netcal 1.2.1

Python Framework to calibrate confidence estimates of classifiers like Neural Networks

Calibration Framework

Calibration framework in Python 3 for Neural Networks. For full API reference documentation, visit https://fabiankueppers.github.io/calibration-framework.

Copyright (C) 2019-2021 Ruhr West University of Applied Sciences, Bottrop, Germany AND Elektronische Fahrwerksysteme GmbH, Gaimersheim, Germany

This Source Code Form is subject to the terms of the Apache License 2.0. If a copy of the APL2 was not distributed with this file, You can obtain one at https://www.apache.org/licenses/LICENSE-2.0.txt.

Important: updated references! If you use this framework (classification or detection) or parts of it for your research, please cite it by:

@InProceedings{Kueppers_2020_CVPR_Workshops,
   author = {Küppers, Fabian and Kronenberger, Jan and Shantia, Amirhossein and Haselhoff, Anselm},
   title = {Multivariate Confidence Calibration for Object Detection},
   booktitle = {The IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR) Workshops},
   month = {June},
   year = {2020}
}

If you use Bayesian calibration methods with uncertainty, please cite it by:

@InProceedings{Kueppers_2021_IV,
       author = {Küppers, Fabian and Kronenberger, Jan and Schneider, Jonas and Haselhoff, Anselm},
       title = {Bayesian Confidence Calibration for Epistemic Uncertainty Modelling},
       booktitle = {Proceedings of the IEEE Intelligent Vehicles Symposium (IV)},
       month = {July},
       year = {2021},
    }

Table of Contents

• Calibration Framework

• Overview

  • Update on version 1.2

  • Update on version 1.1

• Installation

  • Requirements

• Calibration Metrics

• Methods

  • Binning

  • Scaling

  • Regularization

• Visualization

• Examples

  • Classification

  • Detection

  • Uncertainty in Calibration

• References

Overview

This framework is designed to calibrate the confidence estimates of classifiers like neural networks. Modern neural networks are likely to be overconfident with their predictions. However, reliable confidence estimates of such classifiers are crucial especially in safety-critical applications.

For example: given 100 predictions with a confidence of 80% of each prediction, the observed accuracy should also match 80% (neither more nor less). This behaviour is achievable with several calibration methods.

Update on version 1.2

TL;DR: - Bayesian confidence calibration: train and infer scaling methods using variational inference (VI) and MCMC sampling - New metrics: MMCE [13] and PICP [14] (netcal.metrics.MMCE and netcal.metrics.PICP) - New regularization methods: MMCE [13] and DCA [15] (netcal.regularization.MMCEPenalty and netcal.regularization.DCAPenalty) - Updated examples - Switched license from MPL2 to APL2

Now you can also use Bayesian methods to obtain uncertainty within a calibration mapping mainly in the netcal.scaling package. We adapted Markov-Chain Monte-Carlo sampling (MCMC) as well as Variational Inference (VI) on common calibration methods. It is also easily possible to bring the scaling methods to CUDA in order to speed-up the computations. We further provide new metrics to evaluate confidence calibration (MMCE) and to evaluate the quality of prediction intervals (PICP). Finally, we updated our framework by new regularization methods that can be used during model training (MMCE and DCA).

Update on version 1.1

This framework can also be used to calibrate object detection models. It has recently been shown that calibration on object detection also depends on the position and/or scale of a predicted object [12]. We provide calibration methods to perform confidence calibration w.r.t. the additional box regression branch. For this purpose, we extended the commonly used Histogram Binning [3], Logistic Calibration alias Platt scaling [10] and the Beta Calibration method [2] to also include the bounding box information into a calibration mapping. Furthermore, we provide two new methods called the Dependent Logistic Calibration and the Dependent Beta Calibration that are not only able to perform a calibration mapping w.r.t. additional bounding box information but also to model correlations and dependencies between all given quantities [12]. Those methods should be preffered over their counterparts in object detection mode.

