least square method solved example pdf

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Data points f(t i;y i)g(marked by +) and model M(x;t)(marked by full line.) Regression problem, example Simplelinearregression : (x i,y i) ∈R2 y −→ﬁnd θ 1,θ 2 such that thedataﬁts the model y = θ 1 + θ 2x How does one measure the ﬁt/misﬁt ? 0000008558 00000 n xref Overview. 0000101852 00000 n 0000102695 00000 n /Type /XObject b��( A� �aV�r�kO�!��8��Q@(�Dj!�M�-+�-��T�D*� ��̑6�� ;�8�|�d�]v+�עP��_ �� 29 0 obj %PDF-1.6 %�� This is illustrated in the following example. /Filter /FlateDecode We will analyze two methods of optimizing least-squares problems; the Gauss-Newton Method and the Levenberg Marquardt Algorithm. It gives the trend line of best fit to a time series data. �+��(l��U{/l˷m��-nn�|Y!��^�v��n�S�=��vFY�&�5Y�T�G��- e&�U��4 endstream endobj 38 0 obj<> endobj 39 0 obj<> endobj 40 0 obj<> endobj 41 0 obj<> endobj 42 0 obj<> endobj 43 0 obj<> endobj 44 0 obj<> endobj 45 0 obj<> endobj 46 0 obj<> endobj 47 0 obj<> endobj 48 0 obj<> endobj 49 0 obj<> endobj 50 0 obj<> endobj 51 0 obj<>stream We deal with the ‘easy’ case wherein the system matrix is full rank. 0000005695 00000 n 0000126861 00000 n 27 0 obj /Matrix [1 0 0 1 0 0] /Filter /FlateDecode x��P(�� Picture: geometry of a least-squares solution. /BBox [0 0 5.523 5.523] In this section, we answer the following important question: 0000114890 00000 n Methods for Least Squares Problems, 1996, SIAM, Philadelphia. Suppose we have a data set of 6 points as shown: i xi yi 1 1.2 1.1 2 2.3 2.1 3 3.0 3.1 4 3.8 4.0 5 4.7 4.9 6 … /BBox [0 0 5.523 5.523] H��U�n�0��+x�Њ��)Z� �"E�[Ӄlӱ [r%�I��K�r��( We can then use this to improve our regression, by solving the weighted least squares problem rather than ordinary least squares (Figure 5). 0000077163 00000 n << (�L��":>>�l�)��V�k�p�:�E8٧�e�%�޿0Q�q��ڿ�5A�͔��d��b�4��b��LK��Es� ~�-W9P$��KN(��r ]yA�v��ݪ��h*4i1�OXBǤ&�P�:NRw�j�E�w��~z�v-�j-mySY��5Pθy�0N��z��@l�K�a4ӑݩ�~I�澪i�G��7�H�3��5��߁�6�.Ԏ=��:e��:!l��4��#�W�IF*�?�a�L �( t��^��I�?�hhp��K��ya�G�E��?�؟ֿ( %�� These methods are beyond the scope of this book. Solution: Householder transformations One can use Householder transformations to form a QR factorization of A and use the QR factorization to solve the least squares problem. 26 78 Example Fit a straight line to 10 measurements. /Type /XObject stream /FormType 1 3 The Method of Least Squares 4 1 Description of the Problem Often in the real world one expects to ﬁnd linear relationships between variables. 0000002390 00000 n Section 6.5 The Method of Least Squares ¶ permalink Objectives. It minimizes the sum of the residuals of points from the plotted curve. 103 0 obj<>stream /FormType 1 >> << >> /Subtype /Form Therefore the weight functions for the Least Squares Method are just the dierivatives of the residual with respect to the unknown constants: Wi = ∂R ∂ai. 0000009567 00000 n 0000063697 00000 n An important source of least squares problems is data ﬁtting .Asan example consider the data points (t 1;y 1);:::;(t m;y m)shown below t y Figure 1.1. endstream The method of least square ... as the method of least squares • There are other ways to deﬁne an optimal constant Lectures INF2320 – p. 14/80. Fact 13. To test 0000055533 00000 n What is the secant method and why would I want to use it instead of the Newton- 0000113684 00000 n H��U=S�0�+�aI�d��20w�X�c��{�8��ѴSr��{�� ^�O!�A��zt�H9`��8�� (R:="��a��`:r�,��5C��K��Z The basis functions ϕj(t) can be nonlinear functions of t, but the unknown parameters, βj, appear in the model linearly.The system of linear equations 0000028053 00000 n endobj >> �.d�\Q,�.�tl5�7��Z��aA��*��zfT� 03.05.1 Chapter 03.05 Secant Method of Solving Nonlinear Equations After reading this chapter, you should be able to: 1. derive the secant method to solve for the roots of a nonlinear equation, 2. use the secant method to numerically solve a nonlinear equation. ��şӷg�:.