| We study dynamical systems which exhibit chaotic dynamics with a focus on sources of real and apparent randomness including sensitivity to perturbation, dynamical noise, measurement uncertainty and finite data samples for inference. This choice of topics is motivated by a desire to adapt established theoretical tools such as the Perron-Frobenius operator and symbolic dynamics to experimental data.; First, we study prediction of chaotic time series when a perfect model is available but the initial condition is measured with uncertainty. A common approach for predicting future data given these circumstances is to apply the model despite the uncertainty. In systems with fold dynamics, we find prediction is improved over this strategy by recognizing this behavior. A systematic study of the logistic map demonstrates prediction can be extended three time steps using an approximation of the relevant Perron-Frobenius operator dynamics.; Next, we show how to infer kth order Markov chains from finite data by applying Bayesian methods to both parameter estimation and model-order selection. In this pursuit, we connect inference to statistical mechanics through information-theoretic (type theory) techniques and establish a direct relationship between Bayesian evidence and the partition function. This allows for straightforward calculation of the expectation and variance of the conditional relative entropy and the source entropy rate. Also, we introduce a method that uses finite data-size scaling with model-order comparison to infer the structure of out-of-class processes.; Finally, we study a binary partition of time series data from the logistic map with additive noise, inferring optimal, effectively generating partitions and kth order Markov chain models. Here we adapt Bayesian inference, developed above, for applied symbolic dynamics. We show that reconciling Kolmogorov's maximum-entropy partition with the methods of Bayesian model selection requires the use of two separate optimizations. First, instrument design produces a maximum-entropy symbolic representation of time series data. Second, Bayesian model comparison with a uniform prior selects a minimum-entropy model of the symbolic data.; This works takes small steps towards the goal stated in the first paragraph. In the concluding chapter we discuss future directions for the development and application of these ideas. |
