A markov chain problems and solutions pdf optimization-based technique for bootstrapping and simulating Markov chains is proposed. The relevant states and memory of a Markov chain are identified as minimum information loss solution. Numerical applications are provided to validate the theoretical results. Markov chain theory is proving to be a powerful approach to bootstrap and simulate highly nonlinear time series.
In particular, the choice of memory lags and the aggregation of irrelevant states are obtained by looking for regularities in the transition probabilities. Our approach is based on an optimization model. A discussion based on information theory is developed to define the desirable properties for such optimal criteria. Two numerical tests are developed to verify the effectiveness of the proposed method. Check if you have access through your login credentials or your institution. APT-MCMC is a free software package .
APT-MCMC is designed to solve challenging ODE parameter estimation inverse problems. Optimization benchmarks verify APT-MCMC performance and guide algorithm tuning. It combines affine-invariant ensemble of samplers and parallel tempering MCMC techniques to improve the simulation efficiency. Simulations use Bayesian inference to provide probability distributions of parameters, which enable analysis of multiple minima and parameter correlation. Several MCMC hyperparameters were analyzed: number of temperatures, ensemble size, step size, and swap attempt frequency.
Heuristic tuning guidelines are provided for setting these hyperparameters. Agenda: Markov Chains – Structure – 1st Order Markov Chains – Higher Order Markov Chains Hidden Markov Model – Structure – Why using HMMs ? This article assumes that you have a pretty good knowledge of some Machine Learning techniques such as Naive Bayes and have a background about Recursion and Dynamic Programming. Markov Chains is a probabilistic model that consists of finite states. The states are connected with each other through edges, and there are values that are associated with each of these edges. Sunny and Rainy are 2 states. Given the above diagram, we can say that the probability to go from the Sunny state to the Rainy state is 0.
1, while going to Sunny again is 0. We can represent the previous chain using Stochastic Transition Matrix, in which each row describes the transition from a state to the other states, so, the sum of each row must be equal to 1. This is called a Right Stochastic Transition Matrix. From the current state, we want to predict the next state. This could be done by multiplying the current state vector with the transition matrix, the result of the multiplication will be a probabilities vector and each element of the vector shows the probability to be in a state in the next time. The resultant vector shows that the probability to be Sunny is much higher than to change to state Sunny.