7. Consider the time-series (3,1, 1, 3, 5, 7, ). Here, a missing entry is denoted by . What would be the estimated value of the missing entry using linear interpolation on a window of size 3 ?

[Hint] Using linear interpolation over the last three values of 3, 5, and 7, one can derive the relationship y = a x + b, where x is the time-stamp. The values a and b are typically solved using methods such as least squares regression.

[More hints] In a time-series dataset (3,1, 1, 3, 5, 7, ),the * value can be inferred using a linear interpolation,y = a x + b.

Using least squares regression, a and b are estimated as followings:, where N= the number of data points used for buidling the linear model.

When you present the dataset with (x, y)s,x values can be any ordered values. In that case, y is a value measured at a time point x.

(1, -3), (2, -1) , (3, 1) , (4, 3), (5, 5) , (6, 7), (7,*)

The question asks to consider previous three values (window size = 3) to estimate *. Thus you can also consider as followings:

(1, 3), (2, 5) , (3, 7), (4,*)

In either way, you will have the same answer for *.