ES 240: Scientific and Engineering Computation. ES 240: Scientific and Engineering Computation. Chapter 13: Linear Regression Chapter 13: Linear Regression Measure of Location Measure of Location � Arithmetic mean : the sum of the individual data points ( y i ) divided by the number of points n : ∑ y i y = n � Median : the midpoint of a group of data. Uchechukwu Ofoegbu Temple University � Mode : the value that occurs most frequently in a group of 13. 1: Statistical Review data. ES 240: Scientific and Engineering Computation. Chapter 13: Linear Regression ES 240: Scientific and Engineering Computation. Chapter 13: Linear Regression Measures of Spread Measures of Spread Normal Distribution Normal Distribution � Standard deviation : S t s y = n − 1 where S t is the sum of the squares of the data residuals: and n -1 is referred to as the degrees of freedom . � Sum of squares : ∑ ( ) 2 S t = y i − y � Variance : 2 − ( ) ∑ ∑ ∑ 2 ( ) 2 y i − y y i y i / n 2 = = s y n − 1 n − 1 c.v. = s y y × 100% ES 240: Scientific and Engineering Computation. Chapter 13: Linear Regression ES 240: Scientific and Engineering Computation. Chapter 13: Linear Regression Descriptive Statistics in MATLAB Descriptive Statistics in MATLAB Histograms in MATLAB Histograms in MATLAB � MATLAB has several built-in commands to compute and display � [ n , x ] = hist( s , x ) descriptive statistics. Assuming some column vector, s : – Determine the number of elements in each bin of data in s . x is a vector containing the center values of the bins. – mean( s ), median( s ), mode( s ) � [ n , x ] = hist( s , m ) • Calculate the mean, median, and mode of s . mode is a part of the statistics toolbox. – Determine the number of elements in each bin of data in s using m bins. – min( s ), max( s ) x will contain the centers of the bins. The default case is m =10 • Calculate the minimum and maximum value in s . – var( s ), std( s ) � hist( s , x ) or hist( s , m ) or hist( s ) • Calculate the variance and standard deviation of s – With no output arguments, hist will actually produce a histogram. � Note - if a matrix is given, the statistics will be returned for each column. 1
ES 240: Scientific and Engineering Computation. ES 240: Scientific and Engineering Computation. Chapter 13: Linear Regression Chapter 13: Linear Regression Linear Least Linear Least- -Squares Regression Squares Regression � Linear least-squares regression is a method to determine the “best” coefficients in a linear model for given data set. � “Best” for least-squares regression means minimizing the sum of the squares of the estimate residuals. For a straight line model, this gives: n n ( ) ∑ ∑ = = − − 2 2 S e y a a x Uchechukwu Ofoegbu r i i 0 1 i = = i 1 i 1 Temple University = a int ercept 13. 2: 0 Linear Least Squares Regression = a slope 1 � This method will yield a unique line for a given set of data. ES 240: Scientific and Engineering Computation. Chapter 13: Linear Regression ES 240: Scientific and Engineering Computation. Chapter 13: Linear Regression Least- Least -Squares Fit of a Straight Line Squares Fit of a Straight Line Example Example � Using the model: ∑ ∑ ∑ − ( ) − 360 ( ) 5135 ( ) V F n x i y i x i y i = 8 312850 a 1 = = 19.47024 (m/s) (N) ( ) y = a 0 + a 1 x ∑ ∑ ( ) − 360 ( ) 2 2 − 2 8 20400 n x i x i i x i y i (x i ) 2 x i y i ( ) = − 234.2857 a 0 = y − a 1 x = 641.875 − 19.47024 45 the slope and intercept producing the best fit can be 1 10 25 100 250 est = − 234.2857 + 19.47024 v F found using: 2 20 70 400 1400 3 30 380 900 11400 ∑ ∑ ∑ 4 40 550 1600 22000 − n x i y i x i y i a 1 = 5 50 610 2500 30500 ( ) ∑ ∑ 2 − 2 n x i x i 6 60 1220 3600 73200 a 0 = y − a 1 x 7 70 830 4900 58100 8 80 1450 6400 116000 Σ 360 5135 20400 312850 ES 240: Scientific and Engineering Computation. Chapter 13: Linear Regression ES 240: Scientific and Engineering Computation. Chapter 13: Linear Regression Quantification of Error Quantification of Error Standard Error of the Estimate Standard Error of the Estimate � Regression data showing (a) the spread of data around the mean of � Recall for a straight line, the sum of the squares of the estimate the dependent data and (b) the spread of the data around the best residuals: fit line: n n ∑ ∑ ( ) S r = = y i − a 0 − a 1 x i 2 2 e i i = 1 i = 1 � Standard error of the estimate: � The reduction in spread represents the improvement due to linear regression. S r s y / x = n − 2 2
