Kickstart your Probabilistic Forecasting with Level Set & Quantile Regression Forests

1. Kickstart your Probabilistic Forecasting with Level Set & Quantile Regression Forests Inge van den Ende, Forecasting Tech Lead 26 September 2025
2. 2 Understanding uncertainty is essential in decision making
3. Point forecasts donʼt give any information about this uncertainty t 0 Time Actuals Forecast Point forecast ?Uncertainty? Energy production
4. 4 A probabilistic forecast provides information about the uncertainty t 0 Time Actuals Forecast Point forecast t 1 Time y Actuals Forecast Point forecast q80 q60 q40 q20 slice q80 q20 Energy production
5. 5 A probability density function describes the uncertainty Energy MWh Probability density Actual Forecast
6. 6 A probability density function describes the uncertainty Energy MWh Probability density Actual Forecast
7. Core trade-off: calibration vs. sharpness Energy MWh Probability density Actual Theoretical quantile Sample quantile q99 q99 0 Calibration Sharpness q20 q30
8. 8 A probabilistic forecast provides information about the uncertainty t 0 Time Actuals Forecast Point forecast t 1 Time y Actuals Forecast Point forecast q80 q60 q40 q20 slice q80 q20 Energy production
9. 9 Where to start?
10. 10 Practical probabilistic models to the rescue: Level Set Forecaster Quantile regression forest
11. 11 Practical probabilistic models to the rescue: Level Set Forecaster Quantile regression forest
12. 12 Train → predict: the standard ML workflow Train Train Predict Predict f LGBMRegressor().fit(X_train, y_train) model.predict(X_test)
13. 13 The Level Set Forecaster follows the same workflow Hasson, 2021 Train Predict
14. 14 The first step is training the model as usual ## Training # 1. Train a point forecast model model = LGBMRegressor().fit(X_train, y_train) # This can be any point forecast model Train Train Predict Hasson, 2021
15. 15 In sample prediction to have a prediction per observation In-sample predict f Train Train Predict # 2. Predict for the train set point_predictions_train_set = model.predict(X_train) Hasson, 2021
16. 16 Level sets group observations of the same prediction level In-sample predict f Train Train Bin to level sets In-sample predictions Actual values Level set Hasson, 2021 q80 q20 Predict
17. 17 Actual values Level set Hasson, 2021 q80 q20 In-sample prediction 1.3 2.1 2.1 Level sets group observations of the same prediction level
18. 18 In-sample predict f Train Train Predict Bin to level sets # 3. Group the training samples based on the prediction sorting_index = np.argsort(point_predictions_train_set) point_predictions_sorted = point_predictions_train_set[sorting_index] target_sorted_by_prediction = y_train[sorting_index] bin_size = 500 # This is a hyperparameter of LSF num_bins = int(np.ceil(len(y_train) / bin_size)) left_thresholds = [-np.inf] + list(point_predictions_sorted[bin_size::bin_size]) target_per_bin = [target_sorted_by_prediction[i * bin_size : (i + 1) * bin_size] for i in range(num_bins)] Hasson, 2021 Level sets group observations of the same prediction level
19. 19 The usual point prediction enables the next step In-sample predict f Train Train Predict Bin to level sets Predict f ## Predicting # 1. Predict with the point forecast model point_predictions = model.predict(X_test) Hasson, 2021
20. 20 The samples in the level set are our probabilistic prediction In-sample predict f Train Train Predict Bin to level sets Predict f # 2. Select the bin probabilistic_predictions = [] for i, prediction in enumerate(point_predictions): bin_index = np.argmax(left_thresholds >= prediction) probabilistic_predictions.append([target_per_bin[bin_index]]) Retrieve level set Hasson, 2021
21. 21 Your first probabilistic forecast! q80 q20 In-sample predict f Train Train Predict Bin to level sets Predict f Retrieve level set Note: you can find the LSF in gluonts
22. 22 The Level Set Forecaster workflow In-sample predict f Train Train Predict Bin to level sets Predict f Retrieve level set
23. Downside of the Level Set Forecaster Example of time series forecast Level set forecaster Hasson, 2021
24. 24 Quantile regression forest Level Set Forecaster Quantile regression forest
