%PDF-1.2 In this section the situation is just the opposite. ���,y'�,�WҐ0���0U�"y�Ұ�PNK�Tah /Widths[1062.5 531.3 531.3 1062.5 1062.5 1062.5 826.4 1062.5 1062.5 649.3 649.3 1062.5 Least Squares Approximations 221 Figure 4.7: The projection p DAbx is closest to b,sobxminimizes E Dkb Axk2. /LastChar 196 /BaseFont/DKEPNY+CMR8 /Type/Font /LastChar 196 Section 6.5 The Method of Least Squares ¶ permalink Objectives. 31 0 obj /Type/Font 13.1. 639.7 565.6 517.7 444.4 405.9 437.5 496.5 469.4 353.9 576.2 583.3 602.5 494 437.5 593.8 500 562.5 1125 562.5 562.5 562.5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 /Type/Encoding Vocabulary words: least-squares solution. /Encoding 11 0 R Orthogonal polynomials • General orthogonal polynomials — Space: polynomials over domain D — Weighting function: w(x) > 0 888.9 888.9 888.9 888.9 666.7 875 875 875 875 611.1 611.1 833.3 1111.1 472.2 555.6 — Objective: Find a function g(x) from a class G that best approximates f(x), i.e., g =argmin g∈G f −g 2 5. /FirstChar 33 /Subtype/Type1 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. 791.7 777.8] >> >> /FontDescriptor 30 0 R 343.8 593.8 312.5 937.5 625 562.5 625 593.8 459.5 443.8 437.5 625 593.8 812.5 593.8 Here we describe continuous least-square approximations of a function f(x) by using polynomials. /BaseFont/YYYEYA+CMEX10 17 0 obj endobj 295.1 826.4 531.3 826.4 531.3 559.7 795.8 801.4 757.3 871.7 778.7 672.4 827.9 872.8 Figure 1: Least squares polynomial approximation. /LastChar 196 544 516.8 380.8 386.2 380.8 544 516.8 707.2 516.8 516.8 435.2 489.6 979.2 489.6 489.6 The least squares approach puts substantially more weight on a point that is out of line with the rest of the data but will not allow that point to completely dominate the approximation. /FontDescriptor 33 0 R 687.5 312.5 581 312.5 562.5 312.5 312.5 546.9 625 500 625 513.3 343.8 562.5 625 312.5 • Least squares approximation — Data: A function, f(x). In such situations, the least squares solution to a linear system is one means of getting as The optimal linear approximation is given by p(x) = hf,P 0i hP 0,P 0i P 0(x)+ hf,P 1i hP 1,P 1i P 1(x). >> Usually the function φ(x)does not go through points [x i 298.4 878 600.2 484.7 503.1 446.4 451.2 468.8 361.1 572.5 484.7 715.9 571.5 490.3 PDF | On Jan 1, 2020, Ling Guo and others published Constructing Least-Squares Polynomial Approximations | Find, read and cite all the research you need on ResearchGate 4 = 8. x. /BaseFont/LLQVLW+CMMI8 Ͽ=o$����n_7�WOF_����R�P�;��v������������ޞ~�;�i�������/�#��z.�����G��n�����U�2R��)���}5�ʆ�-^�ć3CDW��CIÑo�Ϛ$�L"ҔI v�V�+�ёa�A��.�LK���u3��~>%��k���fu��*��?mTn�ו�p�߬��� �R� Z�3�R���7ED�Ga��@I�+/`w���c�y3�;���!8s��/������r�]�%�,�n�v>�l�/��~%;����j�,kܷ��Β �sG�'?�(��Ki3+�{��"���K�o�G��p%��D�>̑�e�1�h����6�}a�̓��yn1��%-1�܂��k?��˙���}uMA��VJ�. /Filter[/FlateDecode] /FirstChar 33 /Widths[622.5 466.3 591.4 828.1 517 362.8 654.2 1000 1000 1000 1000 277.8 277.8 500 /Subtype/Type1 i x i y i 1 0 1.0000 2 0.25 1.2480 3 0.50 1.6487 4 0.75 2.1170 5 1.00 2.7183 Soln: Let the quadratic polynomial be P 2(x) = a 2x2 +a 1x+a 0. 531.3 531.3 413.2 413.2 295.1 531.3 531.3 649.3 531.3 295.1 885.4 795.8 885.4 443.6 >> Two such data- tting techniques are polynomial interpolation and piecewise polynomial interpolation. 40 0 obj It should be noted, however, that the proposed method is general and can be applied to any integer linear least-square problem. 472.2 472.2 472.2 472.2 583.3 583.3 0 0 472.2 472.2 333.3 555.6 577.8 577.8 597.2 675.9 1067.1 879.6 844.9 768.5 844.9 839.1 625 782.4 864.6 849.5 1162 849.5 849.5 /FirstChar 33 >> /Type/Encoding ��y�� ��&�u���7�`��m����f�� /Widths[660.7 490.6 632.1 882.1 544.1 388.9 692.4 1062.5 1062.5 1062.5 1062.5 295.1 /FirstChar 33 >> 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.. >> 2 Chapter 5. 