least square method example pdf

Abstract. >>/Font << /TT2 32 0 R/TT0 33 0 R/TT1 34 0 R/C2_0 35 0 R/TT3 36 0 R>> 1.Graphical method 2.Method of group averages 3.Method of moments 4.Method of least squares. stream Example of a Straight LineFit a straight line to the x and y values in thefollowing Table:5.119=∑ ii yx28=∑ ix 0.24=∑ iy1402=∑ ixxi yi xiyi xi21 0.5 0.5 12 2.5 5 43 2 6 94 4 16 165 3.5 17.5 256 6 36 367 5.5 38.5 4928 24 119.5 140 /Filter /FlateDecode The fundamental equation is still A TAbx DA b. Itissupposedthat x isan independent (orpredictor)variablewhichisknownexactly, while y is a dependent (or response) variable. Least Squares Fit (1) The least squares ﬁt is obtained by choosing the α and β so that Xm i=1 r2 i is a minimum. The most commonly used method for ﬁnding a model is that of least squares estimation. Weighted least squares play an important role in the parameter estimation for generalized linear models. /Filter /FlateDecode /PTEX.InfoDict 30 0 R y�H5�[@�z!��;#��݃Y��G�':A��NE^"��瀓��@9�w�9YKI�2�N8�F��Dla&Ǎ�p/Tw��X*�Ȧ?��~h�"�R3k�J�v�)��a`Y��4}H��L��cJE2�^vvR gH�*G��UR��RY��rvv. x��\K�$�q�ϯ蛫�R� �/&)J�C2)j��a��w��n��4ŕ��7]�眙((�t/7D^��Ǘ �v3�Bn�?5�o��^��}�z��/�� W��+AiT��R��o��lwC��A��3�Kh&H)�Gl*��vO۝�W�t��ni��{��݉z��i /Length 705 In this example, let m = 1, n = 2, A = £ 1 1 ⁄, and b = £ 2 ⁄. /CS0 31 0 R stream p + 1 coefﬁcients. It minimizes the sum of the residuals of points from the plotted curve. 2 Generalized and weighted least squares 2.1 Generalized least squares Now we have the model Here is a short unofﬁcial way to reach this equation: When Ax Db has no solution, multiply by AT and solve ATAbx DATb: Example 1 A crucial application of least squares is ﬁtting a straight line to m points. Least squares method is one of the important method of estimating the trend value. The method of least square • Above we saw a discrete data set being approximated by a continuous function • We can also approximate continuous functions by simpler functions, see Figure 3 and Figure 4 Lectures INF2320 – p. 5/80 /Length 3970 We deal with the ‘easy’ case wherein the system matrix is full rank. 3 The Method of Least Squares 4 1 Description of the Problem Often in the real world one expects to ﬁnd linear relationships between variables. Thus, we are seeking to solve Ax = b; >> 2.3 Algebra of least squares xڕ[ۮ%7}�_я)=��-E#�I�@ /ProcSet [ /PDF /Text ] The least squares (LS) estimates for β 0 and β 1 are /FormType 1 6 Least Squares Adjustment and ﬁnd the partial derivatives of ϵ with respect to the intercept θ0 and the slope θ1 ∂ϵ ∂θ0 ∑ n i=1 (yi −(θ0 +θ1xi))(−1) = −∑n i=1 yi +nθ0 +θ1 i=1 xi (23) ∂ϵ ∂θ1 ∑n i=1 (yi −(θ0 +θ1xi))(−xi) = −∑ n i=1 xiyi +θ0 ∑n i=1 xi +θ1 i=1 x2 i. 13 0 obj << >> stream 8 0 obj /Resources 11 0 R Suppose that we performed m measurements, i.e. /Filter /FlateDecode They are connected by p DAbx. /PTEX.PageNumber 1 �U��^R�S�N��=ұ��o��ex��Tw��5�x��̳�'��n��|P�+@+�e�r�͂C��Qp�R�u�0 ��y�DX%�翏hRV�IYލF �@O�l�_�-�#��@�C\ǨP2 ;���ɧ�و�-ا�� ٦��C耳u�5L*�1v[ek�"^h��<6�L�G�H�s��8�{��W� ΒW@=��~su��ra$�r The method of least squares is a standard approach in regression analysis to approximate the solution of overdetermined systems (sets of equations in which there are more equations than unknowns) by minimizing the sum of the squares of the residuals made in the results of every single equation.. Modi cations include the following. /Font << /F17 6 0 R /F15 9 0 R >> stream Let us consider a simple example. %�쏢 14 0 obj << Problem: Suppose we measure a distance four times, and obtain the following results: 72, 69, 70 and 73 units 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 These points are illustrated in the next example. 