Evaluation
Submissions are evaluated using the multi-class logarithmic loss. Each id in the test set had a single true class label, Status. For each id, you must submit a set of predicted probabilities for each of the three possible outcomes, e.g., Status_C, Status_CL, and Status_D.

The metric is calculated as follows:

\[ 
\text{log loss} = -\frac{1}{N}\sum_{i=1}^N\sum_{j=1}^M y_{ij}\log(p_{ij}), 
\]

where \( N \) is the number of rows in the test set, \( M \) is the number of outcomes (i.e., 3), \(\log\) is the natural logarithm, \( y_{ij} \) is 1 if row \( i \) has the ground truth label \( j \) and 0 otherwise, and \( p_{ij} \) is the predicted probability that observation \( i \) belongs to class \( j \).

The submitted probabilities for a given row are not required to sum to one because they are rescaled prior to being scored (each row is divided by the row sum). In order to avoid the extremes of the log function, predicted probabilities are replaced with

\[
\text{max}(\text{min}(p, 1 - 10^{-15}), 10^{-15}).
\]

Submission File
For each id row in the test set, you must predict probabilities of the three outcomes Status_C, Status_CL, and Status_D. The file should contain a header and have the following format:

```
id,Status_C,Status_CL,Status_D
7905,0.628084,0.034788,0.337128
7906,0.628084,0.034788,0.337128
7907,0.628084,0.034788,0.337128
...
```

Dataset Description
The dataset for this competition (both train and test) was generated from a deep learning model trained on the Cirrhosis Patient Survival Prediction dataset. Feature distributions are close to, but not exactly the same, as the original. Feel free to use the original dataset as part of this competition, both to explore differences as well as to see whether incorporating the original in training improves model performance.

Files

- `train.csv` - the training dataset; Status is the categorical target; C (censored) indicates the patient was alive at N_Days, CL indicates the patient was alive at N_Days due to a liver transplant, and D indicates the patient was deceased at N_Days.
- `test.csv` - the test dataset; your objective is to predict the probability of each of the three Status values, e.g., Status_C, Status_CL, Status_D.
- `sample_submission.csv` - a sample submission file in the correct format.