Ch E 310 - Fall 10 - Lecture 16

Ch E 310 - Fall 10 - Lecture 16 - Lecture 16 October 19,...

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Lecture 16 – October 19, 2010 Agenda: General Linear Least Squares Regression Linearizing equations (for those than can be) Non-linear Regression: fminsearch Homework 4 assigned (assemble groups of two)
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General Linear Regression Last time we talked about linear regression “General linear regression” refers to the model’s fitting parameters , not the model equation itself Examples: y = a 0 + a 1 x linear in a 0 and a 1 y = a 0 + a 1 x + a 2 x 2 linear in a 0 , a 1 , and a 2 y = a 0 + a 1 exp( x ) linear in a 0 and a 1 y = a 0 + a 1 exp( a 2 x ) non-linear (not valid) The previous method was specific for a linear function, but we can extend this to any arbitrary function for which the parameters are linear (general regression)
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General Linear Regression The general linear least squares model is: y = a 0 z 0 + a 1 z 1 + a 2 z 2 + a 3 z 3 + … + a n z n + e Here, the z i are basis functions (any arbitrary function) and a i are the parameters (constants to be determined) Examples: y = a 0 + a 1 cos( w x ) + a 2 sin( x ) where: z 0 = 1, z 1 = cos( x ), z 2 = sin( x ) y = a 0 + a 1 x + a 2 x 2 where: z 0 = 1, z 1 = x , z 2 = x 2 The equations must follow the form shown at the top
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General Linear Regression The model is: y = a 0 + a 1 cos( w x ) + a 2 sin( x ) The matrix form of this for fitting purposes is: The frequency must be defined (not fitted a parameter)                         1 1 1 2 2 2 0 3 3 3 1 4 4 4 2 5 5 5 6 6 6 1 cos sin 1 cos sin 1 cos sin 1 cos sin 1 cos sin 1 cos sin y x x y x x a y x x a y x x a y x x y x x ww    data function values unknown fitting parameters
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General Linear Regression In matrix notation, we can re-write the equations as: { y } = [ Z ]{ a } + { e } (note the linear form) where { } are columns and [ ] are matrices [ Z ] is a matrix of the calculated values of the basis functions at the measured values of the independent variables { y } contains the observed values of the dependent variable { a } contains the unknown coefficients (parameters) { e } contains the residuals (errors) [ Z ] is m × ( n + 1) : m data points and n + 1 basis functions
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[ Z ] is m × ( n+ 1) : m data points and n + 1 basis functions [ Z ] is populated with calculated values as follows: z ij is the value of the j th basis function calculated at the i th data point For example, row 1 has calculated values for all n + 1 basis functions at the first data point only   mn m m n n z z z z z z z z z Z
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This note was uploaded on 09/21/2011 for the course CH E 310 taught by Professor Staff during the Spring '08 term at Iowa State.

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Ch E 310 - Fall 10 - Lecture 16 - Lecture 16 October 19,...

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