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net:cal - Uncertainty Calibration

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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 © 2019-2023 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.

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Important: updated references! If you use the net:cal 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:

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:

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:

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:

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

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:

Scaling

Implemented scaling methods are:

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:

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:

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.

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")
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References

<a name="ref1">[1]</a> Naeini, Mahdi Pakdaman, Gregory Cooper, and Milos Hauskrecht: "Obtaining well calibrated probabilities using bayesian binning." Twenty-Ninth AAAI Conference on Artificial Intelligence, 2015.

<a name="ref2">[2]</a> 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.

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