Efficient minimax strategies for square loss games

We consider online prediction problems where the loss between the prediction and the outcome is measured by the squared Euclidean distance and its generalization, the squared Mahalanobis distance. We derive the minimax solutions for the case where the prediction and action spaces are the simplex (this setup is sometimes called the Brier game) and the \ell_2 ball (this setup is related to Gaussian density estimation). We show that in both cases the value of each sub-game is a quadratic function of a simple statistic of the state, with coefficients that can be efficiently computed using an explicit recurrence relation. The resulting deterministic minimax strategy and randomized maximin strategy are linear functions of the statistic.

Quantifying uncertainty in parameter estimates for stochastic models of collective cell spreading using approximate Bayesian computation

Wound healing and tumour growth involve collective cell spreading, which is driven by individual motility and proliferation events within a population of cells. Mathematical models are often used to interpret experimental data and to estimate the parameters so that predictions can be made. Existing methods for parameter estimation typically assume that these parameters are constants and often ignore any uncertainty in the estimated values. We use approximate Bayesian computation (ABC) to estimate the cell diffusivity, D, and the cell proliferation rate, λ, from a discrete model of collective cell spreading, and we quantify the uncertainty associated with these estimates using Bayesian inference. We use a detailed experimental data set describing the collective cell spreading of 3T3 fibroblast cells. The ABC analysis is conducted for different combinations of initial cell densities and experimental times in two separate scenarios: (i) where collective cell spreading is driven by cell motility alone, and (ii) where collective cell spreading is driven by combined cell motility and cell proliferation. We find that D can be estimated precisely, with a small coefficient of variation (CV) of 2–6%. Our results indicate that D appears to depend on the experimental time, which is a feature that has been previously overlooked. Assuming that the values of D are the same in both experimental scenarios, we use the information about D from the first experimental scenario to obtain reasonably precise estimates of λ, with a CV between 4 and 12%. Our estimates of D and λ are consistent with previously reported values; however, our method is based on a straightforward measurement of the position of the leading edge whereas previous approaches have involved expensive cell counting techniques. Additional insights gained using a fully Bayesian approach justify the computational cost, especially since it allows us to accommodate information from different experiments in a principled way.

Gaming well: Links between videogames and flourishing mental health

This paper is a review of the state of play of research linking videogaming and flourishing, and explores the role of videogames and technology to improve mental health and well-being. Its purpose is to develop understandings about the positive intersection of gaming and well-being, to document evidence regarding links between videogames and positive mental health, and to provide guidelines for use by other researchers as they design and use tools and games to improve mental health and well-being. Using Huppert's (Huppert and So, 2013) proposition that to flourish is more than the absence of mental disorder but rather a combination of feeling good and functioning effectively, resulting in high levels of mental well-being, and Seligman's (Seligman, 2011) PERMA theory of well-being, the paper identifies strengths in existing games that generate positive affect, positive functioning, and positive social functioning, contributing to, and supporting mental health and well-being.

Curiosity driven reinforcement learning for motion planning on humanoids

Most previous work on artificial curiosity (AC) and intrinsic motivation focuses on basic concepts and theory. Experimental results are generally limited to toy scenarios, such as navigation in a simulated maze, or control of a simple mechanical system with one or two degrees of freedom. To study AC in a more realistic setting, we embody a curious agent in the complex iCub humanoid robot. Our novel reinforcement learning (RL) framework consists of a state-of-the-art, low-level, reactive control layer, which controls the iCub while respecting constraints, and a high-level curious agent, which explores the iCub's state-action space through information gain maximization, learning a world model from experience, controlling the actual iCub hardware in real-time. To the best of our knowledge, this is the first ever embodied, curious agent for real-time motion planning on a humanoid. We demonstrate that it can learn compact Markov models to represent large regions of the iCub's configuration space, and that the iCub explores intelligently, showing interest in its physical constraints as well as in objects it finds in its environment.

