This is a question from the book "Introduction to Statistical Learning" and I don't know how to solve it.
5. Consider the fitted values that result from performing linear regression without an intercept. In this setting, the i-th fitted value takes the form
. . . . .\(\displaystyle \hat{y}_i\, =\, x_i\, \hat{\beta}\)
where
. . . . .\(\displaystyle \displaystyle \hat{\beta}\, =\, \left(\sum_{i=1}^n\, x_i\, y_i\right)\, /\, \left(\sum_{i' = 1}^n\, x_{i'}^2\right)\)
Show that we can write
. . . . .\(\displaystyle \displaystyle \hat{y}_i\, =\, \sum_{i' =1}^n\, a_{i'}\, y_{i'}\)
What is \(\displaystyle a_{i'}\)?
Note: We interpret this result by saying that the fitted values from linear regression are linear combinations of the response values.
I generally have problems with these types of questions (especially when summation signs are involved), is there a text book that explains these concepts with applications anyone can recommend?
If you could help me with this question I'd be very thankful!
5. Consider the fitted values that result from performing linear regression without an intercept. In this setting, the i-th fitted value takes the form
. . . . .\(\displaystyle \hat{y}_i\, =\, x_i\, \hat{\beta}\)
where
. . . . .\(\displaystyle \displaystyle \hat{\beta}\, =\, \left(\sum_{i=1}^n\, x_i\, y_i\right)\, /\, \left(\sum_{i' = 1}^n\, x_{i'}^2\right)\)
Show that we can write
. . . . .\(\displaystyle \displaystyle \hat{y}_i\, =\, \sum_{i' =1}^n\, a_{i'}\, y_{i'}\)
What is \(\displaystyle a_{i'}\)?
Note: We interpret this result by saying that the fitted values from linear regression are linear combinations of the response values.
I generally have problems with these types of questions (especially when summation signs are involved), is there a text book that explains these concepts with applications anyone can recommend?
If you could help me with this question I'd be very thankful!
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