Explanation The concept of linear regression states that two continuous variables representing a data set, and the sum of the squares of the deviations of the actual y -values from those on the line is minimum. For example, consider the data set, x 1 2 4 5 y 1 3 3 4 The data points are ( 1 , 1 ) , ( 2 , 3 ) , ( 4 , 3 ) and ( 5 , 4 ) . According to the least-squares assumption, the line which best fits is the line for which the sum of the squares of the y -deviations is minimum. For that, the y -deviations are y 1 − 1 , y 2 − 3 , y 3 − 3 , y 4 − 4 The sum of the squares of y -deviations has to be minimised. Thus, consider the sum of the squares is S , S = ( y 1 − 1 ) 2 + ( y 2 − 3 ) 2 + ( y 3 − 3 ) 2 + ( y 4 − 4 ) 2 …… (1) The points ( 1 , y 1 ) , ( 2 , y 3 ) , ( 4 , y 3 ) and ( 5 , y 4 ) must be solutions of the equation y = m x + b , thus, y 1 = m ( 1 ) + b y 2 = m ( 2 ) + b y 3 = m ( 4 ) + b y 4 = m ( 5 ) + b Substitute the values of y 1 , y 2 , y 3 and y 4 in the equation (1), S = ( m + b − 1 ) 2 + ( 2 m + b − 3 ) 2 + ( 4 m + b − 3 ) 2 + ( 5 m + b − 4 ) 2 Differentiate the both sides with respect to b , d S d b = d ( ( m + b − 1 ) 2 + ( 2 m + b − 3 ) 2 + ( 4 m + b − 3 ) 2 + ( 5 m + b − 4 ) 2 ) d b Apply the chain rule of differentiation, d S d b = d d b ( m + b − 1 ) 2 + d d b ( 2 m + b − 3 ) 2 + d d b ( 4 m + b − 3 ) 2 + d d b ( 5 m + b − 4 ) 2 = 2 ( m + b − 1 ) ( 0 + b 1 − 1 − 0 ) + 2 ( 2 m + b − 2 ) ( b 1 − 1 ) + 2 ( 4 m + b − 3 ) ( 0 + b 1 − 1 − 0 ) + 2 ( 5 m + b − 4 ) ( 0 + b 1 − 1 − 0 ) = 2 m + 2 b − 2 + 4 m + 2 b − 6 + 8 m + 2 b − 6 + 10 m + 2 b − 8 = 24 m + 8 b − 22 Differentiate the both sides with respect to m , d S d m = d ( ( m + b − 1 ) 2 + ( 2 m + b − 3 ) 2 + ( 3 m + b − 3 ) 2 + ( 5 m + b − 4 ) 2 ) d m Apply the chain rule of differentiation, d S d m = d d m ( m + b − 1 ) 2 + d d m ( 2 m + b − 3 ) 2 + d d m ( 4 m + b − 3 ) 2 + d d m ( 5 m + b − 4 ) 2 = 2 ( m + b − 1 ) ( m 1 − 1 + 0 − 0 ) <

Calculus and Its Applications Plus MyLab Math with Pearson eText -- Access Card Package (11th Edition) (Bittinger, Ellenbogen & Surgent, The Calculus and Its Applications Series)

11th Edition

ISBN: 9780133795561

Author: Marvin L. Bittinger, David J. Ellenbogen, Scott J. Surgent

Publisher: PEARSON

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Calculus and Its Applications Plus MyLab Math with Pearson eText -- Access Card Package (11th Edition) (Bittinger, Ellenbogen & Surgent, The Calculus and Its Applications Series)

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Solution Summary: The author explains the concept of linear regression, which states that two continuous variables represent a data set, and the sum of the squares of y -deviations is minimum.

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