Question

In the derivation of the formulas for the least squares linear regression, we consider the function [ S(a, b)=sum_{i=1}^{n}left(a x_{i}+b-y_{i} ight)^{2} ] where ( left(x_{i}, y_{i} ight) ) with ( i=1,2,3, ldots, n ), are given constants representing ( n ) data points in the ( x y- ) plane, and ( a ) and ( b ) are variables of the function ( S ) which ultimately correspond to the slope ( ( a ) ) and ( y )-intercept ( (b) ) of the line of best fit for the given data points. Find each of the following partial derivatives of ( S ). Note, these partials are used to derive a formula for the line of best fit for a given set of data. [ S_{a}= ] [ S_{b}= ] [ S_{a a}= ] [ S_{b b}= ] [ S_{a b}= ] Be sure to simplify ( S_{b b} )

          In the derivation of the formulas for the least squares linear regression, we consider the function
[
S(a, b)=sum_{i=1}^{n}left(a x_{i}+b-y_{i}
ight)^{2}
]
where ( left(x_{i}, y_{i}
ight) ) with ( i=1,2,3, ldots, n ), are given constants representing ( n ) data points in the ( x y- ) plane, and ( a ) and ( b ) are variables of the function ( S ) which ultimately correspond to the slope ( ( a ) ) and ( y )-intercept ( (b) ) of the line of best fit for the given data points.
Find each of the following partial derivatives of ( S ). Note, these partials are used to derive a formula for the line of best fit for a given set of data.
[
S_{a}=
]
[
S_{b}=
]
[
S_{a a}=
]
[
S_{b b}=
]
[
S_{a b}=
]
Be sure to simplify ( S_{b b} )

Show more…

In the derivation of the formulas for the least squares linear regression, we consider the function
[
S(a, b)=sumi=1^nleft(a xi+b-yi
ight)^2
]
where ( left(xi, yi
ight) ) with ( i=1,2,3, ldots, n ), are given constants representing ( n ) data points in the ( x y- ) plane, and ( a ) and ( b ) are variables of the function ( S ) which ultimately correspond to the slope ( ( a ) ) and ( y )-intercept ( (b) ) of the line of best fit for the given data points.
Find each of the following partial derivatives of ( S ). Note, these partials are used to derive a formula for the line of best fit for a given set of data.
[
Sa=
]
[
Sb=
]
[
Sa a=
]
[
Sb b=
]
[
Sa b=
]
Be sure to simplify ( Sb b )

Added by John B.

College Algebra: Real Mathematics, Real People

Ron Larson 7th Edition

Chapter 5

Instant Answer

Solved by Expert Khoi V

03/11/2023

Step 1

Step 1: Find the partial derivative with respect to a: \[ \frac{\partial s}{\partial a} = 2(x_i a - y_i) \] Show more…

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Transcript

00:01 So this problem has you define the partial derivative and also the second partial derivative.

00:05 Please know that the x -i are constant based on whatever they say, it's in the formula.

00:11 So when you take derivative, it's going to be a constant.

00:14 Now, for s of a, i'm going to keep this in the summation notation from 1 to n.

00:20 And think of this like you're doing with respect to a, with respect to a, and you take derivative, of think of it like a times a number constant, c, plus also all these are constant, right? and then you squared it.

00:34 So what's the derivative of that going to be? well, it's just going to be, first of all, it's going to be the whole thing because this is power two, so you're going to be the whole thing, but it's now just power one.

00:45 Right.

00:46 There's a two, right? minus y, i.

00:50 And it's just power one.

00:51 And then you times the derivative of that, times the derivative.

00:54 Of this which is just going to be derivative with respect to a that's just going to be the constant c which is just the xi does that make sense all right and now we do the same thing for s of b i'm gonna i'm gonna keep the summation notation and again it is a square in term of b so you're gonna have a two and then a xi plus b minus y i right and then times the derivative of b would respect it to b that's just one.

01:25 That's constant goal with b.

01:27 It's just one.

01:27 Let's see this is one time b.

01:30 Okay.

01:31 So that's the first derivative.

01:33 Now the next one that i'll ask you is saa, which means it's derivative as a with respect it to a...

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