Revision NotesExam QuestionsFlashcards
IBMaths: AI HLDPRevision Notes4. Statistics & Probability4.3 Further Correlation & Regression4.3.1 Non-linear Regression

Non-linear Regression (DP IB Maths: AI HL)

Revision Note

Dan

Author

Dan

Last updated

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:
    • Linear: space y equals a x plus b
    • Quadratic: space y equals a x squared plus b x plus c
    • Cubic: space y equals a x cubed plus b x squared plus c x plus d
    • Exponential: space y equals a b to the power of x or y equals a straight e to the power of b x end exponent
    • Power: space y equals a x to the power of b
    • Sine: space y equals a sin left parenthesis b x plus c right parenthesis plus d

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
  • You can use logarithms to linearise exponential and power relationships
    • Power:  space y equals a x to the power of b then ln y equals ln a plus b ln x
      • ln y and ln x will have a linear relationship
    • Exponential: space y equals a b to the power of x then ln y equals ln a plus x ln b
      • ln y and x will have a linear relationship

Examiner Tip

  • 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

Worked example

Scarlett and Violet collect data on the length of a film (x minutes) and the audience rating (y %).

x

75

93

101

107

115

124

132

140

171
y

83

75

51

38

47

56

76

91

70

a)
Scarlett claims that there is a cubic relationship. Find the equation of a cubic regression model of the form space y equals a x cubed plus b x squared plus c x plus d.

4-3-1-ib-ai-hl-non-linear-regression-a-we-solution

b)
Violet claims that there is a sine relationship. Find the equation of a sine regression model of the form space y equals a sin left parenthesis b x plus c right parenthesis plus d.

4-3-1-ib-ai-hl-non-linear-regression-b-we-solution

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

4-3-1-ib-ai-hl-non-linear-regression-c-we-solution

Least Squares Regression Curves

What is a residual?

  • Given a set of n pairs of data and a regression model y = f(x)
  • A residual is the actual y-value (from the data) minus the predicted y-value (using the regression model)
    • space y subscript i minus f open parentheses x subscript i close parentheses
  • The sum of the square residuals is denoted by S S subscript r e s end subscript
    • S S subscript r e s end subscript equals sum from i equals 1 to n of open parentheses y subscript i minus f open parentheses x subscript i close parentheses close parentheses squared
  • If you have two regression models using the same data then the one with the smaller S S subscript r e s end subscript fits the data better

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
  • If two models use different number of pairs of data then S S subscript r e s end subscript is not always a good measure of fit
    • The sum will increase with more pairs of data and so can no longer be compared against a data set with a different number of pairs
    • Compare the two scenarios
      • 10 pairs of data and the absolute value of each residual is 15 then S S subscript r e s end subscript equals 10 cross times 15 squared equals 2250
      • 2250 pairs of data and the absolute value of each residual is 1 then S S subscript r e s end subscript equals 2250 cross times 1 squared equals 2250
    • They have the same value of S S subscript r e s end subscript but the residuals in the second scenario are much smaller
  • Your GDC may give you the mean squared error
    • M S e equals 1 over n S S subscript r e s end subscript equals 1 over n sum from i equals 1 to n of open parentheses y subscript i minus f open parentheses x subscript i close parentheses close parentheses squared
    • This is a better measure of fit
    • You do not need to know this for your exam but it might help with your understanding

Worked example

Jet is the owner of a gym and he is testing different prices options. The table below shows the number of new members per month (M) and the price of a monthly membership (£ p).

p

10

20

30
M

97

68

55

Jet believes that he can fit the data with either the model M subscript 1 left parenthesis p right parenthesis equals fraction numerator 2700 over denominator p plus 20 end fraction or the model M subscript 2 left parenthesis p right parenthesis equals fraction numerator 2100 over denominator p plus 10 end fraction.

Jet wants to choose the model with the smallest value for the sum of square residuals.

Determine which model Jet should choose.

4-3-1-ib-ai-hl-least-squares-regression-we-solution

The Coefficient of Determination

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
  • The coefficient of determination is denoted by R²
    • R² ≤ 1
    • R² = 1 means the model is a perfect fit for the data
    • The closer to 1 the better the fit
    • R² is usually greater than or equal to zero
      • R² can be negative but this is outside the scope of this course
  • If the regression model is linear then the coefficient of determination is equal to square of the PMCC
    • R squared equals r squared for linear models
    • Some GDCs will simply denote R² as r² due to its connection to the PMCC for linear models

How do I calculate the coefficient of determination?

  • When finding the constants for regression models your GDC might give you the value of R squared
    • You will only be asked to calculate the coefficient of determination for models for which GDCs give the value of R²
  • The coefficient of determination can be calculated by
    • R squared equals 1 minus fraction numerator S S subscript r e s end subscript over denominator S S subscript t o t end subscript end fraction
      • Where S S subscript t o t end subscript equals sum from i equals 1 to n of open parentheses y subscript i minus y with bar on top close parentheses squared
    • You do not need to know this formula but it might help with your understanding

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 points and 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

Worked example

Data is collected on the lengths of cheetahs (x metres) and their average running speeds (y ms-1).

x

1.21

1.33

1.12

1.45

1.42

1.39

1.24

1.19

1.32
y

24.3

25.1

22.2

35.1

35.1

33.4

27.1

23.1

24.8

a)
Find the equation of the least squares regression curve using:
(i)
a quadratic model space y equals a x squared plus b x plus c.
(ii)
an exponential model space y equals a b to the power of x.

4-3-1-ib-ai-hl-coefficient-determination-a-we-solution

b)
Based solely on the coefficients of determination, suggest which model is better fit for the data.

4-3-1-ib-ai-hl-coefficient-determination-b-we-solution

You've read 0 of your 5 free revision notes this week

Sign up now. It’s free!

Join the 100,000+ Students that ❤️ Save My Exams

the (exam) results speak for themselves:

    Test yourselfFlashcards

    Did this page help you?

    Previous:4.2.3 Linear RegressionNext:4.3.2 Logarithmic Scales
    Dan

    Author: Dan

    Expertise: Maths

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.