The framework is structured as follows:

netcal
  .binning         # binning methods
  .scaling         # scaling methods
  .regularization  # regularization methods
  .presentation    # presentation methods
  .metrics         # metrics for measuring miscalibration

examples           # example code snippets

Installation

The installation of the calibration suite is quite easy with setuptools. You can either install this framework using PIP:

pip3 install netcal

Or simply invoke the following command to install the calibration suite:

python3 setup.py install

Requirements

• numpy>=1.17

• scipy>=1.3

• matplotlib>=3.1

• scikit-learn>=0.21

• torch>=1.4

• torchvision>=0.5.0

• tqdm>=4.40

• pyro-ppl>=1.3

• tikzplotlib>=0.9.8

• tensorboard>=2.2

Calibration Metrics

The most common metric to determine miscalibration in the scope of classification is the Expected Calibration Error (ECE) [1]. This metric divides the confidence space into several bins and measures the observed accuracy in each bin. The bin gaps between observed accuracy and bin confidence are summed up and weighted by the amount of samples in each bin. The Maximum Calibration Error (MCE) denotes the highest gap over all bins. The Average Calibration Error (ACE) [11] denotes the average miscalibration where each bin gets weighted equally. For object detection, we implemented the Detection Calibration Error (D-ECE) [12] that is the natural extension of the ECE to object detection tasks. The miscalibration is determined w.r.t. the bounding box information provided (e.g. box location and/or scale). For this purpose, all available information gets binned in a multidimensional histogram. The accuracy is then calculated in each bin separately to determine the mean deviation between confidence and accuracy.

• (Detection) Expected Calibration Error [1], [12] (netcal.metrics.ECE)

• (Detection) Maximum Calibration Error [1], [12] (netcal.metrics.MCE)

• (Detection) Average Calibration Error [11], [12] (netcal.metrics.ACE)

• Maximum Mean Calibration Error (MMCE) [13] (netcal.metrics.MMCE) (no position-dependency)

• Prediction interval coverage probability (PICP) (netcal.metrics.PICP) - this score is not a direct measure of confidence calibration but rather to measure the quality of uncertainty prediction intervals.

Methods

The post-hoc calibration methods are separated into binning and scaling methods. The binning methods divide the available information into several bins (like ECE or D-ECE) and perform calibration on each bin. The scaling methods scale the confidence estimates or logits directly to calibrated confidence estimates - on detection calibration, this is done w.r.t. the additional regression branch of a network.

Important: if you use the detection mode, you need to specifiy the flag “detection=True” in the constructor of the according method (this is not necessary for netcal.scaling.LogisticCalibrationDependent and netcal.scaling.BetaCalibrationDependent).

Most of the calibration methods are designed for binary classification tasks. For binning methods, multi-class calibration is performed in “one vs. all” by default.

Some methods like “Isotonic Regression” utilize methods from the scikit-learn API [9].

Another group are the regularization tools which are added to the loss during the training of a Neural Network.

Binning

Implemented binning methods are:

• Histogram Binning for classification [3], [4] and object detection [12] (netcal.binning.HistogramBinning)

• Isotonic Regression [4], [5] (netcal.binning.IsotonicRegression)

• Bayesian Binning into Quantiles (BBQ) [1] (netcal.binning.BBQ)

• Ensemble of Near Isotonic Regression (ENIR) [6] (netcal.binning.ENIR)

Scaling

Implemented scaling methods are:

• Logistic Calibration/Platt Scaling for classification [10], [12] and object detection [12] (netcal.scaling.LogisticCalibration)

• Dependent Logistic Calibration for object detection [12] (netcal.scaling.LogisticCalibrationDependent) - on detection, this method is able to capture correlations between all input quantities and should be preferred over Logistic Calibration for object detection

• Temperature Scaling for classification [7] and object detection [12] (netcal.scaling.TemperatureScaling)

• Beta Calibration for classification [2] and object detection [12] (netcal.scaling.BetaCalibration)

• Dependent Beta Calibration for object detection [12] (netcal.scaling.BetaCalibrationDependent) - on detection, this method is able to capture correlations between all input quantities and should be preferred over Beta Calibration for object detection

