John C. Mayer Introduction to Mathematical Modeling 1 Curve Fitting.

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Presentation transcript:

John C. Mayer Introduction to Mathematical Modeling 1 Curve Fitting

John C. Mayer Introduction to Mathematical Modeling 2 Source of Example The chow/vizsla example is from “Lessons for A First Course in System Dynamics Modeling v1.0” by Diana M. Fisher. Summer Creek Press, 1998.

John C. Mayer Introduction to Mathematical Modeling 3 Procedure Given paired data points (x i,y i ). Produce a scatterplot of the paired data points. Fit a linear equation Y = aX+b and its graph (curve) to the data. Evaluate the fit. Analyze the result.

John C. Mayer Introduction to Mathematical Modeling 4 Research Question Can the appearance of a chow- vizsla hybrid be used to predict its disposition? Dogs are rated based upon their chow-like and vizsla-like characteristics. Scale 0 (most like vizsla) to 10 (most like chow).

John C. Mayer Introduction to Mathematical Modeling 5 Table of Data

John C. Mayer Introduction to Mathematical Modeling 6 Scatterplot of Data

John C. Mayer Introduction to Mathematical Modeling 7 Plotting the Means (x,y)=(5,5.82)

John C. Mayer Introduction to Mathematical Modeling 8 Adding a Trendline e1e1 e3e3 e2e2

John C. Mayer Introduction to Mathematical Modeling 9 Least Squares Fit Minimize the error sum of squares Q =  (e i ) 2 The smaller Q, the closer the correlation coefficient R 2 is to 1. A perfect fit (all points on the line) has R 2 =1.

John C. Mayer Introduction to Mathematical Modeling 10 Trendline and Equation

John C. Mayer Introduction to Mathematical Modeling 11 No Correlation This graph shows essentially no correlation between the variables X and Y.

John C. Mayer Introduction to Mathematical Modeling 12 High Correlation This graph shows a high degree of correlation between X and Y.

John C. Mayer Introduction to Mathematical Modeling 13 Outliers Outliers affect the degree of correlation. Outliers affect the fitted curve.