3. Quantification Methods#

Quantification methods can be categorized as belonging to aggregative, non-aggregative, and meta-learning groups. Aggregative quantifiers rely on a surrogate classifier as an intermediate step, and devise different aggregation functions over the classifier outputs. By contrast, non-aggregative methods perform quantification without requiring an underlying classifier. Meta-learning refers to quantification methods that are constructed over simpler quantification methods, and implement high-level orchestration functions.

Beyond these three traditional categories of methods, we here present an additional, orthogonal one: Bayesian quantifiers, i.e., quantification methods that do not simply return point-estimates of class prevalence, but are also able to provide a measure of uncertaintly around them.

Any quantifier in QuaPy shoud extend the class BaseQuantifier, and implement some abstract methods:

    @abstractmethod
    def fit(self, X, y): ...

    @abstractmethod
    def predict(self, X): ...

The meaning of those functions should be familiar to those used to work with scikit-learn since the class structure of QuaPy is directly inspired by scikit-learn’s Estimators. Functions fit and predict (for which there is an alias quantify) are used to train the model and to provide class estimations. Quantifiers also extend from scikit-learn’s BaseEstimator, in order to simplify the use of set_params and get_params used in model selection.

3.1. Aggregative Methods#

All quantification methods are implemented as part of the qp.method package. In particular, aggregative methods are defined in qp.method.aggregative, and extend AggregativeQuantifier(BaseQuantifier). The methods that any aggregative quantifier must implement are:

    @abstractmethod
    def aggregation_fit(self, classif_predictions, labels):

    @abstractmethod
    def aggregate(self, classif_predictions): ...

The argument classif_predictions is whatever the method classify returns. QuaPy comes with default implementations that cover most common cases, but you can override classify in case your method requires further or different information to work.

These two functions replace the fit and predict methods, which come with default implementations. For instance, the fit function is provided and amounts to:

    def fit(self, X, y):
        self._check_init_parameters()
        classif_predictions, labels = self.classifier_fit_predict(X, y)
        self.aggregation_fit(classif_predictions, labels)
        return self

Note that this function fits the classifier, and generates the predictions. This is assumed to be a routine common to all aggregative quantifiers, and is provided by QuaPy. What remains ahead is to define the aggregation_fit function, that takes as input the classifier predictions and the original training data (this latter is typically unused). The classifier predictions can be:

  • confidence scores: quantifiers inheriting directly from AggregativeQuantifier

  • crisp predictions: quantifiers inheriting from AggregativeCrispQuantifier

  • posterior probabilities: quantifiers inheriting from AggregativeSoftQuantifier

  • anything: custom quantifiers overriding the classify method

Note also that the fit method also calls _check_init_parameters; this function is meant to be overriden (if needed) and allows the method to quickly raise any exception based on any inconsistency found in the __init__ arguments, thus avoiding to break after training the classifier and generating predictions.

Similarly, the function predict (alias quantify) is provided, and amounts to:

def predict(self, X):
    classif_predictions = self.classify(X)
    return self.aggregate(classif_predictions)

in which only the function aggregate is required to be overriden in most cases.

Aggregative quantifiers are expected to maintain a classifier (which is accessed through the @property classifier). This classifier is given as input to the quantifier, and will be trained by the quantifier’s fit (default). Alternatively, the classifier can be already fit on external data; in this case, the fit_learner argument in the __init__ should be set to False (see 4.using_pretrained_classifier.py for a full code example).

The above patterns (in training: (i) fit the classifier, then (ii) fit the aggregation; in test: (i) classify, then (ii) aggregate) allows QuaPy to optimize many internal procedures, on the grounds that steps (i) are slower than steps (ii). In particular, the model selection routing takes advantage of this two-step process and generates classifiers only for the valid combinations of hyperparameters of the classifier, and then clones these classifiers and explores the combinations of hyperparameters that are specific to the quantifier (this can result in huge time savings). Concerning the inference phase, this two-step process allow the evaluation of many standard protocols (e.g., the artificial sampling protocol) to be carried out very efficiently. The reason is that the entire set can be pre-classified once, and the quantification estimations for different samples can directly reuse these predictions, without requiring to classify each element every time. QuaPy leverages this property to speed-up any procedure having to do with quantification over samples, as is customarily done in model selection or in evaluation.

3.1.1. The Classify & Count variants#

QuaPy implements the four CC variants, i.e.:

  • CC (Classify & Count), the simplest aggregative quantifier; one that classifies all instances and computes the prevalence of the predicted labels. This baseline is discussed, among others, in Forman (2008).

  • ACC (Adjusted Classify & Count), the adjusted variant of CC, originally proposed in Forman (2008).

  • PCC (Probabilistic Classify & Count), the probabilistic variant of CC that relies on the posterior probabilities returned by a probabilistic classifier, introduced in Bella et al. (2010).

  • PACC (Probabilistic Adjusted Classify & Count), the adjusted variant of PCC, also introduced in Bella et al. (2010).

