9 Windmill Power

EXAMPLE 11-9 Windmill Power

  A research engineer is investigating the use of a windmill to

  data. The regression model is

  generate electricity and has collected data on the DC output from this windmill and the corresponding wind velocity. The

  y ˆ ⫽ 0.1309 ⫹ 0.2411 x

  data are plotted in Figure 11-14 and listed in Table 11-5 (p.439).

  Inspection of the scatter diagram indicates that the rela-

  The summary statistics for this model are R 2 ⫽ 0.8745,

  tionship between DC output Y and wind velocity (x) may be

  MS

  E ⫽␴ ˆ ⫽ 0.0557 , and F 0 ⫽ 160.26 (the P-value is

  nonlinear. However, we initially fit a straight-line model to the

  ⬍0.0001).

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  CHAPTER 11 SIMPLE LINEAR REGRESSION AND CORRELATION

  DC output,

  Wind velocity, x

  Figure 11-14 Plot of DC output y versus wind velocity x

  – 0.6

  for the windmill data.

  0.4 0.8 1.2 1.6 2.0 2.4 y

  Figure 11-15 Plot of residuals e i versus fitted values yˆ i for the windmill data.

  A plot of the residuals versus is shown in Figure 11-15. y ˆ i

  Figure 11-16 is a scatter diagram with the transformed variable

  This residual plot indicates model inadequacy and implies that

  ⫽1 Ⲑ x . This plot appears linear, indicating that the reciprocal

  the linear relationship has not captured all of the information

  transformation is appropriate. The fitted regression model is

  in the wind speed variable. Note that the curvature that was ap-

  y ˆ

  ⫽ 2.9789 ⫺ 6.9345

  x¿

  parent in the scatter diagram of Figure 11-14 is greatly

  amplified in the residual plots. Clearly some other model form

  The summary statistics for this model are R 2 ⫽ 0.9800,

  must be considered.

  MS

  E ⫽␴ ˆ ⫽ 0.0089 , and F 0 ⫽ 1128.43 (the P value is

  We might initially consider using a quadratic model such as

  ⬍0.0001).

  y ⫽␤ 0 ⫹␤ 1 x ⫹␤ 2 x 2 ⫹⑀

  A plot of the residuals from the transformed model ver-

  sus y ˆ is shown in Figure 11-17. This plot does not reveal any

  to account for the apparent curvature. However, the scatter di-

  serious problem with inequality of variance. The normal prob-

  agram of Figure 11-14 suggests that as wind speed increases,

  ability plot, shown in Figure 11-18, gives a mild indication

  DC output approaches an upper limit of approximately 2.5.

  that the errors come from a distribution with heavier tails than

  This is also consistent with the theory of windmill operation.

  the normal (notice the slight upward and downward curve at

  Since the quadratic model will eventually bend downward as

  the extremes). This normal probability plot has the z-score

  wind speed increases, it would not be appropriate for these

  value plotted on the horizontal axis. Since there is no strong

  data. A more reasonable model for the windmill data that in-

  signal of model inadequacy, we conclude that the transformed

  corporates an upper asymptote would be

  model is satisfactory.

  1 y ⫽␤ 0 ⫹␤ 1 a xb⫹⑀

  DC output,

  Figure 11-17 Plot of residuals versus

  Figure 11-16 Plot of D C output versus x¿ ⫽1 Ⲑ x for the

  fitted values yˆ i for the transformed model

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  11-9 REGRESSION ON TRANSFORMED VARIABLES

  Observation

  Wind Velocity DC Output,

  Number, i

  (mph), x i y i

  Figure 11-18 Normal probability plot of

  the residuals for the transformed model for

  the windmill data.

  17 7.85 Table 11-5 Observed Values y 2.179

  i and Regressor Variable x i

  for Example 11-9

  Wind Velocity

  DC Output,

  Number, i

  (mph), x i