API Reference of net:cal¶
This is the detailed API reference for the net:cal calibration framework. This library can be used to obtain well-calibrated confidence or uncertainty estimates from biased estimators such as neural networks. The API reference contains a detailed description of all available methods and their parameters. For miscellaneous examples on how to use these methods, see readme below.
Available packages¶
Binning Methods for Confidence Calibration This package consists of several methods for confidence calibration which use binning to approximate confidence estimates to the measured accuracy. The most common calibration methods within this package are netcal.binning.HistogramBinning and netcal.binning.IsotonicRegression. |
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Scaling Methods for Confidence Calibration This package consists of several methods for confidence calibration which use confidence scaling to approximate confidence estimates to observed accuracy. The most common scaling methods are netcal.scaling.TemperatureScaling, netcal.scaling.LogisticCalibration, and netcal.scaling.BetaCalibration. Note that all methods can also be applied to object detection and are capable of additional influenting factors such as object position and/or shape. The advanced methods netcal.scaling.LogisticCalibrationDependent and netcal.scaling.BetaCalibrationDependent are able to better represent possible correlations as the underlying probability distributions are joint multivariate distributions with possible correlations. |
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Regularization Methods for Confidence Calibration Regularization methods which are applied during model training. These methods should achieve a confidence calibration during model training. For example, the Confidence Penalty penalizes confident predictions and prohibits over-confident estimates. Use the functions to obtain the regularization and callback instances with prebuild parameters. |
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Metrics Package to Measure Miscalibration Methods for measuring miscalibration in the context of confidence calibration and regression uncertainty calibration. The common methods for confidence calibration evaluation are given with the netcal.metrics.confidence.ECE (ECE), netcal.metrics.confidence.MCE (MCE), and netcal.metrics.confidence.ACE (ACE). Each method bins the samples by their confidence and measures the accuracy in each bin. The ECE gives the mean gap between confidence and observed accuracy in each bin weighted by the number of samples. The MCE returns the highest observed deviation. The ACE is similar to the ECE but weights each bin equally. The common methods for regression uncertainty evaluation are netcal.metrics.regression.PinballLoss (Pinball loss), the netcal.metrics.regression.NLL (NLL), and the netcal.metrics.regression.QCE (M-QCE and C-QCE). The Pinball loss as well as the Marginal/Conditional Quantile Calibration Error (M-QCE and C-QCE) evaluate the quality of the estimated quantiles compared to the observed ground-truth quantile coverage. The NLL is a proper scoring rule to measure the overall quality of the predicted probability distributions. For a detailed description of the available metrics within regression calibration, see the module doc of netcal.regression. |
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Visualization Package Methods for the visualization of miscalibration in the scope of confidence calibration and regression uncertainty calibration. This package consists of a netcal.presentation.ReliabilityDiagram (Reliability Diagram) method used to visualize the calibration properties for confidence calibration in the scope of classification, object detection (semantic label confidence) or segmentation. Similar to the ACE or ECE, this method bins the samples in equally sized bins by their confidence and displays the gap between confidence and observed accuracy in each bin. For regression uncertainty calibration, this package also holds the netcal.presentation.ReliabilityRegression method that is able to visualize the quantile calibration properties of probabilistic regression models, e.g., within probabilistic regression or object detection (spatial position uncertainty). A complementary diagram is the netcal.presentation.ReliabilityQCE that visualizes the computation of the netcal.metrics.regression.QCE (C-QCE) metric. |
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Probabilistic Regression Calibration Package Methods for uncertainty calibration of probabilistic regression tasks. A probabilistic regression model does not only provide a continuous estimate but also an according uncertainty (commonly a Gaussian standard deviation/variance). The methods within this package are able to recalibrate this uncertainty by means of quantile calibration [1], distribution calibration [2], or variance calibration [3], [4]. Quantile calibration [1] requires that a predicted quantile for a quantile level t covers approx. 100t% of the ground-truth samples. Methods for quantile calibration: IsotonicRegression [1]. Distribution calibration [2] requires that a predicted probability distribution should be equal to the observed error distribution. This must hold for all statistical moments. Methods for distribution calibration: GPBeta [2]. GPNormal [5]. GPCauchy [5]. Variance calibration [3], [4] requires that the predicted variance of a Gaussian distribution should match the observed error variance which is equivalent to the root mean squared error. Methods for variance calibration: VarianceScaling [3], [4]. GPNormal [5]. References [1] Volodymyr Kuleshov, Nathan Fenner, and Stefano Ermon: "Accurate uncertainties for deep learning using calibrated regression." International Conference on Machine Learning. PMLR, 2018. Get source online [2] Hao Song, Tom Diethe, Meelis Kull and Peter Flach: "Distribution calibration for regression." International Conference on Machine Learning. PMLR, 2019. Get source online [3] Levi, Dan, et al.: "Evaluating and calibrating uncertainty prediction in regression tasks." arXiv preprint arXiv:1905.11659 (2019). Get source online [4] Laves, Max-Heinrich, et al.: "Well-calibrated regression uncertainty in medical imaging with deep learning." Medical Imaging with Deep Learning. PMLR, 2020. Get source online [5] Küppers, Fabian, Schneider, Jonas, and Haselhoff, Anselm: "Parametric and Multivariate Uncertainty Calibration for Regression and Object Detection." ArXiv preprint arXiv:2207.01242, 2022. Get source online |
Each calibration method must inherit the base class AbstractCalibration. If you want to write your own method and
include into the framework, include this class as the base class.
