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I'm reading the book "Fault-Diagnosis Systems" by Isermann in the par. 9.2.1a. The author explains how to estimate the parameter of a linear difference equation using Least Squares. We start with a standard ordinary linear equation with an error term:

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the equation is called 9.15. Then,

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The first thing i don't understand is the syntax used by the author. What's the meaning of the | inside $\hat{y}(k|k-1)$ ?

ps: I'm sorry if what i ask is trivial, but when i have a kind of doubt like this, i don't know how to solve it if not asking it in this site...

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  • $\begingroup$ Hi: that notation denotes the prediction ( also referred to as estimate ) of $y(k)$ at time $(k-1)$. $\endgroup$ – mark leeds Jan 26 '19 at 13:45
  • $\begingroup$ Is the conditional probability ? $\endgroup$ – Jhdoe Jan 26 '19 at 14:15
  • $\begingroup$ Hi: No. Assume the model is estimated which means that the coefficients are estimated. Then, that notation denotes the prediction of the next value in the series at time $k$, given that one is at time $k-1$. Maybe check out a book or a discussion on arima models if this is not clear. $\endgroup$ – mark leeds Jan 27 '19 at 4:01
  • $\begingroup$ So it's like the author is saying : "suppose we use a model to predict y. Than we get something like this" .... :S ? $\endgroup$ – Jhdoe Jan 27 '19 at 10:23
  • $\begingroup$ yes. but y ONE step ahead. time matters a lot. $\endgroup$ – mark leeds Jan 27 '19 at 19:09
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The notation $ \hat{y} \left( k \mid k - 1 \right) $ usually means this is an estimated value of $ y \left( k \right) $ given all the available data up to time index $ k - 1 $.
So generally speaking, this is a prediction of one step in time of the data.

The case above also suggests linear estimation.
Namely, $ \hat{y} \left( k \right) $ is built using linear combination of all the given data.

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