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Tree-based machine learning methods with non-life insurance applications
Stockholm University, Faculty of Science, Department of Mathematics.
2024 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

Non-life insurance is a field which has been data-driven for a long time, with the statistical framework behind modern-day actuarial sciences laid out at the beginning of the 20th century. Problems regarding the estimation and prediction of risk are relevant to the insurance industry specifically, but also for society as a whole. The rise of machine learning methods has created a new set of tools that can be used to solve these problems. This thesis contains five individual papers, all of which are related to developing machine learning- or data-driven methods and algorithms that can be applied to, but are not limited to, non-life insurance applications.

Paper I takes an existing probabilistic model for claims reserving, the Collective Reserving Model (CRM), and replaces the linear modeling approach of the original paper with non-linear machine learning methods. The paper addresses issues in these applications and provides a framework for how to implement and evaluate machine learning models in a reserving setting. It also discusses how to implement early stopping methods given different levels of data granularity. The models are evaluated on a series of simulated data sets with promising results.

Paper II does not use a machine learning method per se but instead develops the CRM used in Paper I by adding the openness status of the claims to the dynamics and presents the CRM with Openness (CRMO), as a means to model the non-linear effects implied in Paper I. The paper presents how the model can be estimated using regression methods, and provides recursive formulas for the moments of the predicted reserve. The algorithm is evaluated in terms of accuracy on the same data set as in Paper I and shows results that are comparable to the machine learning implementations of the CRM model.

Paper III presents a new boosting algorithm called the Cyclic Gradient Boosting Machine (CGBM). The algorithm extends the classical gradient boosting machine to provide multi-dimensional function approximation. The paper shows how the CGBM can be used to estimate entire probability distributions rather than just the mean of the distribution. The paper also discusses potential problems with hyperparameter tuning in this higher-dimensional hyperparameter space and provides a dimension-wise early stopping method, which is proven useful to avoid overfitting. Numerical illustrations show accurate results on simulated and real data sets.

Paper IV is a paper that is not directly related to non-life insurance but rather to so-called decision trees used for classification and regression. The paper presents the trinary tree algorithm, which is a new way to handle missing input data for tree-based models, meant to provide a more regularized model than other suggested methods. The algorithm is benchmarked against standard methods for missing data-handling and shows promising results even for high rates of missing data.

Paper V presents a generalized linear model with non-linear effects induced by varying coefficients, with the varying coefficients estimated using the CGBM from Paper III. This is a special case of a varying coefficient model (VCM). The model that can handle highly non-linear effects while maintaining local interpretability. The paper also shows how tuning, feature selection, and evaluation of interaction effects can be simplified as compared to other VCMs. The model is evaluated on the same data set as in Paper III and shows promising results in terms of accuracy and interpretability.

Place, publisher, year, edition, pages
Stockholm: Department of Mathematics, Stockholm University , 2024. , p. 65
National Category
Probability Theory and Statistics
Research subject
Mathematical Statistics
Identifiers
URN: urn:nbn:se:su:diva-226748ISBN: 978-91-8014-677-7 (print)ISBN: 978-91-8014-678-4 (electronic)OAI: oai:DiVA.org:su-226748DiVA, id: diva2:1838795
Public defence
2024-04-12, hörsal 4, hus 2, Campus Albano, Greta Arwidssons väg 28, Stockholm, 13:00 (English)
Opponent
Supervisors
Available from: 2024-03-20 Created: 2024-02-19 Last updated: 2024-03-12Bibliographically approved
List of papers
1. Machine Learning, Regression Models, and Prediction of Claims Reserves
Open this publication in new window or tab >>Machine Learning, Regression Models, and Prediction of Claims Reserves
2020 (English)In: Casualty Actuarial Society E-Forum, Summer 2020, Arlington: Casualty Actuary Society , 2020Conference paper, Published paper (Refereed)
Abstract [en]

The current paper introduces regression based reserving models that allow for separate RBNS and IBNR reserves based on aggregated discrete time data containing information about accident years, reporting years, and payment delay, since reporting. All introduced models will be closely related to the cross-classified over-dispersed Poisson (ODP) chain-ladder model. More specifically, two types of models are introduced (i) models consisting of an explicit claim count part, where payments, in a second step, are modelled conditionally on claim counts, and (ii) models defined directly in terms of claim payments without using claim count information. Further, these general ODP models will be estimated using regression functions defined by (i) tree-based gradient boosting machines (GBM), and (ii) feed-forward neural networks (NN). This will provide us with machine learning based reserving models that have interpretable output, and that are easy to bootstrap from. In the current paper we will give a brief introduction to GBMs and NNs, including calibration and model selection. All of this is illustrated in a longer numerical simulation study, which shows the benefits that can be gained by using machine learning based reserving models. 

