Stochastic Projection Research Articles

Producing the Dutch and Belgian mortality projections: a stochastic multi-population standard

The quantification of longevity risk in a systematic way requires statistically sound forecasts of mortality rates and their corresponding uncertainty. Actuarial associations have a long history and continue to play an important role in the development, application and dispersion of mortality projections for the countries they represent. This paper gives an in depth presentation and discussion of the mortality projections as published by the Dutch (in 2014) and Belgian (in 2015) actuarial associations. The goal of these institutions was to publish a stochastic mortality projection model in line with both rigorous standards of state-of-the-art academic work as well as the requirements of practical work such as robustness and transparency. Constructed by a team of authors from both academia and practice, the developed mortality projection standard is a Li and Lee type multi-population model. To project mortality, a global Western European trend and a country-specific deviation from this trend are jointly modelled with a bivariate time series model. We motivate and document all choices made in the model specification, calibration and forecasting process as well as the model selection strategy. We show the model fit and mortality projections and illustrate the use of the model in several pension-related applications.

Probabilistic Population Projections Using Monte Carlo Methods: Spain, 2016-2066

In this paper we implement a methodology that generates population projections combining the cohort-component method with Monte Carlo simulation of its main inputs: fertility rates by age, survival probabilities by age and gender and immigration flows by age and gender. The Monte Carlo simulation is based on a parsimonious parameterization of the corresponding curves and a multivariate time series model (Vector of Autoregressions, VAR) used to simulate future scenarios. The model considers separately the dynamics of the national and foreign population, allowing for a detailed analysis of the migration phenomenon. The paper includes a detailed case study based on the Spanish population projections for the period 2016-2066. Both our deterministic and stochastic demographic projections are fully replicable as the methodology, coded in MATLAB and Visual Basic, is freely accessible and all the assumptions are made explicit. The methodology is coded in MATLAB and Visual Basic, making explicit and replicable all the data and assumptions used to generate the population projections, deterministic as well as stochastic. The paper includes a detailed case study based on the Spanish population projections, 2016-2065.

An imputation/copula-based stochastic individual tree growth model\xa0for mixed species Acadian forests: a case study using the Nova Scotia permanent sample plot network

BackgroundA novel approach to modelling individual tree growth dynamics is proposed. The approach combines multiple imputation and copula sampling to produce a stochastic individual tree growth and yield projection system.Methods The Nova Scotia, Canada permanent sample plot network is used as a case study to develop and test the modelling approach. Predictions from this model are compared to predictions from the Acadian variant of the Forest Vegetation Simulator, a widely used statistical individual tree growth and yield model.ResultsDiameter and height growth rates were predicted with error rates consistent with those produced using statistical models. Mortality and ingrowth error rates were higher than those observed for diameter and height, but also were within the bounds produced by traditional approaches for predicting these rates. Ingrowth species composition was very poorly predicted. The model was capable of reproducing a wide range of stand dynamic trajectories and in some cases reproduced trajectories that the statistical model was incapable of reproducing. ConclusionsThe model has potential to be used as a benchmarking tool for evaluating statistical and process models and may provide a mechanism to separate signal from noise and improve our ability to analyze and learn from large regional datasets that often have underlying flaws in sample design.

A Dynamic BI–Orthogonal Field Equation Approach to Efficient Bayesian Inversion

Abstract This paper proposes a novel computationally efficient stochastic spectral projection based approach to Bayesian inversion of a computer simulator with high dimensional parametric and model structure uncertainty. The proposed method is based on the decomposition of the solution into its mean and a random field using a generic Karhunen-Loève expansion. The random field is represented as a convolution of separable Hilbert spaces in stochastic and spatial dimensions that are spectrally represented using respective orthogonal bases. In particular, the present paper investigates generalized polynomial chaos bases for the stochastic dimension and eigenfunction bases for the spatial dimension. Dynamic orthogonality is used to derive closed-form equations for the time evolution of mean, spatial and the stochastic fields. The resultant system of equations consists of a partial differential equation (PDE) that defines the dynamic evolution of the mean, a set of PDEs to define the time evolution of eigenfunction bases, while a set of ordinary differential equations (ODEs) define dynamics of the stochastic field. This system of dynamic evolution equations efficiently propagates the prior parametric uncertainty to the system response. The resulting bi-orthogonal expansion of the system response is used to reformulate the Bayesian inference for efficient exploration of the posterior distribution. The efficacy of the proposed method is investigated for calibration of a 2D transient diffusion simulator with an uncertain source location and diffusivity. The computational efficiency of the method is demonstrated against a Monte Carlo method and a generalized polynomial chaos approach.