3 The Method of Least Squares 5 1 Description of the Problem Often in the real world one expects to ﬁnd linear relationshi psbetween variables. To test /Type /Page Least Squares with Examples in Signal Processing1 Ivan Selesnick March 7, 2013 NYU-Poly These notes address (approximate) solutions to linear equations by least squares. /Length 196 '\�;\eP��-��[j��qj#D�� Z��_i��VZ /Matrix [0.00000000 -1.00000000 1.00000000 0.00000000 127.55906700 656.70867900] �T��9Y��K!&��_�-YM9 v�R(��;PxFN.Я�]�;�ābZ04�2$��^�ݞi�x�J��Q�q�K�2��kIl��d�� ۝Yx:� So it's the least squares solution. The method of least squares gives a way to find the best estimate, assuming that the errors (i.e. Σx 2 is the sum of squares of units of all data pairs. values y were measured for specified values of t: Our aim is to model y(t) … ��ۛ��ޓĨPQ��Po�Z�i��ۘ8��pڍ5κ��ۿ@Hh�ʔ��8Sq�2`/L��>l��x�~��]�3/4�r#��Bu,�Uݞ-n�V��8O�쭿��6�L��/;p��w�|GKB�p��Z;z��kR8�}��ԉJ��Dz�-��2�4HH�s(��>�p�,�=w}�ƪۀ{F^��C]u;�V�D�,��x(��k��;g��Y�녴�C:��{ ��: .��ɘ4d��:��{�c/��b�G�k��ٗ5%k�l��H�Gr��AW�sҫ�rʮ�� Ol��=%�"kt�֝e"{�%��Իe�|�Lx:V��|��Y��R-Ƒ`�u@EY��4�H� S��VMi��*�lSM��3닾I��6ݼ�� '-S�f� This is done by finding the partial derivative of L, equating it to 0 and then finding an expression for m and c. After we do the math, we are left with these equations: If the system matrix is rank de cient, then other methods are �(� ��Ͱ6� GkmD�g�}��0ԅ�U��[��Q�u�q߃�ڑ̦��6�$�}��D��Vk>�u&'6A�b`dA�ĴP0-�~��;r3��:��F��q�5��i�A$~"�x�0 e3t�>�^(��t�s|G_ it is indeed the case that the least squares solution can be written as x = A0t, and in fact the least squares solution is precisely the unique solution which can be written this way. >> endobj ��Z{��}0�h�B�F�C�� +N��Q`B/�� [�L�@�Fx��ۄ>Xi5~��{�6;ߪ��k�FK��(�Ԫ��>�`m7"!Z��$n��r i� Example Method of Least Squares The given example explains how to find the equation of a straight line or a least square line by using the method of least square, which is very useful in statistics as well as in mathematics. We will analyze two methods of optimizing least-squares problems; the Gauss-Newton Method and the Levenberg Marquardt Algorithm. %PDF-1.4 2 Chapter 5. The minimum requires ∂ρ ∂α ˛ ˛ ˛ ˛ β=constant =0 and ∂ρ ∂β ˛ ˛ ˛ ˛ α=constant =0 NMM: Least Squares … /ExtGState << Not surprisingly there is typically some orthogonality or the Pythagoras theorem behind them. stream Least Square is the method for finding the best fit of a set of data points. time, and y(t) is an unknown function of variable t we want to approximate. >> /Font << /F17 6 0 R /F15 9 0 R >> We discuss the method of least squares in the lecture. by the method of least squares General problem: In our all previous examples, our problem reduces to nding a solution to a system of n linear equations in m variables, with n > m. Using our traditional notations for systems of linear equations, we translate our problem into matrix notation. We can then use this to improve our regression, by solving the weighted least squares problem rather than ordinary least squares (Figure 5). <> endobj /Type /Page /Contents 3 0 R ∑y = na + b∑x ∑xy = ∑xa + b∑x² Note that through the process of elimination, these equations can be used to determine the values of a and b. et'�#��J�4ψ�Qfh��b]�8˃m��hB��1w�1��X3r�2��fףt�\�r�m�vH}�>�@��h�f� ��oŰ]Št�2�n:�u��OT��FYZ��ٍ�e��ō��w�"��\�(y'N��JD=o /MediaBox [0 0 612 792] Let ρ = r 2 2 to simplify the notation. �. /Type /XObject Least Squares The symbol ≈ stands for “is approximately equal to.” We are more precise about this in the next section, but our emphasis is on least squares approximation. ]f �t�D��[f��o�rT{�� W$�Fő��(��7�_�J��+*��dޖ�+��B��F�pf��a�b�ɠ3��e6��\+��إb��k�?e��)2FD�A�ʜ~��t$P-�T˵1�� >~'��+OwS( y��L�~8�� /5�K ��嵊��8Fendstream We call it the least squares solution because, when you actually take the length, or when you're minimizing the length, you're minimizing the squares of the differences right there. /Resources 15 0 R the differences from the true value) are random and unbiased. /ProcSet [ /PDF /Text ] c��6�� -�a��6tw��Ƃq��ހ�� h�q�3�|�{@ Now, to find this, we know that this has to be the closest vector in our subspace to b. endobj Least-square method Let t is an independent variable, e.g. /Resources 1 0 R 1 0 obj << For example, the force of a spring linearly depends on the displacement of the spring: y = kx (here y is the force, x is the displacement of the spring from rest, and k is the spring constant). 11 0 obj << Least Squares Regression Line Example Suppose we wanted to estimate a score for someone who had spent exactly 2.3 hours on an essay. To test I’m sure most of us have experience in drawing lines of best fit , where we line up a ruler, think “this seems about right”, and draw some lines from the X to the Y axis. >> /Resources << 2 0 obj << /BBox [218.26774600 90.70867900 566.00000000 780.00000000] Least Squares method. square of the usual Pearson correlation of xand y. 3.1.1 Introduction More than one explanatory variable In the foregoing chapter we considered the simple regression model where the dependent variable is related to one explanatory variable. /ProcSet [ /PDF /Text ] 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. PART I: Least Square Regression 1 Simple Linear Regression Fitting a straight line to a set of paired observations (x1;y1);(x2;y2);:::;(xn;yn). /Contents 17 0 R ��& ��Otm�:�Ag�q�t��3�'D��a��)� �?��P",� @��D��9��`��&��q�,1a�\5Ƹ� y҉�~ֲ!w�8T{��$A��d�AVʒ&��i07��U!� �0��/�)�x��R8��ܼ+X�T��B��-. The determination of the relative orientation using essential or fundamental matrix from the observed coordinates of the corresponding points in two images. Find α and β by minimizing ρ = ρ(α,β). And we call this the least squares solution. This document describes least-squares minimization algorithms for tting point sets by linear structures or quadratic structures. >> endobj ɃX�zl�)r4�Cу��Nm�m��.��T�n@�6l.C��|C)��$^�W��ۙ +h��d�1.�Ɏ�A�2��b��D�'��qF��Ɛ��-}�c�n��B˪TS�;�w��i��6��y��B�4T��m�o6k��K�d��^��ԩ��f��QY��HHznmM*i�16�I坢�[��xg�Ͼ�mYe��UV�'�^�],Na`��xb��vӑRl��Q��1��3E�9:T*%*��j�rU��sX��0o�9� bu[ʟbT�� S�v�Ŧ�6�"�� i��)��0�>��l��o�":��!��&hbe ;D�\��6I�i�Su�� ÈNB��}K��6!�FN�&�I%t�̉�0�Ca� a��YHR#�4R-�Z �ڀZ��v��3��-��de8�*]t�� N � /Length 1949 1��j�kG�c��^JN�An�o��V��6NI�-� ;L�J��7��?�� "��qc�E't�Zyr��I}�F��(U�R��W/m ?��R�j ��XixȠܿ{̮'v��O~c�Y. Maths reminder Find a local minimum - gradient algorithm When f : Rn −→R is differentiable, a vector xˆ satisfying ∇f(xˆ) = 0 and ∀x ∈Rn,f(xˆ) ≤f(x) can be found by the descent algorithm : given x 0, for each k : 1 select a direction d k such that ∇f(x k)>d k <0 2 select a step ρ k, such that x k+1 = x k + ρ kd k, satisﬁes (among other conditions) >> endobj >>>> The organization is somewhat di erent from that of the previous version of the document. /Contents 13 0 R endobj It gives the trend line of best fit to a time series data. Nonetheless, formulas for total fixed costs (a) and variable cost per unit (b)can be derived from the above equations. 12 0 obj << >> endobj This method is most widely used in time series analysis. The advantages and dis- In order to compare the two methods, we will give an explanation of each methods’ steps, as well as show examples of two di erent function types. The following example based on the same data as in high-low method illustrates the usage of least squares linear regression method to split a mixed cost into its fixed and variable components. 