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I am now trying to fit a line to some data with Excel. I do this in order to estimate some values within the data range, e.g. $X=16.$ As you can see, a 5-th grade polynomial fits very well.

However, when running the regression analysis in Minitab, this same 5-th grade polynomial model gives a PREDICTED $R^2 = 0.$ According to Minitab, if this "pred $R^2$" is $0,$ then the model has "no power of prediction, even for predictions inside the data range". But looking at the plot it looks like this line can predict very well inside the data range. Am I missing something?

enter image description here

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    $\begingroup$ I don't think a 5th order polynomial fits well. the first 9 terms are nearly zero. Can you make the y-axis log-scaled? What physics or "physics" generated this data, and how do they speak to the 5th order model, or whatever model? You can use spline fit to estimate locally. You can use something like AIC to find the best spline parameter/s, and then you can use the spline to interpolate. You could use local linear or quadratic to interpolate as well. Can you supply a table of actual values? Are you using excel? Why do you care about f(16)? $\endgroup$
    – EngrStudent
    Commented Apr 11, 2017 at 12:59
  • $\begingroup$ Is grade the same as order/degree? $\endgroup$
    – Michael R. Chernick
    Commented Apr 11, 2017 at 15:56
  • $\begingroup$ Any details on how that "Predicted $R^2$" is calculated? $\endgroup$
    – David
    Commented Mar 12, 2019 at 7:46

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Predicted R^2 in Minitab is based on predicting an observation not fitted by the model. You're using a 5th degree polynomial (why you selected 5 is another question) to estimate approximately 10 data points. Removing one of the points will greatly change your estimated coefficients. In other words, you are incredibly overfitting your data and your model has no prediction power for samples not fitted with this 5th degree polynomial. Try a simpler model, such as quadratic.

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