Claims Reserving with a Stochastic Vector Projection

In the last three decades, a variety of stochastic reserving models has been proposed in the general insurance literature mainly using (or reproducing) the eminent Chain-Ladder claims reserving estimates. In practice, when the data doesn’t satisfy the Chain-Ladder assumptions, high prediction errors might occur. Thus, in this paper, a combined methodology is proposed which is based on the stochastic vector projection method and uses the regression through the origin approach of Murphy (1994), but with heteroscedastic errors instead, and different to those that used by Mack (1993, 1994). Furthermore, the Mack (1993) distribution-free model appears to have higher prediction errors when it is compared with the pro-posed one, particularly, for data sets with increasing (regular) trends. Finally, three empirical examples with irregular and regular data sets illustrate the theoretical findings, and the concepts of best estimate and risk margin are reported.

Robust scheduling for flexible processing networks

Abstract Modern processing networks often consist of heterogeneous servers with widely varying capabilities, and process job flows with complex structure and requirements. A major challenge in designing efficient scheduling policies in these networks is the lack of reliable estimates of system parameters, and an attractive approach for addressing this challenge is to design robust policies, i.e. policies that do not use system parameters such as arrival and/or service rates for making scheduling decisions. In this paper we propose a general framework for the design of robust policies. The main technical novelty is the use of a stochastic gradient projection method that reacts to queue-length changes in order to find a balanced allocation of service resources to incoming tasks. We illustrate our approach on two broad classes of processing systems, namely the flexible fork-join networks and the flexible queueing networks, and prove the rate stability of our proposed policies for these networks under nonrestrictive assumptions.

The calculation of the Stackelberg–Nash equilibrium as a fixed point problem in static hierarchical games

In this paper, a regularized version of the Extraproximal Method together with a Stochastic Approach is suggested to calculate the Stackelberg–Nash equilibrium in a N-person finite game. In this game, two levels of hierarchy in decision making are considered: one leader and (\(\hbox {N}-1\)) followers. Here, the followers playing according to the Nash equilibrium concept among themselves to the leaders announced strategy. An Extraproximal Technique is used to realize the application of a two-step procedure for finding the Nash equilibrium corresponding for the followers providing the loss function for a leader: at the first (or preliminary) step some predictive approximation of the a current approximation is calculated, at the second step (the main step of the iteration) this prediction is used to complete the current iteration. The next step, applying the stochastic gradient projection technique suggested, the Stackelberg–Nash equilibrium is found. The optimality conditions for a strategy being an equilibrium in participants game are derived based on the strong convexity of the \(\delta \)-regularized loss function. Simulation results illustrate the feasibility and the performance of this method.

Approximate Conditional Gradient Descent on Multi-Class Classification

Conditional gradient descent, aka the Frank-Wolfe algorithm,regains popularity in recent years. The key advantage of Frank-Wolfe is that at each step the expensive projection is replaced with a much more efficient linear optimization step. Similar to gradient descent, the loss function of Frank-Wolfe scales with the data size. Training on big data poses a challenge for researchers. Recently, stochastic Frank-Wolfe methods have been proposed to solve the problem, but they do not perform well in practice. In this work, we study the problem of approximating the Frank-Wolfe algorithm on the large-scale multi-class classification problem which is a typical application of the Frank-Wolfe algorithm. We present a simple but effective method employing internal structure of data to approximate Frank-Wolfe on the large-scale multiclass classification problem. Empirical results verify that our method outperforms the state-of-the-art stochastic projection free methods.

