Exploiting statistical dependencies of time series with hierarchical correlation reconstruction
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While we are usually focused on predicting future values of time series, it is often valuable to additionally predict their entire probability distributions, for example to evaluate risk or Monte Carlo simulations. On example of time series of ≈ 30000 Dow Jones Industrial Averages, there will be shown application of hierarchical correlation reconstruction for this purpose: mean-square fitting polynomial as joint density for (current value, context), where context is for example a few previous values. Then substituting the currently observed context and normalizing density to 1, we get predicted probability distribution for the current value. In contrast to standard machine learning approaches like neural networks, optimal coefficients here can be inexpensively directly calculated, are unique and independent, each has a specific cumulant-like interpretation, and such approximation can approach complete description of any joint distribution - providing a perfect tool to quantitatively describe and exploit statistical dependencies in time series.
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