Lecture-5-2-Least square-slides

Lecture-5-2-Least square-slides - BME 5020 Computer and...

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BME5020 2006, Fall Semester |Biomedical Engineering, Wayne State University Least Squares Regression BME 5020 Computer and Mathematical Application in Bioengineering © 2006 Jingwen Hu October 5, 2006
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BME5020 2006, Fall Semester |Biomedical Engineering, Wayne State University Curve Fitting • There are two general approaches for curve fitting: Data exhibit a significant degree of scatter . The strategy is to derive a single curve that represents the general trend of the data. Data is very precise. The strategy is to pass a curve or a series of curves through each of the points. • In engineering two types of applications are encountered: • Trend analysis. Predicting values of dependent variable, may include extrapolation beyond data points or interpolation between data points. • Hypothesis testing. Comparing existing mathematical model with measured data.
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BME5020 2006, Fall Semester |Biomedical Engineering, Wayne State University Curve Fitting
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BME5020 2006, Fall Semester |Biomedical Engineering, Wayne State University Mathematical Background Simple Statistics/ • In course of engineering study, if several measurements are made of a particular quantity, additional insight can be gained by summarizing the data in one or more well chosen statistics that convey as much information as possible about specific characteristics of the data set. • These descriptive statistics are most often selected to represent • The location of the center of the distribution of the data, • The degree of spread of the data.
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BME5020 2006, Fall Semester |Biomedical Engineering, Wayne State University Mathematical Background
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BME5020 2006, Fall Semester |Biomedical Engineering, Wayne State University Mathematical Background Arithmetic mean . The sum of the individual data points (y i ) divided by the number of points (n). Standard deviation . The most common measure of a spread for a sample. or n i n y y i , , 1 , K = = = = 2 ) ( 1 y y S n S S i t t y ( ) 1 / 2 2 2 = n n y y S i i y
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BME5020 2006, Fall Semester |Biomedical Engineering, Wayne State University Mathematical Background Variance . Representation of spread by the square of the standard deviation. Coefficient of variation . Has the utility to quantify the spread of data. 1 ) ( 2 2 = n y y S i y Degrees of freedom % 100 . . y S v c y =
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BME5020 2006, Fall Semester |Biomedical Engineering, Wayne State University Mathematical Background
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BME5020 2006, Fall Semester |Biomedical Engineering, Wayne State University Least Squares Regression Linear Regression • Fitting a straight line to a set of paired observations: (x 1 , y 1 ), (x 2 , y 2 ),…,(x n , y n ). y=a 0 +a 1 x+e a 1 - slope a 0 - intercept e- error, or residual, between the model and the observations
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BME5020 2006, Fall Semester |Biomedical Engineering, Wayne State University Least Squares Regression Criteria for a “Best” Fit/ • Minimize the sum of the residual errors for all available data: n = total number of points • However, this is an inadequate criterion, so is the sum of the absolute values = = = n i i o i n i i x a a y e 1 1 1 ) ( = = = n i i i n i i x a a y e 1 1 0 1
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This note was uploaded on 04/04/2008 for the course BME 5020 taught by Professor King during the Fall '06 term at Wayne State University.

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Lecture-5-2-Least square-slides - BME 5020 Computer and...

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