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Assuming that I have data with repeated measures (or, in other words, multiple time series of different realizations of the same process), can I train a Gaussian Process on this data? In fact, is there a good way of achieving this, e.g. by training a GP for each day and averaging the parameters or something like that?

Thank you in advance!

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You could use a (locally) periodic covariance function. Effectively, this provides a way of saying that you expect data points from similar times of the day to have similar values, all else equal.

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A bit late but an alternative to David's suggestion would be to combine your daily data sets.

Assuming the mean function is constant from one set of measurements to the next you could create a combined data set in which you store: the unique observed covariate values, $\vec x_{i}$; the number of times each unique combination of covariate values has occurred, $m_{i}$; the average value of the response, $\bar y_i$, for observations with these covariate values.

The combined data can then be used as the training data in Gaussian Process Regression but where the known constant noise variance, $\sigma^2$, is divided by $m_i$.

In other words, the sample average process is the same as the process you are interested in but where the noise variance is inversely proportional to the number of measurements at each unique covariate value.

For inference regarding $\sigma^2$ you could also store the average squared response, $\bar{y^2}_i$.

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