Base class¶
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Abstract base class for all calibration methods. |
net:cal - Uncertainty Calibration¶
The net:cal calibration framework is a Python 3 library for measuring and mitigating miscalibration of uncertainty estimates, e.g., by a neural network. For full API reference documentation, visit https://efs-opensource.github.io/calibration-framework.
Copyright © 2019-2022 Ruhr West University of Applied Sciences, Bottrop, Germany AND e:fs TechHub 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},
}
If you use Regression calibration methods, please cite it by:
@InProceedings{Kueppers_2022_ECCV_Workshops,
author = {Küppers, Fabian and Schneider, Jonas and Haselhoff, Anselm},
title = {Parametric and Multivariate Uncertainty Calibration for Regression and Object Detection},
booktitle = {European Conference on Computer Vision (ECCV) Workshops},
year = {2022},
month = {October},
publisher = {Springer},
}
Table of Contents¶
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 1.3¶
TL;DR:
Regression calibration methods: train and infer methods to rescale the uncertainty of probabilistic regression models
New package: netcal.regression with regression calibration methods:
Isotonic Regression (netcal.regression.IsotonicRegression)
Variance Scaling (netcal.regression.VarianceScaling)
GP-Beta (netcal.regression.GPBeta)
GP-Normal (netcal.regression.GPNormal)
GP-Cauchy (netcal.regression.GPCauchy)
Implement netcal.regression.GPNormal method with correlation estimation and recalibration
Restructured netcal.metrics package to distinguish between (semantic) confidence calibration in netcal.confidence and regression uncertainty calibration in netcal.regression:
Expected Calibration Error (ECE - netcal.confidence.ECE)
Maximum Calibration Error (MCE - netcal.confidence.MCE)
Average Calibration Error (ACE - netcal.confidence.ACE)
Maximum Mean Calibration Error (MMCE - netcal.confidence.MMCE)
Negative Log Likelihood (NLL - netcal.regression.NLL)
Prediction Interval Coverage Probability (PICP - netcal.regression.PICP)
Pinball loss (netcal.regression.PinballLoss)
Uncertainty Calibration Error (UCE - netcal.regression.UCE)
Expected Normalized Calibration Error (ENCE - netcal.regression.ENCE)
Quantile Calibration Error (QCE - netcal.regression.QCE)
Added new types of reliability diagrams to visualize regression calibration properties:
Reliability Regression diagram to visualize calibration for different quantile levels (preferred - netcal.presentation.ReliabilityRegression)
Reliability QCE diagram to visualize QCE over stddev (netcal.presentation.QCE)
Updated examples
Minor bugfixes
Use library tikzplotlib within the netcal.presentation package to enable a direct conversion of matplotlib.Figure objects to Tikz-Code (e.g., can be used for LaTeX figures)
Within this release, we provide a new package netcal.regression to enable recalibration of probabilistic regression tasks. Within probabilistic regression, a regression model does not output a single score for each prediction but rather a probability distribution (e.g., Gaussian with mean/variance) that targets the true output score. Similar to (semantic) confidence calibration, regression calibration requires that the estimated uncertainty matches the observed error distribution. There exist several definitions for regression calibration which the provided calibration methods aim to mitigate (cf. README within the netcal.regression package). We distinguish the provided calibration methods into non-parametric and parametric methods. Non-parametric calibration methods take a probability distribution as input and apply recalibration in terms of quantiles on the cumulative (CDF). This leads to a recalibrated probability distribution that, however, has no analytical representation but is given by certain points defining a CDF distribution. Non-parametric calibration methods are netcal.regression.IsotonicRegression and netcal.regression.GPBeta.
