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Non-linear Regression (DP IB Applications & Interpretation (AI))

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    What is a non-linear regression model?

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  • What is a non-linear regression model?

    A non-linear regression model is a curve that is used to model bivariate data rather than a straight-line.

  • True or False?

    You need to find the equation of any non-linear regression model by hand using algebra.

    False.

    You do not need to find the equation of any non-linear regression model by hand using algebra.

    You have to use your GDC to find equations.

  • f open parentheses x close parentheses is a regression model which is used to predict the values of y.

    What is meant by a residual?

    f open parentheses x close parentheses is a regression model which is used to predict the values of y.

    A residual is the difference between an actual value and the predicted value.

    Residual = y subscript i minus f open parentheses x subscript i close parentheses.

  • What is denoted by S S subscript r e s end subscript?

    S S subscript r e s end subscript denotes the sum of the square residuals of a regression model.

    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.

    This is not given in your exam formula booklet.

  • True or False?

    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.

    True.

    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?

    For a given type of model, a least squares regression curve is the curve that minimises the sum of the square residuals.

  • What is the coefficient of determination (R squared)?

    The coefficient of determination (R squared) is a measure of fit for a model.

    It is the proportion of the variance of the dependent variable that can be explained by the variance of the independent variable.

  • What is the value of the coefficient of determination for a model that is a perfect fit for some data?

    The coefficient of determination is equal to 1 for a model that is a perfect fit for some data.

  • True or False?

    For a linear regression model, the coefficient of determination is equal to the square of the PMCC, i.e. R squared equals r squared.

    True.

    For a linear regression model, the coefficient of determination is equal to the square of the PMCC, i.e. R squared equals r squared.

  • If you have to decide on a model based solely on the coefficients of determination, should you choose the one with the smaller or bigger coefficient?

    If you have to decide on a model based solely on the coefficients of determination, you choose the one with the bigger coefficient.

    A bigger coefficient of determination implies a better fit with the data.

  • True or False?

    Logarithmic scales increase exponentially.

    True.

    Logarithmic scales increase exponentially.

  • What is a log-log graph?

    A log-log graph uses logarithmic scales for both axes.

  • What is a semi-log graph?

    A semi-log graph only uses a logarithmic scale for the y-axis.

  • Which type of graph appears as a straight line when plotted on a log-log graph?

    A power graph (y equals a x to the power of b) will appear as a straight line when plotted on a log-log graph.

  • Which type of graph appears as a straight line when plotted on a semi-log graph?

    An exponential graph (y equals a b to the power of x) will appear as a straight line when plotted on a semi-log graph.

  • The point open parentheses 3 comma space 1.74 close parentheses lies on the graph where base 10 is used for the logarithms. How would you find the corresponding pair of data open parentheses x comma space y close parentheses?

    Log-log plot with a curve extending from bottom left to top right, labelled log y on the vertical axis and log x on the horizontal axis. The y axis goes from 0 to 2 and the x axis from 0 to 3.

    To find the corresponding pair of data, you would raise both values as powers of 10, i.e. open parentheses 10 cubed comma space 10 to the power of 1.74 end exponent close parentheses.

  • Which variables should you use to linearise the graph of an exponential graph y equals a b to the power of x?

    To linearise the graph of an exponential graph y equals a b to the power of x, you should use the variables x and log y using any base, (although the ln is frequently used).

  • Which variables should you use to linearise the graph of a power graph y equals a x to the power of b?

    To linearise the graph of a power graph y equals a x to the power of b, you should use the variables log x and log y using any base, (although the ln is frequently used).

  • If y equals a b to the power of x, write an expression for log y in terms of x.

    If y equals a b to the power of x, then log y equals log a plus x log b.

  • If y equals a x to the power of b, then write an expression for log y in terms of log x.

    If y equals a x to the power of b, then log y equals log a plus b log x.

  • If log y equals a x plus b then which type of function is y: exponential or power?

    If log y equals a x plus b then y is an exponential graph.

  • If you know the equation of the straight line that is produced when a power graph is plotted on a log-log graph, how can you find the equation of the power function?

    For example, if you know ln y equals 2 plus 3 ln x, how can you find the values of a and b where y equals a x to the power of b?

    If you know the equation of the straight line that is produced when a power graph is plotted on a log-log graph, then to find the equation of the power function:

    • linearise the general form of a power function, ln y equals ln a plus b ln x,

    • equate coefficients with the equation of the straight line,

    • solve to find the values of a and b.

    For example, if you know ln y equals 2 plus 3 ln x, then ln a equals 2 rightwards double arrow a equals e squared and b equals 3.

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      Number Toolkit

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      Correlation & Regression

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      Non-linear Regression

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      Probability Distributions

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      Random Variables

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      Binomial Distribution

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      Differential Equations

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      Coupled & Second Order Differential Equations