The BrownExponentialSmoothing class in the hana_ml.algorithms.pal.tsa.exponential_smoothing module is used to model time series data with a trend but without seasonality, offering both non-adaptive and adaptive brown linear exponential smoothing, with various parameters to customize the smoothing process and forecast.
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Here is a Python code template for the `BrownExponentialSmoothing` class:

```python
from hana_ml.algorithms.pal.tsa.exponential_smoothing import BrownExponentialSmoothing

# Create BrownExponentialSmoothing instance
brown_exp_smooth = BrownExponentialSmoothing(alpha=0.1,
                                             delta=0.2,
                                             forecast_num=6,
                                             adaptive_method=False,
                                             accuracy_measure='mse',
                                             ignore_zero=0,
                                             expost_flag=1)

# Assume that df is your input DataFrame
# Perform fit_predict() on the given data
brown_exp_smooth.fit_predict(data=df)

# Print forecast values
print(brown_exp_smooth.forecast_.collect())

# Print statistics
print(brown_exp_smooth.stats_.collect())
```

Please replace `df` with your actual DataFrame. The `fit_predict()` method fits the model to the data and then makes predictions. The `forecast_` attribute holds the forecast values and the `stats_` attribute holds the statistics of the model.