# Linear Difference Equation and Method of Least Squares

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:

the equation is called 9.15. Then,

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...

• Hi: that notation denotes the prediction ( also referred to as estimate ) of $y(k)$ at time $(k-1)$. – mark leeds Jan 26 '19 at 13:45
• Is the conditional probability ? – Jhdoe Jan 26 '19 at 14:15
• 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. – mark leeds Jan 27 '19 at 4:01
• So it's like the author is saying : "suppose we use a model to predict y. Than we get something like this" .... :S ? – Jhdoe Jan 27 '19 at 10:23
• yes. but y ONE step ahead. time matters a lot. – mark leeds Jan 27 '19 at 19:09

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 $$k - 1$$.
Namely, $$\hat{y} \left( k \right)$$ is built using linear combination of all the given data.
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