Gradient boosting for Dirichlet regression models

Balzer, Michael; Bergherr, Elisabeth; Hutter, Swen; Hepp, Tobias Article — Published Version Gradient boosting for Dirichlet regression models AStA Advances in Statistical Analysis Provided in Cooperation with: WZB Berlin Social Science Center Suggested Citation: Balzer, Michael; Bergherr, Elisabeth; Hutter, Swen; Hepp, Tobias (2025) : Gradient boosting for Dirichlet regression models, AStA Advances in Statistical Analysis, ISSN 1863-818X, Springer Nature, Berlin, Iss. Online first articles, https://doi.org/10.1007/s10182-025-00526-5 This Version is available at: https://hdl.handle.net/10419/318277 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0 Vol.:(0123456789) AStA Advances in Statistical Analysis https://doi.org/10.1007/s10182-025-00526-5 ORIGINAL PAPER Gradient boosting forDirichlet regression models MichaelBalzer1 · ElisabethBergherr2· SwenHutter3,4· TobiasHepp2,5 Received: 19 April 2024 / Accepted: 19 March 2025 © The Author(s) 2025 Abstract In various real-world applications, researchers often work with compositional data which appears as proportions, amounts or rates. As a framework for dealing with the unique nature of compositional data, Dirichlet regression models have been introduced. In this article, we propose a novel model-based gradient boosting approach for Dirichlet regression models embedded in the framework of generalized additive models for location, scale and shape. This approach allows for data-driven variable selection in lowas well as high-dimensional data settings. Moreover, the implementation enables the direct calculation of marginal effects for different predictor variables. Thus, it provides an alternative estimation procedure besides the wellestablished approach based on the maximum likelihood principle. After conducting detailed simulation studies to evaluate the performance of the estimation procedure regarding prediction accuracy and variable selection in lowand high-dimensional settings, we present a real-world application concerning the changes in election results in the Great Recession utilizing a large-scale European dataset. Using our proposed approach, we investigate the effect of protests on voting proportions of distinct party families while identifying important socioeconomic variables and their effect on those voting proportions via variable selection. Keywords Dirichlet regression models· Gradient boosting· Statistical learning· Great Recession· Elections 1 Introduction In many real-world applications, applied researchers have to deal with compositional data. For instance, in ecology, economics and medicine compositional data occurs as proportions, amounts or rates. The data matrix often has at least two columns and the entries are bounded in the interval (0,1) which are also sum-constrained such that each row sums up to one representing one observation. A common approach to deal with compositional data is to transform the columns such that log-ratio analysis is possible (Aitchison 2003; Maier 2014). Lately, the framework of Dirichlet Extended author information available on the last page of the article M.Balzer et al. regression models has been established as an alternative for modeling compositional data on the original scale. Specifically, Dirichlet regression models can be described in the framework of multivariate generalized additive models with the Dirichlet distribution providing the unknown parameters to be estimated through the additive predictors (McCullagh and Nelder 1989; Maier 2014). Considering that the number of additive predictors in Dirichlet regression models directly depends on the cardinality of the set of proportions to be analyzed, strategies regarding model choice and variable selection play a very important role in the application of these models. To this end, different criteria have been developed in order to obtain a quantifiable value for deciding between multiple candidate models of varying complexity. Particularly, if model estimation is based on maximum likelihood (ML), a common and popular option is to calculate the Akaike information criterion via the likelihood principle. Instead of a criteria-based approach, however, variable selection and model choice can also be achieved using regularizing constraints. Such techniques include ridge regression and the least absolute shrinkage and selection operator (lasso) as well as gradient boosting algorithms (Fahrmeir etal. 2013; Hepp etal. 2016). Originally established in the domain of machine learning, the concept behind boosting algorithms has been quickly adapted to the framework of typical statistical regression models under the names of component-wise, model-based or statistical boosting (Mayr etal. 2014). The implementation of the statistical boosting algorithm is rather straightforward and starts with the specification of a suitable loss function for the estimation problem. Then, parameter estimation reduces to iteratively fitting (often regression-type) base-learners to the negative gradient of the loss and selecting only the best performing one in each iteration. Following this iterative procedure yields interpretable results and, in combination with early stopping, the possibility for variable selection. The gradient boosting algorithm is also particularly useful in a high-dimensional setting, where the number of predictors exceeds the number of observations (Bühlmann and Hothorn 2007). Even though gradient boosting algorithms have been proposed for a variety of regression models and applications (see, for example, Robinzonov etal. (2012); Zumeta-Olaskoaga etal. (2023)), to the best of our knowledge, there does not exist a gradient boosting algorithm for the framework of Dirichlet regression models. Therefore, we aim to extend the algorithms in the gamboostLSS (Hofner etal. 2023) package for boosting generalized additive models for location, scale and shape (GAMLSS) to Dirichlet regression models in the programming language R (R Core Team 2023). To this end, we conduct simulation studies in lowand high-dimensional linear settings, where the focus is on the evaluation of the performance of estimation, variable selection and prediction. Additionally, the performance is evaluated with respect to nonlinear base-learners. Finally, we use the implemented algorithm to examine changes in voting proportions in the European Union. We take up the study by Bremer etal. (2020) who found that the level of economic protests triggered by the economic crisis had a significant impact on election results in the Great Recession where the authors mainly focus on the effect of protests on the changes in electoral loss of incumbent parties. Based on the Dirichlet regression framework, we aim to extend the scope of the analysis to examine the effect of protests on election results for all major European party families, enabling us a more differentiated view Gradient boosting forDirichlet regression models on how the Great Recession affected the structure of party systems across the European Union. Our structure for this article is as follows: We first describe the theoretical background regarding the Dirichlet distribution, regression and gradient boosting. Furthermore, we document the implementation of gradient boosting in R in Sect.2. We then perform simulation studies in Sect.3. After an in-depth description of the context, dataset and variables, we perform estimation by gradient boosting for Dirichlet regression models with the real-world dataset where we comment on the impact of protests on election results in the Great Recession in Sect.4. Finally, we provide a discussion in Sect.5. 