## 2 Materials and Methods
### Calibration Transfer  
Calibration transfer without standards was performed to investigate the feasibility of transferring the NIR models from one fruit to another using two distinct methods: PLS correction [34] and SS-PFCE [33]. These methods are able to transfer a model developed for one fruit (in this case, strawberry) to accommodate the spectra of other fruits (grape and apple) with only a few spectra and reference values, thereby minimizing the need for extensive calibration efforts for each individual fruit. The PLS correction method [34] was employed to correct the bias of the strawberry model when applied to grape and apple samples. This approach is based on the assumption of a linear relationship between the spectral difference and the prediction error of the primary model, which is developed using spectra from a large calibration set of strawberry. Specifically, the PLS model of strawberry, represented by \( \begin{bmatrix}b_{0}\\ \mathbf{b}\end{bmatrix} \), was used to predict the spectra of the transfer set of grape or apple, denoted as \(\mathbf{X}\), using Equation (1),  
\[\mathbf{\hat{y}}=[\mathbf{1}\mathbf{X}] \begin{bmatrix}b_{0}\\ \mathbf{b}\end{bmatrix}  \tag{1}\]  
where \(\mathbf{1}\) denotes a vector consisting of a column of numbers 1 of sample size on the standard set, and \(b_{0}\) and \(\mathbf{b}\) are the intercept and coefficients of the model, respectively. The prediction error of grape or apple spectra with direct usage of the strawberry model, represented by \(\mathbf{e}\), was obtained as Equation (2),  
\[\mathbf{e}=\mathbf{y}-\mathbf{\hat{y}} \tag{2}\]  
Subsequently, the prediction error of grape or apple spectra was regarded as the dependent variable, and the spectra in the corresponding transfer set as the independent variable, to establish the correction models, \( \begin{bmatrix}b_{0}^{*}\\ \mathbf{b}^{*}\end{bmatrix} \), using PLS algorithm, as shown in Equation (3),  
\[\mathbf{e}=[\mathbf{1}\mathbf{X}] \begin{bmatrix}b_{0}^{*}\\ \mathbf{b}^{*}\end{bmatrix} +\mathbf{r} \tag{3}\]  
where, \(\mathbf{r}\) denotes the fitting residual of the correction model. During the prediction process with PLS correction, the strawberry model was initially used to predict the spectra of grape or apple, resulting in a biased prediction. Next, the corresponding correction models were used to obtain the estimated prediction error of the direct usage of the strawberry model on other fruits. Finally, the accurate prediction of grape and apple was achieved through the addition of biased prediction and the estimated prediction error. In contrast to the aforementioned PLS correction strategy, the SS-PFCE method [33] proposes a different approach to achieve the transferability of prediction models across different fruit types. This method employs a few spectra of new fruits with reference values, in order to directly adjust the PLS model originally developed for strawberry to accommodate grape and apple spectra. To implement this method, the strawberry model \( \begin{bmatrix}b_{0}\\ \mathbf{b}\end{bmatrix} \) was utilized as the initial values, and the spectra (\(\mathbf{X}\)) and the reference values (\(\mathbf{y}\)) in the standard sets of grape and apple were fed into Equation (4) to obtain the transferred model \( \begin{bmatrix}b_{0,s}\\ \mathbf{b}_{s}\end{bmatrix} \). \[\min_{b_{0,s},\ \mathbf{b}_{s}}\left(\left\|y-[\mathbf{1}\mathbf{X}] \begin{bmatrix}b_{ 0,s}\\ \mathbf{b}_{s}\end{bmatrix} \right\|^{2}\right) \tag{4}\]  
\[\text{s. t. corr}(\mathbf{b}_{s},\ \mathbf{b}_{m})>r_{th}\]  
where corr(\(\mathbf{b}_{s},\ \mathbf{b}_{m})>r_{th}\) represents that during the establishment of a transferred model, the correlation coefficients of model coefficients, excluding the intercept term, must not exceed a predefined threshold of 0.98, as recommended by [33]. This constraint ensures that the transferred model is not overfitted to the training data and maintains a reasonable level of generalizability to new data.