Regression with a straight line with a piecewise

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= 70 fitting the data quite well. Regression with a straight line with a piecewise linear function PAGE 43
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2.3 Qualitative Independent Variables c circlecopyrt HYON-JUNG KIM, 2017 Polynomial Regression In many cases the usual linear relationship between the dependent variable and any independent variable we have assumed may not be adequate. One way to account for such a nonlinear relationship is through a polynomial or trigonometric regression model. Such models are linear in the parameters and the least squares method can be used for estimation of the parameters as long as the usual assumptions on the errors are appropriate. For example, a model for a single predictor, X , is: Y = β 0 + β 1 X + β 2 X 2 + ... + β p X p + ǫ , where p is the degree of the polynomial. For lower degrees, the relationship has a specific name (i.e., p = 2 is called quadratic, p = 3 is called cubic, p = 4 is called quartic, and so on). However, polynomial regression models may have other predictor variables in them as well, which could lead to interaction terms and the model can grow depending on your application. The design matrix for the second-degree polynomial model is: X = 1 X 1 X 2 1 1 X 2 X 2 2 . . . . . . . . . 1 X n X 2 n . For the polynomial regression in one independent variable, an important aspect that distinguishes it from other multiple regression models is that the mean of the dependent variable is a function of a single independent variable. The fact that the independent vari- ables in a simple polynomial model are functions of a single independent variable affects the interpretation of the parameters. In the second-degree model, the parameter β 1 is the slope only at X = 0. The parameter β 2 is half the rate of change in the slope of E ( Y ). The higher-degree polynomial models provide greatly increased flexibility in the response surface. Although it is unlikely that any complex process will be truly polynomial in form, the flexibility of the higher-degree polynomials allows any true model to be approximated to any desired degree of precision. PAGE 44
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2.3 Qualitative Independent Variables c circlecopyrt HYON-JUNG KIM, 2017 The full second-degree polynomial model in two variables is Y i = β 0 + β 1 X i 1 + β 2 X i 2 + β 3 X 2 i 1 + β 4 X i 1 X i 2 + β 5 X 2 i 2 + ǫ i . The degree (or order) of an individual term in a polynomial is defined as the sum of the powers of the independent variables in the term. The degree of the entire polynomial is defined as the degree of the highest-degree term. The squared terms allow for a curved response in each variable and the product term allows for the surface to be twisted. A quadratic response surface will have a maximum, a minimum, or a saddle point, depending on the coefficients in the regression equation. (Refer to Box and Draper (1987) for a further discussion of the analysis of the properties of quadratic response surfaces.) Due to its extreme flexibility, some caution is needed in the use of polynomial models; it is easy to overfit a set of data with polynomial models. Extrapolation is particularly dangerous
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