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Figure 6.7.1 Linear plot of Cp versus time illustrating error between observed data and calculated line
Again, usually the residual or error is assumed to be in the vertical direction although there are programs available that are capable of looking at oblique error in both the x and y direction. For the rest of our modeling discussion we will assume that the error is in the y axis variable only.
Looking at an individual data point and the calculated value with the same x value the residual can be expressed as a simple subtraction.
Equation 6.7.1 Residual in the y direction
The problem is that over all the data points there might be high positive and high negative residuals that might cancel out. An absolute difference would solve this problem but squaring the residual is better statistically and achieves the same result.
Equation 6.7.2 Residual in the y direction squared
This gives us an equation of the residual for one data points. To complete the calculation we need to include the residuals for all the data points. This is called the sum of the squared residuals (SS).
Equation 6.7.3 Sum of the squared residuals
Finally we need to take the error in each data point as a separate value. That is the error may be different for each measured, observed data point. We can compensate for this by applying a weight to each residual thus the usual criteria for a best fit is a minimum sum of the weighted, squared residuals (WSS).
Equation 6.7.4 Weighted sum of squared residuals
The job of the computer program is produce a minimum value for WSS which represents the best fit according to the least squares criteria. Inspection of Equation 6.7.4 leads to the conclusion that this can be achieved by changing the calculated values (Ycalculated,i) by changing the parameter values.
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