2 Methodology 2.1 Boosting Dirichlet regression models In general, boosting can be seen as the steepest descent in a function space (Friedman 2001), that is, the algorithm aims to iteratively reduce the empirical risk using so-called base-learners. While the original algorithms utilized the same prediction function as base-learners on reweighted observations, component-wise boosting algorithms mainly rely on using regression functions for the separate input variables in the data, thereby resulting in an additive model formula (see Hofner etal. (2014) for a comprehensive overview of available base-learners). Moreover, the loss function can be expressed in terms of a negative log-likelihood (NLL) function, which allows to describe the outcome in a very similar way as generalized linear models (Hastie and Tibshirani 1990; Hastie etal. 2009; Mayr etal. 2014). The model is then estimated by starting the algorithm with an empty model and fitting all provided base-learners to the negative gradient of the specified loss function. In principle, the model is only updated by a small proportion of the best performing base-learner. In the next iteration, the gradient is reevaluated using the updated model and again fitted by all base-learners. This process is then repeated until the desired number of boosting iterations is reached. Using the L2 -loss then results in the boosting algorithm to converge to the same solution as ordinary least squares (Bühlmann and Hothorn 2007). Although there seems to be no proof for more general loss functions, empirical results using the NLL as a loss function indicate a very similar behavior with respect to the maximum likelihood solution of many generalized linear models. Appendix 2 provides a demonstration of this property for our proposed algorithm. A very convenient advantage of using boosting algorithms to estimate regression models is that by fitting separate base-learners for all input variables in the data, the algorithm can in fact be used in high-dimensional settings where the number of variables exceeds the number of observations. In this context, Bühlmann (2006) proved consistency of boosting with squared error loss for high-dimensional linear models. Moreover, a thorough statistical perspective on the boosting framework is given in Bühlmann and Hothorn (2007). Another important connection can be drawn between boosting and the lasso (Tibshirani 1996), as the stepwise model-building process performed by boosting in fact results in the estimation of a regularized regression model. M.Balzer et al. Furthermore, Hastie etal. (2007) demonstrated that boosting approximates the solution path of a strictly monotone L1 -regularized regression model for general convex loss functions. This means that the boosting solutions are optimal with respect to the socalled L1 -arc-length, which is the reason for a rather “greedy” selection behavior due to the inability to deselect a variable once it has been added to the active set of covariates, which results in a higher number of false positive selections compared to the L1 -norm optimal solutions of the lasso (Hepp etal. 2016). Where the lasso controls the amount of regularization with an explicit penalty term, regularization via boosting algorithms works by stopping the algorithm deliberately during model fitting before convergence. Therefore, the number of iterations mstop acts as the main tuning parameter that prevents overfitting and improves prediction accuracy of the overall model by controlling the bias-variance trade-off (Mayr etal. 2012a). If mstop is chosen sufficiently large, the algorithm eventually reaches a solution where additional updates have no meaningful impact on the loss (that is, ordinary least squares for L2 -loss). By stopping the updates early, sparser models and additional variable selection can be achieved. If the distribution has multiple parameters, a multiple dimension version of the gradient boosting algorithm has to be utilized since the aim is often to boost all parameters of the distribution. In this case, the default algorithm is the so-called cyclical component-wise gradient boosting algorithm, in which the grid search for identifying the optimal stopping criteria scales exponentially with the absolute number of these criteria (Schmid etal. 2010; Mayr etal. 2012a). To circumvent the problems associated with increased complexity and computational demand, the noncyclical component-wise boosting algorithm is an appropriate alternative. It has been noted that this method is more flexible in variable selection and time efficient in comparison with the cyclical algorithm (Thomas etal. 2018). For the implementation of model-based gradient boosting in the framework of Dirichlet regression models, derivations of the NLL and the negative gradient vector are necessary. Therefore, characteristics of the Dirichlet distribution are described and necessary expressions for the ingredients in the boosting algorithms are provided. Let y be a random vector representing a set of proportions. Then, yi is the proportion of item or category i∈{1, …,K} with K being the maximum number of items or categories in the dataset. In general, a random vector is called Dirichlet distributed where 𝜶=(𝛼1,…,𝛼K) is the parameter vector with 𝛼i > 0 if and only if following three assumptions hold Sets of proportions can additionally be combined, leading to a matrix D={y1,…,yN} where N is the number of observations in the respective dataset. y=(y1,…,yK)∼Dir(𝜶) y=(y 1 ,…,y K ) yi∈[0, 1] K ∑ i=1 yi=1. Gradient boosting forDirichlet regression models The sum of all 𝛼i is called the precision parameter and is given by 𝛼0 = ∑K i=1 𝛼 i with which the expectations of the proportions can be quantified via In principle, the precision parameter governs the position of the probability density in the (K−1) -simplex. More density is concentrated near the expectation vector, if the precision is high. In general, the probability density function for the common parametrization as introduced in this paper is written as Applying the logarithm to the probability density function, utilizing logarithm rules and negating yields the NLL as The negative gradient vector can be obtained as the partial derivatives of the NLL with respect to each parameter 𝛼i which is given by where 𝜓( ⋅ ) is the digamma function (Narayanan 1991; Minka 2000; Gueorguieva etal. 2008). Based on the introduced expressions, Dirichlet regression models can be implemented which rely on the notion that the random vector y=(y1,…,yK) has a conditional Dirichlet distribution with density p(y|x)=p(y|𝜶(x)) and distributional parameters 𝛼i(x) which are dependent on variables. These distributional parameters are connected to the additive predictors 𝜂 𝛼 i via a log link function g(𝛼 i (x)) = log(𝛼 i (x)) = 𝜂 𝛼 i . The corresponding response function is thus given by g−1 (𝜂𝛼 i )=exp(𝜂𝛼 i )=𝛼i(x ) which ensures restrictions on the parameter space of the variable dependent distributional parameters. The additive predictors can be expressed as in which 𝛽i0 are the intercepts, fij,j={1, …,Qi} are functional effects of independent variables x and Qi the