ES 240: Scientific and Engineering Computation. ES 240: Scientific and Engineering Computation. Chapter 13: Linear Regression Chapter 13: Linear Regression Goodness of Fit Goodness of Fit Example Example � C oefficient of determination – the difference between the sum of the squares of the data residuals and V F est = − 234.2857 + 19.47024 v F the sum of the squares of the estimate residuals, normalized by the sum (m/s) (N) ∑ of the squares of the data residuals: ( ) S t = y i − y 2 = 1808297 (y i - ȳ ) 2 i x i y i a 0 +a 1 x i (y i -a 0 -a 1 x i ) 2 r 2 = S t − S r ∑ ( ) S r = y i − a 0 − a 1 x i = 216118 1 10 25 -39.58 380535 4171 2 S t � r 2 represents the percentage of the original uncertainty 2 20 70 155.12 327041 7245 1808297 s y = = 508.26 explained by the model. 3 30 380 349.82 68579 911 8 − 1 4 40 550 544.52 8441 30 216118 � For a perfect fit, S r =0 and r 2 =1. s y / x = = 189.79 8 − 2 5 50 610 739.23 1016 16699 r 2 = 1808297 − 216118 � If r 2 =0, there is no improvement over simply picking the mean. 6 60 1220 933.93 334229 81837 = 0.8805 1808297 7 70 830 1128.63 35391 89180 � If r 2 <0, the model is worse than simply picking the mean! 88.05% of the original uncertainty 8 80 1450 1323.33 653066 16044 has been explained by the Σ 360 5135 1808297 216118 linear model ES 240: Scientific and Engineering Computation. Chapter 13: Linear Regression ES 240: Scientific and Engineering Computation. Chapter 13: Linear Regression Nonlinear Relationships Nonlinear Relationships � Linear regression is predicated on the fact that the relationship between the dependent and independent variables is linear - this is not always the case. � Three common examples are: Uchechukwu Ofoegbu β = α x exponentia l : y e 1 Temple University 1 13. 3: Linearization of Nonlinear Relationships Linearization of Nonlinear Relationships = α β power : y x 2 2 x = α saturation - growth - rate : y β + 3 x 3 ES 240: Scientific and Engineering Computation. Chapter 13: Linear Regression ES 240: Scientific and Engineering Computation. Chapter 13: Linear Regression Linearization Linearization Transformation Examples Transformation Examples � One option for finding the coefficients for a nonlinear fit is to linearize it. For the three common models, this may involve taking logarithms or inversion: Model Nonlinear Linearized y = α 1 e β 1 x ln y = ln α 1 + β 1 x exponential : y = α 2 x β 2 log y = log α 2 + β 2 log x power : + β 3 x y = 1 1 1 y = α 3 saturation -growth - rate : β 3 + x α 3 α 3 x 3
ES 240: Scientific and Engineering Computation. ES 240: Scientific and Engineering Computation. Chapter 13: Linear Regression Chapter 13: Linear Regression Example Example � Ex 13.7 Uchechukwu Ofoegbu Temple University 13. 4: Linear Regression in Matlab Linear Regression in Matlab ES 240: Scientific and Engineering Computation. Chapter 13: Linear Regression ES 240: Scientific and Engineering Computation. Chapter 13: Linear Regression Lab Lab MATLAB Functions MATLAB Functions � Ex 13.5 � Linregr function � MATLAB has a built-in function polyfit that fits a least-squares nth – Solve the problem by hand and using the linregr function order polynomial to data : and compare – p = polyfit( x , y , n ) • x : independent data • y : dependent data • n : order of polynomial to fit • p : coefficients of polynomial f ( x )= p 1 x n + p 2 x n -1 +…+ p n x + p n +1 � MATLAB’s polyval command can be used to compute a value using the coefficients. – y = polyval( p , x ) 4
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