The Frequency Distribution of the Difference Between Two Poisson Variates Belonging to Different Populations

Skellam, J. G.

doi:10.2307/2981372

Cited by 363 publications

(289 citation statements)

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“…If we ignore repeated diffusion of the same molecule, the net change is exactly described by the Skellam distribution, and the mean and variance are given by a 1 τ − a 2 τ and a 1 τ + a 2 τ, respectively. 40 The simplest way to generate leap sizes with the right mean and variance is to generate the forward j and backward j ′ leaps using (11) and use the difference as the effective leap size. The expected number of firings is…”

Section: Population Change In a Leapmentioning

confidence: 99%

Asynchronous τ-leaping

Jędrzejewski-Szmek

Blackwell

2016

The Journal of Chemical Physics

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Stochastic simulation of cell signaling pathways and genetic regulatory networks has contributed to the understanding of cell function; however, investigation of larger, more complicated systems requires computationally efficient algorithms. τ-leaping methods, which improve efficiency when some molecules have high copy numbers, either use a fixed leap size, which does not adapt to changing state, or recalculate leap size at a heavy computational cost. We present a hybrid simulation method for reaction-diffusion systems which combines exact stochastic simulation and τ-leaping in a dynamic way. Putative times of events are stored in a priority queue, which reduces the cost of each step of the simulation. For every reaction and diffusion channel at each step of the simulation the more efficient of an exact stochastic event or a τ-leap is chosen. This new approach removes the inherent trade-off between speed and accuracy in stiff systems which was present in all τ-leaping methods by allowing each reaction channel to proceed at its own pace. Both directions of reversible reactions and diffusion are combined in a single event, allowing bigger leaps to be taken. This improves efficiency for systems near equilibrium where forward and backward events are approximately equally frequent. Comparison with existing algorithms and behaviour for five test cases of varying complexity shows that the new method is almost as accurate as exact stochastic simulation, scales well for large systems, and for various problems can be significantly faster than τ-leaping. C 2016 AIP Publishing LLC. [http://dx

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Section: Population Change In a Leapmentioning

confidence: 99%

Asynchronous τ-leaping

Jędrzejewski-Szmek

Blackwell

2016

The Journal of Chemical Physics

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“…Win probability: Suppose we are given the offence and defence skills for team i and j, we can estimate the distributions over performance difference variables of x and y (c.f., Figure 2), and compute the Poisson parameters for s i and s j by using λ i = exp(x) and λ j = exp(y). To compute the winning probability of team i, i.e., p(s i > s j ), we first construct a new variable s = s i − s j , the difference variable between two Poisson distributions, which proves to be a Skellam distribution in [10]. Thus, we can compute the win probability of P (s > 0) of team i, according to the probability mass function for the Skellam distribution…”

Section: Inference In Poisson-od Modelmentioning

confidence: 99%

Score-Based Bayesian Skill Learning

Guo

Sanner

Graepel

et al. 2012

Machine Learning and Knowledge Discovery in Databases

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Abstract. We extend the Bayesian skill rating system of TrueSkill to accommodate score-based match outcomes. TrueSkill has proven to be a very effective algorithm for matchmaking -the process of pairing competitors based on similar skill-level -in competitive online gaming. However, for the case of two teams/players, TrueSkill only learns from win, lose, or draw outcomes and cannot use additional match outcome information such as scores. To address this deficiency, we propose novel Bayesian graphical models as extensions of TrueSkill that (1) model player's offence and defence skills separately and (2) model how these offence and defence skills interact to generate score-based match outcomes. We derive efficient (approximate) Bayesian inference methods for inferring latent skills in these new models and evaluate them on three real data sets including Halo 2 XBox Live matches. Empirical evaluations demonstrate that the new score-based models (a) provide more accurate win/loss probability estimates than TrueSkill when training data is limited, (b) provide competitive and often better win/loss classification performance than TrueSkill, and (c) provide reasonable score outcome predictions with an appropriate choice of likelihood -prediction for which TrueSkill was not designed, but which can be useful in many applications.

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“…An example is the contribution by Jorgensen et al (1999), who proposed to model Poisson counts by a state space model driven by a latent gamma Markov process. The Skellam distribution is a natural extension to this literature, as it was originally introduced as the difference of two independent Poisson random variables; see Irwin (1937) and Skellam (1946). However, it is not immediately clear how the treatment by Jorgensen et al (1999) can be extended for the difference of Poisson random variables as it requires an analytical expression of a conditional distribution for a gamma variable given a Skellam variable.…”

Section: Introductionmentioning

confidence: 99%

Intraday Stochastic Volatility in Discrete Price Changes: The Dynamic Skellam Model

Koopman

Lit

Lucas

2017

Journal of the American Statistical Association

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We study intraday stochastic volatility for four liquid stocks traded on the New York Stock Exchange using a new dynamic Skellam model for high-frequency tick-by-tick discrete price changes. Since the likelihood function is analytically intractable, we rely on numerical methods for its evaluation. Given the high number of observations per series per day (1000 to 10,000), we adopt computationally efficient methods including Monte Carlo integration. The intraday dynamics of volatility and the high number of trades without price impact require nontrivial adjustments to the basic dynamic Skellam model. In-sample residual diagnostics and goodness-of-fit statistics show that the final model provides a good fit to the data. An extensive dayto-day forecasting study of intraday volatility shows that the dynamic modified Skellam model provides accurate forecasts compared to alternative modeling approaches. Supplementary materials for this article are available online.

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The Frequency Distribution of the Difference Between Two Poisson Variates Belonging to Different Populations

Cited by 363 publications

References 0 publications

Asynchronous τ-leaping

Asynchronous τ-leaping

Score-Based Bayesian Skill Learning

Intraday Stochastic Volatility in Discrete Price Changes: The Dynamic Skellam Model

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