The following code serves as a complete example using CC equipped with a SVM as the classifier:

import quapy as qp
import quapy.functional as F
from sklearn.svm import LinearSVC

training, test = qp.datasets.fetch_twitter('hcr', pickle=True).train_test
Xtr, ytr = training.Xy

# instantiate a classifier learner, in this case a SVM
svm = LinearSVC()

# instantiate a Classify & Count with the SVM
# (an alias is available in qp.method.aggregative.ClassifyAndCount)
model = qp.method.aggregative.CC(svm)
model.fit(Xtr, ytr)
estim_prevalence = model.predict(test.X)

The same code could be used to instantiate an ACC, by simply replacing the instantiation of the model with:

model = qp.method.aggregative.ACC(svm)

Note that the adjusted variants (ACC and PACC) need to estimate some parameters for performing the adjustment (e.g., the true positive rate and the false positive rate in case of binary classification) that are estimated on a validation split of the labelled set. In this case, the __init__ method of ACC defines an additional parameter, val_split. If this parameter is set to a float in [0,1] representing a fraction (e.g., 0.4) then that fraction of labelled data (e.g., 40%) will be used for estimating the parameters for adjusting the predictions. This parameters can also be set with an integer, indicating that the parameters should be estimated by means of k-fold cross-validation, for which the integer indicates the number k of folds (the default value is 5). Finally, val_split can be set to a specific held-out validation set (i.e., an tuple (X,y)).

The following code illustrates the case in which PCC is used:

model = qp.method.aggregative.PCC(svm)
model.fit(Xtr, ytr)
estim_prevalence = model.predict(Xte)
print('classifier:', model.classifier)

In this case, QuaPy will print:

The learner LinearSVC does not seem to be probabilistic. The learner will be calibrated.
classifier: CalibratedClassifierCV(base_estimator=LinearSVC(), cv=5)

The first output indicates that the learner (LinearSVC in this case) is not a probabilistic classifier (i.e., it does not implement the predict_proba method) and so, the classifier will be converted to a probabilistic one through calibration. As a result, the classifier that is printed in the second line points to a CalibratedClassifierCV instance. Note that calibration can only be applied to hard classifiers if fit_learner=True; an exception will be raised otherwise.

Lastly, everything we said about ACC and PCC applies to PACC as well.

A Bayesian counterpart of the ACC family is also available; see the Bayesian Quantification Methods section for BayesianCC.

New in v0.1.9: quantifiers ACC and PACC now have three additional arguments: method, solver and norm:

3.1.2. Threshold Optimization methods#

QuaPy implements Forman’s threshold optimization methods; see, e.g., (Forman 2006) and (Forman 2008). These include: T50, MAX, X, Median Sweep (MS), and its variant MS2.

These methods are binary-only and implement different heuristics for improving the stability of the denominator of the ACC adjustment (tpr-fpr). The methods are called “threshold” since said heuristics have to do with different choices of the underlying classifier’s threshold.

3.1.3. Expectation Maximization (EMQ) / Maximum Likelihood for Label Shift (MLLS)#

The Expectation Maximization Quantifier (EMQ) (also known as SLD after the name of the proponets, or Maximum Likelihood for Label Shift, MLLS) , is available at qp.method.aggregative.EMQ or via the alias qp.method.aggregative.ExpectationMaximizationQuantifier. The method is described in:

Saerens, M., Latinne, P., and Decaestecker, C. (2002). Adjusting the outputs of a classifier to new a priori probabilities: A simple procedure. Neural Computation, 14(1):21–41.

EMQ works with a probabilistic classifier (if the classifier given as input is a hard one, a calibration will be attempted). Although this method was originally proposed for improving the posterior probabilities of a probabilistic classifier, and not for improving the estimation of prior probabilities, EMQ ranks almost always among the most effective quantifiers in the experiments we have carried out.

An example of use can be found below:

import quapy as qp
from sklearn.linear_model import LogisticRegression

train, test = qp.datasets.fetch_twitter('hcr', pickle=True).train_test

model = qp.method.aggregative.EMQ(LogisticRegression())
model.fit(*train.Xy)
estim_prevalence = model.predict(test.X)

EMQ accepts additional parameters in the construction method:

  • exact_train_prev: set to True for using the true training prevalence as the departing prevalence estimation (default behaviour), or to False for using an approximation of it as suggested by Alexandari et al. (2020)

  • calib: allows to indicate a calibration method, among those proposed by Alexandari et al. (2020), including the Bias-Corrected Temperature Scaling (bcts), Vector Scaling (bcts), No-Bias Temperature Scaling (nbvs), or Temperature Scaling (ts); default is None (no calibration).

  • on_calib_error: indicates the policy to follow in case the calibrator fails at runtime. Options include raise (default), in which case a RuntimeException is raised; and backup, in which case the calibrator is silently skipped.

You can use the class method EMQ_BCTS to effortlessly instantiate EMQ with the best performing heuristics found by Alexandari et al. (2020). See the API documentation for further details.

