A noise trader model as a generator of apparent financial power laws and long memory

In various agent-based models the stylized facts of financial markets (unit-roots, fat tails and volatility clustering) have been shown to emerge from the interactions of agents. However, the complexity of these models often limits their analytical accessibility. In this paper we show that even a very simple model of a financial market with heterogeneous interacting agents is capable of reproducing these ubiquitous statistical properties. The simplicity of our approach permits to derive some analytical insights using concepts from statistical mechanics. In our model, traders are divided into two groups: fundamentalists and chartists, and their interactions are based on a variant of the herding mechanism introduced by Kirman [1993]. The statistical analysis of simulated data points toward long-term dependence in the auto-correlations of squared and absolute returns and hyperbolic decay in the tail of the distribution of raw returns, both with estimated decay parameters in the same range like those of empirical data. Theoretical analysis, however, excludes the possibility of ‘true’ scaling behavior because of the Markovian nature of the underlying process and the boundedness of returns. The model, therefore, only mimics power law behavior. Similarly as with the phenomenological volatility models analyzed in LeBaron [2001], the usual statistical tests are not able to distinguish between true or pseudo-scaling laws in the dynamics of our artificial market --Herd Behavior,Speculative Dynamics,Fat Tails,Volatility Clustering

Iterated filtering methods for Markov process epidemic models

Dynamic epidemic models have proven valuable for public health decision makers as they provide useful insights into the understanding and prevention of infectious diseases. However, inference for these types of models can be difficult because the disease spread is typically only partially observed e.g. in form of reported incidences in given time periods. This chapter discusses how to perform likelihood-based inference for partially observed Markov epidemic models when it is relatively easy to generate samples from the Markov transmission model while the likelihood function is intractable. The first part of the chapter reviews the theoretical background of inference for partially observed Markov processes (POMP) via iterated filtering. In the second part of the chapter the performance of the method and associated practical difficulties are illustrated on two examples. In the first example a simulated outbreak data set consisting of the number of newly reported cases aggregated by week is fitted to a POMP where the underlying disease transmission model is assumed to be a simple Markovian SIR model. The second example illustrates possible model extensions such as seasonal forcing and over-dispersion in both, the transmission and observation model, which can be used, e.g., when analysing routinely collected rotavirus surveillance data. Both examples are implemented using the R-package pomp (King et al., 2016) and the code is made available online.Comment: This manuscript is a preprint of a chapter to appear in the Handbook of Infectious Disease Data Analysis, Held, L., Hens, N., O'Neill, P.D. and Wallinga, J. (Eds.). Chapman \& Hall/CRC, 2018. Please use the book for possible citations. Corrected typo in the references and modified second exampl

Compound Markov counting processes and their applications to modeling infinitesimally over-dispersed systems

We propose an infinitesimal dispersion index for Markov counting processes. We show that, under standard moment existence conditions, a process is infinitesimally (over-) equi-dispersed if, and only if, it is simple (compound), i.e. it increases in jumps of one (or more) unit(s), even though infinitesimally equi-dispersed processes might be under-, equi- or over-dispersed using previously studied indices. Compound processes arise, for example, when introducing continuous-time white noise to the rates of simple processes resulting in Lévy-driven SDEs. We construct multivariate infinitesimally over dispersed compartment models and queuing networks, suitable for applications where moment constraints inherent to simple processes do not hold.Continuous time, Counting Markov process, Birth-death process, Environmental stochasticity, Infinitesimal over-dispersion, Simultaneous events

Fisher-Wright model with deterministic seed bank and selection

Seed banks are a common characteristics to many plant species, which allow storage of genetic diversity in the soil as dormant seeds for various periods of time. We investigate an above-ground population following a Fisher-Wright model with selection coupled with a deterministic seed bank assuming the length of the seed bank is kept constant and the number of seeds is large. To assess the combined impact of seed banks and selection on genetic diversity, we derive a general diffusion model. The applied techniques outline a path of approximating a stochastic delay differential equation by an appropriately rescaled stochastic differential equation, which is a common issue in statistical physics. We compute the equilibrium solution of the site-frequency spectrum and derive the times to fixation of an allele with and without selection. Finally, it is demonstrated that seed banks enhance the effect of selection onto the site-frequency spectrum while slowing down the time until the mutation-selection equilibrium is reached