maximum number of these functional effects for every distributional parameter (Fahrmeir etal. 2013; Maier 2014). (1) 𝔼 (yi)= 𝛼 i 𝛼0 . (2) p(y � 𝜶)= Γ �∑ K i=1𝛼i � ∏ K i=1 Γ(𝛼 i ) K � i=1 y𝛼i−1 i . (3) l (y | 𝜶)=−log p(y | 𝜶)=−log Γ (K ∑ i=1 𝛼i ) + K ∑ i=1 log Γ(𝛼i)− K ∑ i=1 (𝛼i−1)log yi . (4) − 𝜕 𝜕𝛼i l(y | 𝜶)=𝜓 (K ∑ i=1 𝛼i ) −𝜓(𝛼i)+log yi∀i∈{1, 2, …,K} , 𝜂 𝛼i=𝛽i0+ Q i ∑ j=1 fij(x), for i=1, …,K , M.Balzer et al. The construction of the model-based gradient boosting algorithms for Dirichlet regression models is based on the extension of the algorithms for GAMLSS proposed in Mayr et al. (2012a) and Thomas et al. (2018). In principle, only minor adjustments to the former algorithms are necessary to ensure proper functionality. First, the algorithms for GAMLSS are designed for a maximum number of four distributional parameters. However, in the context of Dirichlet regression models, the number of distributional parameters depends on the cardinality of the set of proportions given by the response matrix. Thus, the algorithms are extended to accommodate more than four distributional parameters. Second, in the initialization of the algorithms, offset values for the additive predictors  𝜼[0] =(𝜂 [0] 𝛼1 ,…,𝜂 [0] 𝛼K ) have to be specified. The algorithms are quite general in this context and even allow for userspecific inputs. However, in Dirichlet regression models the offset values are set as the minimum of the available proportions in the response matrix for each distributional parameter (Ronning 1989). Aside from adjustments for the number of distributional parameters and offset values, the algorithmic outline remains unchanged. The adjusted algorithms can be found in the Appendix 1. For more general details on the specific components in the algorithms, we refer to Mayr etal. (2012a) and Thomas etal. (2018). 2.2 Stability selection forboosted Dirichlet regression models In general, model choice and variable selection in model-based gradient boosting is performed via early stopping of the algorithms by means of the stopping criterion mstop . However, the choice of the stopping criterion via k-fold cross-validation, subsampling or bootstrapping tends to include too many (non-informative) variables resulting in a non-parsimonious model (Mayr etal. 2012b; Thomas etal. 2018). To mitigate the consequences of the inclusion of too many non-informative variables in the final model, Meinshausen and Bühlmann (2010) propose a generic algorithm called stability selection which can be applied to any possible variable selection technique, including model-based gradient boosting. The goal of stability selection is to gain error control for the non-informative variables. In principle, denote by V the number of non-informative (false positive) variables, then the error control is obtained by running the stability selection algorithm on multiple subsamples of the data where the so-called per-family error rate (PFER) 𝔼(V) is controlled. In all subsamples, the informative base-learners should almost always be selected by the algorithm (Meinshausen and Bühlmann 2010; Shah and Samworth 2013; Hofner etal. 2015). The stability selection algorithm for boosted Dirichlet regression models is identical to the algorithm developed for component-wise gradient boosting for the GAMLSS. The general algorithmic outline as proposed by Thomas etal. (2018) follows three steps: 1. Draw random subsets B of the data with size ⌊n∕2⌋ where n is the original number of observations in the data and fit a boosted model to each subset until the number Gradient boosting forDirichlet regression models of chosen base-learners reaches the maximum q or the iterations the stopping criterion mstop . 2. Calculate the selection frequency 𝜋 j for each base-learner defined as where  Sb is the set of base-learners chosen in iteration b. 3. Include variables in a stable set (and thus in the model) if the calculated selection frequency 𝜋 j is greater or equal to a user-specified threshold 𝜋thr An upper-bound for the PFER is then obtained via where P is the number of variables. A considerable challenge lies in the specification of q,𝜋thr and PFER since the choice is not trivial for the practitioner although only two out of three parameters have to be specified. Typically, q has to be chosen at least as large as the number of informative variables in the dataset whereas the threshold 𝜋thr should be between (0.6,0.9) (Meinshausen and Bühlmann 2010). Then, the PFER can be computed based on Eq.5. Unfortunately, the informative variables in a practical setting are generally unknown. To circumvent the problem, q should be chosen intuitively by the practitioner. Afterward, the idea is to look for a break-point in the decreasing order of values of the selection frequencies 𝜋 j . Finally, a threshold is chosen where all variables are included if the selection frequencies are larger than the break-point (Thomas etal. 2018). Alternatively, it is possible to set an upper-bound for the PFER which then is the expected value of false positive base-learners (Hofner etal. 2015). 3 Simulation study To evaluate the performance of our implemented model-based gradient boosting approach for Dirichlet regression models, simulation studies for the trivariate and septivariate Dirichlet families with ingredients as stated in Sect.2 are conducted. Specifically, the focus is on the evaluation of the performance regarding estimation, variable selection and prediction. Although the implemented Dirichlet family can handle any number of distributional parameters, the choice of the septivariate Dirichlet family in the simulation study is motivated based on the application in the case study in Sect.4. Since the functional form of the variables in the application is assumed to be strictly linear, nonlinear base-learners are evaluated using the trivariate Dirichlet family, which eases the computational burden in estimation. The learning rate is set to value of 𝜈=sl =0.1 because that is the usual practice (see, for 𝜋 j=1 B B ∑ b=1 𝕀{j∈ Sb }  Sstable ={j∶𝜋 j ≥𝜋 thr } . (5) 𝔼 (V)≤q 2 (2 𝜋 thr −1)P M.Balzer et al. example, Schmid and Hothorn (2008); Mayr etal. (2012b); Hofner etal. (2014). Regarding the choice of the updating strategy, the adjusted noncyclical algorithm is utilized due to the advantages stated in Sect.2. Nevertheless, we point out that the cyclical algorithm can also be easily utilized instead. The search for the optimal main tuning parameter vector mstop is conducted by minimizing the empirical risk via 25-fold subsampling in the linear setting and via an additional validation set with Nval =1000 observations in the nonlinear setting. In every simulation setting, a total of 100 repetitions are performed. Regarding the evaluation of the variable selection, the focus is on the criteria based on the confusion matrix (Stehman 1997). Particularly, the choice falls on the true positive rate (TPR), which is the proportion of correctly selected variables out of all true informative variables, true negative rate (TNR), which is the proportion of correctly nonselected variables out of all true non-informative variables, the false discovery rate (FDR), which is the proportion of non-informative variables in the set of all selected variables, the positive predictive value (PPV), which is the proportion of correctly selected variables out of all selected variables and the negative predictive value (NPV), which is the correctly nonselected variables out of all nonselected variables. Furthermore, for the evaluation of the predictive performance, an additional test dataset with the same number of observations as the train dataset from the same Dirichlet distribution is drawn. Regarding the criteria for the predictive performance, the NLL which is calculated on the test dataset is utilized. Furthermore, the marginal root mean squared error of prediction (RMSEP) defined as is additionally provided. For both NLL and RMSEP, lower values are always preferred. The ML estimation of the Dirichlet regression model in the low-dimensional linear setting is performed via the DirichletReg package based on theoretical result and derivations discussed in Maier (2014). The flexible boosting approaches for GAMLSS can be found in the corresponding gamboostLSS package which is extended to Dirichlet regression models. An implementation for any Dirichlet distribution via the corresponding Dirichlet family and the corresponding R code for reproducibility of all simulation studies can be found in the GitHub repository https:// github. com/ micba lz/ Diric hletR egBoo st. 3.1 Study design For the linear setting, the number of observations in the simulated datasets is fixed at N=150 . In contrast, the number of independent variables is varied between P=10 and P=300 which yields two simulation settings. Particularly, a low- ( N>P ) and high-dimensional ( N<P ) linear setting is considered in which the true predictors in the data generating process are chosen as RMSEP (Yk)= √ √ √ √ 1 Ntest Ntest ∑ i =1 (yki −yki) 2 Gradient boosting forDirichlet regression models the true functional form. Furthermore, the previously not reached extrema are now approximated much better. This suggests that application settings involving highly nonlinear functional effects of variables (for example, square-root transformations) can be effectively estimated using suitable base-learners within gradient boosting. Fig. 4 Paths of the true nonlinear effects (red) and estimated paths by the additive boosting model for the nonlinear setting with N = 200, P = 10 and 100 repetitions (colour figure online) Fig. 5 Paths of the true nonlinear effects (red) and estimated paths by the additive boosting model for the nonlinear setting with N = 1000, P = 10 and 100 repetitions (colour figure online) M.Balzer et al. 4 Case study: Changes inelection results intheGreat Recession intheEuropean union The Great Recession lasting from the years 2007 to 2009 was accompanied with severe and devastating economic downturn in countries in the European Union where macroeconomic indicators changed significantly. Particularly, the fall in aggregate demand caused a recession which in turn led to a decreased output and thus a stagnating economic growth in several countries around the world. Furthermore, the recession led to severe financial and sovereign debt crises (Ball 2014). Looking at the labor market, the recession caused a sharp increase in unemployment (van Ours 2015). Sociologists inferred that negative economic effects of involuntarily unemployment would spill over to well-being and health of the individuals (Burgard and Kalousova 2015). Macroeconomic theory suggests that policymakers can react to these exogenous shocks in order to cushion them. Although the governments and central banks bailed out industries, provided relief programs, introduced tax cuts and other fiscal stimuli, they had to pay a price in doing so. Particularly, these policies are all connected to a sharp increase in public debt due to an increase in government spending (Danziger 2013). As a response to these policies and the worsening economic performance, people took it massively to the streets, voicing their economic and political dissatisfaction. (Bremer etal. 2020; Kriesi etal. 2020; Portos and Carvalho 2022). In our application, we build on the study of Bremer etal. (2020) who aimed to show if changes in the economic conditions in the Great Recession had an effect on the protest landscape. Furthermore, the authors elaborated if the intensity of these economic effects were as strong as in the case of the electoral politics. Moreover, the aim was to establish to what extent protest politics had driven the massive electoral punishment at that time. To this end, a large-scale real-world European dataset is utilized, combining electoral results with the protest event data from the European Observatory for Political Conflict and Democracy (PolDem) (Kriesi and etal.. 2020). Although Bremer etal. (2020) already established that economic protests have an effect on elections using the dataset, the authors focus mainly on electoral loss of incumbents, that is, the difference in the voting proportion of the prime minister’s party at time t and t+1 as their response variable. In particular, they showed that poor economic performance led to electoral losses and an increase in protests using ordinary least squares estimation. In this case, incumbents were stronger punished when the number of protests was large (Bremer etal. 2020). To get a more differentiated picture on how the Great Recession has affected the structure of Europe’s party systems, it is also possible to establish the effect of independent variables on voting proportions of all party families simultaneously. We can view the voting proportions as the dependent variables and enable the possibility to work in the framework of Dirichlet regression models. This approach is much more flexible since individual changes in party families can be observed and it allows for an interpretation of marginal effects of protests on individual parties performance (Maier 2014). In order to maintain comparability between the results of Bremer etal. (2020) and our analysis, we utilize the exact same dataset. The original dataset consists Gradient boosting forDirichlet regression models of 900 observations, where each observation corresponds to the voting proportion of an individual party in an election. Moreover, 57 variables are additionally included which are information on voting proportions, number of protests, number of participants in the protest events, parties, debt change, gross domestic product (GDP) change, unemployment change as well as dummy variables that indicate if the country is in Eastern Europe, was bailed out or was in the Great Recession. Although a diverse range of variables can be utilized, the listed variables are of main interest in our analysis. In Tables4and 5, the description of dependent and independent variables can be observed. During the examination period lasting from the years 2000 until the end of 2015, we have access to 117 elections. In order to maintain a distinct temporal relationship to the Great Recession, the elections are classified as follows: Elections before October 2008 are classified as precrisis elections, the first election after October 2008— when the US-American bank Lehman Brothers collapsed—as first-crisis election and all elections afterward as later-crisis elections. The dataset includes all parties in 30 countries1 that have obtained at least three percent in voting proportions and have at least one seat in the respective countries parliament. Since the original dataset is on the level of individual parties per election, aggregation and combination of parties into party families are necessary to obtain the