For a Bayesian label-shift counterpart based on the same general family of ideas, see the Bayesian Quantification Methods section for BayesianMAPLS.

3.1.4. Regularized Learning under Label Shift (RLLS)#

RLLS is available at qp.method.aggregative.RLLS and ports the regularized importance-weight estimation procedure of Azizzadenesheli, K., Liu, A., Yang, F., and Anandkumar, A. (2019). Regularized Learning for Domain Adaptation under Label Shifts. ICLR 2019 to QuaPy’s aggregative interface. The method estimates the label-shift importance weights w = q(y)/p(y) from the classifier’s validation posteriors (or, in mode='hard', its argmax predictions) and the corresponding source labels, regularizing the estimation by an amount controlled by alpha (scaled by a finite-sample confidence term governed by delta). The resulting weights are then used to rescale the training prevalence into the target prevalence estimate.

Like ACC and PACC, RLLS requires validation predictions and therefore expects val_split to be set (as an integer for k-fold cross-validation, a float for a held-out split, or an explicit (X, y) tuple) whenever fit_classifier=True. This method relies on the optional cvxpy dependency, which must be installed separately ($ pip install cvxpy).

import quapy as qp
from quapy.method.aggregative import RLLS
from sklearn.linear_model import LogisticRegression

train, test = qp.datasets.fetch_UCIBinaryDataset('haberman').train_test

model = RLLS(LogisticRegression(max_iter=2000), val_split=5)
model.fit(*train.Xy)
estim_prevalence = model.predict(test.X)

3.1.5. Distribution Matching#

Distribution Matching (DM) methods search for the mixture parameter (the sought class prevalence values) yielding the mixture between the class-wise representations that best matches the test distribution. Different criteria for deciding how this matching is assessed, and different ways for modelling the distributions give rise to different instantiations of DM methods.

The following methods are here discussed because they rely on a surrogate classifier for representing the distributions, albeit different non-aggregative variants of them do often exist. Aside from this, the formulation of DM methods is flexible enough as to accomodate methods that were proposed under a different framework; examples include ACC and PACC.

See the frameworks by Firat, Bunse, Garg et al., or Dussap, for more details.

3.1.5.1. Hellinger Distance y (HDy)#

Implementation of the method based on the Hellinger Distance y (HDy) proposed by González-Castro, V., Alaiz-Rodríguez, R., and Alegre, E. (2013). Class distribution estimation based on the Hellinger distance. Information Sciences, 218:146-164.

It is implemented in qp.method.aggregative.HDy (also accessible through the allias qp.method.aggregative.HellingerDistanceY). This method works with a probabilistic classifier (hard classifiers can be used as well and will be calibrated) and requires a validation set to estimate parameter for the mixture model. Just like ACC and PACC, this quantifier receives a val_split argument in the constructor that can either be a float indicating the proportion of training data to be taken as the validation set (in a random stratified split), or the validation set itself (i.e., an tuple (X,y)).

HDy was proposed as a binary classifier and the implementation provided in QuaPy accepts only binary datasets.

The following code shows an example of use:

import quapy as qp
from sklearn.linear_model import LogisticRegression

# load a binary dataset
dataset = qp.datasets.fetch_reviews('hp', pickle=True)
qp.data.preprocessing.text2tfidf(dataset, min_df=5, inplace=True)

model = qp.method.aggregative.HDy(LogisticRegression())
model.fit(*dataset.training.Xy)
estim_prevalence = model.predict(dataset.test.X)

3.1.5.2. Generalized Distribution Matching y (DMy)#

QuaPy also provides a generalized posterior-space distribution-matching quantifier for binary or multiclass problems, implemented as qp.method.aggregative.DMy. This class follows the generic distribution matching view discussed by Firat (2016): it represents class-conditional posterior distributions by histograms and then searches for the prevalence vector whose mixture best matches the test distribution.

DMy is intentionally flexible and exposes three main design choices: the number of histogram bins (nbins), the divergence to minimize (divergence, e.g., 'HD' or 'topsoe'), and whether to match PDFs or CDFs (cdf). The optimization routine can also be selected through search; the default 'optim_minimize' works for multiclass problems, while 'linear_search' and 'ternary_search' are binary-only. A multiclass HDy-like instance can be obtained as:

multiclass_hdy = qp.method.aggregative.DMy(
    classifier=LogisticRegression(),
    divergence='HD',
    cdf=False,
)

3.1.5.3. DyS#

QuaPy implements the binary DyS framework proposed by Maletzke et al. (2020) as qp.method.aggregative.DyS. Conceptually, DyS can be seen as a generalization of HDy in which the prevalence is found by ternary search over a distribution-matching objective. In QuaPy, the user can select the number of histogram bins (n_bins), the divergence (divergence), and the optimization tolerance (tol).

3.1.5.4. Energy Distance y (EDy)#

QuaPy also adapts EDy from quantificationlib, which is available as qp.method.aggregative.EDy.

This method replaces histogram matching with an energy-distance formulation defined directly on posterior-probability vectors and solves the resulting optimization problem by quadratic programming. The method is proposed in Castaño et al.’s (2024) paper.

