## 2 Methodology
### Numerical Simulation of Heat and Mass Transfer
#### 2.2.2 Data Amplification  
The principle of data amplification drew inspiration from electroencephalogram (EEG) signal processing. The EEG signal can be regarded as a time-dependent amplitude function that is expressed as a spectrum in different frequencies [30; 31]. It was associated with five typical brain activities, including relaxed awareness without any attention (alpha waves), active thinking (beta waves), deep sleep (delta waves), deep meditation (theta waves), and some certain brain diseases (gamma waves) [32]. An analogy was made between the EEG signal processing and the macro-environment data processing. The data can be decomposed into many components to represent the data features. Each key feature can be regarded as a factor that affects the macro-environmental fluctuations. The Fourier transform (FT) is a tool to achieve data transformation between the time domain (original data) and the frequency domain (features). It can be expressed by Equation (8) [33].  
\[X(k)=\sum_{j=1}^{n}x(j)W_{n}^{(j-1)(k-1)} \tag{8}\]  
And the inverse FT can be expressed by Equation (9).  
\[x(j)=\frac{1}{n}\sum_{k=1}^{n}X(k)W_{n}^{(j-1)(k-1)} \tag{9}\]  
where \(X(k)\) and \(x(j)\) are the FT pairs in time and frequency domains. \(X(k)\) is the macro-environmental data with \(k\) points sequence. \(x(j)\) can be regarded as \(j\) data features with successive amplitude at specific frequencies. \(W_{n}=e^{(-2\pi i)/n}\) is a primitive \(n\)-th root of 1. \(i=\sqrt{-1}\), the imaginary unit.  
The daily RH data, \(X(k)\), was decomposed to the data features, \(x(j)\), by using the FT function tool in MATLAB 2024a Win. A subset of key frequency components wasintentionally selected to represent the key features of the original data. Once the set of key features was determined, the decomposition process became repeatable. These components were combined by using the inverse FT to reconstruct an approximate version of the original data. The accuracy of this reconstruction depended on the number of selected components. The selection threshold was set at a high R-squared value, R\({}^{2}\)\(\geq\) 0.95, between the original data and the represented data. Sequentially, the selected components \(x(j)\) were scaled by an amplification factor and inversely transferred back to the time domain to obtain the data with large fluctuations. This synchronous amplification was based on the conservative assumption that all the selected components experienced maximum amplification simultaneously, representing a worst-case scenario for collection care.  
The trial-and-error method was used to amplify the fluctuations gradually until the micro-environment did not comply with the specifications of preventive conservation in the risk assessment.  
The data amplification of macro-environment applied to RH only, excluding temperature, has two reasons. First, as a useful approximation in RH control for preventive conservation, a coupled RH change of 3% occurs for each degree of temperature change [34]. A slight temperature change possibly could cause the RH out of the control band or allowable range of 24 h fluctuation. The setting of temperature change is dependent on the RH control requirement in the precision air conditioning with humidity priority because the RH change in paper-based collection storage directly impacts paper stability and preservation. Second, the thermal mass of the library's storage room can ensure a small fluctuation for indoor temperature. Throughout the two-year dataset, the macro-environmental temperature remained consistently within the range of 17-20 \({}^{\circ}\)C, with fluctuations of \(\leq\)0.2 \({}^{\circ}\)C occurring for 95% of the time. The relaxation of temperature control was not considered.