In contrast, parametric calibration methods also take a probability distribution as input and provide a recalibrated distribution that has an analytical expression (e.g., Gaussian). Parametric calibration methods are netcal.regression.VarianceScaling, netcal.regression.GPNormal, and netcal.regression.GPCauchy.
The calibration methods are designed to also work with multiple independent dimensions. The methods netcal.regression.IsotonicRegression and netcal.regression.VarianceScaling apply a recalibration of each dimension independently of each other. In contrast, the GP methods netcal.regression.GPBeta, netcal.regression.GPNormal, and netcal.regression.GPCauchy use a single GP to apply recalibration. Furthermore, the GP-Normal netcal.regression.GPNormal is can model possible correlations within the training data to transform multiple univariate probability distributions of a single sample to a joint multivariate (normal) distribution with possible correlations. This calibration scheme is denoted as correlation estimation. Additionally, the GP-Normal is also able to take a multivariate (normal) distribution with correlations as input and applies a recalibration of the whole covariance matrix. This is referred to as correlation recalibration.
Besides the recalibration methods, we restructured the netcal.metrics package which now also holds several metrics for regression calibration (cf. netcal.metrics package documentation for detailed information). Finally, we provide several ways to visualize regression miscalibration within the netcal.presentation package.
All plot-methods within the netcal.presentation package now support the option “tikz=True” which switches from standard matplotlib.Figure objects to strings with Tikz-Code. Tikz-code can be directly used for LaTeX documents to render images as vector graphics with high quality. Thus, this option helps to improve the quality of your reliability diagrams if you are planning to use this library for any type of publication/document
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 (confidence calibration)
.scaling # scaling methods (confidence calibration)
.regularization # regularization methods (confidence calibration)
.presentation # presentation methods (confidence/regression calibration)
.metrics # metrics for measuring miscalibration (confidence/regression calibration)
.regression # methods for regression uncertainty calibration (regression calibration)
examples # example code snippets
Installation¶
The installation of the calibration suite is quite easy as it registered in the Python Package Index (PyPI). You can either install this framework using PIP:
$ python3 -m pip install netcal
Or simply invoke the following command to install the calibration suite when installing from source:
$ git clone https://github.com/EFS-OpenSource/calibration-framework
$ cd calibration-framework
$ python3 -m pip install .
Note: with update 1.3, we switched from setup.py to pyproject.toml according to PEP-518. The setup.py is only for backwards compatibility.
Requirements¶
According to requierments.txt:
numpy>=1.18
scipy>=1.4
matplotlib>=3.3
scikit-learn>=0.24
torch>=1.9
torchvision>=0.10.0
tqdm>=4.40
pyro-ppl>=1.8
tikzplotlib>=0.9.8
tensorboard>=2.2
gpytorch>=1.5.1
Calibration Metrics¶
We further distinguish between onfidence calibration which aims to recalibrate confidence estimates in the [0, 1] interval, and regression uncertainty calibration which addresses the problem of calibration in probabilistic regression settings.
Confidence 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.
Regression Calibration Metrics¶
In regression calibration, the most common metric is the Negative Log Likelihood (NLL) to measure the quality of a predicted probability distribution w.r.t. the ground-truth:
Negative Log Likelihood (NLL) (netcal.metrics.NLL)
The metrics Pinball Loss, Prediction Interval Coverage Probability (PICP), and Quantile Calibration Error (QCE) evaluate the estimated distributions by means of the predicted quantiles. For example, if a forecaster makes 100 predictions using a probability distribution for each estimate targeting the true ground-truth, we can measure the coverage of the ground-truth samples for a certain quantile level (e.g., 95% quantile). If the relative amount of ground-truth samples falling into a certain predicted quantile is above or below the specified quantile level, a forecaster is told to be miscalibrated in terms of quantile calibration. Appropriate metrics in this context are
Pinball Loss (netcal.metrics.PinballLoss)
Prediction Interval Coverage Probability (PICP) [14] (netcal.metrics.PICP)
Quantile Calibration Error (QCE) [15] (netcal.metrics.QCE)
Finally, if we work with normal distributions, we can measure the quality of the predicted variance/stddev estimates. For variance calibration, it is required that the predicted variance mathes the observed error variance which is equivalent to then Mean Squared Error (MSE). Metrics for variance calibration are
Methods¶
We further give an overview about the post-hoc calibration methods for (semantic) confidence calibration as well as about the methods for regression uncertainty calibration.