In QuaPy, EDy works for binary and multiclass problems and lets the user choose the pairwise distance through the distance parameter ('manhattan', 'euclidean', or a custom callable). Because the optimization relies on quadprog, this method requires the optional dependency pip install quadprog.

3.1.5.5. SMM#

QuaPy also includes the binary SMM method of Hassan et al. (2019), available as qp.method.aggregative.SMM. This is a very lightweight distribution-matching variant in which the posterior representation is reduced to class-wise means rather than full histograms, making it conceptually close to PACC.

3.1.5.6. Kernel Density Estimation methods (KDEy)#

QuaPy provides implementations for the three variants of KDE-based methods proposed in Moreo, A., González, P. and del Coz, J.J.. Kernel Density Estimation for Multiclass Quantification. Machine Learning. Vol 114 (92), 2025 (a preprint is available online). The variants differ in the divergence metric to be minimized:

  • KDEy-HD: minimizes the (squared) Hellinger Distance and solves the problem via a Monte Carlo approach

  • KDEy-CS: minimizes the Cauchy-Schwarz divergence and solves the problem via a closed-form solution

  • KDEy-ML: minimizes the Kullback-Leibler divergence and solves the problem via maximum-likelihood

These methods are specifically devised for multiclass problems (although they can tackle binary problems too).

All KDE-based methods depend on the hyperparameter bandwidth of the kernel. Typical values that can be explored in model selection range in [0.01, 0.25]. Previous experiments reveal the methods’ performance varies smoothly at small variations of this hyperparameter.

A Bayesian counterpart is available as well; see the Bayesian Quantification Methods section for BayesianKDEy.

3.1.6. Explicit Loss Minimization#

The Explicit Loss Minimization (ELM) represent a family of methods based on structured output learning, i.e., quantifiers relying on classifiers that have been optimized targeting a quantification-oriented evaluation measure. The original methods are implemented in QuaPy as classify & count (CC) quantifiers that use Joachim’s SVMperf as the underlying classifier, properly set to optimize for the desired loss.

In QuaPy, this can be more achieved by calling the functions:

the last two methods (SVM(AE) and SVM(RAE)) have been implemented in QuaPy in order to make available ELM variants for what nowadays are considered the most well-behaved evaluation metrics in quantification.

3.1.6.1. Installing the SVMperf backend#

These methods rely on Joachim’s SVMperf, patched with quantification-oriented losses. QuaPy provides the script prepare_svmperf.sh, which downloads the original sources, applies the patch, and compiles the resulting binary. In practice, this amounts to running:

./prepare_svmperf.sh

This creates a directory svm_perf_quantification/. Once this is available, you can point QuaPy to it with:

qp.environ['SVMPERF_HOME'] = './svm_perf_quantification'

The patch extends the one originally released for Esuli and Sebastiani (2015) and also covers the Q, AE, and RAE losses used by QuaPy’s ELM wrappers.

All ELM methods are binary because SVMperf itself is binary. They can still be wrapped in a one-vs-all scheme for single-label multiclass problems, though this strategy is generally considered inappropriate under prior probability shift. See the examples on explicit loss minimization and on one versus all quantification for minimal working code.

3.2. Non-Aggregative Methods#

Non-aggregative methods are quantifiers that do not follow the two-step (classify, then aggregate) pattern described above for aggregative methods. These methods are implemented in the qp.method.non_aggregative module and extend BaseQuantifier directly, implementing fit and predict on their own terms.

3.2.1. Maximum Likelihood Prevalence Estimation (MLPE)#

MaximumLikelihoodPrevalenceEstimation (MLPE) is a lazy baseline quantifier that assumes the IID assumption holds, i.e., that there is no prior probability shift between the training and the test distributions. Its fit method simply computes and stores the training prevalence, and its predict method returns that same training prevalence for any test sample, irrespective of the sample itself. MLPE is considered a lower-bound quantifier: any quantification method worth using should outperform it.

import quapy as qp
from quapy.method.non_aggregative import MaximumLikelihoodPrevalenceEstimation

dataset = qp.datasets.fetch_UCIBinaryDataset('haberman')
train, test = dataset.train_test

model = MaximumLikelihoodPrevalenceEstimation()
model.fit(*train.Xy)
estim_prevalence = model.predict(test.X)  # always equals train.prevalence()

3.2.2. Distribution Matching x (DMx) and Hellinger Distance x (HDx)#

DMx is the covariate-space counterpart of the DMy distribution-matching quantifier described in the Hellinger Distance y (HDy) section: instead of matching distributions built from the classifier’s predictions, DMx matches distributions built directly from the (discretized) feature space, and thus requires no classifier at all. For each class, DMx builds one histogram per feature from the training instances of that class; at prediction time, it searches for the mixture of these class-conditional histograms that best matches the (also histogram-based) representation of the test sample, in terms of a chosen divergence.