desired data. Particularly, the parties are classified into distinct party families on which the analysis is conducted. The classification of individual parties into party families leads to a discrepancy in the observations of the original dataset and our final dataset since in some countries more than one individual party can be assigned to the exact same party family for which the proportions are summed up. Regarding the classification of the parties in different countries, we follow the proposed ParlGov classification by Döring and Manow (2019), allowing us to differentiate the major European parties families from both right and left. This limits the results to this particular definition of the party families. Following a different definition and classification for the party families leads to a change in aggregation of the parties in different countries and thus to potentially varying results in the analysis. We do not omit any election and thus consider a holistic approach. Further preprocessing is also necessary to obtain a dataset which is suitable for the analysis. Particularly, for nonexistent party families in elections, the voting proportions are assigned to zero. Furthermore, voting proportions are originally in percent which are divided by 100 to obtain a proportional scale. Finally, row sums do not perfectly sum up to one such that we follow the normalization and transformation suggested by Smithson and Verkuilen (2006) and generalized by Maier (2014) which is defined as y ∗ i= yi(N−1)+ 1 K N . 1 Austria, Belgium, Bulgaria, Cyprus, Czech Republic, Denmark, Estonia, Finland, France, Germany, Greece, Hungary, Iceland, Ireland, Italy, Latvia, Lithuania, Luxembourg, Malta, Norway, Netherlands, Poland, Portugal, Romania, Slovakia, Slovenia, Spain, Sweden, Switzerland and the UK. M.Balzer et al. Table 4 Data type and description of dependent and independent variables Variable Type Description Vote Compositional Voting proportion for party families Protests Numeric Number of protest events in the legislative period before the election scaled by the duration of the legislative period Unemployment Numeric Change in the quarterly unemployment rate (in percent) between elections Debt Numeric Change in Government debt between elections GDP Numeric Change in annual GDP (in percent) between elections Crisis Binary Dummy variable that measures whether the election took place during the Great Recession Bailout Binary Dummy variable that measures whether a country was bailed out by the International Monetary Fund or the European Union East Binary Dummy variable that measures whether a country is in Eastern Europe Gradient boosting forDirichlet regression models In general, the results are presented with a focus on the estimation via model-based gradient boosting. Since Bremer etal. (2020) worked in a strictly linear setting in the ordinary least square estimation, simple linear models are utilized as base-learners for comparability. Furthermore, all presented variables in Table4 are included in the analysis and additionally an intercept for every distributional parameter. Regarding the learning rate, the narrative in the simulation study in Sect.3.1 is followed and a steplength of 0.1 is chosen. As for the optimization, the search for the mstop is conducted by minimizing the empirical risk via 25-fold subsampling. After obtaining the optimal stopping criterion, the following results which can be seen in Table6 are extracted. It can be observed that every 𝛼i depends on different variables, even though the model equations often share the same variables. Particularly, important for the analysis is the fact that the Protests variable is almost always included except in the case of 𝛼1 , that is, the social democrats. In contrast to the results presented in Bremer etal. (2020), the distributional parameters in the Dirichlet regression framework 𝛼i depend on socioeconomic variables, namely GDP,Unemployment and Debt. Thus, the results presented in this case study underline the difference in the estimation framework with additional variable selection technique. Unfortunately, the model equations imply practical restrictions with regard to the interpretation of the effects of ceteris paribus changes in variables. Particularly, since Table 5 Descriptive statistics of voting proportions for party families before normalization and transformation Party family Zero count Median Interquartile range 𝛼1 Social Democratic 0 0.268 0.156 𝛼2 Conservative 1 0.343 0.2419 𝛼3 Liberals 29 0.098 0.221 𝛼4 Greens 66 0 0.073 𝛼5 Radical left 45 0.0399 0.0965 𝛼6 Radical right 49 0.0437 0.139 𝛼7 Others 5 0.0663 0.07 Table 6 Estimation results via model-based gradient boosting for Dirichlet regression models in the case study 𝛼1 𝛼2 𝛼3 𝛼4 𝛼5 𝛼6 𝛼7 Intercept 0.84 0.78 − 0.22 − 0.65 − 0.59 − 0.67 − 0.24 Protests 0.011 − 0.010 − 0.032 0.029 0.0073 0.026 Bailout − 0.24 Crisis − 0.049 0.024 0.1 Unemployment − 0.0069 0.0029 Debt − 0.008 0.0039 − 0.0023 GDP 0.0062 East 0.2 − 0.439 − 0.484 0.084 M.Balzer et al. the voting proportions are not directly modeled, the effect of changes in the variables can only be interpreted on the distributional parameters 𝛼i (Maier 2014). Given the results of the boosted Dirichlet regression model, the interpretation of the coefficients is now straightforward. In general, low coefficient values for most of the variables for all distributional parameters are observed. Regarding the absolute magnitude, higher coefficients can be observed for the variables Bailout and East, whereas socioeconomic variables have rather low positive to negative coefficient values. For instance, if Protests increases by one scaled event ceteris paribus then the 𝛼2 increases by 100 ⋅(exp(𝛽 (𝛼 2 ) Protests −1)=100 ⋅(exp(0.011 −1)≈ 1.106 percent on average. Similar interpretations can be made for every distributional parameter. Thus, with an increasing number of economic protests the distributional parameter 𝛼2 increases in absolute value. In contrast, 𝛼3 will decrease due to the negative sign of the coefficient when the number of economic protests increases. Although implications can be obtained through this interpretation, that is, the expectation of the proportions of the greens will decrease with an increasing number of protests, it would be convenient to obtain a method to get a general overview of changes in proportions for all party families directly. An alternative approach relies on generating a test dataset for eligible values of Protests or any other variable of interest. Afterward, marginal predictions for distributional parameters 𝛼i are computed. The general idea for the computation of the marginal effects is provided by the gamboostLSS package where the effects of a corresponding base-learner for a variable of interest are returned while the other continuous variables are fixed at their mean and categorical variables at their mode. Utilizing the expression given by Eq.1, the marginal effects of the variables on the expectations of the proportions for every observation in the test dataset can be quantified. In Fig.6, the progress of the expectations of the voting proportions for increasing number of economic protests can be observed. Particularly, the expectations for the greens and the liberals are declining almost linearly and strongly if the number of economic protests increases. Thus, the model suggests that both party families