Stochastic Population Forecast for Germany and Its Consequence for the German Pension System

Population forecasts are crucial for many social, political and economic decisions. Official population projections rely in general on deterministic models which use different scenarios for future vital rates to indicate uncertainty. However, this technique shows substantial weak points such as assuming absolute correlations between the demographic components. In this paper, we argue that a stochastic projection alternative, with no a priori assumptions provides point forecasts and probabilistic prediction intervals for demographic parameters in addition. Age-sex specific population forecast for Germany is derived through a stochastic population renewal process using forecasts of mortality, fertility and migration. Time series models with demographic restrictions are used to describe immigration, emigration and time varying indices of mortality and fertility rates. These models are then used in the simulation of future vital rates to obtain age-specific population forecast using the cohort-component method. The consequence for the German pension system is discussed. To maintain the actual average pension level the premium rate of the present system rises at least by 50% as the old-age ratio nearly doubles by 2040.

Nonintrusive Polynomial Chaos Expansions for Sensitivity Analysis in Stochastic Differential Equations

A Galerkin polynomial chaos (PC) method was recently proposed to perform variance decomposition and sensitivity analysis in stochastic differential equations (SDEs), driven by Wiener noise and involving uncertain parameters. The present paper extends the PC method to nonintrusive approaches enabling its application to more complex systems hardly amenable to stochastic Galerkin projection methods. We also discuss parallel implementations and the variance decomposition of the derived quantity of interest within the framework of nonintrusive approaches. In particular, a novel hybrid PC-sampling-based strategy is proposed in the case of nonsmooth quantities of interest (QoIs) but smooth SDE solution. Numerical examples are provided that illustrate the decomposition of the variance of QoIs into contributions arising from the uncertain parameters, the inherent stochastic forcing, and joint effects. The simulations are also used to support a brief analysis of the computational complexity of the method, providing...

Multiclass, Multicriteria Dynamic Traffic Assignment with Path-Dependent Link Cost and Entropy-Based Risk Preference

Challenges arise in dynamic traffic assignment (DTA) when heterogeneous users evaluate choices on the basis of multiple interrelated criteria such as travel time and travel time uncertainty. This paper proposes a density-based formulation along with a stochastic quasigradient projection (SQGP) solution scheme with the aid of a traffic simulator. Path-dependent link cost is proposed to allow for the objective function formulation and more tractable analysis. The criteria in the discussion and the case study are travel time (link-additive), monetary cost (non additive), and travel time uncertainty (path-dependent link-additive). An information entropy-based uncertainty measure is proposed because of concerns about using conventional measures such as variability and reliability. The case study shows stochastic and efficient convergence, demonstrates the ability of SQGP to bypass local optima, and exemplifies the significant effect of using path-independent and path-dependent link costs to forecast traffic pattern and toll revenue. The results also suggest that a pricing strategy aimed at optimizing travel time and reliability for different user classes should consider travel time correlations between toll segments and the adjacent no-toll segments if enumerating paths is practically infeasible.

Iterative Projection approach for solving the Territorial Business Sales optimization problem

A well designed territory enhances customer coverage, increases sales, fosters fair performance and rewards systems and lower travel cost. This paper considers a real life case study to design a sales territory for a business sales plan. The business plan consists in assigning the optimal quantity of sellers to a territory including the scheduling and routing plans for each seller. The problem is formulated as a combination of assignment, scheduling and routing optimization problems. The solution approach considers a meta-heuristic using stochastic iterative projection method for large systems. Several real life instances of different sizes were tested with stochastic data to represent raise/fall in the customers demand as well as the appearance/loss of customers.

Household projections by the headship rates method: The case of Serbia

Household projections by the headship rates method: The case of Serbia

Accounting for uncertainty in dormant life stages in stochastic demographic models