A semi-alternating direction method for a 2-D fractional FitzHugh–Nagumo monodomain model on an approximate irregular domain

A FitzHugh-Nagumo monodomain model has been used to describe the propagation of the electrical potential in heterogeneous cardiac tissue. In this paper, we consider a two-dimensional fractional FitzHugh-Nagumo monodomain model on an irregular domain. The model consists of a coupled Riesz space fractional nonlinear reaction-diffusion model and an ordinary differential equation, describing the ionic fluxes as a function of the membrane potential. Secondly, we use a decoupling technique and focus on solving the Riesz space fractional nonlinear reaction-diffusion model. A novel spatially second-order accurate semi-implicit alternating direction method (SIADM) for this model on an approximate irregular domain is proposed. Thirdly, stability and convergence of the SIADM are proved. Finally, some numerical examples are given to support our theoretical analysis and these numerical techniques are employed to simulate a two-dimensional fractional Fitzhugh-Nagumo model on both an approximate circular and an approximate irregular domain.

Numerical algorithms for time-fractional subdiffusion equation with second-order accuracy

This article aims to fill in the gap of the second-order accurate schemes for the time-fractional subdiffusion equation with unconditional stability. Two fully discrete schemes are first proposed for the time-fractional subdiffusion equation with space discretized by finite element and time discretized by the fractional linear multistep methods. These two methods are unconditionally stable with maximum global convergence order of $O(\tau+h^{r+1})$ in the $L^2$ norm, where $\tau$ and $h$ are the step sizes in time and space, respectively, and $r$ is the degree of the piecewise polynomial space. The average convergence rates for the two methods in time are also investigated, which shows that the average convergence rates of the two methods are $O(\tau^{1.5}+h^{r+1})$. Furthermore, two improved algorithms are constrcted, they are also unconditionally stable and convergent of order $O(\tau^2+h^{r+1})$. Numerical examples are provided to verify the theoretical analysis. The comparisons between the present algorithms and the existing ones are included, which show that our numerical algorithms exhibit better performances than the known ones.

High order unconditionally stable difference schemes for the Riesz space-fractional telegraph equation

In this paper, a class of unconditionally stable difference schemes based on the Pad´e approximation is presented for the Riesz space-fractional telegraph equation. Firstly, we introduce a new variable to transform the original dfferential equation to an equivalent differential equation system. Then, we apply a second order fractional central difference scheme to discretise the Riesz space-fractional operator. Finally, we use (1, 1), (2, 2) and (3, 3) Pad´e approximations to give a fully discrete difference scheme for the resulting linear system of ordinary differential equations. Matrix analysis is used to show the unconditional stability of the proposed algorithms. Two examples with known exact solutions are chosen to assess the proposed difference schemes. Numerical results demonstrate that these schemes provide accurate and efficient methods for solving a space-fractional hyperbolic equation.

A fast semi-implicit difference method for a nonlinear two-sided space-fractional diffusion equation with variable diffusivity coefficients

In this paper, we derive a new nonlinear two-sided space-fractional diffusion equation with variable coefficients from the fractional Fick’s law. A semi-implicit difference method (SIDM) for this equation is proposed. The stability and convergence of the SIDM are discussed. For the implementation, we develop a fast accurate iterative method for the SIDM by decomposing the dense coefficient matrix into a combination of Toeplitz-like matrices. This fast iterative method significantly reduces the storage requirement of O(n2)O(n2) and computational cost of O(n3)O(n3) down to n and O(nlogn)O(nlogn), where n is the number of grid points. The method retains the same accuracy as the underlying SIDM solved with Gaussian elimination. Finally, some numerical results are shown to verify the accuracy and efficiency of the new method.