16 0 obj << /MediaBox [0 0 612 792] xڅXK��6��z�јE==�h��I�$�͵��+��l~}�EI�YD$g83��7�u�?�1�E��BI�"X%l�$ /PTEX.FileName (figura3.pdf) /Parent 10 0 R /ColorSpace << Let us discuss the Method of Least Squares … %PDF-1.3 Title: Abdi-LeastSquares-pretty.dvi Created Date: 9/23/2003 5:46:46 PM /Parent 10 0 R 3.1 Least squares in matrix form E Uses Appendix A.2–A.4, A.6, A.7. ANOVA decompositions split a variance (or a sum of squares) into two or more pieces. Least squares and linear equations minimize kAx bk2 solution of the least squares problem: any xˆ that satisﬁes kAxˆ bk kAx bk for all x rˆ = Axˆ b is the residual vector if rˆ = 0, then xˆ solves the linear equation Ax = b if rˆ , 0, then xˆ is a least squares approximate solution of the equation in most least squares applications, m > n and Ax = b has no solution /Type /Page x��n�0��~ Nonlinear Least-Squares Problems with the Gauss-Newton and Levenberg-Marquardt Methods Alfonso Croeze1 Lindsey Pittman2 Winnie Reynolds1 1Department of Mathematics Louisiana State University Baton Rouge, LA 2Department of Mathematics University of Mississippi Oxford, MS July 6, 2012 Croeze, Pittman, Reynolds LSU&UoM /Parent 10 0 R An example of the least squares method is an analyst who wishes to test the relationship between a company’s stock returns, and the returns of … x�m�?� ��~�a ��mbٌC��O�Fۺ��=ڪ�60ܽw��E��tl/��)E��c2��F�^MC2Y��H��}L�R/�1vk6;�٭�j.��X�7aI9��ң�f��dת.�'~v�.�[�"��ˆ��;Տ��z��d>�D��D�'W|��̭��Zi��~GD>��zSH�endstream /MediaBox [0 0 612 792] 17 0 obj << Some examples of using homogenous least squares adjustment method are listed as: The determination of the camera pose parameters by the Direct Linear Transformation (DLT). �7~~zi�ǳ��#�ȿv#&�0b2=FS.�*u�x�'ʜ��t돑i��L�}o��B�&��a��wy̘��82:q��I��T��ʔ4h��6��&::�O��m8��&1cR 3&sZ�Nr�d��y>�.nڼ$�ҙ~�i�ٲ��IyC�`� �j &��`2'$�q��1鷲��Ů]�/]�e��U^�5!�Fn�'i!R�v[��8��D:s��Bs�5)6�:1��W��&0endstream Example 24: Use least-squares regression to fit a straight line to x 1 3 5 7 10 12 13 16 18 20 y 4 5 6 5 8 7 6 9 12 11 a 7.3 - 0.3725 *10.5 3.3888 0.3725 10 *1477 105 10 *906 105 *73 n x ( x ) n (x y ) x y a 0 2 i 2 i i i i i 1 ¦ ¦ ¦ ¦ ¦ Exercise 24: It is always a good idea to plot the data points and the regression line to see /GS0 37 0 R The following are standard methods for curve tting. Least square method 1. For example, the force of a spring linearly depends on the displacement of the spring: y = kx (here y is the force, x is the displacement of the spring from rest, and k is the spring constant). /Filter /FlateDecode >> endobj Equation (2.7) is an example of an ANOVA (short for analysis of variance) decomposition. least squares solution). Now that we have determined the loss function, the only thing left to do is minimize it. squares which is an modiﬁcation of ordinary least squares which takes into account the in-equality of variance in the observations. A section on the general formulation for nonlinear least-squares tting is now available. /Subtype /Form P. Sam Johnson (NIT Karnataka) Curve Fitting Using Least-Square Principle February 6, … ]��3�O|��aB��̓�#��P/�l��*Y��>��ͅ��j��!��T��(a[��n�E��>vOU��*���(5��@��+qqn��8d��Z0r��Hم�j�ݧH'�d��,&:W�݄)�o�:�d��=�}չ{,��Mj+�|��EN:�,zz�z�!u�Ul�]S9� 1%�a� �Keb��ϳw=.L��"4��{'1t�#�^\��k��3k�ᦑf�~��p~]�d�WlMi�u�q�E�]��BN�N2�uc��Q��)�Af��3M��Jq��v ��Ę��B�g��;�Hn��=؀��Lb��$R�(^ �Zy�՘��;%�2��z�!CMKD_h�$%pqbG��J�~�`+��C;U�r��/,��.&[��p�r��Mwn��S� �8�@�{��z�� o#�|V��t��h �R�;�n� 3 0 obj << Mathematical expression for the straight line (model) y = a0 +a1x where a0 is the intercept, and a1 is the slope.
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