Dormant life stages are often critical for population viability in stochastic environments, but accurate field data characterizing them are difficult to collect. Such limitations may translate into uncertainties in demographic parameters describing these stages, which then may propagate errors in the examination of population‐level responses to environmental variation. Expanding on current methods, we 1) apply data‐driven approaches to estimate parameter uncertainty in vital rates of dormant life stages and 2) test whether such estimates provide more robust inferences about population dynamics. We built integral projection models (IPMs) for a fire‐adapted, carnivorous plant species using a Bayesian framework to estimate uncertainty in parameters of three vital rates of dormant seeds – seed‐bank ingression, stasis and egression. We used stochastic population projections and elasticity analyses to quantify the relative sensitivity of the stochastic population growth rate (log λs) to changes in these vital rates at different fire return intervals. We then ran stochastic projections of log λs for 1000 posterior samples of the three seed‐bank vital rates and assessed how strongly their parameter uncertainty propagated into uncertainty in estimates of log λs and the probability of quasi‐extinction, Pq(t). Elasticity analyses indicated that changes in seed‐bank stasis and egression had large effects on log λs across fire return intervals. In turn, uncertainty in the estimates of these two vital rates explained > 50% of the variation in log λs estimates at several fire‐return intervals. Inferences about population viability became less certain as the time between fires widened, with estimates of Pq(t) potentially > 20% higher when considering parameter uncertainty. Our results suggest that, for species with dormant stages, where data is often limited, failing to account for parameter uncertainty in population models may result in incorrect interpretations of population viability.

Parameter risk in time-series mortality forecasts

The projection of mortality rates is an essential part of valuing liabilities in life insurance portfolios and pension schemes. An important tool for risk management and solvency purposes is a stochastic projection model. We show that ARIMA models can be better representations of mortality time-series than simple random-walk models. We also consider the issue of parameter risk in time-series models from the point of view of an insurer using them for regulatory risk reporting – formulae are given for decomposing overall risk into undiversifiable trend risk (parameter uncertainty) and diversifiable volatility. Particular attention is given to the contrasts in how academic researchers might view these models and how insurance regulators and practitioners in life offices might use them. Using a bootstrap method we find that, while certain kinds of parameter risk are negligible, others are too material to ignore. We also find that an objective model selection criterion, such as goodness of fit to past data, can result in the selection of a model with unstable parameter values. While this aspect of the model is superficially undesirable, it also leads to slightly higher capital requirements and thus makes the model of keen interest to regulators. Our conclusions have relevance to insurers using value-at-risk capital assessments in the European Union under Solvency II, but also territories using conditional tail expectations such as Australia, Canada and Switzerland.

From regulatory life tables to stochastic mortality projections: The exponential decline model

Often in actuarial practice, mortality projections are obtained by letting age-specific death rates decline exponentially at their own rate. Many life tables used for annuity pricing are built in this way. The present paper adopts this point of view and proposes a simple and powerful mortality projection model in line with this elementary approach, based on the recently studied mortality improvement rates. Two main applications are considered. First, as most reference life tables produced by regulators are deterministic by nature, they can be made stochastic by superposing random departures from the assumed age-specific trend, with a volatility calibrated on market or portfolio data. This allows the actuary to account for the systematic longevity risk in solvency calculations. Second, the model can be fitted to historical data and used to produce longevity forecasts. A number of conservative and tractable approximations are derived to provide the actuary with reasonably accurate approximations for various relevant quantities, available at limited computational cost. Besides applications to stochastic mortality projection models, we also derive useful properties involving supermodular, directionally convex and stop-loss orders.

To exclose nests or not: structured decision making for the conservation of a threatened species

AbstractDecisions regarding endangered species recovery often face sparse data and multiple sources of uncertainty about the effects of management. Structured decision making (SDM) provides a framework for assembling knowledge and expert opinion and evaluating the tradeoffs between different objectives while formally incorporating uncertainty. The Atlantic Coast piping plover provides an illustrative case for the utility of SDM in endangered species management because its population growth is simple to model, most populations are monitored, decision alternatives are well defined, and many managers are open to recovery recommendations. We built a model to evaluate the decision to use nest exclosures to protect piping plover eggs from predators, where the objective was to maximize λ and the tradeoff was between nest survival and adult survival. The latter can be reduced by exclosures. We used a novel mixed multinomial logistic exposure model to predict daily nest fates and incorporated the results into a stochastic projection matrix that included renesting after nest failure, and adult mortality associated with abandonment. In our test data set (n = 329 nests from 28 sites over four years), the mean nest survival over 34 days was markedly higher for exclosed nests (0.76 ± 0.03 SE) than for unexclosed nests (0.37 ± 0.07). Abandonment rates were also higher for exclosed nests (0.092 ± 0.017) than for unexclosed nests (0.045 ± 0.017), but the difference was not statistically significant and the loss rate to “other sources” (mostly predators) was much lower for exclosed nests (0.15 ± 0.03) than for unexclosed nests (0.58 ± 0.07). Population growth rate (λ) was clearly improved by exclosure use at the sites with high background nest loss rates, but λ was still <1 with exclosure use. Where the background nest loss rates were low, the decision to use exclosures was ambiguous, and λ could benefit from reducing uncertainty in vital rates. Our process demonstrated that geographic and temporal variation in nest mortality determines whether exclosures will be useful in attaining positive population growth rates and that other management options must be considered where the background nest mortality rates are high.

