The net:cal calibration framework is a Python 3 library for measuring and mitigating miscalibration of uncertainty estimates, e.g., by a neural network.
Project description
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://efsopensource.github.io/calibrationframework.
Copyright © 20192023 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/LICENSE2.0.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
 Installation
 Requirements
 Calibration Metrics
 Methods
 Visualization
 Examples
 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 safetycritical 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)
 GPBeta (netcal.regression.GPBeta)
 GPNormal (netcal.regression.GPNormal)
 GPCauchy (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 TikzCode (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 nonparametric and parametric methods. Nonparametric 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. Nonparametric 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 GPNormal 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 GPNormal 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 plotmethods within the netcal.presentation package now support the option "tikz=True" which switches from standard matplotlib.Figure objects to strings with TikzCode. Tikzcode 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 MarkovChain MonteCarlo 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 speedup 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/EFSOpenSource/calibrationframework
$ cd calibrationframework
$ python3 m pip install .
Note: with update 1.3, we switched from setup.py to pyproject.toml according to PEP518. The setup.py is only for backwards compatibility.
Requirements
According to requierments.txt:
 numpy>=1.18
 scipy>=1.4
 matplotlib>=3.3
 scikitlearn>=0.24
 torch>=1.9
 torchvision>=0.10.0
 tqdm>=4.40
 pyroppl>=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 (DECE) [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 positiondependency)
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 groundtruth:
 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 groundtruth, we can measure the coverage of the groundtruth samples for a certain quantile level (e.g., 95% quantile). If the relative amount of groundtruth 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
 Expected Normalized Calibration Error (ENCE) [17] (netcal.metrics.ENCE)
 Uncertainty Calibration Error (UCE) [18] (netcal.metrics.UCE)
Methods
We further give an overview about the posthoc calibration methods for (semantic) confidence calibration as well as about the methods for regression uncertainty calibration.
Confidence Calibration Methods
The posthoc calibration methods are separated into binning and scaling methods. The binning methods divide the available information into several bins (like ECE or DECE) 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, multiclass calibration is performed in "one vs. all" by default.
Some methods such as "Isotonic Regression" utilize methods from the scikitlearn 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 feedforward with maximum likelihood estimates on the calibration parameters (standard)  'momentum': use nonconvex momentum optimization (e.g. default on dependent beta calibration)  'mcmc': use MarkovChain MonteCarlo 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 posthoc methods to recalibrate the uncertainty of probabilistic regression tasks. We distinguish the calibration methods into nonparametric and parametric methods. Nonparametric 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).
Nonparametric calibration
The common nonparametric 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. Nonparametric calibration methods within the netcal.regression package are
 Isotonic Regression [19] which applies a (marginal) recalibration of the CDF (netcal.regression.IsotonicRegression)
 GPBeta [20] which applies an inputdependent recalibration of the CDF using a Gaussian process for parameter estimation (netcal.regression.GPBeta)
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)
 GPNormal [16] which applies an inputdependent 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).
 GPCauchy [16] is similar to GPNormal 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 xposition 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 groundtruth 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 plotmethods within the netcal.presentation package now support the option "tikz=True" which switches from standard matplotlib.Figure objects to strings with TikzCode. Tikzcode 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 1D array with ground truth digits between 09  shape: (n_samples,)
confidences # this is a NumPy 2D array with confidence estimates between 01  shape: (n_samples, n_classes)
Posthoc 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 xposition of the predicted bounding boxes. Our groundtruth provided to the calibration algorithm denotes if a bounding box has matched a groundtruth box with a certain IoU and the correct class label.
matched # binary NumPy 1D 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 1D array with confidence estimates between 01  shape: (n_samples,)
relative_x_position # NumPy 1D array with relative centerx position between 01 of each prediction  shape: (n_samples,)
Posthoc 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 DECE:
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 1D 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 1D array with confidence estimates between 01  shape: (n_samples,)
relative_x_position # NumPy 1D array with relative centerx position between 01 of each prediction  shape: (n_samples,)
Posthoc 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 MarkovChain MonteCarlo sampling with 250 warmup 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 DECE 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("DECE:", 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("DECE:", 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 posthoc 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 groundtruth score. Thus, the following information is required to construct the calibration methods:
mean # NumPy nD array holding the estimated mean of shape (n, d) with n samples and d dimensions
stddev # NumPy nD array holding the estimated stddev (independent) of shape (n, d) with n samples and d dimensions
ground_truth # NumPy nD array holding the groundtruth scores of shape (n, d) with n samples and d dimensions
Posthoc 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 GPNormal 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 GPNormal  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)
Posthoc Calibration (NonParametric)
We can also use nonparametric 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 GPNormal 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 GPBeta  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 nd 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: xvalues of the PDF/CDF with shape (t, n, d)
#  pdf_isotonic/pdf_gpbeta: yvalues of the PDF with shape (t, n, d)
#  cdf_isotonic/cdf_gpbeta: yvalues of the CDF with shape (t, n, d)
You can visualize the nonparametric 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 GPNormal 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 GPNormal 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 GPNormal
# 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
[1] Naeini, Mahdi Pakdaman, Gregory Cooper, and Milos Hauskrecht: "Obtaining well calibrated probabilities using bayesian binning." TwentyNinth AAAI Conference on Artificial Intelligence, 2015.
[2] Kull, Meelis, Telmo Silva Filho, and Peter Flach: "Beta calibration: a wellfounded and easily implemented improvement on logistic calibration for binary classifiers." Artificial Intelligence and Statistics, PMLR 54:623631, 2017.
[3] Zadrozny, Bianca and Elkan, Charles: "Obtaining calibrated probability estimates from decision trees and naive bayesian classifiers." In ICML, pp. 609–616, 2001.
[4] 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: "Nearlyisotonic 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.: "Scikitlearn: Machine Learning in Python." In Journal of Machine Learning Research, volume 12 pp 28252830, 2011.
[10] 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] Neumann, Lukas, Andrew Zisserman, and Andrea Vedaldi: "Relaxed Softmax: Efficient Confidence AutoCalibration for Safe Pedestrian Detection." Conference on Neural Information Processing Systems (NIPS) Workshop MLITS, 2018.
[12] 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] 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 DoshiVelez: "Quality of Uncertainty Quantification for Bayesian Neural Network Inference." Workshop on Uncertainty and Robustness in Deep Learning, ICML, 2019
[15] Liang, Gongbo, et al.: "Improved trainable calibration method for neural networks on medical imaging classification." arXiv preprint arXiv:2009.04057 (2020)
[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, MaxHeinrich, et al.: "Wellcalibrated 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.
[20] Hao Song, Tom Diethe, Meelis Kull and Peter Flach: "Distribution calibration for regression." International Conference on Machine Learning. PMLR, 2019.
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