## 2 Materials and Methods
### Statistical Analysis and Machine Learning Model Development
#### 2.4.2 Prediction of Early BCR Using Auto-Encoder Methods  
As explained in Section 2.4.1, prediction of early BCR can be done by supervised learning with classification capabilities [16]. However, in some special cases, supervised learning may show its limitations. One example, such as our BCR data set, is an imbalanced data set in which data are concentrated in only certain classes, whereas other classes are rare. In the case of an unbalanced set of data, as input data are more likely to be classified into classes of training data on many sides, the classification accuracy of classes with a small number of training data becomes worse. Namely, it could provide a biased prediction model with good classification performance of a specific class and a huge imbalance sensitivity and specificity. In our case, the ratio of early BCR patients and non-BCR patients is about 8 to 2, which is a somewhat imbalanced data set. This may lead to imbalanced diagnostic performance if we use a general supervised learning method. In order to overcome the imbalanced prediction performance, we apply an auto-encoder for identification of early BCR patients with prostate cancer after RP. An auto-encoder is one of the famous deep unsupervised learning methods [15]. As you can see in Figure 1, it has two parts, the encoder and the decoder. The role of the encoder part compresses the input data information to extract the essential features, and the number of neurons in the encoder part gradually decreases. In addition, the decoder part restores compressed data through an increasing number of neurons. The main uses of the auto-encoder are noise reduction and identification. 1) As data are compressed in the encoder part, meaningless information in the original data such as noise are eliminated. In this point of view, the auto-encoder can be used for noise reduction, such as denoising of medical imaging [ref]. 2) The final output of the auto-encoder is trained to be the original input data, so if the test input differs from the training data set, the automatic encoder cannot recover the original input data well. That means we can find abnormal data that differ from training data, and we can then use the auto-encoder as identification with abnormal detection. For example, like in this paper, we first train the auto-encoder using only one class of data that is easy to acquire [ref]. When new data is put into the trained auto-encoder, if the result data are similar to the original data, it can be determined to be the same class as that used in the auto-encoder for training, and if not similar, it can be determined to be a different class of training data. Namely, it is possible to distinguish between the same and different data used in training, and we call this method identification. In this paper, we use an auto-encoder as a perspective of identification. For the classification by an auto-encoder to predict early BCR, when data of the same class used for training are entered, it can be restored to the same value as the original data, but when data not used for training are entered, it is restored to a different value from the input data. Thus, we make an auto-encoder with a non-BCR patient data set. When a new input is applied to an auto-encoder that has completed training, if the non-BCR patient data are entered, the output is almost the same as the original input data, but if the early BCR data are entered, the output has different values from the original input data. Based on input-output difference, we can classify the early BCR and non-BCR patients. Therefore, we can classify the data based on the difference between the input and output as below. \[if\left(\sum\left(\text{input-output}\right)^{2}>\text{threshold}\right) \quad then\text{ input data is early BCR Patient}\] \[\quad else\text{ input data is non-BCR Patient}\]  
As this method has the advantage of using only one group for training, we can achieve unbiased performance.