Numerical analysis for the time distributed-order and Riesz space fractional diffusions on bounded domains

Subdiffusion equations with distributed-order fractional derivatives describe some important physical phenomena. In this paper, we consider the time distributed-order and Riesz space fractional diffusions on bounded domains with Dirichlet boundary conditions. Here, the time derivative is defined as the distributed-order fractional derivative in the Caputo sense, and the space derivative is defined as the Riesz fractional derivative. First, we discretize the integral term in the time distributed-order and Riesz space fractional diffusions using numerical approximation. Then the given equation can be written as a multi-term time–space fractional diffusion. Secondly, we propose an implicit difference method for the multi-term time–space fractional diffusion. Thirdly, using mathematical induction, we prove the implicit difference method is unconditionally stable and convergent. Also, the solvability for our method is discussed. Finally, two numerical examples are given to show that the numerical results are in good agreement with our theoretical analysis.

Stability and convergence of a new finite volume method for a two-sided space-fractional diffusion equation

In this paper, we consider a two-sided space-fractional diffusion equation with variable coefficients on a finite domain. Firstly, based on the nodal basis functions, we present a new fractional finite volume method for the two-sided space-fractional diffusion equation and derive the implicit scheme and solve it in matrix form. Secondly, we prove the stability and convergence of the implicit fractional finite volume method and conclude that the method is unconditionally stable and convergent. Finally, some numerical examples are given to show the effectiveness of the new numerical method, and the results are in excellent agreement with theoretical analysis.

A Novel High Order Space-Time Spectral Method for the Time Fractional Fokker--Planck Equation

The fractional Fokker-Planck equation is an important physical model for simulating anomalous diffusions with external forces. Because of the non-local property of the fractional derivative an interesting problem is to explore high accuracy numerical methods for fractional differential equations. In this paper, a space-time spectral method is presented for the numerical solution of the time fractional Fokker-Planck initial-boundary value problem. The proposed method employs the Jacobi polynomials for the temporal discretization and Fourier-like basis functions for the spatial discretization. Due to the diagonalizable trait of the Fourier-like basis functions, this leads to a reduced representation of the inner product in the Galerkin analysis. We prove that the time fractional Fokker-Planck equation attains the same approximation order as the time fractional diffusion equation developed in [23] by using the present method. That indicates an exponential decay may be achieved if the exact solution is sufficiently smooth. Finally, some numerical results are given to demonstrate the high order accuracy and efficiency of the new numerical scheme. The results show that the errors of the numerical solutions obtained by the space-time spectral method decay exponentially.

Weather variability and influenza A (H7N9) transmission in Shanghai, China: A Bayesian spatial analysis

Abstract Background A novel avian influenza A (H7N9) virus was first found in humans in Shanghai, and infected over 433 patients in China. To date, very little is known about the spatiotemporal variability or environmental drivers of the risk of H7N9 infection. This study explored the spatial and temporal variation of H7N9 infection and assessed the effects of temperature and rainfall on H7N9 incidence. Methods A Bayesian spatial conditional autoregressive (CAR) model was used to assess the spatiotemporal distribution of the risk of H7N9 infection in Shanghai, by district and fortnight for the period 19th February–14th April 2013. Data on daily laboratory-confirmed H7N9 cases, and weather variability including temperature (°C) and rainfall (mm) were obtained from the Chinese Information System for Diseases Control and Prevention and Chinese Meteorological Data Sharing Service System, respectively, and aggregated by fortnight. Results High spatial variations in the H7N9 risk were mainly observed in the east and centre of Shanghai municipality. H7N9 incidence rate was significantly associated with fortnightly mean temperature (Relative Risk (RR): 1.54; 95% credible interval (CI): 1.22–1.94) and fortnightly mean rainfall (RR: 2.86; 95% CI: 1.47–5.56). Conclusion There was a substantial variation in the spatiotemporal distribution of H7N9 infection across different districts in Shanghai. Optimal temperature and rainfall may be one of the driving forces for H7N9.

