Given two series $(x_1,...x_n)$ and $(y_1,...y_n)$, and assume that we know $x_{n+1}$. Given the fact that the pearson correlation won't change in the next observation of $y_{n+1}$, can we bound the error of predicting $y_{n+1}$ in terms of pearson correlation of two series?
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kjetil b halvorsen
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Raba Poco
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1At the very least you would need extra assumptions, e.g. about the relationship between $x$ and $y$. For instance, without any assumptions, $y_{n+1}$ could be anything and hence the error can be arbitrarily large. – JohnA May 15 '15 at 22:17
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Right. I edited my question – Raba Poco May 16 '15 at 11:43
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As said in comments, your question is not entirely clear. You say "Given the fact that the pearson correlation won't change in the next observation of $y_{n+1}$", but correlation do not apply to a single observation, it applies to some well-defined population (or model). But maybe you try to say that the model do not change, that still do not give information about which model you have in mind.
If what you have in mind is a standard linear model (which would be my guess), then you can find an answer here: Obtaining a formula for prediction limits in a linear model which tells you how to calculate prediction intervals in linear regression. The same principles (but not formulas) can be used for other kinds of models. For the difference between confidence intervals and prediction intervals see Difference between confidence intervals and prediction intervals or Linear regression prediction interval
kjetil b halvorsen
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