This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS: SYSTEMS 1 An Incremental Probability Model for Dynamic Systems Wolf Kohn, Member, IEEE, Philip C. Placek , Zelda B. Zabinsky, and Jonathan Cross Abstract—In this paper, we present an incremental probabil- ity model for dynamic systems. This model combines historical data and new, real-time sequential data as it becomes available. Traditionally, models and algorithms assume the data follows a Gaussian distribution or other specified form. Instead, we propa- gate the transition probabilities directly which allows us to build the probability distribution from data. This provides a more real- istic algorithm for probabilistic forecasting. To address large scale problems, our method is made computationally efficient by using an incremental model to construct probabilities relative to the nominal (or mean) of the state which is allowed to change over time. A mean-field approach further reduces computation while preserving statistical dependencies. Index Terms—Forecasting, modeling, system analysis and design. I. I NTRODUCTION I N THIS paper we consider a stochastic and dynamic model for forecasting. This model combines historical data and new data as it becomes available. Most probabilistic forecasts propagate the mean and the covariance directly, which assumes that the data follows a cer- tain distribution [1], [2]. Well-developed algorithms, such as maximum likelihood and Kalman filters, make assumptions about the distribution of data. There is a need for a forecaster, that is, stochastic and dynamic without assuming a probabil- ity distribution. We propagate the probabilities directly which allows us to build the probability distribution over time. In order to propagate the probabilities to make accurate forecasts we require that the mean of the state can be easily calculated when new data arrives. Problems in price fore- casting [1], chemistry [3], networked systems [4], [5], and electrodynamics and quantum mechanics [6] all contain this feature. An observable, or estimated mean, allows us to build the probability distribution instead of assuming it a priori. Our method is derived from the differential Chapman– Kolmogorov equation for a continuous-time Markov process Manuscript received August 5, 2017; revised November 22, 2017; accepted January 11, 2018. This work was supported by Microsoft Dynamics. This paper was recommended by Associate Editor Z. Wang. (Corresponding author: Philip C. Placek.) W. Kohn is with the Department of Research, Atigeo LLC, Bellevue, WA 98004 USA. P. C. Placek is with the Department of Data Science, CityBldr, Seattle, WA 98122 USA (e-mail: pcplacek@gmail.com). Z. B. Zabinsky is with the Department of Industrial and Systems Engineering, University of Washington, Seattle, WA 98195-0005 USA. J. Cross is with the Department of Research, Google, Seattle, WA 98103 USA. Digital Object Identifier 10.1109/TSMC.2018.2797119 which describes the evolution of the unconditioned prob- ability. One benefit of the Chapman–Kolmogorov equation is that an underlying distribution of the state does not need to be assumed. However, propagating the probabili- ties using just the Chapman–Kolmogorov equation is com- putationally intractable. To address this issue, we use the Fokker–Planck equation which is a linearized form of the Chapman–Kolmogorov equation. However, the Fokker–Planck equation is a second-order, partial differential equation that is not easy to work with. To address this issue we discretize the Fokker–Planck equation with respect to state and time. This only serves as an approxi- mation but allows the unconditioned probabilities to be written as a finite space, finite time Markov chain. The discrete time transition probabilities of this Markov chain have parameters that explain how the system changes over time. Estimating these parameters may require a substantial amount of computational time, especially if the state space is large. Our approach to create a computationally efficient algorithm is to formulate the Fokker–Planck equation relative to the mean of the state. This creates a finite set of transition probabilities which depend on the difference between the cur- rent state and the mean. The state space is then divided up into mutually exclusive intervals called the incremental levels. The model forecasts the future incremental level instead of just calculating the expected value. The mean of the state is allowed to change over time as new data is obtained, and as a result the incremental levels may change as well. The dynami- cally changing levels allow the forecasts to change over time in accordance with the new data. It is also possible to interpolate the resulting probabilities to approximate a distribution. The number of parameters to estimate for our incremental model grows exponentially as the number of components of the state increases. We use a mean-field approach to approx- imate the dependencies between the components of the state space in a computationally efficient manner. This reduces the number of parameters that need to be estimated from expo- nential to polynomial. Recent advances in mean-field theory have been used to solve high-dimensional problems in particle physics, nanotechnology, and parallel programming [7], [8]. The rest of this paper is organized as follows. In Section II, we derive our incremental probability model around the mean as a discrete time Markov chain. Section III derives necessary conditions to show that the Markov chain is a good approx- imation to the Fokker–Planck equation. Then in Section IV, we explain the use of the mean-field for generating a compu- tationally feasible method for forecasting. In Section V, we 2168-2216 c  2018 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.