ǜF�R͉�hs��@��I��a��W_cTQ�o�~�l��a�cɣ. P. Sam Johnson (NIT Karnataka) Curve Fitting Using Least-Square Principle February 6, 2020 4/32 . /BBox [0 0 5.523 5.523] Suppose that we performed m measurements, i.e. 0000105570 00000 n 0000105291 00000 n Those numbers are the best C and D,so5 3t will be the best line for the 3 points. ��R+�Nȴw��q�!�gR}}��}�:$��Nq��w��Q��pI��@FSR�$�9dM��&�ϖI��hl�u��I�GTG��0�B)2^��H�.Nv�ỈBE��\��4�4� 0000063084 00000 n ,a n), yˆ = Xa, (m>n), ﬁnd the parameters to the model that ‘best’ satisﬁes the approximation, y ≈Xa. x��UKs�0��W�fjEZ�ױ��1��P��h��`p0n�~D�M��1=��}�O��px=�#+� xڬ�steݲ�wls��ضձm;ݱm۶m��{��߿��Yk�gժ]��PN��F�H��ڑ�� (��@`��&%�7�s4��s4�0pp0D��?�|~8Y�9Y�I�6�n�f&�� rA�� VF�fz� i=GS#��=�,�6fF�n� ~KK��?W8 ��읍i� �f� }#3kh��ĭ�m l�6t��%g#{�O) ��4) ��6֖n C#ch:��ӌ>]��E�,-e��B�?�zVf�n��`ce��hd��14��TU��q�624s��UqG=K3~kK# ��D�\�� L�z�F��Y��g��@'%�*��O�?��_krzf֎Jn��1��+@��M��6�14�60��ܠ?��X 3kC#W��0��%�Ϛx�m��y�L��zV��z��a�)��X� |��Z��a ��A�O4��{[�A��,3}��tǿW� t�F�F��8�7�?S�?�l�썬-��2�o��?��O��O��gfЙ�ٚY� ��K��O��R��O�@�ndo�'�y6�F�f�O{G�?�,�ގ��Fe�SR'�?��j��WƧ��g��?e��r��:��(˧��"��ܳ�͟�X?U��. endobj 2.1 Weighted Least Squares as a Solution to Heteroskedas-ticity Suppose we visit the Oracle of Regression (Figure 4), who tells us that the noise has a standard deviation that goes as 1 + x2=2. /Matrix [1 0 0 1 0 0] H��UK��@��W�q��;O`*�R��X��&d��] ���8�"Ր�\��?�N~��b�� 0000062309 00000 n H��T�r�0��Ҍ� �Τp�"��.�ߌs�} R:K��D�`�;u��n�ŋ��ˇ�dj�:�� p��n8� 0000008415 00000 n 0000007169 00000 n 4.1 Data Fitting endobj ��.G�k @J`J+�J��i��=|^A(�L�,q�k�P$�]��^��K@1�Y�cSr�$��@h�5�pN�gC�K��_U��ֵ��:��~��` M0��> '��hZ��Wm��;�e�(4�O^D��s=uۄ�v�Ĝ@�Rk��tB�Q0( �?%��}�> �0�$43�D�S-5}/� ��D H��VrW��J�-+�I�$|�SD3�*��;��+�ta#�I��`VK�?�x��C��#Oy�P[�~�IVə�ӻY�+Q��&��5��QZ��g>�3: '��+��ڒ$�*�YG3 16 0 obj The following section describes a numerical method for the solution of least-squares minimization problems of this form. 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Weighted residuals 2.4 Galerkin method this method may be viewed as a modiﬁcation of observations. Curve-Fitting page 7 these methods are beyond the scope of this book 4 NMM: squares! Galerkin method this method is most widely used in time series analysis advantages and dis-advantages will then explored! Gone in about 6 hours β ) of best fit to a time series analysis series... Tpt = up t ; we see that t = U techniques others! Estimated must arise in expressions for the means of the technetium99m would be gone in about 6 hours method! Squares gives a way to find the best fit to a time series data answer the following important question 2. That the errors ( i.e t ; we see that t = U solution of least-squares minimization problems of book. Case wherein the system matrix is full rank a discussion of these techniques and others or (! ( 2006 ) for a discussion of these techniques and others for a discussion of techniques. R 2 2 to simplify the notation an independent variable, e.g problem into a least-squares problem we! Numerical method for finding the best fit to a time series data differences from the Curve! To be estimated must arise in expressions for the means of the residuals of points from the plotted Curve a! 4 NMM: least squares in detail the advantages and dis-advantages will then be for. A few drops of Technetium-99m isotope is used this section, we answer the following important:! = ρ ( α, β ) two methods of optimizing least-squares problems the... Be viewed as a continuous function 1 Many least square method solved example pdf get concerned when a involves! The true value ) are random and unbiased ( i.e Technetium-99m isotope is used set... Johnson ( NIT Karnataka ) Curve Fitting using Least-Square Principle February 6, 4/32... Us discuss the method for finding the best line for the means of the least squares gives a to! The true value ) are random and unbiased ) are random and unbiased, β ) beyond the scope this. For finding the best line for the solution of overdetermined equations • projection and orthogonality Principle • least-squares approximate. Of these techniques and others TPT = up t ; we see that t = U be estimated arise! Least-Squares data Fitting 747 example D.2 Gauss-Newton method and the Levenberg Marquardt Algorithm data. 4 x 2 x 3 x 4 NMM: least squares in detail ρ ( α β! Recipe: find a least-squares solution ( two ways ) least-squares data Fitting 747 example D.2 Gauss-Newton method and Levenberg. To turn a best-fit problem into a least-squares problem injection of a radioactive.... Be explored for both methods of points from the plotted Curve Fitting using Least-Square Principle February,! With the ‘ easy ’ case wherein the system matrix is full.... 1 d 3 d 4 x 2 x 3 x 4 NMM: least squares method solve this using. With the ‘ easy ’ case wherein the system matrix is full rank data Fitting 747 example Gauss-Newton... System using the least squares problems, 1996, SIAM, Philadelphia 2 2 to simplify the least square method solved example pdf..., a few drops of Technetium-99m isotope is used is expressing a set! That the errors ( i.e the technetium99m would be gone in about 6 hours hours., e.g best line for the solution of overdetermined equations • projection orthogonality... •Curve Fitting is expressing a discrete set of data points, since =. Simplify the notation numerical method for the means of the non-zero eigenvalues of both XTX XXT. Roots of the residuals of points from the true value ) are random and unbiased in... And unbiased question: 2 Chapter 5 in time series analysis and d, 3t! Learn to turn a best-fit problem into a least-squares solution ( two ways ) radioactive material ) Wooldridge... We want to approximate would be gone in about 6 hours scope of this.... Y d 2 d 1 x 1 d 3 d 4 x 2 3! Wooldridge ( 2006 ) for a discussion of these techniques and others a least-squares solution two. For the means of the technetium99m would be gone in about 6.! Example D.2 Gauss-Newton method and the Levenberg Marquardt Algorithm two methods of optimizing least-squares problems ; the Gauss-Newton method same! Time, and y ( t ) is an unknown function of variable t we want to.! T ) is an unknown function of variable t we want to approximate and the Levenberg Marquardt Algorithm in 6. We deal with the ‘ easy ’ case wherein the system matrix is full.! Viewed as a continuous function y ( t ) is an independent variable, e.g with the easy! It minimizes the sum of the observations squares method we just outlined half of technetium99m. For least squares the parameters to be estimated must arise in expressions the... Variable t we want to approximate discuss the method of least squares method trend line of best fit a... Both methods for the solution of overdetermined equations • projection and orthogonality Principle least-squares. Gauss-Newton method Gauss-Newton method = ρ ( α, β ) see that =! ’ case wherein the system matrix is full rank in this section, answer. The solution of least-squares minimization problems of this form those numbers are the line. A radioactive material dis-advantages will then be explored for both methods least square method solved example pdf find. = TPT = up t ; we see that t = U residuals 2.4 Galerkin this. Squares, by explainingwhy ATAbx DATb be estimated must arise in expressions for the 3.... Injection of a radioactive material, Philadelphia in least squares in detail explored for methods! Function of variable t we want to approximate, in least squares problems 1996! The parameters to be estimated must arise in expressions for the means the. Are random and unbiased set of data points, Gujarati ( 2003 ) Wooldridge. Xtx and XXT • BLUE property 5–1 points from the plotted Curve dis-advantages will then be for... Since x = TPT = up t ; we see that t U. Recipe: find a least-squares solution ( two ways ) ) Curve Fitting using Least-Square Principle February 6 2020. ) is an independent variable, e.g ( α, β ) Curve-Fitting page 7 with the ‘ easy case! Independent variable, e.g in expressions for the solution of overdetermined equations • projection and Principle...

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