A framework for understanding and generating integrated solutions for residential peak energy demand

Supplying peak energy demand in a cost effective, reliable manner is a critical focus for utilities internationally. Successfully addressing peak energy concerns requires understanding of all the factors that affect electricity demand especially at peak times. This paper is based on past attempts of proposing models designed to aid our understanding of the influences on residential peak energy demand in a systematic and comprehensive way. Our model has been developed through a group model building process as a systems framework of the problem situation to model the complexity within and between systems and indicate how changes in one element might flow on to others. It is comprised of themes (social, technical and change management options) networked together in a way that captures their influence and association with each other and also their influence, association and impact on appliance usage and residential peak energy demand. The real value of the model is in creating awareness, understanding and insight into the complexity of residential peak energy demand and in working with this complexity to identify and integrate the social, technical and change management option themes and their impact on appliance usage and residential energy demand at peak times.

Inferring lung cancer risk factor patterns through joint Bayesian spatio-temporal analysis

Background: Preventing risk factor exposure is vital to reduce the high burden from lung cancer. The leading risk factor for developing lung cancer is tobacco smoking. In Australia, despite apparent success in reducing smoking prevalence, there is limited information on small area patterns and small area temporal trends. We sought to estimate spatio-temporal patterns for lung cancer risk factors using routinely collected population-based cancer data. Methods: The analysis used a Bayesian shared component spatio-temporal model, with male and female lung cancer included separately. The shared component reflected exposure to lung cancer risk factors, and was modelled over 477 statistical local areas (SLAs) and 15 years in Queensland, Australia. Analyses were also run adjusting for area-level socioeconomic disadvantage, Indigenous population composition, or remoteness. Results: Strong spatial patterns were observed in the underlying risk factor exposure for both males (median Relative Risk (RR) across SLAs compared to the Queensland average ranged from 0.48-2.00) and females (median RR range across SLAs 0.53-1.80), with high exposure observed in many remote areas. Strong temporal trends were also observed. Males showed a decrease in the underlying risk across time, while females showed an increase followed by a decrease in the final two years. These patterns were largely consistent across each SLA. The high underlying risk estimates observed among disadvantaged, remote and indigenous areas decreased after adjustment, particularly among females. Conclusion: The modelled underlying exposure appeared to reflect previous smoking prevalence, with a lag period of around 30 years, consistent with the time taken to develop lung cancer. The consistent temporal trends in lung cancer risk factors across small areas support the hypothesis that past interventions have been equally effective across the state. However, this also means that spatial inequalities have remained unaddressed, highlighting the potential for future interventions, particularly among remote areas.

The Lymphedema Evaluation in Gynecological cancer Study (LEGS): Design of a prospective, longitudinal, cohort study

Background The Lymphoedema Evaluation in Gynecological cancer Study (LEGS) was a longitudinal, observational, cohort study prospectively evaluating the incidence and risk factors of lower-limb lymphedema after treatment for gynecological cancer. Here we describe the study protocol and characteristics of the sample. Methods Women with a newly diagnosed gynecological cancer between June 1, 2008 and February 28, 2011, aged 18 years or older, and treated at one of six hospitals in Queensland, Australia, were eligible. Lymphedema was assessed by circumference measurements, bioimpedance spectroscopy, and self-reported swelling. LEGS incorporated a cohort of patients requiring surgery for benign gynecological conditions for comparison purposes. Data were collected prior to surgery and at regular intervals thereafter up to 2-years post-diagnosis. Results 546 women participated (408 cancer, 138 benign), with a 24-month retention rate of 78%. Clinical and treatment characteristics of participants were similar to the Queensland gynecological cancer population, except for a higher proportion of early-stage cervical cancers recruited to LEGS compared with Queensland proportions (89% versus 55%, respectively). Discussion Few imbalances were observed between participants with complete and incomplete follow-up data. The prospective design and collection of objective and patient-reported outcome data will allow comprehensive assessment of incidence and risk factors of lower-limb lymphedema.