Real-time growth rate for general stochastic SIR epidemics on unclustered networks

Networks have become an important tool for infectious disease epidemiology. Most previous theoretical studies of transmission network models have either considered simple Markovian dynamics at the individual level, or have focused on the invasion threshold and final outcome of the epidemic. Here, we provide a general theory for early real-time behaviour of epidemics on large configuration model networks (i.e. static and locally unclustered), in particular focusing on the computation of the Malthusian parameter that describes the early exponential epidemic growth. Analytical, numerical and Monte-Carlo methods under a wide variety of Markovian and non-Markovian assumptions about the infectivity profile are presented. Numerous examples provide explicit quantification of the impact of the network structure on the temporal dynamics of the spread of infection and provide a benchmark for validating results of large scale simulations.Comment: 45 pages, 8 figures, submitted to Mathematical Biosciences on 29/11/2014; Version 2: resubmitted on 15/04/2015; accepted on 17/04/2015. Changes: better explanations in introduction; restructured section 3.3 (3.3.3 added); section 6.3.1 added; more precise terminology; typos correcte

A Nonparametric Bayesian Approach to Uncovering Rat Hippocampal Population Codes During Spatial Navigation

Rodent hippocampal population codes represent important spatial information about the environment during navigation. Several computational methods have been developed to uncover the neural representation of spatial topology embedded in rodent hippocampal ensemble spike activity. Here we extend our previous work and propose a nonparametric Bayesian approach to infer rat hippocampal population codes during spatial navigation. To tackle the model selection problem, we leverage a nonparametric Bayesian model. Specifically, to analyze rat hippocampal ensemble spiking activity, we apply a hierarchical Dirichlet process-hidden Markov model (HDP-HMM) using two Bayesian inference methods, one based on Markov chain Monte Carlo (MCMC) and the other based on variational Bayes (VB). We demonstrate the effectiveness of our Bayesian approaches on recordings from a freely-behaving rat navigating in an open field environment. We find that MCMC-based inference with Hamiltonian Monte Carlo (HMC) hyperparameter sampling is flexible and efficient, and outperforms VB and MCMC approaches with hyperparameters set by empirical Bayes

Performance in population models for count data, part II: a new SAEM algorithm.

International audienceAnalysis of count data from clinical trials using mixed effect analysis has recently become widely used. However, algorithms available for the parameter estimation, including LAPLACE and Gaussian quadrature (GQ), are associated with certain limitations, including bias in parameter estimates and the long analysis runtime. The stochastic approximation expectation maximization (SAEM) algorithm has proven to be a very efficient and powerful tool in the analysis of continuous data. The aim of this study was to implement and investigate the performance of a new SAEM algorithm for application to count data. A new SAEM algorithm was implemented in MATLAB for estimation of both, parameters and the Fisher information matrix. Stochastic Monte Carlo simulations followed by re-estimation were performed according to scenarios used in previous studies (part I) to investigate properties of alternative algorithms (Plan et al., 2008, Abstr 1372 [ http://wwwpage-meetingorg/?abstract=1372 ]). A single scenario was used to explore six probability distribution models. For parameter estimation, the relative bias was less than 0.92% and 4.13% for fixed and random effects, for all models studied including ones accounting for over- or under-dispersion. Empirical and estimated relative standard errors were similar, with distance between them being <1.7% for all explored scenarios. The longest CPU time was 95 s for parameter estimation and 56 s for SE estimation. The SAEM algorithm was extended for analysis of count data. It provides accurate estimates of both, parameters and standard errors. The estimation is significantly faster compared to LAPLACE and GQ. The algorithm is implemented in Monolix 3.1, (beta-version available in July 2009)