I am curious how one can generate simulated time series data.I fount a list of simulated series here and a similar tool for stock market. What is best way to generate domain specific time series data with some desired patterns? How one should approach this problem. I know this question is not very complete yet so please feel free to suggest modifications.
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For instance, if you want to simulate an AR1 process in MATLAB, you could proceed as follows:
This generates M points in time of the model: $$X_k = \alpha X_{k1} + \sigma u_k,$$ where $u_k$ is a standard normal variable. If you want a more sophisticated model for some natural (or synthetic) phenomenon then you would have to specify that model. To find a model for the movement of a stock price is a difficult problem. To calibrate (that is, to find the parameters of) such a model is also a difficult problem. 

