extrapolation

Examples of extrapolation in the following topics:

• Some Pitfalls: Estimability, Multicollinearity, and Extrapolation

  • Some problems with multiple regression include multicollinearity, variable selection, and improper extrapolation assumptions.
  • Typically, the quality of a particular method of extrapolation is limited by the assumptions about the regression function made by the method.
  • If the method assumes the data are smooth, then a non-smooth regression function will be poorly extrapolated.
  • Even for proper assumptions about the function, the extrapolation can diverge strongly from the regression function.
  • This divergence is a specific property of extrapolation methods and is only circumvented when the functional forms assumed by the extrapolation method (inadvertently or intentionally due to additional information) accurately represent the nature of the function being extrapolated.

• Extrapolation is treacherous

  • Applying a model estimate to values outside of the realm of the original data is called extrapolation.
  • If we extrapolate, we are making an unreliable bet that the approximate linear relationship will be valid in places where it has not been analyzed.

• The Regression Method

  • Extrapolation is also frequently used, in which data points beyond the known range of values is predicted.
  • An example of extrapolation, where data outside the known range of values is predicted.
  • The red points are assumed known and the extrapolation problem consists of giving a meaningful value to the blue box at $x=7$.

• Predictions and Probabilistic Models

  • Prediction outside this range of the data is known as extrapolation.
  • Performing extrapolation relies strongly on the regression assumptions.
  • This means that any extrapolation is particularly reliant on the assumptions being made about the structural form of the regression relationship.
  • The implications of this step of choosing an appropriate functional form for the regression can be great when extrapolation is considered.
  • At a minimum, it can ensure that any extrapolation arising from a fitted model is "realistic" (or in accord with what is known).

• Charles' and Gay-Lussac's Law: Temperature and Volume

  • This extrapolation of Charles' Law was the first evidence of the significance of this temperature.

• Enolate Intermediates

  • For common reference, these acidity values have all been extrapolated to water solution, even though the conjugate bases of those compounds having pKas greater than 18 will not have a significant concentration in water solution.

• Acid-Base Reactions

  • Since these studies are generally extrapolated to high dilution, the molar concentration of water (55.5) is constant and may be eliminated from the denominator.
  • Consequently, the value reported here is extrapolated from measurements in much less acidic solvents, such as acetonitrile.
  • Relative acidity measurements in these solvents may be extrapolated to water.
  • For both these groups, the reported pKa values extrapolated to water are approximate, and many have large uncertainties.

• Absolute Zero

  • Note that all of the graphs extrapolate to zero pressure at the same temperature

• Absolute Temperature

  • Note that all of the graphs extrapolate to zero pressure at the same temperature

• Acid Dissociation Constant (Ka)

  • A strong acid is almost completely dissociated in aqueous solution; it is dissociated to the extent that the concentration of the undissociated acid becomes undetectable. pKa values for strong acids can be estimated by theoretical means or by extrapolating from measurements in non-aqueous solvents with a smaller dissociation constant, such as acetonitrile and dimethylsulfoxide.