have not profited from citizens’ voicing their economic grievances in the streets. In contrast, the radical left in particular but also other parties are gaining in expected proportions as the number of protests increases. The finding underlines the emergence of anti-austerity parties from the left during the Great Recession (DellaPorta etal. 2017). However, the extent of economic protests has hardly any effect on the electoral performance of the conservative and the radical right. Additionally, a negative effect is observable in the expected proportions of the social democrats. This is surprising since the coefficient for the distributional parameter is nonexistent such that the marginal prediction is always constant. The behavior can be best explained by looking at Eq.1. Thus, if 𝛼1 remains constant and 𝛼0 increases with the number of economic protests, the expected proportions of social democrats decreases. To provide an in-depth analysis of the marginal effect of socioeconomic variables, graphical results for Debt and Unemployment are additionally presented. Based on Fig.7, we concur that liberals lose in expected voting proportions if Gradient boosting forDirichlet regression models the percent change in debt is positive and high, whereas the radical lefts are winning in expected proportions with an increase in debt change. Regarding other parties, such a conclusive result cannot be given. Particularly, the expected proportions increase slightly for social democrats, conservatives, greens and others with increasing percentage change in debt. For the radical right, the expected proportion slightly decreases. Therefore, the interpretation of marginal effects has to be performed carefully if only a few distributional parameters 𝛼i depend on a variable of interest. Since for four parameters, Debt is not included in the model Fig. 6 Marginal effect of the number of economic protests on the expectations of the voting proportions obtained via the boosted Dirichlet regression model Fig. 7 Marginal effect of debt change in percent on the expectations of the voting proportions obtained via boosted Dirichlet regression model M.Balzer et al. equations, the marginal predictions remain constant regardless of the magnitude in debt change. In turn, this behavior amplifies the consequences of Eq.1 since the absolute increase in 𝛼0 remains overall low. A similar behavior can be observed in the marginal effects of unemployment change where even less distributional parameters depend on the unemployment change variable. In Fig.8, the expected proportions for almost all party families remain constant. Small marginal effects can be observed in the expected voting proportions of the radical rights and others. These effects can be mainly traced back to the corresponding distributional parameters since these depend on the unemployment change variable. Particularly, radical rights lose whereas other parties gain in expected proportions as the percent change in unemployment increases. Although the model indicates a shift of voters toward other parties, the magnitude of change in expected proportions remains generally low. Summing up the obtained results, following conclusion can be drawn from the boosted Dirichlet regression model: First, distinct changes in voting behavior of citizens can be observed compared to prerecession elections. Particularly, the model suggests that an increase in the number of economic protests is accompanied by change of expected voting proportions for party families. Specifically, people are turning their backs on green and liberal parties when the number of economic protests is high, whereas radical left and other parties are profiting electorally under these conditions. Moreover, a shift to the right in voting behavior for different values of economic protests cannot be observed, implying that radical rights and conservatives fail to provide attractive alternatives for situations where the number of economic protests is high. Second, the marginal effects of debt and unemployment change are considerably low over all expected voting proportions. Nevertheless, liberals tend to lose in expected proportions indicating a shift of voters to radical left parties if percent change in debt increases. Once again radical lefts seem to provide desired alternatives for an increasing debt burden in the European Union. Since strong positive increases in absolute debt were observed in the Great Recession, this result implies that people are becoming Fig. 8 Marginal effect of unemployment change in percent on the expectations of the voting proportions obtained via the boosted Dirichlet regression model Gradient boosting forDirichlet regression models increasingly left-leaning when faced with such realities. Likewise, high numbers of unemployment were observed in the Great Recession. Although the effect of unemployment is overall low, voters from radical right parties tend to deflect to other parties in such situations. More insight into the marginal effects of other socioeconomic variables recovered from the boosted Dirichlet regression model can be observed in Appendix 4.1. In the simulation study, we observe that the gradient boosting algorithm exhibits high FDR values when relying solely on cross-validation to determine the optimal stopping criterion. To examine how combining gradient boosting with stability selection affects performance in the case study, we present additional insights in Appendix 4.2. The final model suggested by the chosen parameter combination further emphasizes the importance of the number of protests, while other variables with less impact such as unemployment do not reach the required stability threshold. Nevertheless, it should be noted that, as with statistical hypothesis tests, the noninclusion of a variable in the model should not be taken as evidence that there is no marginal effect of this variable on the change in the expected voting proportions at all. 5 Discussion Our key findings and the main contributions of this paper are: (a) Gradient boosting can be extended to Dirichlet regression by an adjustment of the algorithms by Mayr etal. (2012a) and Thomas etal. (2018). (b) The implementation is performed by extending the package for boosting GAMLSS called gamboostLSS (Hofner etal. 2023). The extension is implemented by writing a new Dirichlet family which can be specified in the estimation procedure. (c) In the simulation studies, the estimation procedure is accompanied with high TPR values in the low-dimensional linear setting. The true coefficients are estimated with a high degree of accuracy and the predictive performance is generally better in the gradient boosting approach than in the ML Dirichlet regression. For the high-dimensional linear setting, the gradient boosting estimation procedure is also feasible. Furthermore, nonlinear complex baselearners can also be utilized in application settings since they estimate the true form of nonlinear effects quite well. (d) In summary, practitioners and applied statisticians have an alternative approach for flexible estimation of Dirichlet regression models. This allows for data-driven variable selection which leads to sparser and much more interpretable models. Although our analysis focuses on the trivariate and septivariate Dirichlet distribution, any Dirichlet distribution is implemented, thereby allowing for a potential