25; 26; 27]. TCA and JDA have been commonly applied to fault diagnosis based on vibration data, but application has not been found on static signal data, such as the ECS data in this paper. It is worth exploring how TCA and JDA perform in the ECS dataset, because differences in the complex boundary conditions for ECS under different conditions is likely to lead to significant and unpredictable distribution discrepancies, which are particularly challenging for TL methods. Hence, this work applies TCA and JDA as the TL algorithm to attempt the cross-condition fault diagnosis problem for the PACK. TCA and JDA follow similar mathematical operations to align the distribution discrepancy between the source and target domains. The difference between TCA and JDA is that while TCA only considers marginal distribution, JDA considers both marginal and conditional distribution alignment between the source and target domains [28].