4950 A common problem in experimental work is to obtaina mathematical relationship y =

Chapter 13, Problem 50

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A common problem in experimental work is to obtain a mathematical relationship \(y=f(x)\) between two variables \(x\) and \(y\) by “fitting” a curve to points in the plane that correspond to experimentally determined values of \(x\) and \(y\), say

                   

                                  \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right), \ldots,\left(x_{n}, y_{n}\right)\)

The curve \(y=f(x)\) is called a mathematical model of the data. The general form of the function \(f\) is commonly determined by some underlying physical principle, but sometimes it is just determined by the pattern of the data. We are concerned with fitting a straight line \(y=mx+b\) to data. Usually, the data will not lie on a line (possibly due to experimental error or variations in experimental conditions), so the problem is to find a line that fits the data “best” according to some criterion. One criterion for selecting the line of best fit is to choose \(m\) and \(b\) to minimize the function

         

                                                   \(g(m, b)=\sum_{i=1}^{n}\left(m x_{i}+b-y_{i}\right)^{2}\)                                  

This is called the method of least squares, and the resulting line is called the regression line or the least squares line of best fit. Geometrically, \(\left|m x_{i}+b-y_{i}\right|\) is the vertical distance between the data point \(\left(x_{i}, y_{i}\right)\) and the line \(y=mx+b\).

                                   

These vertical distances are called the residuals of the data points, so the effect of minimizing \(g(m,b)\) is to minimize the sum of the squares of the residuals. In these exercises, we will derive a formula for the regression line.

Assume that not all the \(x_{i}\) ’s are the same, so that \(g(m,b)\) has a unique critical point at the values of m and b obtained in Exercise 49(c). The purpose of this exercise is to show that g has an absolute minimum value at this point.

(a) Find the partial derivatives \(g_{m m}(m, b), g_{b b}(m, b)\),and \(g_{m b}(m, b)\), and then apply the second partials test to show that g has a relative minimum at the critical point obtained in Exercise 49.

(b) Show that the graph of the equation \(z=g(m,b)\) is a quadric surface.

[Hint: See Formula (4) of Section 11.7.]

(c) It can be proved that the graph of \(z =g(m,b)\) is an elliptic paraboloid. Accepting this to be so, show that this paraboloid opens in the positive \(z \text { - direction }\), and explain how this shows that g has an absolute minimum at the critical point obtained in Exercise 49.

Equation Transcription:

direction

Text Transcription:

y=f(x)

x

x

y

(x_1,y_1),(x_2,y_2),...,(x_n,y_n)

y=f(x)

f

y=mx+b

m

b

g(m,b)=Sum over i=1 ^n(mx_i+b-y_i)^2

|mx_i+b-y_i|

(x_i,y_i)

y=mx +b

g(m,b)

x_i

g(m,b)

m

b

g

g_mm(m,b), g_bb(m,b)

g_mb(m,b)

g

z=g(m,b)

z=g(m,b)

z-direction