Time Series Modelling

The analysis and modeling of time series is of the utmost importance in various fields of application. This Special Issue is a collection of articles on a wide range of topics, covering stochastic models for time series as well as methods for their analysis, univariate and multivariate time series, real-valued and discrete-valued time series, applications of time series methods to forecasting and statistical process control, and software implementations of methods and models for time series. The proposed approaches and concepts are thoroughly discussed and illustrated with several real-world data examples

Modeling count time series following generalized linear models

Count time series are found in many different applications, e.g. from medicine, finance or industry, and have received increasing attention in the last two decades. The class of count time series following generalized linear models is very flexible and can describe serial correlation in a parsimonious way. The conditional mean of the observed process is linked to its past values, to past observations and to potential covariate effects. In this thesis we give a comprehensive formulation of this model class. We consider models with the identity and with the logarithmic link function. The conditional distribution can be Poisson or Negative Binomial. An important special case of this class is the so-called INGARCH model and its log-linear extension.A key contribution of this thesis is the R package tscount which provides likelihood-based estimation methods for analysis and modeling of count time series based on generalized linear models. The package includes methods for model fitting and assessment, prediction and intervention analysis. This thesis summarizes the theoretical background of these methods. It gives details on the implementation of the package and provides simulation results for models which have not been studied theoretically before. The usage of the package is illustrated by two data examples. Additionally, we provide a review of R packages which can be used for count time series analysis. A detailed comparison of tscount to those packages demonstrates that tscount is an important contribution which extends and complements existing software. A thematic focus of this thesis is the treatment of all kinds of unusual effects influencing the ordinary pattern of the data. This includes structural changes and different forms of outliers one is faced with in many time series. Our first study on this topic is concerned with retrospective detection of such changes. We analyze different approaches for modeling such intervention effects in count time series based on INGARCH models. Other authors treated a model where an intervention affects the non-observable underlying mean process at the time point of its occurrence and additionally the whole process thereafter via its dynamics. As an alternative, we consider a model where an intervention directly affects the observation at its occurrence, but not the underlying mean, and then also enters the dynamics of the process. While the former definition describes an internal change of the system, the latter can be understood as an external effect on the observations due to e.g. immigration. For our alternative model we develop conditional likelihood estimation and, based on this, develop tests and detection procedures for intervention effects. Both models are compared analytically and using simulated and real data examples. The procedures for our new model work reliably and we find some robustness against misspecification of the intervention model. The aforementioned methods are applied after the complete time series has been observed. In another study we investigate the prospective detection of structural changes, i.e. in real time. For example in public health, surveillance of infectious diseases aims at recognizing outbreaks of epidemics with only short time delays in order to take adequate action promptly. We point out that serial dependence is present in many infectious disease time series. Nevertheless it is still ignored by many procedures used for infectious disease surveillance. Using historical data, we design a prediction-based monitoring procedure for count time series following generalized linear models. We illustrate benefits but also pitfalls of using dependence models for monitoring.Moreover, we briefly review the literature on model selection, robust estimation and robust prediction for count time series. We also make a first study on robust model identification using robust estimators of the (partial) autocorrelation

Application of Machine Learning Algorithms to Actuarial Ratemaking within Property and Casualty Insurance