New on version 1.2:: you can provide a parameter named “method” to the constructor of each scaling method. This parameter could be one of the following: - ‘mle’: use the method feed-forward with maximum likelihood estimates on the calibration parameters (standard) - ‘momentum’: use non-convex momentum optimization (e.g. default on dependent beta calibration) - ‘mcmc’: use Markov-Chain Monte-Carlo sampling to obtain multiple parameter sets in order to quantify uncertainty in the calibration - ‘variational’: use Variational Inference to obtain multiple parameter sets in order to quantify uncertainty in the calibration

Regularization

With some effort, it is also possible to push the model training towards calibrated confidences by regularization. Implemented regularization methods are:

• Confidence Penalty [8] (netcal.regularization.confidence_penalty and netcal.regularization.ConfidencePenalty - the latter one is a PyTorch implementation that might be used as a regularization term)

• Maximum Mean Calibration Error (MMCE) [13] (netcal.regularization.MMCEPenalty - PyTorch regularization module)

• DCA [15] (netcal.regularization.DCAPenalty - PyTorch regularization module)

Visualization

For visualization of miscalibration, one can use a Confidence Histograms & Reliability Diagrams. These diagrams are similar to ECE, the output space is divided into equally spaced bins. The calibration gap between bin accuracy and bin confidence is visualized as a histogram.

On detection calibration, the miscalibration can be visualized either along one additional box information (e.g. the x-position of the predictions) or distributed over two additional box information in terms of a heatmap.

• Reliability Diagram [1], [12] (netcal.presentation.ReliabilityDiagram)

Examples

The calibration methods work with the predicted confidence estimates of a neural network and on detection also with the bounding box regression branch.

Classification

This is a basic example which uses softmax predictions of a classification task with 10 classes and the given NumPy arrays:

ground_truth  # this is a NumPy 1-D array with ground truth digits between 0-9 - shape: (n_samples,)
confidences   # this is a NumPy 2-D array with confidence estimates between 0-1 - shape: (n_samples, n_classes)

This is an example for netcal.scaling.TemperatureScaling but also works for every calibration method (remind different constructor parameters):

import numpy as np
from netcal.scaling import TemperatureScaling

temperature = TemperatureScaling()
temperature.fit(confidences, ground_truth)
calibrated = temperature.transform(confidences)

The miscalibration can be determined with the ECE:

from netcal.metrics import ECE

n_bins = 10

ece = ECE(n_bins)
uncalibrated_score = ece.measure(confidences)
calibrated_score = ece.measure(calibrated)

The miscalibration can be visualized with a Reliability Diagram:

from netcal.presentation import ReliabilityDiagram

n_bins = 10

diagram = ReliabilityDiagram(n_bins)
diagram.plot(confidences, ground_truth)  # visualize miscalibration of uncalibrated
diagram.plot(calibrated, ground_truth)   # visualize miscalibration of calibrated

Detection

In this example we use confidence predictions of an object detection model with the according x-position of the predicted bounding boxes. Our ground-truth provided to the calibration algorithm denotes if a bounding box has matched a ground-truth box with a certain IoU and the correct class label.

matched                # binary NumPy 1-D array (0, 1) that indicates if a bounding box has matched a ground truth at a certain IoU with the right label - shape: (n_samples,)
confidences            # NumPy 1-D array with confidence estimates between 0-1 - shape: (n_samples,)
relative_x_position    # NumPy 1-D array with relative center-x position between 0-1 of each prediction - shape: (n_samples,)

This is an example for netcal.scaling.LogisticCalibration and netcal.scaling.LogisticCalibrationDependent but also works for every calibration method (remind different constructor parameters):

import numpy as np
from netcal.scaling import LogisticCalibration, LogisticCalibrationDependent

input = np.stack((confidences, relative_x_position), axis=1)

lr = LogisticCalibration(detection=True, use_cuda=False)    # flag 'detection=True' is mandatory for this method
lr.fit(input, matched)
calibrated = lr.transform(input)

lr_dependent = LogisticCalibrationDependent(use_cuda=False) # flag 'detection=True' is not necessary as this method is only defined for detection
lr_dependent.fit(input, matched)
calibrated = lr_dependent.transform(input)