25. The Quantile Regression Forest follows the same workflow Predict Train
26. 26 The first step is training the Random Forest as usual ## Training # 1. Train a point forecast model: Random Forest model = RandomForestRegressor().fit(X_train, y_train) Train Predict Train Meinhausen, 2006
27. 27 Random Forest averages the predictions of every tree 2.3 3.5 1.7 2.5
28. 28 Quantile Regression Forest groups the samples per leaf Meinhausen, 2006
29. 29 # 2. Predict for the train set with every tree of the Random Forest listed_predictions_per_tree = [tree.predict(X_test) for tree in model.estimators_] predictions_per_tree = np.column_stack(listed_predictions_per_tree) # Every column is a tree and every row is an observations In-sample predict f Train Predict Train Meinhausen, 2006 Quantile Regression Forest groups the samples per leaf
30. 30 Resample In-sample predict f Train Predict Bin per leaf Predict f Retrieve leaves Train # 3. Group the true values based on prediction (leaf value) for every tree leaf_values_per_tree: List[Dict[float, np.ndarray]] = [ ( pd.DataFrame( {"predictions": predictions_per_tree[:, i], "true_values": y_train.copy()} ) .sort_values("predictions") .groupby("predictions")["true_values"].apply(list).to_dict() ) for i in range(model.n_estimators) ] # Every dictionary in the list is a tree: # every key a predicted value of a leave, the items are the true values in that leaf Meinhausen, 2006 Quantile Regression Forest groups the samples per leaf
31. 31 Resample In-sample predict f Train Predict Bin per leaf Train # 4. Resample the true values to have a constant leaf weight leaf_size = 100 # the number of values in every leaf after resampling balanced_values_per_leaf = [ { leaf: resample(true_values, sample_size=leaf_size) for leaf, true_values in tree.items() } for tree in leaf_values_per_tree ] Meinhausen, 2006 Quantile Regression Forest groups the samples per leaf
32. 32 The usual point prediction enables the next step Resample In-sample predict f Train Predict Bin per leaf Predict f Train ## Predicting # 1. Predict with every tree of the random forest listed_predictions_per_tree = [tree.predict(X_test) for tree in model.estimators_] predictions_per_tree = np.column_stack(listed_predictions_per_tree) # Every column is a tree and every row is a sample Meinhausen, 2006
33. 33 Selecting the samples per leaf gives us a prediction Resample In-sample predict f Train Predict Bin per leaf Predict f Retrieve leaves Train # 2. Collect all true values of the relevant leave per tree probabilistic_predictions_QRF = np.zeros( [predictions_per_tree.shape[0], predictions_per_tree.shape[1] * leaf_size] ) for sample in range(predictions_per_tree.shape[0]): # Loop over all samples for tree in range(predictions_per_tree.shape[1]): # Loop over all trees start_idx = tree * leaf_size end_idx = start_idx + leaf_size probabilistic_predictions_QRF[sample, start_idx:end_idx] = ( balanced_values_per_leaf[tree][predictions_per_tree[sample, tree]] ) Meinhausen, 2006
34. 34 Your probabilistic forecast! Resample In-sample predict f Train Predict Bin per leaf Predict f Retrieve leaves Train q80 q20 Note: you can find the QRF in gluonts
35. Resample The Quantile Regression Forest has six steps In-sample predict f Train Predict Bin per leaf Predict f Retrieve leaves Train
36. 36 How do the methods compare? Level Set Forecaster Quantile regression forest
37. 37 How do the methods compare? Level Set Forecaster Quantile Regression Forest Grouping Based on predictions Based on features Full distribution 🟢 🟢 No assumptions on distribution 🟢 🟢 Good performance 🟢 🟢 Model agnostic 🟢 🔴 Distinct forecast per sample 🔴 🟢 Fast 🟢 🟠
38. How do the methods compare? Example of time series forecast Level set forecaster Quantile regres- sion forest How valuable for your project? Sufficient information in features? Performance of Random Forest in your project? Which do you choose?
39. Key takeaways about probabilistic forecasting 39 With these two models you can start today Tailoring evaluation to probabilistic metrics is crucial to boost performance Probabilistic forecasts provide valuable information for decision making
40. Thank you! Read my blog about probabilistic forecasts