Confidence Calibration 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 such as “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] 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)
Regression Calibration Methods¶
The netcal library provides post-hoc methods to recalibrate the uncertainty of probabilistic regression tasks. We distinguish the calibration methods into non-parametric and parametric methods. Non-parametric calibration methods take a probability distribution as input and apply recalibration in terms of quantiles on the cumulative (CDF). This leads to a recalibrated probability distribution that, however, has no analytical representation but is given by certain points defining a CDF distribution. In contrast, parametric calibration methods also take a probability distribution as input and provide a recalibrated distribution that has an analytical expression (e.g., Gaussian).
Non-parametric calibration¶
The common non-parametric recalibration methods use the predicted cumulative (CDF) distribution functions to learn a mapping from the uncalibrated quantiles to the observed quantile coverage. Using a recalibrated CDF, it is possible to derive the respective density functions (PDF) or to extract statistical moments such as mean and variance. Non-parametric calibration methods within the netcal.regression package are
Parametric calibration¶
The parametric recalibration methods apply a recalibration of the estimated distributions so that the resulting distribution is given in terms of a distribution with an analytical expression (e.g., a Gaussian). These methods are suitable for applications where a parametric distribution is required for subsequent applications, e.g., within Kalman filtering. We implemented the following parametric calibration methods:
Variance Scaling [17], [18] which is nothing else but a temperature scaling for the predicted variance (netcal.regression.VarianceScaling)
GP-Normal [16] which applies an input-dependent rescaling of the predicted variance (netcal.regression.GPNormal). Note: this method is also able to capture correlations between multiple input dimensions and can return a joint multivariate normal distribution as calibration output (cf. examples section).
GP-Cauchy [16] is similar to GP-Normal but utilizes a Cauchy distribution as calibration output (netcal.regression.GPCauchy)
Visualization¶
For visualization of miscalibration, one can use a Confidence Histograms & Reliability Diagrams for (semantic) confidence calibration as well as for regression uncertainty calibration. Within confidence calibration, 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.
For 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.
For regression uncertainty calibration, the reliability diagram shows the relative prediction interval coverage of the ground-truth samples for different quantile levels.
Reliability Diagram [1], [12] (netcal.presentation.ReliabilityDiagram)
Reliability Diagram for regression calibration (netcal.presentation.ReliabilityRegression)
Reliability QCE Diagram [16] shows the Quantile Calibration Error (QCE) for different variance levels (netcal.presentation.ReliabilityQCE)
New on version 1.3: All plot-methods within the netcal.presentation package now support the option “tikz=True” which switches from standard matplotlib.Figure objects to strings with Tikz-Code. Tikz-code can be directly used for LaTeX documents to render images as vector graphics with high quality. Thus, this option helps to improve the quality of your reliability diagrams if you are planning to use this library for any type of publication/document
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)
Post-hoc Calibration for Classification¶
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)
Measuring Miscalibration for Classification¶
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, ground_truth)
calibrated_score = ece.measure(calibrated, ground_truth)
Visualizing Miscalibration for Classification¶
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
# you can also use this method to create a tikz file with tikz code
# that can be directly used within LaTeX documents:
diagram.plot(confidences, ground_truth, tikz=True, filename="diagram.tikz")
Detection (Confidence of Objects)¶
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,)
Post-hoc Calibration for Detection¶
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)
Measuring Miscalibration for Detection¶
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)
Visualizing Miscalibration for Detection¶
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
# you can also use this method to create a tikz file with tikz code
# that can be directly used within LaTeX documents:
diagram.plot(input, matched, tikz=True, filename="diagram.tikz")