DMx accepts the following hyperparameters in its constructor:

  • nbins: the number of bins used to discretize each feature (default 8)

  • divergence: a string (“HD” for Hellinger Distance, or “topsoe”) or a callable taking two histograms and returning a divergence value (default “HD”)

  • cdf: whether to match cumulative distributions (CDFs) instead of the histograms (PDFs) themselves (default False)

  • search: the strategy used for finding the optimal prevalence; valid options are optim_minimize (default, works for binary and multiclass problems), linear_search, and ternary_search (these last two are binary-only)

  • n_jobs: number of parallel workers (default None)

DMx also offers the class method DMx.HDx (aliased as qp.method.non_aggregative.HDx, and also as HellingerDistanceX) that reproduces the original Hellinger Distance x (HDx) method proposed by González-Castro, Alaiz-Rodríguez, and Alegre (2013), the same paper that introduced HDy. HDx is a binary-only method that computes the matching for nbins ranging over [10, 20, ..., 110] (via a MedianEstimator, taking the median of the resulting estimates) and searches for the best prevalence via a linear search stepping by 0.01, rather than via the optim_minimize search used by DMx by default.

The following code, adapted from the example comparing HDy and HDx (examples/11.comparing_HDy_HDx.py), shows the two methods side-by-side:

from sklearn.linear_model import LogisticRegression
import quapy as qp
from quapy.method.aggregative import HDy
from quapy.method.non_aggregative import DMx

train, test = qp.datasets.fetch_UCIBinaryDataset('haberman').train_test
Xtr, ytr = train.Xy

hdy = HDy(LogisticRegression()).fit(Xtr, ytr)
estim_prevalence_hdy = hdy.predict(test.X)

hdx = DMx.HDx(n_jobs=-1).fit(Xtr, ytr)
estim_prevalence_hdx = hdx.predict(test.X)

Note that, unlike HDy, HDx requires no classifier whatsoever, since it operates directly on the covariates.

3.2.3. Energy Distance x (EDx)#

QuaPy also provides qp.method.non_aggregative.EDx, which is the feature-space counterpart of EDy: it keeps the same energy-distance formulation and quadratic-program solver, but applies them directly to the raw instances instead of first projecting them onto posterior probabilities through a classifier. In this sense, EDx is to EDy what DMx is to DMy.

EDx works for binary and multiclass problems, accepts the same distance options as EDy ('manhattan', 'euclidean', or a custom callable), and requires the optional dependency pip install quadprog.

3.2.4. ReadMe#

ReadMe is a non-aggregative quantification method proposed by Hopkins, D. and King, G. (2007). A method of automated nonparametric content analysis for social science. American Journal of Political Science, 54(1):229-247. The method estimates Q(Y=i) directly from Q(X) = sum_i Q(X|Y=i) Q(Y=i) by solving a (constrained) least-squares regression, thus avoiding the cost of estimating posterior probabilities Q(Y=i|X) altogether.

Since Q(X) and Q(X|Y=i) can be of very high dimension for realistic feature spaces, ReadMe renders the problem tractable by performing bagging in the feature space: many small random subsets of features (of size bagging_range) are drawn, the least-squares problem is solved on each subset, and the resulting estimates are averaged. ReadMe additionally combines this bagging procedure with bootstrap resampling of the training instances in order to derive confidence regions around the point estimate; accordingly, ReadMe implements the WithConfidenceABC interface (see the confidence regions section), and exposes a predict_conf method in addition to predict.

ReadMe accepts the following hyperparameters:

  • prob_model: either "full" (default), the original Hopkins and King formulation, in which Q(X) and Q(X|Y) are modelled empirically and thus require the feature matrix X to be binary (e.g., term presence/absence); or "naive", a much faster approximation that models Q(X) and Q(X|Y) as multinomial (bag-of-words) distributions, and that supports much larger values of bagging_range

  • bootstrap_trials: number of bootstrap resamplings of the training data used for deriving the confidence region (default 300)

  • bagging_trials: number of bagging trials, i.e., random feature subsets, averaged for each point estimate (default 300)

  • bagging_range: number of features kept in each bagging trial (default 15); note that, when prob_model="full", this value should typically be kept small (the authors advise against values above 25) since the empirical distribution requires enumerating 2^bagging_range possible feature configurations

  • confidence_level: the confidence level for the confidence region (default 0.95)

  • region: the type of confidence region to construct, one of "intervals" (default), "ellipse", "ellipse-clr", or "ellipse-ilr" (see the confidence regions section for details)

  • bonferroni: whether to apply Bonferroni correction when region="intervals" (default False); this parameter has no effect for ellipse-based regions

  • random_state: an int for replicability, or None (default)

  • verbose: whether to display progress information (default False)

The following minimal example, adapted from examples/18.ReadMe_for_text_analysis.py, shows ReadMe applied to a binary bag-of-words text quantification problem:

from sklearn.feature_extraction.text import CountVectorizer
from sklearn.pipeline import Pipeline
import quapy as qp
from quapy.method.non_aggregative import ReadMe

reviews = qp.datasets.fetch_reviews('imdb').reduce(n_train=1000, random_state=0)

# ReadMe's "full" model requires a binary feature matrix
encode_0_1 = Pipeline([('0_1_terms', CountVectorizer(min_df=5, binary=True))])
train, test = qp.data.preprocessing.instance_transformation(reviews, encode_0_1, inplace=True).train_test

model = ReadMe(prob_model='full', bootstrap_trials=100, bagging_trials=100, bagging_range=20, random_state=0)
model.fit(*train.Xy)  # lazy: only bootstrap resampling happens here

estim_prevalence, conf_region = model.predict_conf(test.X)

Note that ReadMe is computationally expensive: its cost scales with the product of bootstrap_trials and bagging_trials, each of which requires solving a least-squares problem.

