Gaussian Processes: used in regression and optimisation problems (eg.Markov decision processes: commonly used in Computational Biology and Reinforcement Learning.Random Walk and Brownian motion processes: used in algorithmic trading.Poisson processes: for dealing with waiting times and queues.Some examples of stochastic processes used in Machine Learning are: One of the main application of Machine Learning is modelling stochastic processes. Johnny Rich, The Human Script Introduction “The only simple truth is that there is nothing simple in this complex universe. (Source: ) Stochastic Processes Analysis An introduction to Stochastic processes and how they are applied every day in Data Science and Machine Learning. Auto-Regressive Moving average processes.Random Walk and Brownian motion processes.An introduction to Stochastic processes and how they are applied every day in Data Science and Machine Learning. Having looked on the scenario trees, scenario lattices and stochastic processes, we are now ready to have a look on the stochastic approximation process. It is important to write a function that follows the above criterion as this becomes very important in the stochastic approximation process. Up to this point, the user can now create a random scenario tree and look at its characteristics as well as write a process function that generates trajectories for the stochastic approximation process. In higher dimension, depending on the user, the arrays could be dependent on each other.This is important for the stochastic approximation process. \[M_t = \max \ where N is the number of stages and d is the dimension. The position $X_t$ of a Gaussian random walk after $t$ steps is As stated before, the length of the array and the dimension of the array naturally depends on the characteristics of the scenario tree or scenario lattice that you are using. Generally, a function in d-dimension generates d arrays. What we mean by dimension is that, a function in 1 dimension generates just one array. We have created the first two example functions in 1 and 2 dimensions. We have the following examples in our package: Therefore, knowing the number of stages and the dimension of the states will help the user to design a stochastic function that generates a trajectory with the same length as the number of stages in the scenario tree and with the same dimension as the dimension of the states in the scenario tree. Why is this important? For each iteration, the stochastic approximation algorithm generates one trajectory from the stochastic function that will improve one path in the scenario tree. To be able to do a good approximation, the user of this package should be aware of the number of stages in the scenario tree or scenario lattice that she/he is working on and more importantly, the dimension of the states of nodes in the tree or the scenario lattice. This is the function that will generate trajectories that helps to improve the states of the nodes in the scenario tree or scenario lattice. This is a user-defined function representing the stochastic process in which she/he wants to approximate. With this in mind, in our stochastic approximation process, we approximate a discrete time stochastic process with a scenario tree and a Markovian data process with a scenario lattice. The Markov processes are processes with short memory where the process has memory only at its immediate past. We classify these processes as discrete time stochastic processes and Markovian data processes. However, we find it important to classify the stochastic processes according to their memory. They could also be described with respect to the properties of their distribution functions. For example, they can be classified in terms of boundedness, continuity etc., i.e. There are many ways in which the stochastic processes can be classified. with probability one, all of its trajectories are continuous functions in t. A stochastic process is also called a random process.The stochastic process is said to be continuous, if i.e. Stochastic processes Stochastic processesĪ stochastic process is a sequence of events in which the outcome of any stage depends on some probability.
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