Uncertainty in Confidence 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,)
Post-hoc Calibration with Uncertainty¶
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)
Measuring Miscalibration with Uncertainty¶
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, kind="confidence")
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
Probabilistic Regression¶
The following example shows how to use the post-hoc calibration methods for probabilistic regression tasks. Within probabilistic regression, a forecaster (e.g. with Gaussian prior) outputs a mean and a variance targeting the true ground-truth score. Thus, the following information is required to construct the calibration methods:
mean # NumPy n-D array holding the estimated mean of shape (n, d) with n samples and d dimensions
stddev # NumPy n-D array holding the estimated stddev (independent) of shape (n, d) with n samples and d dimensions
ground_truth # NumPy n-D array holding the ground-truth scores of shape (n, d) with n samples and d dimensions
Post-hoc Calibration (Parametric)¶
These information might result e.g. from object detection where the position information of the objects (bounding boxes) are parametrized by normal distributions. We start by using parametric calibration methods such as Variance Scaling:
from netcal.regression import VarianceScaling, GPNormal
# the initialization of the Variance Scaling method is pretty simple
varscaling = VarianceScaling()
# the GP-Normal requires a little bit more parameters to parametrize the underlying GP
gpnormal = GPNormal(
n_inducing_points=12, # number of inducing points
n_random_samples=256, # random samples used for likelihood
n_epochs=256, # optimization epochs
use_cuda=False, # can also use CUDA for computations
)
# fit the Variance Scaling
# note that we need to pass the first argument as tuple as the input distributions
# are parametrized by mean and variance
varscaling.fit((mean, stddev), ground_truth)
# fit GP-Normal - similar parameters here!
gpnormal.fit((mean, stddev), ground_truth)
# transform distributions to obtain recalibrated stddevs
stddev_varscaling = varscaling.transform((mean, stddev)) # NumPy array with stddev - has shape (n, d)
stddev_gpnormal = gpnormal.transform((mean, stddev)) # NumPy array with stddev - has shape (n, d)
Post-hoc Calibration (Non-Parametric)¶
We can also use non-parametric calibration methods. In this case, the calibrated distributions are defined by their density (PDF) and cumulative (CDF) functions:
from netcal.regression import IsotonicRegression, GPBeta
# the initialization of the Isotonic Regression method is pretty simple
isotonic = IsotonicRegression()
# the GP-Normal requires a little bit more parameters to parametrize the underlying GP
gpbeta = GPBeta(
n_inducing_points=12, # number of inducing points
n_random_samples=256, # random samples used for likelihood
n_epochs=256, # optimization epochs
use_cuda=False, # can also use CUDA for computations
)
# fit the Isotonic Regression
# note that we need to pass the first argument as tuple as the input distributions
# are parametrized by mean and variance
isotonic.fit((mean, stddev), ground_truth)
# fit GP-Beta - similar parameters here!
gpbeta.fit((mean, stddev), ground_truth)
# transform distributions to obtain recalibrated distributions
t_isotonic, pdf_isotonic, cdf_isotonic = varscaling.transform((mean, stddev))
t_gpbeta, pdf_gpbeta, cdf_gpbeta = gpbeta.transform((mean, stddev))
# Note: the transformation results are NumPy n-d arrays with shape (t, n, d)
# with t as the number of points that define the PDF/CDF,
# with n as the number of samples, and
# with d as the number of dimensions.
# The resulting variables can be interpreted as follows:
# - t_isotonic/t_gpbeta: x-values of the PDF/CDF with shape (t, n, d)
# - pdf_isotonic/pdf_gpbeta: y-values of the PDF with shape (t, n, d)
# - cdf_isotonic/cdf_gpbeta: y-values of the CDF with shape (t, n, d)
You can visualize the non-parametric distribution of a single sample within a single dimension using Matplotlib:
from matplotlib import pyplot as plt
fig, (ax1, ax2) = plt.subplots(2, 1)
# plot the recalibrated PDF within a single axis after calibration
ax1.plot(
t_isotonic[:, 0, 0], pdf_isotonic[:, 0, 0],
t_gpbeta[:, 0, 0], pdf_gpbeta[:, 0, 0],
)
# plot the recalibrated PDF within a single axis after calibration
ax2.plot(
t_isotonic[:, 0, 0], cdf_isotonic[:, 0, 0],
t_gpbeta[:, 0, 0], cdf_gpbeta[:, 0, 0],
)
plt.show()
We provide a method to extract the statistical moments expectation and variance from the recalibrated cumulative (CDF). Note that we advise to use one of the parametric calibration methods if you need e.g. a Gaussian for subsequent applications such as Kalman filtering.
