IB DP Maths: AI HL复习笔记4.3.1 Non-linear Regression

What is non-linear regression?

• You have already seen that linear regression is when you can use a straight line to fit bivariate data
• Non-linear regression is when you can use a curve (rather than a straight line) to fit bivariate data
• In your exam the regression could be:

How do I find the equation of the non-linear regression model?

• Using your GDC:
  • Type the two sets of the data into your GDC
  • Select the relevant model
    • The exam question will tell you which model to use
  • Your GDC will calculate the constants

What is a residual?

What is a least squares regression curve?

• The least squares regression curve can be thought of as a “curve of best fit” y = f(x)
• For a given type of model the least squares regression curve minimises the sum of the square residuals
  • Your GDC calculates the constants for the least squares regression curves

Why is the sum of the square residuals not always a good measure of fit?

• If two models are formed using the same number of pairs of data then the sum of the square residuals is a good measure of fit

What is the coefficient of determination?

• The coefficient of determination is a measure of fit for a model
  • If the coefficient of determination is 0.57 this means 57% of the variation of the y-variable can be explained by the variation in the x-variable
  • The other 43% can be explained by other factors
  • The higher this proportion the more the model fits the data

How do I calculate the coefficient of determination?

Does the coefficient of determination determine the validity of a model?

• If R² is close to 1 then the model fits the data well
  • However this alone does not guarantee that it is a good model for the relationship between the two variables
• Consider the scenario where there are big gaps between data pointsand a model which fits the data well
  • The model only fits the data at the data points
  • As there are gaps between the data points the model might not be a good fit for these areas
• Different types of models have different number of parameters
  • Therefore using different types of models to fit the same data will have different levels of accuracy
  • Linear models need at least two pairs of data
  • Quadratic models need at least three pairs of data
  • Cubic models need at least four pairs of data
    • Using four pairs of data will mean the cubic model will have R² = 1
      This is because the cubic graph will go through all four pieces of data – the value is likely to decrease as extra pairs of data are included
    • However this does not mean it is a better fit than the quadratic model
    • The quadratic model could be more accurate as it has one more pair of data than is needed