A scientific pricing assessment is essential for maintaining viable customer relationship management solutions (CRM) for various stakeholders including consumers, insurance intermediaries, and insurers. The thesis aims to examine research problems neighboring the ratemaking process, including relaxing the conventional loss model assumption of homogeneity and independence. The thesis identified three major research scopes within multiperil insurance settings: heterogeneity in consumer behaviour on pricing decisions, loss trending under non-linearity and temporal dependencies, and loss modelling in presence of inflationary pressure. Heterogeneous consumers on pricing decisions were examined using demand and loyalty-based strategy. A hybrid decision tree classification framework is implemented, that includes semi-supervised learning model, variable selection technique, and partitioning approach with different treatment effects in order to achieve adequate risk profiling. Also, the thesis explored a supervised tree learning mechanism under highly imbalanced overlap classes and having a non-linear response-predictors relationship. The two-phase classification framework is applied to an owner’s occupied property portfolio from a personal insurance brokerage powered by a digital platform within the Canadian market. The hybrid three-phase tree algorithm, which includes conditional inference trees, random forest wrapped by the Boruta algorithm, and model-based recursive partitioning under a multinomial generalized linear model, is proposed to study the price sensitivity ranking of digital consumers. The empirical results suggest a well-defined segmentation of digital consumers with differential price sensitivity. Further, with highly imbalanced and overlapped classes, the resampling technique was modelled together with the decision tree algorithm, providing a more scientific approach to overcome classification problems than the traditional multinomial regression. The resulting segmentation was able to identify the high-sensitivity consumers group, where premium rate reductions are recommended to reduce the churn rate. Consumers are classified as an insensitive group for which the price strategy to increase the premium rate is expected to have a slight impact on the closing ratio and retention rate. Insurance loss incurred greatly exhibits abnormal characteristics such as temporal dependence, nonlinear relationship between dependent and independent variables, seasonal variation, and mixture distribution resulting from the implicit claim inflation component. With such abnormal variable characteristics, the severity and frequency components may exhibit an altered trending pattern, that changes over time and never repeats. This could have a profound impact on the experience rating model, where the estimates of the pure premium and the rate relativity of tariff class are likely to be under or over-estimated. A discussion of the pros and cons of the conventional loss trending approach leads to an alternative framework for the loss cost structure. The conventional pure premium is further split into base severity and severity deflator random variables using a do(·) operator within causal inference. The components are separately modelled based on different time basis predictors using the semiparametric generalized additive model (GAM) with a spline curve. To maximize the claim inflation calendar year effect and improve the efficiency of severity trending, this thesis refines the claim inflation estimation by adapting Taylor’s [86] separation method that estimates the inflation index from a loss development triangle. In the second phase of developing the severity trend model, we integrated both the base severity and severity deflator under a new generalized mechanism known as Discount, Model, and Trend (DMT). The two-phase modelling was built to overcome the mixture distribution effect on final trend estimates. A simulation study constructed using the claims paid development triangle from a Canadian Insurtech broker’s houseowners/householders portfolio was used in a severity trend movement prediction analysis. We discovered that the conventional framework understated the severity trends more than the separation cum DMT framework. GAM provides a flexible and effective mechanism for modelling nonlinear time series in studies of the frequency loss trend. However, GAM assumes that residuals are independent and identically distributed (iid), while frequency loss time series can be correlated in adjacent time points. This thesis introduces a new model called Generalized Additive Model with Seasonal Autoregressive term (GAMSAR) that accounts for temporal dependency and seasonal variation in order to improve prediction confidence intervals. Parameters of the GAMSAR model are estimated by maximum partial likelihood using a modified Newton’s method developed by Yang et al. [97], and the goodness-of-fit between GAM, and GAMSAR is demonstrated using a simulation study. Simulation results show that the bias of the mean estimates from GAM differs greatly from their true value. The proposed GAMSAR model shows to be superior, especially in the presence of seasonal variation. Further, a comparison study is conducted between GAMSAR and Generalized Additive Model with Autoregressive term (GAMAR) developed by Yang et al. [97], and the coverage rate of 95% confidence interval confirms that the GAMSAR model has the ability to incorporate the nonlinear trend effects as well as capture the serial correlation between the observations. In the empirical analysis, a claim dataset of personal property insurance obtained from digital brokers in Canada is used to show that the GAMSAR(1)12 captures the periodic dependence structure of the data precisely compared to standard regression models. The proposed frequency severity trend models support the thesis’s goal of establishing a scientific approach to pricing that is robust under different trending processes

Fractional Calculus and the Future of Science

Newton foresaw the limitations of geometry’s description of planetary behavior and developed fluxions (differentials) as the new language for celestial mechanics and as the way to implement his laws of mechanics. Two hundred years later Mandelbrot introduced the notion of fractals into the scientific lexicon of geometry, dynamics, and statistics and in so doing suggested ways to see beyond the limitations of Newton’s laws. Mandelbrot’s mathematical essays suggest how fractals may lead to the understanding of turbulence, viscoelasticity, and ultimately to end of dominance of the Newton’s macroscopic world view.Fractional Calculus and the Future of Science examines the nexus of these two game-changing contributions to our scientific understanding of the world. It addresses how non-integer differential equations replace Newton’s laws to describe the many guises of complexity, most of which lay beyond Newton’s experience, and many had even eluded Mandelbrot’s powerful intuition. The book’s authors look behind the mathematics and examine what must be true about a phenomenon’s behavior to justify the replacement of an integer-order with a noninteger-order (fractional) derivative. This window into the future of specific science disciplines using the fractional calculus lens suggests how what is seen entails a difference in scientific thinking and understanding