Training a Neural Network to Output a Probability Cone for Regression Tasks

Neural networks are widely used for a variety of prediction tasks, but often they provide a single prediction without indicating the uncertainty associated with the forecast. In regression tasks, it is beneficial to output a probability cone, which visually represents the range of possible values. This article will guide you through the process of training a neural network to output such a cone, ensuring it captures both the central estimate and the uncertainty around the predictions.

Understanding the Probability Cone Concept

A probability cone typically represents the uncertainty in the predictions of a model. In regression tasks, instead of predicting a single value, you can predict a range or interval around that value often defined by a lower and upper bound. This can be visualized as a cone or band around the predicted value. Understanding this concept is crucial for accurately interpreting the predictions made by your model.

Model Architecture

To achieve this, you can modify your neural network architecture to output multiple values. For instance, instead of a single output, you can have your network output the mean or median prediction alongside the lower and upper bounds of the prediction interval.

Output Layer Configuration

For a typical regression task, you might use a single output neuron for the mean prediction. To incorporate the lower and upper bounds, configure your output layer to have three neurons:

Output 1: Mean prediction Output 2: Lower bound Output 3: Upper bound

This setup allows your model to provide a comprehensive output that includes the central estimate and the uncertainty around it.

Loss Function

To train the model effectively, you need to define a custom loss function that penalizes the model based on how well the predictions fit within the specified bounds. A common approach is to use a quantile loss or a pinball loss, which allows you to specify different penalties for under- and over-predictions. For instance, if you want to predict the 10th and 90th percentiles, you can use the following custom loss function:

def quantile_loss(y_true, y_pred, quantile):
    e  y_true - y_pred
    return quantile * tf.where(e  0, -e, e)

You would then combine the losses for the lower and upper bounds to optimize your model. This ensures that the model learns to capture the uncertainty in the predictions.

Training the Model

Train the model using your training dataset. Ensure that your dataset includes enough examples that represent the variability in the target variable. This variability helps the model learn to capture the underlying uncertainty. The training process involves:

Providing the training data to the model. Updating the model's weights through backpropagation. Iterating over the dataset until the model's performance improves.

During the training phase, pay attention to the learning rate, batch size, and number of epochs to achieve the best results.

Post-Processing Predictions

After training, you can generate predictions using the model. The model will output the mean prediction and the lower and upper bounds. Here's how you can interpret these outputs:

The mean prediction provides the central estimate. The lower and upper bounds can be used to visualize the uncertainty or confidence interval.

Visualization

To visualize the results, plot the mean predictions along with the lower and upper bounds. This visualization allows you to easily understand the range of possible values and the confidence of the model's predictions. For instance:

mean_pred  predictions[:, 0]
lower_bound  predictions[:, 1]
upper_bound  predictions[:, 2]# Plotting the results(X_test, mean_pred, label'Mean prediction')_between(X_test, lower_bound, upper_bound, alpha0.5, label'Confidence interval')plt.legend()()

Example Implementation

Here's a simplified example of a neural network that outputs a probability cone using TensorFlow/Keras:

import tensorflow as tffrom  import layers, modelsfrom  import Adamfrom  import Huber# Define the modeldef create_model(input_dim):    model  ()    ((64, activation'relu', input_shape(input_dim,)))    ((64, activation'relu'))    ((3))  # Output: mean, lower, upper    return model# Compile the model with a custom loss functiondef custom_loss_function(y_true, y_pred):    q1_loss  quantile_loss(y_true, y_pred, 0.1)    q9_loss  quantile_loss(y_true, y_pred, 0.9)    return (q1_loss   q9_loss) / 2model  create_model(input_dim1)(optimizerAdam(), losscustom_loss_function)# Fit the modelX_train, y_train  ...  # Your training data(X_train, y_train, epochs100, batch_size32)# Make predictionsX_test  ...  # Your test datapredictions  (X_test)mean_pred  predictions[:, 0]lower_bound  predictions[:, 1]upper_bound  predictions[:, 2]# Plot the results(figsize(10, 6))(X_test, mean_pred, label'Mean prediction')_between(X_test, lower_bound, upper_bound, alpha0.5, label'Confidence interval')plt.legend()()

Conclusion

By following these steps, you can successfully train a neural network to output a probability cone for a regression task, capturing both the central tendency and the uncertainty around predictions. This approach enhances the interpretability of your model's predictions and provides a more robust understanding of the underlying data distribution.