Predictive Bayesian selection of multistep Markov chains, applied to the detection of the hot hand and other statistical dependencies in free throws

Consider the problem of modelling memory effects in discrete-state random walks using higher-order Markov chains. This paper explores cross-validation and information criteria as proxies for a model’s predictive accuracy. Our objective is to select, from data, the number of prior states of recent history upon which a trajectory is statistically dependent. Through simulations, I evaluate these criteria in the case where data are drawn from systems with fixed orders of history, noting trends in the relative performance of the criteria. As a real-world illustrative example of these methods, this manuscript evaluates the problem of detecting statistical dependencies in shot outcomes in free throw shooting. Over three National Basketball Association (NBA) seasons analysed, several players exhibited statistical dependencies in free throw hitting probability of various types—hot handedness, cold handedness and error correction. For the 2013–2014 to 2015–2016 NBA seasons, I detected statistical dependencies in 23% of all player-seasons. Focusing on a single player, in two of these three seasons, LeBron James shot a better percentage after an immediate miss than otherwise. Conditioning on the previous outcome makes for a more-predictive model than treating free throw makes as independent. When extended specifically to LeBron James' 2016–2017 season, a model depending on the previous shot (single-step Markovian) does not clearly beat a model with independent outcomes. An error-correcting variable length model of two parameters, where James shoots a higher percentage after a missed free throw than otherwise, is more predictive than either model.

Potential kernel, hitting probabilities and distributional asymptotics

$\mathbb{Z}^{d}$-extensions of probability-preserving dynamical systems are themselves dynamical systems preserving an infinite measure, and generalize random walks. Using the method of moments, we prove a generalized central limit theorem for additive functionals of the extension of integral zero, under spectral assumptions. As a corollary, we get the fact that Green–Kubo’s formula is invariant under induction. This allows us to relate the hitting probability of sites with the symmetrized potential kernel, giving an alternative proof and generalizing a theorem of Spitzer. Finally, this relation is used to improve, in turn, the assumptions of the generalized central limit theorem. Applications to Lorentz gases in finite horizon and to the geodesic flow on Abelian covers of compact manifolds of negative curvature are discussed.

Analytical solution on dosage of self-healing capsules in materials with two-dimensional multi-shaped crack patterns

This article presents a numerical method for determining the dosage of pre-embedded capsules in self-healing materials with complex crack patterns. The crack distribution on the surface of materials is simplified into a two-dimensional (2D) multi-shaped geometrical structure composed of triangles, rhombuses, and hexagons with specified area fractions, and further decomposed into three separate mono-shaped crack systems. Then, the dosage of capsules required to heal the cracks in each mono-shaped crack system is computed. According to the area fraction of each mono-shaped polygon in the whole system, the integrated models of crack-hitting probability by the capsules and the capsule dosage for the multi-shaped crack system are derived. The analytical results reveal that the dosage of capsules significantly depends on the spatial distribution of the cracks and the ratio of the capsule length to the crack size. For a certain fixed crack pattern, the size and dosage of capsules will strongly affect crack healing efficiency.

Markov Processes

This chapter deals with Markov processes. It first defines the “Markov property” and shows that all the relevant information about a Markov process assuming values in a finite set of cardinality n can be captured by a nonnegative n x n matrix known as the state transition matrix, and an n-dimensional probability distribution of the initial state. It then invokes the results of the previous chapter on nonnegative matrices to analyze the temporal evolution of Markov processes. It also estimates the state transition matrix and considers the dynamics of stationary Markov chains, recurrent and transient states, hitting probability and mean hitting times, and the ergodicity of Markov chains.