3.3. Composable Methods#

The quapy.method.composable module integrates qunfold allows the composition of quantification methods from loss functions and feature transformations (thanks to Mirko Bunse for the integration!).

Any composed method solves a linear system of equations by minimizing the loss after transforming the data. Methods of this kind include ACC, PACC, HDx, HDy, and many other well-known methods, as well as an unlimited number of re-combinations of their building blocks.

3.3.1. Installation#

pip install --upgrade pip setuptools wheel
pip install "jax[cpu]"
pip install "qunfold @ git+https://github.com/mirkobunse/qunfold@v0.1.5"

Note: since version 0.2.0, QuaPy is only compatible with qunfold >=0.1.5.

3.3.2. Basics#

The composition of a method is implemented through the quapy.method.composable.ComposableQuantifier class. Its documentation also features an example to get you started in composing your own methods.

from quapy.method.composable import (
    ComposableQuantifier,
    TikhonovRegularized,
    LeastSquaresLoss,
    ClassRepresentation,
)

ComposableQuantifier( # ordinal ACC, as proposed by Bunse et al., 2022
    TikhonovRegularized(LeastSquaresLoss(), 0.01),
    ClassRepresentation(RandomForestClassifier(oob_score=True))
)

More exhaustive examples of method compositions, including hyper-parameter optimization, can be found in the example directory.

To implement your own loss functions and feature representations, follow the corresponding manual of the qunfold package, which provides the back-end of QuaPy’s composable module.

3.3.3. Loss functions#

Hint

You can use the quapy.method.composable.CombinedLoss to create arbitrary, weighted sums of losses and regularizers.

3.3.4. Regularization functions#

3.3.5. Feature transformations#

Hint

The quapy.method.composable.ClassRepresentation requires the classifier to have a property oob_score==True and to produce a property oob_decision_function during fitting. In scikit-learn, this requirement is fulfilled by any bagging classifier, such as random forests. Any other classifier needs to be cross-validated through the quapy.method.composable.CVClassifier.

3.4. Meta Models#

By meta models we mean quantification methods that are defined on top of other quantification methods, and that thus do not squarely belong to the aggregative nor the non-aggregative group (indeed, meta models could use quantifiers from any of those groups). Meta models are implemented in the qp.method.meta module.

3.4.1. Ensembles#

QuaPy implements (some of) the variants proposed in:

The following code shows how to instantiate an Ensemble of 30 Adjusted Classify & Count (ACC) quantifiers operating with a Logistic Regressor (LR) as the base classifier, and using the average as the aggregation policy (see the original article for further details). The last parameter indicates to use all processors for parallelization.

import quapy as qp
from quapy.method.aggregative import ACC
from quapy.method.meta import Ensemble
from sklearn.linear_model import LogisticRegression

dataset = qp.datasets.fetch_UCIBinaryDataset('haberman')
train, test = dataset.train_test

model = Ensemble(quantifier=ACC(LogisticRegression()), size=30, policy='ave', n_jobs=-1)
model.fit(*train.Xy)
estim_prevalence = model.predict(test.X)

Other aggregation policies implemented in QuaPy include:

  • ‘ptr’ for applying a dynamic selection based on the training prevalence of the ensemble’s members

  • ‘ds’ for applying a dynamic selection based on the Hellinger Distance

  • any valid quantification measure (e.g., ‘mse’) for performing a static selection based on the performance estimated for each member of the ensemble in terms of that evaluation metric.

When using any of the above options, it is important to set the red_size parameter, which informs of the number of members to retain.

Please, check the model selection manual if you want to optimize the hyperparameters of ensemble for classification or quantification.

3.4.2. The QuaNet neural network#

QuaPy offers an implementation of QuaNet, a deep learning model presented in:

Esuli, A., Moreo, A., & Sebastiani, F. (2018, October). A recurrent neural network for sentiment quantification. In Proceedings of the 27th ACM International Conference on Information and Knowledge Management (pp. 1775-1778).

