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

Category: IB课程, 教材笔记, 福利干货 Date: 2022年7月21日上午10:14

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:

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

You can use your GDC to plot the scatter diagram and include the graph of a regression model
- This will allow you to get a sense of how well the model fits the data

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c)Whose model predicts a higher audience rating for a film which is 100 minutes long?

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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

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

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

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