# Hilbert spaces and time series

Suppose that $\{Y_{t}: t \in \mathbb{Z} \}$ is a stationary zero mean time series. Consider the Hilbert space $\mathcal{H}$ generated by the random variables $\{Y_t: t \in \mathbb{Z} \}$ with inner product $$\langle X, Y \rangle = E(XY)$$ and norm $$||X||^2 = E|X|^2$$

Consider the subspace $\mathcal{M}$ generated by the random variables $\{Y_u: u \leq t \}$. Why are future values found by projecting onto the subspace $\mathcal{M}$? For example, why is $Y_{t+1}$ found by $\mathcal{P}_{\mathcal{M}}Y_{t+1}$?

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You must mean: "Why is the prediction of $Y_{t+1}$ (based on the information in $\mathcal M$ and relative to the natural loss function induced by the norm) obtained via the projection operator?" Correct? As currently written the question doesn't make much sense. Perhaps there are typos. –  cardinal Jul 6 '12 at 16:26
Well any prediction of Y beyond t would have to involve only the the Yjs in M since that is all you know about the series at time t. –  Michael Chernick Jul 6 '12 at 16:43
@Michael: The question appears to be "why (orthogonal) projection" not "why use the past", though I think an edit and some further clarification from the OP would be ideal. –  cardinal Jul 6 '12 at 16:49
@cardinal from what part of H is he projecting? If he is projecting from all of H into M then wouldn't he be doing it to predict Yt+1 from the past? I think it is a question that does not deserve a why. You project because the set up is that the present is t and predicting the value for the series at t+1 can only use information from M assuming "future values" means prediction of future values. –  Michael Chernick Jul 6 '12 at 16:58
@MichaelChernick: You seem to be implicitly assuming that the element in $\mathcal M$ that is chosen should be $\mathcal P_{\mathcal M} Y_{t+1}$, whereas, I am guessing this is precisely the question of the OP, i.e., what motivates the choice $P_{\mathcal M} Y_{t+1}$ versus some other element of $\mathcal M$. :) –  cardinal Jul 6 '12 at 17:16

Question: "Why are future values found by projecting onto the subspace..."

Answer: Because the projection is the conditional expectation of $Y_{t+1}$ given the sigma-field generated by $Y_1,\dots,Y_t$, and it is known in time series analysis that this conditional expectation is, in a specific sense, the best predictor.

Here is a sketch of the geometry behind the first claim:

Take $Y$ as being $Y_{t+1}$, and $\mathscr{G}=\sigma(Y_1,\dots,Y_t)$.

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(+1) Depending on exactly is intended by "Consider the subspace $\mathcal M$ generated by the random variables $\{Y_u: u \leq t\}$", $\mathcal M$ may not be (directly) associated with a $\sigma$-algebra and yet the problem makes sense. For example, $\mathcal M$ is often taken to be the set of finite linear combinations of $\{Y_u: u \leq t\}$. –  cardinal Jul 9 '12 at 3:00