from netcal import cumulative_moments
# extract the expectation (mean) and the variance from the recalibrated CDF
ymean_isotonic, yvar_isotonic = cumulative_moments(t_isotonic, cdf_isotonic)
ymean_gpbeta, yvar_gpbeta = cumulative_moments(t_gpbeta, cdf_gpbeta)
# each of these variables has shape (n, d) and holds the
# mean/variance for each sample and in each dimension
Correlation Estimation and Recalibration¶
With the GP-Normal netcal.regression.GPNormal, it is also possible to detect possible correlations between multiple input dimensions that have originally been trained/modelled independently from each other:
from netcal.regression import GPNormal
# the GP-Normal requires a little bit more parameters to parametrize the underlying GP
gpnormal = GPNormal(
n_inducing_points=12, # number of inducing points
n_random_samples=256, # random samples used for likelihood
n_epochs=256, # optimization epochs
use_cuda=False, # can also use CUDA for computations
correlations=True, # enable correlation capturing between the input dimensions
)
# fit GP-Normal
# note that we need to pass the first argument as tuple as the input distributions
# are parametrized by mean and variance
gpnormal.fit((mean, stddev), ground_truth)
# transform distributions to obtain recalibrated covariance matrices
cov = gpnormal.transform((mean, stddev)) # NumPy array with covariance - has shape (n, d, d)
# note: if the input is already given by multivariate normal distributions
# (stddev is covariance and has shape (n, d, d)), the methods works similar
# and simply applies a covariance recalibration of the input
Measuring Miscalibration for Regression¶
Measuring miscalibration is as simple as the training of the methods:
import numpy as np
from netcal.metrics import NLL, PinballLoss, QCE
# define the quantile levels that are used to evaluate the pinball loss and the QCE
quantiles = np.linspace(0.1, 0.9, 9)
# initialize NLL, Pinball, and QCE objects
nll = NLL()
pinball = PinballLoss()
qce = QCE(marginal=True) # if "marginal=False", we can also measure the QCE by means of the predicted variance levels (realized by binning the variance space)
# measure miscalibration with the initialized metrics
# Note: the parameter "reduction" has a major influence to the return shape of the metrics
# see the method docstrings for detailed information
nll.measure((mean, stddev), ground_truth, reduction="mean")
pinball.measure((mean, stddev), ground_truth, q=quantiles, reduction="mean")
qce.measure((mean, stddev), ground_truth, q=quantiles, reduction="mean")
Visualizing Miscalibration for Regression¶
Example visualization code block using the netcal.presentation.ReliabilityRegression class:
from netcal.presentation import ReliabilityRegression
# define the quantile levels that are used for the quantile evaluation
quantiles = np.linspace(0.1, 0.9, 9)
# initialize the diagram object
diagram = ReliabilityRegression(quantiles=quantiles)
# visualize miscalibration with the initialized object
diagram.plot((mean, stddev), ground_truth)
# you can also use this method to create a tikz file with tikz code
# that can be directly used within LaTeX documents:
diagram.plot((mean, stddev), ground_truth, tikz=True, filename="diagram.tikz")
References¶
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[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
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[16] Fabian Küppers, Jonas Schneider, Jonas, and Anselm Haselhoff: “Parametric and Multivariate Uncertainty Calibration for Regression and Object Detection.” In: Proceedings of the European Conference on Computer Vision (ECCV) Workshops, Springer, October 2022
[17] Levi, Dan, et al.: “Evaluating and calibrating uncertainty prediction in regression tasks.” arXiv preprint arXiv:1905.11659 (2019).
[18] Laves, Max-Heinrich, et al.: “Well-calibrated regression uncertainty in medical imaging with deep learning.” Medical Imaging with Deep Learning. PMLR, 2020.
[19] Volodymyr Kuleshov, Nathan Fenner, and Stefano Ermon: “Accurate uncertainties for deep learning using calibrated regression.” International Conference on Machine Learning. PMLR, 2018.
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