Engineering Analysis ENG 3420 Fall 2009Lecture 21Statistics built-in functionsHistogramsHistogram ExampleLinear Least-Squares RegressionLeast-Squares Fit of a Straight LineExampleNonlinear modelsLinearization of nonlinear modelsTransformation ExamplesLinear Regression ProgramPolynomial least-fit squaresPolynomial RegressionProcess and Measures of FitEngineering Analysis ENG 3420 Fall 2009Dan C. MarinescuOffice: HEC 439 BOffice hours: Tu-Th 11:00-12:0022Lecture 21Lecture 21Last time: RelaxationNon-linear systemsRandom variables, probability distributions, Matlab support for random variablesTodayHistogramsLinear regressionLinear least squares regressionNon-linear data modelsNext TimeMultiple linear regressionGeneral linear squaresStatistics built-in functionsBuilt-in statistics functions for a column vector s: mean(s), median(s), mode(s)Calculate the mean, median, and mode of s. mode is a part of the statistics toolbox.min(s), max(s)Calculate the minimum and maximum value in s.var(s), std(s)Calculate the variance and standard deviation of sIf a matrix is given, the statistics will be returned for each column.Histograms [n, x] = hist(s, x)Determine the number of elements in each bin of data in s. x is a vector containing the center values of the bins.[n, x] = hist(s, m)Determine the number of elements in each bin of data in s using m bins. x will contain the centers of the bins. The default case is m=10 hist(s, x ) or hist(s, m) or hist(s)With no output arguments, hist will actually produce a histogram.Histogram ExampleLinear Least-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:This method will yield a unique line for a given set of data.Sr ei2i1n yi a0 a1xi 2i1nLeast-Squares Fit of a Straight LineUsing the model:the slope and intercept producing the best fit can be found using:y a0 a1xa1n xiyi xiyin xi2 xi 2a0y  a1xExampleV(m/s)F(N)i xiyi(xi)2xiyi1 10 25 100 2502 20 70 400 14003 30 380 900 114004 40 550 1600 220005 50 610 2500 305006 60 1220 3600 732007 70 830 4900 581008 80 1450 6400 116000360 5135 20400 312850a1n xiyi xiyin xi2 xi 28 312850  360 5135 8 20400  360 219.47024a0y  a1x 641.875  19.47024 45  234.2857 Fest 234.285 7 19.47 02 4vNonlinear models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:exp onential : y 1e1xpower : y 2x2saturation - growth - rate : y 3x3 xLinearization of nonlinear modelsxyxxyxyxyxyeyx111:rate-growth-saturationloglog log:powerln ln:lexponentiaLinearizedNonlinearModel3333322211121Transformation ExamplesLinear Regression ProgramPolynomial least-fit squaresMATLAB has a built-in function polyfit that fits a least-squares n-th order polynomial to data:p = polyfit(x, y, n)x: independent datay: dependent datan: order of polynomial to fitp: coefficients of polynomialf(x)=p1xn+p2xn-1+…+pnx+pn+1MATLAB’s polyval command can be used to compute a value using the coefficients.y = polyval(p, x)Polynomial RegressionThe least-squares procedure from can be extended to fit data to a higher-order polynomial. The idea is to minimize the sum of the squares of the estimate residuals.The figure shows the same data fit with:a) A first order polynomialb) A second order polynomialProcess and Measures of FitFor a second order polynomial, the best fit would mean minimizing:In general, this would mean minimizing: The standard error for fitting an mth order polynomial to n data points is:because the mth order polynomial has (m+1) coefficients.The coefficient of determination r2 is still found using:Sr ei2i1n yi a0 a1xi a2xi2 2i1n Sr ei2i1n yi a0 a1xi a2xi2 L  amxim 2i1nsy/ xSrn  m 1 r2St