This model requires torch to be installed. QuaNet also requires a classifier that can provide embedded representations of the inputs. In the original paper, QuaNet was tested using an LSTM as the base classifier. In the following example, we show an instantiation of QuaNet that instead uses CNN as a probabilistic classifier, taking its last layer representation as the document embedding:

import quapy as qp
from quapy.method.meta import QuaNet
from quapy.classification.neural import NeuralClassifierTrainer, CNNnet

# use samples of 100 elements
qp.environ['SAMPLE_SIZE'] = 100

# load the kindle dataset as text, and convert words to numerical indexes
dataset = qp.datasets.fetch_reviews('kindle', pickle=True)
qp.data.preprocessing.index(dataset, min_df=5, inplace=True)

# the text classifier is a CNN trained by NeuralClassifierTrainer
cnn = CNNnet(dataset.vocabulary_size, dataset.n_classes)
learner = NeuralClassifierTrainer(cnn, device='cuda')

# train QuaNet
model = QuaNet(learner, device='cuda')
model.fit(*dataset.training.Xy)
estim_prevalence = model.predict(dataset.test.X)

3.5. Quantifiers with Uncertainty Quantification#

(New in v0.2.0!) Some quantification methods go beyond providing a single point estimate of class prevalence values and also produce confidence regions, which characterize the uncertainty around the point estimate. In QuaPy, two such families are currently implemented: bootstrap methods and Bayesian methods.

Confidence regions are constructed around a point estimate, which is typically computed as the mean value of a set of samples.

The confidence region can be instantiated in four ways:

  • Confidence intervals: are standard confidence intervals generated for each class independently (method=”intervals”). Since confidence intervals are independently derived for each class, Bonferroni correction can be applied.

  • Confidence ellipse in the simplex: an ellipse constructed around the mean point; the ellipse lies on the simplex and takes into account possible inter-class dependencies in the data (method=”ellipse”).

  • Confidence ellipse in the Centered-Log Ratio (CLR) space: the underlying assumption of the ellipse is that the components are normally distributed. However, we know elements from the simplex have an inner structure. A better approach is to first transform the components into an unconstrained space (the CLR), and then construct the ellipse in such space (method=”ellipse-clr”).

  • Confidence ellipse in the Isometric-Log Ratio (ILR) space: analogous to the CLR-based ellipse, but built in the ILR space (method=”ellipse-ilr”).

3.5.1. Aggregative Bootstrap#

The Aggregative Bootstrap method extends any aggregative quantifier by generating confidence regions for class prevalence estimates through bootstrapping. The method is described in the paper Moreo, A., Salvati, N. An Efficient Method for Deriving Confidence Intervals in Aggregative Quantification. Learning to Quantify: Methods and Applications (LQ 2025), co-located at ECML-PKDD 2025. pp 12-33, Porto (Portugal).

This implementation is optimized for aggregative quantifiers. The bootstrap is applied to pre-classified instances, significantly speeding up training and inference. During training, bootstrap repetitions are performed only after training the classifier once. These repetitions are used to train multiple aggregation functions. During inference, bootstrap is applied over pre-classified test instances.

Aggregative Bootstrap can be applied to any aggregative quantifier. For further information, check the example provided. A minimal working example is:

import quapy as qp
from quapy.method.aggregative import PACC
from quapy.method.confidence import AggregativeBootstrap

train, test = qp.datasets.fetch_UCIMulticlassDataset('molecular').train_test

model = AggregativeBootstrap(
    PACC(),
    n_test_samples=200,
    confidence_level=0.95,
    region='ellipse-clr',  # choose among: intervals, ellipse, ellipse-clr, ellipse-ilr
    random_state=0,
)
model.fit(*train.Xy)
point_estimate, conf_region = model.predict_conf(test.X)

Here region makes the type of uncertainty region explicit. In practice, intervals is often the simplest default, while ellipse, ellipse-clr, and ellipse-ilr provide coupled regions over the simplex. If region='intervals', you can additionally set bonferroni=True to apply Bonferroni correction; this flag has no effect for ellipse-based regions.

Beyond aggregative quantifiers, Bootstrap sampling can be applied to any type of quantification method, although the speedup procedure described above is not applied.

3.5.2. Bayesian Quantification Methods#

QuaPy also provides a number of Bayesian quantifiers. While these methods are usually related to an existing point-estimation family (e.g., ACC, EMQ/MLLS, HDy, or KDEy), they differ enough in their goals and outputs to deserve a separate presentation. In particular, Bayesian quantifiers typically return a posterior mean rather than a single optimization result, expose posterior samples or confidence regions, and often require additional probabilistic inference dependencies.

The optional dependencies needed for these methods can be installed with:

pip install quapy[bayes]

3.5.2.1. BayesianCC (a Bayesian implementation of ACC)#

The BayesianCC is a variant of ACC introduced in Ziegler, A. and Czyż, P. “Bayesian quantification with black-box estimators”, arXiv (2023), which models the probabilities q = Mp using latent random variables with weak Bayesian priors, rather than plug-in probability estimates. In particular, it uses Markov Chain Monte Carlo sampling to find the values of p compatible with the observed quantities. The aggregate method returns the posterior mean and the get_prevalence_samples method can be used to find uncertainty around p estimates (conditional on the observed data and the trained classifier) and is suitable for problems in which the q = Mp matrix is nearly non-invertible.

