4. Quantification Methods

Quantification methods can be categorized as belonging to aggregative and non-aggregative groups. Most methods included in QuaPy at the moment are of type aggregative (though we plan to add many more methods in the near future), i.e., are methods characterized by the fact that quantification is performed as an aggregation function of the individual products of classification.

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

    @abstractmethod
    def fit(self, data: LabelledCollection): ...

    @abstractmethod
    def quantify(self, instances): ...

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 quantify are used to train the model and to provide class estimations (the reason why scikit-learn’ structure has not been adopted as is in QuaPy responds to the fact that scikit-learn’s predict function is expected to return one output for each input element –e.g., a predicted label for each instance in a sample– while in quantification the output for a sample is one single array of class prevalences). Quantifiers also extend from scikit-learn’s BaseEstimator, in order to simplify the use of set_params and get_params used in model selection.

4.1. 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: LabelledCollection, data: LabelledCollection):

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

These two functions replace the fit and quantify methods, since those come with default implementations. The fit function is provided and amounts to:

def fit(self, data: LabelledCollection, fit_classifier=True, val_split=None):
    self._check_init_parameters()
    classif_predictions = self.classifier_fit_predict(data, fit_classifier, predict_on=val_split)
    self.aggregation_fit(classif_predictions, data)
    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 quantify is provided, and amounts to:

def quantify(self, instances):
    classif_predictions = self.classify(instances)
    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 can be already fit on external data (in which case, the fit_learner argument should be set to False), or be fit by the quantifier’s fit (default).

The above patterns (in training: fit the classifier, then fit the aggregation; in test: classify, then aggregate) allows QuaPy to optimize many internal procedures. 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.

4.1.1. The Classify & Count variants

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

  • CC (Classify & Count), the simplest aggregative quantifier; one that simply relies on the label predictions of a classifier to deliver class estimates.

  • ACC (Adjusted Classify & Count), the adjusted variant of CC.

  • PCC (Probabilistic Classify & Count), the probabilistic variant of CC that relies on the soft estimations (or posterior probabilities) returned by a (probabilistic) classifier.

  • PACC (Probabilistic Adjusted Classify & Count), the adjusted variant of PCC.

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

# 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(training)
estim_prevalence = model.quantify(test.instances)

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 instance of LabelledCollection).

The specification of val_split can be postponed to the invokation of the fit method (if val_split was also set in the constructor, the one specified at fit time would prevail), e.g.:

model = qp.method.aggregative.ACC(svm)
# perform 5-fold cross validation for estimating ACC's parameters
# (overrides the default val_split=0.4 in the constructor)
model.fit(training, val_split=5)

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

model = qp.method.aggregative.PCC(svm)
model.fit(training)
estim_prevalence = model.quantify(test.instances)
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 CalibratedClassifier instance. Note that calibration can only be applied to hard classifiers when fit_learner=True; an exception will be raised otherwise.

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

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

4.1.1.1. BayesianCC (New in v0.1.9!)

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.

4.1.2. Expectation Maximization (EMQ)

The Expectation Maximization Quantifier (EMQ), also known as the SLD, 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

dataset = qp.datasets.fetch_twitter('hcr', pickle=True)

model = qp.method.aggregative.EMQ(LogisticRegression())
model.fit(dataset.training)
estim_prevalence = model.quantify(dataset.test.instances)

New in v0.1.7: EMQ now accepts two new parameters in the construction method, namely exact_train_prev which allows to use the true training prevalence as the departing prevalence estimation (default behaviour), or instead an approximation of it as suggested by Alexandari et al. (2020) (by setting exact_train_prev=False). The other parameter is recalib which allows to indicate a calibration method, among those proposed by Alexandari et al. (2020), including the Bias-Corrected Temperature Scaling, Vector Scaling, etc. See the API documentation for further details.

4.1.3. Hellinger Distance y (HDy)

Implementation of the method based on the Hellinger Distance y (HDy) proposed by González-Castro, V., Alaiz-Rodrı́guez, 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 (or in the fit method, in which case the previous value is overridden) 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 a validation set (i.e., an instance of LabelledCollection) itself.