application in real-world settings with an arbitrarily number of categories and thus distributional parameters. (e) Additionally, the new implemented method is applied in a real-world application setting. Our proposed estimation procedure allows for a more differentiated view on the electoral consequences of the Great Recession. However, we want to indicate that further research is necessary in this application setting and we only viewed a limiting case. Particularly, different economic questions can be investigated using the dataset and gradient boosting for Dirichlet regression. Naturally, there are potential improvements and extensions beyond the scope of this paper. Although simulation studies have been conducted in lowand highdimensional settings as well as for the estimation nonlinear effects, the scope of M.Balzer et al. our settings is of course by no means exhaustive. Further interesting scenarios may include correlated noise variables or different types of base-learners, for example, spatial or random effects. Moreover, the implementation of the gradient boosting algorithm is based on the so-called common parametrization of the Dirichlet distribution and, as a consequence, the effect of variables on the expectations of dependent proportions cannot be interpreted directly (Maier 2014). This becomes particularly visible in the application. Although we could conclude some results due to marginal effects, a direct quantifiable value for any change in protests on the expectation of voting proportions for different party families is still missing. Looking at the average selection rates, our algorithm reveals a relatively high FDR, indicating that many non-informative variables with minor importance are selected. The FDR is even higher when the optimal stopping criterion is found via tenfold crossvalidation instead of 25-fold subsampling as can be seen in appendix 3.1.1, 3.1.2 and 3.1.3. Therefore, we conducted additional simulations to assess the applicability of stability selection which allows control over the amount of false positive selections (Meinshausen and Bühlmann 2010; Hofner etal. 2015). Using permutation techniques also allows the computation of p values for individual base-learners without the error control provided by stability selection Hepp etal. (2019). Alternatively, implementing a recently proposed strategy to deselect base-learners with low impact in order to obtain sparser models can be considered (Strömer etal. 2022). Furthermore, the potentially increasing number of distributional parameters due to varying Dirichlet distributions leads to an increased demand in computational resources for running the algorithms. Therefore, alternatives for early stopping could be considered. Probing is already such an approach to early stopping for univariate location models, where the algorithm is stopped when the first randomly shuffled version of the observed variables is added to the dataset, thereby also resulting in relatively sparse solutions (Thomas etal. 2017). An important aspect noted in boosting algorithms for distributional regression models is that using the same fixed step-length for every predictor function may result in imbalances that prevent certain parameters from being updated. Suggested strategies involve scaling the outcome variables or the negative gradient vector (Hofner etal. 2016; Strömer etal. 2023 or using adaptive step-lengths Zhang etal. (2022). However, since the scale of the gradients for all 𝛼i in the Dirichlet regression framework remains identical regardless of i, these issues should be of minor concern for the implementation discussed in this article. Thus, the next goal would be to implement a model-based gradient boosting algorithm for the alternative parametrization of the Dirichlet distribution. This approach enables the direct modeling of expected voting proportions using a logit link function (Ferrari and Cribari-Neto 2004; Maier 2014). Regarding the case study, further research is needed to examine the drivers of voting proportions during crises in countries outside the European Union. Additionally, modeling random effects remains a key area of interest. An updated dataset covering more recent recessions could provide new insights into protest effects, allowing for meaningful comparisons with previous findings. Moreover, reconsidering definitions for party family classifications could refine the analysis. Beyond the sociology context, the newly implemented gradient boosting method for Dirichlet regression models has broad applicability in fields such as economics, medicine and ecology. We encourage practitioners and applied Gradient boosting forDirichlet regression models Table 12 Results for the average selection rates in the highdimensional linear setting (N = 150, P = 300) for 50 repetitions and trivariate Dirichlet family Reported are the true positive (TPR) false discovery (FDR), true negative rate (TNR), positive predictive (PPV) and negative predictive value (NPV) TPR (%) TNR (%) FDR (%) PPV (%) NPV (%) 𝛼1 100 94.53 83.41 16.59 100 𝛼2 100 94.49 84.16 15.84 100 𝛼3 99.33 96.07 77.47 22.53 99.99 Table 13 Results for the predictive performance on independent test dataset in the high-dimensional linear setting (N = 150, P = 300) for 50 repetitions and trivariate Dirichlet family Reported are the root mean squared error of prediction (RMSEP) and negative log-likelihood (NLL) Boosting Dirichlet regression RMSEP( Y1 ) 0.015135604 – RMSEP( Y2 ) 0.010652941 – RMSEP( Y3 ) 0.009215892 – NLL − 408.3796 – Fig. 11 Results for the estimated linear effects of the first ten variables in the high-dimensional linear setting (N = 150, P = 300) with trivariate Dirichlet family for 50 repetitions. Horizontal red lines represent the true values (colour figure online) M.Balzer et al. (A) Balanced case: Every 𝛼i depends on two informative variables but all of them are shared. In total, we have 7⋅2=14 informative variables. This leads to (B) Unbalanced case: Every 𝛼i depends on two informative variables but none of them are shared. In total, we have 7⋅2=14 informative variables. This corresponds to: log (𝛼1)=𝛽 (1) 1X1−𝛽 (1) 2X 2 log (𝛼2)=𝛽(2) 1X1+𝛽(2) 2X 2 log (𝛼3)=𝛽(3) 1X1−𝛽(3) 2X 2 log (𝛼4)=𝛽(4) 1X1−𝛽(4) 2X 2 log (𝛼5)=𝛽(5) 1X1+𝛽(5) 2X 2 log (𝛼6)=𝛽(6) 1X1−𝛽(6) 2X 2 log (𝛼 7 )=𝛽(7) 1 X 1 +𝛽(7) 2 X 2 Fig. 12 Results for the investigation of the regularization behavior with changing signs of the true coefficients in the trivariate Dirichlet family for 𝐦stop = 𝟏𝟓𝟎 . Red line represents the central coefficients of interest (colour figure online) Gradient boosting forDirichlet regression models Table 14 Results for the average selection rates in the lowdimensional linear setting (N = 150, P = 49) for 50 repetitions and septivariate Dirichlet family Reported are the true positive (TPR) false discovery (FDR), true negative rate (TNR), positive predictive (PPV) and negative predictive value (NPV) TPR (%) TNR (%) FDR (%) PPV (%) NPV (%) 𝛼1 100 81.61 72.56 27.44 100 𝛼2 100 95.26 37.14 62.86 100 𝛼3 94.00 94.39 42.03 57.97 99.60 𝛼4 100 95.57 34.30 65.70 100 𝛼5 95.33 93.39 47.00 53.00 99.68 𝛼6 92.67 96.87 28.68 71.32 99.52 𝛼7 100 81.00 72.81 27.19 100 Table 15 Results for the predictive performance on independent test dataset in the low-dimensional linear setting (N = 150, P = 49) for 50 repetitions and septivaraite Dirichlet family Reported are the root mean squared error of prediction (RMSEP) and negative log-likelihood (NLL) Boosting Dirichlet regression RMSEP( Y1 ) 0.06856661 0.09017981 RMSEP( Y2 ) 0.04404223 0.07814164 RMSEP( Y3 ) 0.03376675 0.04628070 RMSEP( Y4 ) 0.03275246 0.05141709 RMSEP( Y5 ) 0.04976701 0.07501433 RMSEP( Y6 ) 0.02130379 0.04965494 RMSEP( Y7 ) 0.07476324 0.09988373 NLL − 3385.321 83745487845 Fig. 13 Results for the estimated linear effects for the (N = 150, P = 49) linear setting for 50 repetitions and septivariate Dirichlet family. Horizontal red lines represent the true values (colour figure online) M.Balzer et al. All variables are independently drawn from the uniform distribution U(0,1). The criteria for the performance evaluation remain identical to the main text. For the threshold 𝜋thr , varying values between 0.55 and 0.99 in 0.01 steps are considered. For the maximum number of included base-learners, three different values, namely q∈{15, 25, 50} are chosen. Finally, the number of noise variables is varied between P=50, 100 and 150. Starting with the balanced case (Fig. 14), results are reported for the different combinations of P and q. Generally, the results suggest that the number of TP is high if the threshold is low and gradually decreases as the threshold increases. Unfortunately, the number of informative variables of 14 is not attained in any setting. The closest case achieving a number of TP of 13 is q=50 and P=50 which is rather unlikely in practical applications. However, the number of FP is low or outright zero for most of the considered simulations. For instance, only for an increasing number of noise variables P in combination with a huge number of base-learners q as well a low values of 𝜋thr , the number of FP achieves a maximum value of two. Thus, the setting attaining the highest number of FP is P=150 and q=50 which is a high-dimensional case in combination with a low threshold. Afterward, with an increasing value of 𝜋thr the number of FP reaches zero starting from a threshold of 𝜋thr =0.7 . The results for the unbalanced case (Fig.15) are generally in line with the results for the balanced case, although small changes can be noted. Particularly, the number of TP is higher across every simulation setting compared to the balanced case. Furthermore, even with an increasing threshold 𝜋thr , the number of TP decreases slowly, sometimes even remaining constant (see, for example, q=50 and P=15 ). Additionally, the closest case achieving a number of TP of 13 is again q=50 and P=50 . Regarding the FP, the results suggest overall low to outright zero inclusion of additional non-informative variable. Particularly, even for large value of noise variables and base-learners in combination with low values for the threshold, the FP remains zero. Only for the edge case of q=50 and P=50 , the maximum number of FP of two is attained. log (𝛼1)=𝛽 (1) 1X1−𝛽 (1) 2X2 log (𝛼2)=𝛽(2) 1X3+𝛽(2) 2X4 log (𝛼3)=𝛽(3) 1X5−𝛽(3) 2X6 log (𝛼4)=𝛽(4) 1X7−𝛽(4) 2X8 log (𝛼5)=𝛽(5) 1X9+𝛽(5) 2X10 log (𝛼6)=𝛽(6) 1X11 −𝛽(6) 2X 12 log (𝛼 7 )=𝛽(7) 1 X 13 +𝛽(7) 2 X 14 Gradient boosting forDirichlet regression models Appendix4: Further results inthecase study Appendix4.1: Marginal effects ofsocioeconomic variables See Figs.16, 17, 18 and 19. Appendix4.2: Coefficients andselection frequencies instability selection In the main text, we present the results for the application setting where the optimal stopping is found via 25-fold subsampling. However, as evidenced in Sect.3.2.1 model-based gradient boosting with the stopping criterion found by means of crossvalidation yields a high FDR. To improve the FDR, we propose to utilize stability selection which shows considerable improvement when applied to the artificial data in the simulation study. Consequently, the results in the application might also include many non-informative variables. Therefore, we investigate the results of the case study under consideration of stability selection. To this end, we set the maximum number of base-learners q=40 and the PFER =5 . However, since the specific choice of the three parameters q, 𝜋thr and PFER is not trivial and depends on the practitioner, exploring different combinations is always an option. The maximum selection frequencies of the first 15 variables for the specified parameters can be seen in Fig.20. In general, the results indicate that the number of chosen variables has decreased substantially in comparison with the model obtained by cross-validation. Particularly, the Protests variable is still present in the final model for the conservatives, greens, radical lefts and others, whereas most of the socioeconomic variables have been dropped. Nevertheless, the GDP variable for conservatives still remains in the model. Furthermore, the binary dummy variable East is included for greens and radical Lefts. The updated results for the coefficients can be now seen in Table16. Fig. 14 Stability selection in the balanced Case (Scenario A) with septivariate Dirichlet family. Black solid lines represent the progress of the number of true positive variables (TP). Blue dashed lines represent the progress of the number of false positive variables (FP) (colour figure online) M.Balzer et al. In principle, these results are fairly unsurprising because the presented marginal effects already hint toward variables which have been removed by stability selection. Particularly, the leftover coefficients for Protests are exhibiting large changes in the voting proportions as evidenced by the marginal effects in Fig.6. Additionally, only the conservatives exhibit substantial increases in voting proportions as the percent change of GDP increases in Fig.16. Finally, the largest changes in marginal effects for the East variable are seen in greens and radical lefts as indicated by Fig.19. Fig. 15 Stability selection in the unbalanced case (Scenario B) with septivariate Dirichlet family. Black solid lines represent the progress of the number of true positive variables (TP). Blue dashed lines represent the progress of the number of false positive variables (FP) (colour figure online) Fig. 16 Marginal effect of GDP change in percent on the expectations of the voting proportions obtained via the boosted Dirichlet regression model Gradient boosting forDirichlet regression models Fig. 17 Marginal effect of crisis elections on the expectations of the voting proportions obtained via the boosted Dirichlet regression model Fig. 18 Marginal effect of bailouts on the expectations of the voting proportions obtained via the boosted Dirichlet regression model M.Balzer et al. Fig. 19 Marginal effect if countries are in Eastern Europe on the expectations of the voting proportions obtained via the boosted Dirichlet regression model Fig. 20 Maximum selection frequencies of the first 15 variables after stability selection under parameter choices of q = 40 and PFER =5 Gradient boosting forDirichlet regression models Acknowledgements We greatly thank Benjamin Hofner for his support in integrating our algorithms into the gamboostLSS package. The work on this article was supported by the Volkswagen Foundation (Freigeist Fellowship) and Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) within projects 51701299 and 492988838. Funding Open Access funding enabled and organized by Projekt DEAL. Declarations Conflict of interest The authors declare that they have no conflict of interest. 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