Place, publisher, year, edition, pages
Arlington: Casualty Actuary Society, 2020
Keywords
Claims reserving, Reported But Not Settled Claims, Incurred But Not Reported Claims, Gradient Boosting Machines, Neural Networks
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:su:diva-190526 (URN)
Conference
Casualty Actuary Society, E-Forum, Summer 2020
Available from: 2021-02-22 Created: 2021-02-22 Last updated: 2024-02-19Bibliographically approved
2. A Collective Reserving Model with Claim Openness
Open this publication in new window or tab >>A Collective Reserving Model with Claim Openness
2022 (English)In: Astin Bulletin: Actuarial Studies in Non-Life Insurance, ISSN 0515-0361, E-ISSN 1783-1350, Vol. 52, no 1, p. 117-143Article in journal (Refereed) Published
Abstract [en]

The present paper introduces a simple aggregated reserving model based on claim count and payment dynamics, which allows for claim closings and re-openings. The modelling starts off from individual Poisson process claim dynamics in discrete time, keeping track of accident year, reporting year and payment delay. This modelling approach is closely related to the one underpinning the so-called double chain-ladder model, and it allows for producing separate reported but not settled and incurred but not reported reserves. Even though the introduction of claim closings and re-openings will produce new types of dependencies, it is possible to use flexible parametrisations in terms of, for example, generalised linear models (GLM) whose parameters can be estimated based on aggregated data using quasi-likelihood theory. Moreover, it is possible to obtain interpretable and explicit moment calculations, as well as having consistency of normalised reserves when the number of contracts tend to infinity. Further, by having access to simple analytic expressions for moments, it is computationally cheap to bootstrap the mean squared error of prediction for reserves. The performance of the model is illustrated using a flexible GLM parametrisation evaluated on non-trivial simulated claims data. This numerical illustration indicates a clear improvement compared with models not taking claim closings and re-openings into account. The results are also seen to be of comparable quality with machine learning models for aggregated data not taking claim openness into account.

Keywords
Aggregated reserving, open claim dynamics, RBNS and IBNR reserves
National Category
Mathematics
Identifiers
urn:nbn:se:su:diva-201424 (URN)10.1017/asb.2021.33 (DOI)000725892800001 ()
Available from: 2022-02-01 Created: 2022-02-01 Last updated: 2024-02-19Bibliographically approved
3. On Cyclic Gradient Boosting Machines
Open this publication in new window or tab >>On Cyclic Gradient Boosting Machines
(English)Manuscript (preprint) (Other academic)
Abstract [en]

The present paper introduces a general multi-parametric gradient boosting machine (GBM) approach. The starting point is a standard univariate GBM, which is generalised to higher dimensions by using cyclic coordinate descent. This allows for different covariate dependencies in different dimensions.

Given weak assumptions, the method can be shown to converge for convex negative log-likelihood functions, which is the case, e.g., for d-parameter exponential families. Further, for this type of distribution functions it is important to design appropriate early stopping schemes. A simple dimension-wise early stopping procedure is introduced, and more advanced schemes are discussed.

The flexibility of the method is illustrated both on simulated and real data examples using different multi-parametric distributions.

Keywords
gradient boosting machines, cyclic coordinate descent, multi-parameter exponential family, early stopping
National Category
Probability Theory and Statistics
Research subject
Mathematical Statistics
Identifiers
urn:nbn:se:su:diva-226742 (URN)
Available from: 2024-02-19 Created: 2024-02-19 Last updated: 2024-02-26Bibliographically approved
4. Trinary Decision Trees for handling missing data
Open this publication in new window or tab >>Trinary Decision Trees for handling missing data
(English)Manuscript (preprint) (Other academic)
Abstract [en]

This paper introduces the Trinary decision tree, an algorithm designed to improve the handling of missing data in decision tree regressors and classifiers. Unlike other approaches, the Trinary decision tree does not assume that missing values contain any information about the response. Both theoretical calculations on estimator bias and numerical illustrations using real data sets are presented to compare its performance with established algorithms in different missing data scenarios (Missing Completely at Random (MCAR), and Informative Missingness (IM)). Notably, the Trinary tree outperforms its peers in MCAR settings, especially when data is only missing out-of-sample, while lacking behind in IM settings. A hybrid model, the TrinaryMIA tree, which combines the Trinary tree and the Missing In Attributes (MIA) approach, shows robust performance in all types of missingness. Despite the potential drawback of slower training speed, the Trinary tree offers a promising and more accurate method of handling missing data in decision tree algorithms.

National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:su:diva-226745 (URN)
Available from: 2024-02-19 Created: 2024-02-19 Last updated: 2024-02-26Bibliographically approved
5. A tree-based varying coefficient model
Open this publication in new window or tab >>A tree-based varying coefficient model
(English)Manuscript (preprint) (Other academic)
Abstract [en]

The paper introduces a tree-based varying coefficient model (VCM) where the varying coefficients are modelled using the cyclic gradient boosting machine (CGBM) from Delong et al. (2023). Modelling the coefficient functions using a CGBM allows for dimension-wise early stopping and feature importance scores. The dimension-wise early stopping not only reduces the risk of dimension-specific overfitting, but also reveals differences in model complexity across dimensions. The use of feature importance scores allows for simple feature selection and easy model interpretation. The model is evaluated on the same simulated and real data examples as those used in Richman and Wüthrich (2023), and the results show that it produces results in terms of out of sample loss that are comparable to those of their neural network-based VCM called LocalGLMnet. 

National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:su:diva-226747 (URN)
Available from: 2024-02-19 Created: 2024-02-19 Last updated: 2024-02-26Bibliographically approved

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