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8. In this problem we develop the rudiments of the theory of linear regression. Suppose that associated with an experiment there are two random variables y and x. If the outcomes of severa] measurements of y and x are plotted on a two-dimensional graph, the result may look somewhat like that shown in Figure 4.6. These results could be effectively summarized by saying that y is approximately a linear func- tion of x. So y would be described by the equation y = a + bx which LEAST-SQUARES ESTIMATION is represented by the dashed line in the figure. A natural way to choose the appropriate dashed line is to choose that line which minimizes the total sum of the squared errors E, e;? where e, = y, – (a + bx) is the vertical distance between an observation point on the graph and the dashed line. Figure 4.6 Regression

8. In this problem we develop the rudiments of the theory of linear regression. Suppose that associated with an experiment there are two random variables y and x. If the outcomes of severa] measurements of y and x are plotted on a two-dimensional graph, the result may look somewhat like that shown in Figure 4.6. These results could be effectively summarized by saying that y is approximately a linear func- tion of x. So y would be described by the equation y = a + bx which LEAST-SQUARES ESTIMATION is represented by the dashed line in the figure. A natural way to choose the appropriate dashed line is to choose that line which minimizes the total sum of the squared errors E, e;? where e, = y, – (a + bx) is the vertical distance between an observation point on the graph and the dashed line. Figure 4.6 Regression

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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8. In this problem we develop the rudiments of the theory of linear regression. Suppose that associated with an experiment there are two random variables y and x. If the outcomes of severaļ measurements of y and x are plotted on a two-dimensional graph, the result may look somewhat like that shown in Figure 4.6. These results could be effectively summarized by saying that y is approximately a linear func- tion of x. So y would be described by the equation y = a + bx which LEAST-SQUARES ESTIMATION. is represented by the dashed line in the figure. A natural way to choose the appropriate dashed line is to choose that line which minimizes the total sum of the squared errors E, e;? where e, = y, - (a + bx;) is the vertical distance between an observation point on the graph and the dashed line. Figure 4.6 Regression (a) Show that the best linear approximation is given by y = ỹ + b(x- x) where E* b= y = (b) Show that b may be alternatively expressed as E(v - )(x – X) N
Transcribed Image Text:8. In this problem we develop the rudiments of the theory of linear regression. Suppose that associated with an experiment there are two random variables y and x. If the outcomes of severaļ measurements of y and x are plotted on a two-dimensional graph, the result may look somewhat like that shown in Figure 4.6. These results could be effectively summarized by saying that y is approximately a linear func- tion of x. So y would be described by the equation y = a + bx which LEAST-SQUARES ESTIMATION. is represented by the dashed line in the figure. A natural way to choose the appropriate dashed line is to choose that line which minimizes the total sum of the squared errors E, e;? where e, = y, - (a + bx;) is the vertical distance between an observation point on the graph and the dashed line. Figure 4.6 Regression (a) Show that the best linear approximation is given by y = ỹ + b(x- x) where E* b= y = (b) Show that b may be alternatively expressed as E(v - )(x – X) N
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