The miscalibration can be determined with the D-ECE:

from netcal.metrics import ECE

n_bins = [10, 10]
input_calibrated = np.stack((calibrated, relative_x_position), axis=1)

ece = ECE(n_bins, detection=True)           # flag 'detection=True' is mandatory for this method
uncalibrated_score = ece.measure(input, matched)
calibrated_score = ece.measure(input_calibrated, matched)

The miscalibration can be visualized with a Reliability Diagram:

from netcal.presentation import ReliabilityDiagram

n_bins = [10, 10]

diagram = ReliabilityDiagram(n_bins, detection=True)    # flag 'detection=True' is mandatory for this method
diagram.plot(input, matched)                # visualize miscalibration of uncalibrated
diagram.plot(input_calibrated, matched)     # visualize miscalibration of calibrated

Uncertainty in Calibration

We can also quantify the uncertainty in a calibration mapping if we use a Bayesian view on the calibration models. We can sample multiple parameter sets using MCMC sampling or VI. In this example, we reuse the data of the previous detection example.

matched                # binary NumPy 1-D array (0, 1) that indicates if a bounding box has matched a ground truth at a certain IoU with the right label - shape: (n_samples,)
confidences            # NumPy 1-D array with confidence estimates between 0-1 - shape: (n_samples,)
relative_x_position    # NumPy 1-D array with relative center-x position between 0-1 of each prediction - shape: (n_samples,)

This is an example for netcal.scaling.LogisticCalibration and netcal.scaling.LogisticCalibrationDependent but also works for every calibration method (remind different constructor parameters):

import numpy as np
from netcal.scaling import LogisticCalibration, LogisticCalibrationDependent

input = np.stack((confidences, relative_x_position), axis=1)

# flag 'detection=True' is mandatory for this method
# use Variational Inference with 2000 optimization steps for creating this calibration mapping
lr = LogisticCalibration(detection=True, method'variational', vi_epochs=2000, use_cuda=False)
lr.fit(input, matched)

# 'num_samples=1000': sample 1000 parameter sets from VI
# thus, 'calibrated' has shape [1000, n_samples]
calibrated = lr.transform(input, num_samples=1000)

# flag 'detection=True' is not necessary as this method is only defined for detection
# this time, use Markov-Chain Monte-Carlo sampling with 250 warm-up steps, 250 parameter samples and one chain
lr_dependent = LogisticCalibrationDependent(method='mcmc',
                                            mcmc_warmup_steps=250, mcmc_steps=250, mcmc_chains=1,
                                            use_cuda=False)
lr_dependent.fit(input, matched)

# 'num_samples=1000': although we have only sampled 250 different parameter sets,
# we can randomly sample 1000 parameter sets from MCMC
calibrated = lr_dependent.transform(input)

You can directly pass the output to the D-ECE and PICP instance to measure miscalibration and mask quality:

from netcal.metrics import ECE
from netcal.metrics import PICP

n_bins = 10
ece = ECE(n_bins, detection=True)
picp = PICP(n_bins, detection=True)

# the following function calls are equivalent:
miscalibration = ece.measure(calibrated, matched, uncertainty="mean")
miscalibration = ece.measure(np.mean(calibrated, axis=0), matched)

# now determine uncertainty quality
uncertainty = picp.measure(calibrated, matched, uncertainty="mean")

print("D-ECE:", miscalibration)
print("PICP:", uncertainty.picp) # prediction coverage probability
print("MPIW:", uncertainty.mpiw) # mean prediction interval width

If we want to measure miscalibration and uncertainty quality by means of the relative x position, we need to broadcast the according information:

# broadcast and stack x information to calibrated information
broadcasted = np.broadcast_to(relative_x_position, calibrated.shape)
calibrated = np.stack((calibrated, broadcasted), axis=2)

n_bins = [10, 10]
ece = ECE(n_bins, detection=True)
picp = PICP(n_bins, detection=True)

