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Consider a photo copier. The relationship between an input variable X and the response variable Y is given as follows:
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Input X |
0.01
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0.12
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0.54
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1.52
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5.60
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25.30
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552.00
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Response Y
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0.0072
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0.0719
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0.3720
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0.7080
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0.9260
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0.9750
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0.9800
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The engineer wants to define this relationship between x and y so that the electronic system can be controlled by a programmable chip. An initial relationship (obtained graphically or otherwise) is fitted with the constants A, B, C and D as follows:
Y = A + ( 1 - A - B ) / ( 1 + C X D )
Our objective is now to refine the values of the constants A, B, C and D so the fit is nearly perfect. How?
>
How do we do it?
Our strategy is as follows:
Linear programming is not likely to be of any use since the non linearly is quite marked. Our approach is to create a disturbance about each constant A, B, C and D, fit the equation at each of a set of disturbance and calculate the residual sum of squares. From the perturbations, we decide if the constants should be increased or decreased. We keep on repeating this procedure until the error sum of squares is below a set limit.
Observe the difference!
The data points are shown as black dots The magenta line indicates the initial fit. The blue line indicates the final fit. Note the Breakthrough Improvement in the data fit.
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