Assessing the sustainability of African lion trophy hunting, with recommendations for policy.

While trophy hunting provides revenue for conservation, it must be carefully managed to avoid negative population impacts, particularly for long-lived species with low natural mortality rates. Trophy hunting has had negative effects on lion populations throughout Africa, and the species serves as an important case study to consider the balance of costs and benefits, and to consider the effectiveness of alternative strategies to conserve exploited species. Age-restricted harvesting is widely recommended to mitigate negative effects of lion hunting, but this recommendation was based on a population model parameterized with data from a well-protected and growing lion population. Here, we used demographic data from lions subject to more typical conditions, including source-sink dynamics between a protected National Park and adjacent hunting areas in Zambia's Luangwa Valley, to develop a stochastic population projection model and evaluate alternative harvest scenarios. Hunting resulted in population declines over a 25-yr period for all continuous harvest strategies, with large declines for quotas >1 lion/concession (~0.5 lion/1,000km2 ) and hunting of males younger than seven years. A strategy that combined periods of recovery, an age limit of ≥7yr, and a maximum quota of ~0.5 lions shot/1,000km2 yielded a risk of extirpation <10%. Our analysis incorporated the effects of human encroachment, poaching, and prey depletion on survival, but assumed that these problems will not increase, which is unlikely. These results suggest conservative management of lion trophy hunting with a combination of regulations. To implement sustainable trophy hunting while maintaining revenue for conservation of hunting areas, our results suggest that hunting fees must increase as a consequence of diminished supply. These findings are broadly applicable to hunted lion populations throughout Africa and to inform global efforts to conserve exploited carnivore populations.

Population viability analysis of plant and animal populations with stochastic integral projection models.

Integral projection models (IPM) make it possible to study populations structured by continuous traits. Recently, Vindenes et al. (Ecology 92:1146-1156, 2011) proposed an extended IPM to analyse the dynamics of small populations in stochastic environments, but this model has not yet been used to conduct population viability analyses. Here, we used the extended IPM to analyse the stochastic dynamics of IPM of small size-structured populations in one plant and one animal species (evening primrose and common lizard) including demographic stochasticity in both cases and environmental stochasticity in the lizard model. We also tested the accuracy of a diffusion approximation of the IPM for the two empirical systems. In both species, the elasticity for λ was higher with respect to parameters linked to body growth and size-dependent reproduction rather than survival. An analytical approach made it possible to quantify demographic and environmental variance to calculate the average stochastic growth rate. Demographic variance was further decomposed to gain insights into the most important size classes and demographic components. A diffusion approximation provided a remarkable fit to the stochastic dynamics and cumulative extinction risk, except for very small populations where stochastic growth rate was biased upward or downward depending on the model. These results confirm that the extended IPM provides a powerful tool to assess the conservation status and compare the stochastic demography of size-structured species, but should be complemented with individual based models to obtain unbiased estimates for very small populations of conservation concern.

Stochastic projection methods and applications to some nonlinear inverse problems of phase retrieving

In this short paper we present a stochastic projection based Monte Carlo algorithm for solving a nonlinear ill-posed inverse problem of recovering the phase of a complex-valued function provided its absolute value is known, under some additional information. The method is developed here for retrieving the step structure of the epitaxial films from the X-ray diffraction analysis. We suggest to extract some additional information from the measurements which makes the problem well-posed, and with this information, the method suggested works well even for noisy measurements. Results of simulations for a layer structure recovering problem with 26 sublayers are presented.