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)
estim_prevalence = model.quantify(dataset.test.instances)

New in v0.1.7: QuaPy now provides an implementation of the generalized “Distribution Matching” approaches for multiclass, inspired by the framework of Firat (2016). One can instantiate a variant of HDy for multiclass quantification as follows:

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

New in v0.1.7: QuaPy now provides an implementation of the “DyS” framework proposed by Maletzke et al (2020) and the “SMM” method proposed by Hassan et al (2019) (thanks to Pablo González for the contributions!)

4.1.4. Threshold Optimization methods

New in v0.1.7: QuaPy now 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.

4.1.5. 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.

In order to make these models work, you would need to run the script prepare_svmperf.sh (distributed along with QuaPy) that downloads SVMperf’ source code, applies a patch that implements the quantification oriented losses, and compiles the sources.

If you want to add any custom loss, you would need to modify the source code of SVMperf in order to implement it, and assign a valid loss code to it. Then you must re-compile the whole thing and instantiate the quantifier in QuaPy as follows:

# you can either set the path to your custom svm_perf_quantification implementation
# in the environment variable, or as an argument to the constructor of ELM
qp.environ['SVMPERF_HOME'] = './path/to/svm_perf_quantification'

# assign an alias to your custom loss and the id you have assigned to it
svmperf = qp.classification.svmperf.SVMperf
svmperf.valid_losses['mycustomloss'] = 28

# instantiate the ELM method indicating the loss
model = qp.method.aggregative.ELM(loss='mycustomloss')

All ELM are binary quantifiers since they rely on SVMperf, that currently supports only binary classification. ELM variants (any binary quantifier in general) can be extended to operate in single-label scenarios trivially by adopting a “one-vs-all” strategy (as, e.g., in Gao, W. and Sebastiani, F. (2016). From classification to quantification in tweet sentiment analysis. Social Network Analysis and Mining, 6(19):1–22). In QuaPy this is possible by using the OneVsAll class.

There are two ways for instantiating this class, OneVsAllGeneric that works for any quantifier, and OneVsAllAggregative that is optimized for aggregative quantifiers. In general, you can simply use the newOneVsAll function and QuaPy will choose the more convenient of the two.

import quapy as qp
from quapy.method.aggregative import SVMQ

# load a single-label dataset (this one contains 3 classes)
dataset = qp.datasets.fetch_twitter('hcr', pickle=True)

# let qp know where svmperf is
qp.environ['SVMPERF_HOME'] = '../svm_perf_quantification'

model = newOneVsAll(SVMQ(), n_jobs=-1)  # run them on parallel
model.fit(dataset.training)
estim_prevalence = model.quantify(dataset.test.instances)

Check the examples on explicit_loss_minimization and on one versus all quantification for more details.

4.1.6. Kernel Density Estimation methods (KDEy)

New in v0.1.8: QuaPy now provides implementations for the three variants of KDE-based methods proposed in Moreo, A., González, P. and del Coz, J.J., 2023. Kernel Density Estimation for Multiclass Quantification. arXiv preprint arXiv:2401.00490.. 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]. The methods’ performance vary smoothing with smooth variations of this hyperparameter.

4.2. Composable Methods

The quapy.method.composable module allows the composition of quantification methods from loss functions and feature transformations. 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.

4.2.1. Installation

pip install --upgrade pip setuptools wheel
pip install "jax[cpu]"
pip install "qunfold @ git+https://github.com/mirkobunse/qunfold@v0.1.4"

4.2.2. 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.

ComposableQuantifier( # ordinal ACC, as proposed by Bunse et al., 2022
  TikhonovRegularized(LeastSquaresLoss(), 0.01),
  ClassTransformer(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.

4.2.3. Loss functions

Hint

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

4.2.4. Regularization functions

4.2.5. Feature transformations

Hint

The quapy.method.composable.ClassTransformer 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.

4.3. 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.

4.3.1. 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')

model = Ensemble(quantifier=ACC(LogisticRegression()), size=30, policy='ave', n_jobs=-1)
model.fit(dataset.training)
estim_prevalence = model.quantify(dataset.test.instances)

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.

4.3.2. 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)
estim_prevalence = model.quantify(dataset.test.instances)