# the following function calls are equivalent:
miscalibration = ece.measure(calibrated, matched, uncertainty="mean")
miscalibration = ece.measure(np.mean(calibrated, axis=0), matched)

# now determine uncertainty quality
uncertainty = picp.measure(calibrated, matched, uncertainty="mean")

print("D-ECE:", miscalibration)
print("PICP:", uncertainty.picp) # prediction coverage probability
print("MPIW:", uncertainty.mpiw) # mean prediction interval width

References

[1] (1,2,3,4,5)

Naeini, Mahdi Pakdaman, Gregory Cooper, and Milos Hauskrecht: “Obtaining well calibrated probabilities using bayesian binning.” Twenty-Ninth AAAI Conference on Artificial Intelligence, 2015.

[2] (1,2)

Kull, Meelis, Telmo Silva Filho, and Peter Flach: “Beta calibration: a well-founded and easily implemented improvement on logistic calibration for binary classifiers.” Artificial Intelligence and Statistics, PMLR 54:623-631, 2017.

[3] (1,2)

Zadrozny, Bianca and Elkan, Charles: “Obtaining calibrated probability estimates from decision trees and naive bayesian classifiers.” In ICML, pp. 609–616, 2001.

[4] (1,2)

Zadrozny, Bianca and Elkan, Charles: “Transforming classifier scores into accurate multiclass probability estimates.” In KDD, pp. 694–699, 2002.

[5]

Ryan J Tibshirani, Holger Hoefling, and Robert Tibshirani: “Nearly-isotonic regression.” Technometrics, 53(1):54–61, 2011.

[6]

Naeini, Mahdi Pakdaman, and Gregory F. Cooper: “Binary classifier calibration using an ensemble of near isotonic regression models.” 2016 IEEE 16th International Conference on Data Mining (ICDM). IEEE, 2016.

[7]

Chuan Guo, Geoff Pleiss, Yu Sun and Kilian Q. Weinberger: “On Calibration of Modern Neural Networks.” Proceedings of the 34th International Conference on Machine Learning, 2017.

[8]

Pereyra, G., Tucker, G., Chorowski, J., Kaiser, L. and Hinton, G.: “Regularizing neural networks by penalizing confident output distributions.” CoRR, 2017.

[9]

Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., Vanderplas, J., Passos, A., Cournapeau, D., Brucher, M., Perrot, M. and Duchesnay, E.: “Scikit-learn: Machine Learning in Python.” In Journal of Machine Learning Research, volume 12 pp 2825-2830, 2011.

[10] (1,2)

Platt, John: “Probabilistic outputs for support vector machines and comparisons to regularized likelihood methods.” Advances in large margin classifiers, 10(3): 61–74, 1999.

[11] (1,2)

Neumann, Lukas, Andrew Zisserman, and Andrea Vedaldi: “Relaxed Softmax: Efficient Confidence Auto-Calibration for Safe Pedestrian Detection.” Conference on Neural Information Processing Systems (NIPS) Workshop MLITS, 2018.

[12] (1,2,3,4,5,6,7,8,9,10,11,12,13,14)

Fabian Küppers, Jan Kronenberger, Amirhossein Shantia and Anselm Haselhoff: “Multivariate Confidence Calibration for Object Detection”.” The IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR) Workshops, 2020

[13] (1,2,3,4)

Kumar, Aviral, Sunita Sarawagi, and Ujjwal Jain: “Trainable calibration measures for neural networks from _kernel mean embeddings.” International Conference on Machine Learning. 2018

[14]

Jiayu Yao, Weiwei Pan, Soumya Ghosh, and Finale Doshi-Velez: “Quality of Uncertainty Quantification for Bayesian Neural Network Inference.” Workshop on Uncertainty and Robustness in Deep Learning, ICML, 2019

[15] (1,2)

Liang, Gongbo, et al.: “Improved trainable calibration method for neural networks on medical imaging classification.” arXiv preprint arXiv:2009.04057 (2020)