Note that this quantification method requires val_split to be a float and installation of additional dependencies ($ pip install quapy[bayes]) needed to run Markov chain Monte Carlo sampling. Markov Chain Monte Carlo is is slower than matrix inversion methods, but is guaranteed to sample proper probability vectors, so no clipping strategies are required. An example presenting how to run the method and use posterior samples is available in examples/bayesian_quantification.py.

3.5.2.2. BayesianMAPLS (a Bayesian implementation of EMQ/MLLS)#

BayesianMAPLS is a Bayesian variant of EMQ/MLLS proposed by Ye, C. et al. (2024). Label shift estimation for class-imbalance problem: A Bayesian approach. Proceedings of the IEEE/CVF Winter Conference on Applications of Computer Vision (WACV 2024). QuaPy’s implementation is adapted from the authors’ reference code. Rather than returning a single point estimate for the class prevalence, it places a Dirichlet prior over the sought prevalence vector (in an unconstrained, Isometric-Log-Ratio-transformed space) and samples from the resulting posterior via Markov Chain Monte Carlo (using numpyro/jax), conditioned on a preliminary MAP estimate obtained via the underlying mapls routine. Like BayesianCC, its aggregate method returns the posterior mean, while predict_conf additionally returns a confidence region (intervals, ellipse, ellipse-clr, or ellipse-ilr) built from the posterior samples. For interval regions, all Bayesian methods also accept bonferroni=True to apply Bonferroni correction; this flag has no effect for ellipse-based regions.

This method requires installation of additional dependencies ($ pip install quapy[bayes]) needed to run MCMC sampling; parameters num_warmup and num_samples control the length of the chain, and prior allows choosing between a uniform Dirichlet prior (default) or one of the data-dependent priors (“map”/”map2”) proposed in the original paper.

import quapy as qp
from quapy.method._bayesian import BayesianMAPLS
from sklearn.linear_model import LogisticRegression

train, test = qp.datasets.fetch_UCIBinaryDataset('haberman').train_test

model = BayesianMAPLS(LogisticRegression())
model.fit(*train.Xy)
estim_prevalence, conf_region = model.predict_conf(test.X)

3.5.2.3. PQ: Precise Quantifier (a Bayesian implementation of HDy)#

PQ (Precise Quantifier), available at qp.method.confidence.PQ, is a Bayesian distribution-matching variant of HDy proposed in Igiraneza, A.B., Fraser, C., and Hinch, R. (2025). Estimating prevalence with precision and accuracy. Rather than matching a single test histogram against a mixture of two class-conditional histograms via a divergence measure (as HDy does), PQ places the histogram-matching problem in a Bayesian setting and samples the full posterior distribution over the (binary) prevalence value via Markov Chain Monte Carlo (using stan). Its aggregate method returns the posterior mean, while predict_conf additionally returns a confidence region built from the posterior samples (intervals, ellipse, or ellipse-clr).

PQ accepts nbins (the number of histogram bins, quantile-based by default, or uniform if fixed_bins=True), and the usual MCMC controls num_warmup, num_samples, and stan_seed. This method relies on the optional stan dependency, installed via $ pip install quapy[bayes].

import quapy as qp
from quapy.method.confidence import PQ
from sklearn.linear_model import LogisticRegression

train, test = qp.datasets.fetch_UCIBinaryDataset('haberman').train_test

model = PQ(LogisticRegression())
model.fit(*train.Xy)
estim_prevalence, conf_region = model.predict_conf(test.X)

3.5.2.4. BayesianKDEy (a Bayesian implementation of KDEyML)#

BayesianKDEy, available at qp.method._bayesian.BayesianKDEy, is a Bayesian version of KDEy proposed by Moreo et al. 2026. Instead of solving for the single prevalence vector that minimizes a divergence between the test distribution and a KDE-based mixture model (as the KDEy variants above do), BayesianKDEy places a Dirichlet prior over the prevalence vector and samples its posterior via Markov Chain Monte Carlo (using numpyro/jax), conditioned on the same KDE mixture components. Its aggregate method returns the posterior mean, while predict_conf additionally returns a confidence region built from the posterior samples.

In addition to the kernel and bandwidth hyperparameters (with the same gaussian/aitchison/ilr kernel choice, and shrinkage regularization for the latter two, available in KDEyML), BayesianKDEy exposes the usual MCMC controls: num_warmup, num_samples, mcmc_seed, a temperature for posterior calibration, and prior for choosing the Dirichlet prior ('uniform' by default, or a custom scalar/array). This method relies on the optional MCMC dependencies, installed via $ pip install quapy[bayes].

import quapy as qp
from quapy.method._bayesian import BayesianKDEy
from sklearn.linear_model import LogisticRegression

train, test = qp.datasets.fetch_UCIBinaryDataset('haberman').train_test

model = BayesianKDEy(LogisticRegression(), bandwidth=0.1)
model.fit(*train.Xy)
estim_prevalence, conf_region = model.predict_conf(test.X)