1062.5 1062.5 826.4 288.2 1062.5 708.3 708.3 944.5 944.5 0 0 590.3 590.3 708.3 531.3 Problem: Given these measurements of the two quantities x and y, find y 7: x 1 = 2. x. /Subtype/Type1 << /BaseFont/GYIZGA+CMCSC10 Least-squares data fitting we are given: The sample times are assumed to be increasing: s 0 < s 1 < ::: < s m.A B-spline curve that ts the data is parameterized The least-squares line. 795.8 795.8 649.3 295.1 531.3 295.1 531.3 295.1 295.1 531.3 590.3 472.2 590.3 472.2 277.8 305.6 500 500 500 500 500 750 444.4 500 722.2 777.8 500 902.8 1013.9 777.8 >> The least squares approximation for otherwise unsolvable equations If you're seeing this message, it means we're having trouble loading external resources on our website. 160/space/Gamma/Delta/Theta/Lambda/Xi/Pi/Sigma/Upsilon/Phi/Psi 173/Omega/arrowup/arrowdown/quotesingle/exclamdown/questiondown/dotlessi/dotlessj/grave/acute/caron/breve/macron/ring/cedilla/germandbls/ae/oe/oslash/AE/OE/Oslash/suppress/dieresis] endobj P. Sam Johnson (NIT Karnataka) Curve Fitting Using Least-Square Principle February 6, 2020 11/32. FINDING THE LEAST SQUARES APPROXIMATION We solve the least squares approximation problem on only the interval [−1,1]. )�g�������Q�y��6;�/d��R��� ��B^��ʋ��6����+�9�LK�"8�6�� ~�#8w��'��F��eH&�O�E d1�9���[�� [+n�HJ�c�S�r 5"��d��J0�!d�9�Sǃ-��>Ǜ�epf9o�!7um��rs��S��^6�G��� lׂ�.��x������b�p�Ц�ݖ���@u]����8f����0�Aӓ����·��O���H��.Xp����9��jM�j�̨�ȷJm(b ����Z"��Ds[�cF�B2m׆@��BcM� �jU����9qk�2��L��$�R��[&�^1��|�D�V� FcH�R��ѝ�NY�̌K��bev�Tq2�cĺƗ�al���`�[���2}H�*�C؇����]������wi��&��3�!����b��wI__<0)@�}p8Cq �G�+3���G*���� oH�%X'`��b�����Y����R;Z�L+�ꢥ�a2��9�����N��b ���⛫T+pX�L8 0��%�p������_�d~�'�]p�A�{xP�����L+ډ��O?v�dާ�56���x[� �U#uS%��Yw��;�1G�L'v���Wq�f8��_+E� ��&N`^A��e���!�nKh U38�w��:T~aU���QB�n볓`#xl��M_=�f^ݵ�#��m���2����-�����ʂ��zFٜ�m�,7�}�*�U��.wTE�p��. >> /Encoding 28 0 R Figure 2: The continuous least squares approximation of order 2 for f(x) = cos(πx) on [-1,1]. Least Squares Interpolation 1. endobj 750 708.3 722.2 763.9 680.6 652.8 784.7 750 361.1 513.9 777.8 625 916.7 750 777.8 /BaseFont/XCEACZ+CMR12 endobj Figure 4.3 shows the big picture for least squares. /Encoding 11 0 R The problem can be stated as follows: /LastChar 196 << 324.7 531.3 590.3 295.1 324.7 560.8 295.1 885.4 590.3 531.3 590.3 560.8 414.1 419.1 /Name/F4 /Subtype/Type1 Least-Squares Approximation of a Function We have described least-squares approximation to t a set of discrete data. 10.1.1 Least-Squares Approximation ofa Function We have described least-squares approximation to fit a set of discrete data. /Filter /FlateDecode Fit the data in the table using quadratic polynomial least squares method. To test /Differences[0/minus/periodcentered/multiply/asteriskmath/divide/diamondmath/plusminus/minusplus/circleplus/circleminus/circlemultiply/circledivide/circledot/circlecopyrt/openbullet/bullet/equivasymptotic/equivalence/reflexsubset/reflexsuperset/lessequal/greaterequal/precedesequal/followsequal/similar/approxequal/propersubset/propersuperset/lessmuch/greatermuch/precedes/follows/arrowleft/arrowright/arrowup/arrowdown/arrowboth/arrownortheast/arrowsoutheast/similarequal/arrowdblleft/arrowdblright/arrowdblup/arrowdbldown/arrowdblboth/arrownorthwest/arrowsouthwest/proportional/prime/infinity/element/owner/triangle/triangleinv/negationslash/mapsto/universal/existential/logicalnot/emptyset/Rfractur/Ifractur/latticetop/perpendicular/aleph/A/B/C/D/E/F/G/H/I/J/K/L/M/N/O/P/Q/R/S/T/U/V/W/X/Y/Z/union/intersection/unionmulti/logicaland/logicalor/turnstileleft/turnstileright/floorleft/floorright/ceilingleft/ceilingright/braceleft/braceright/angbracketleft/angbracketright/bar/bardbl/arrowbothv/arrowdblbothv/backslash/wreathproduct/radical/coproduct/nabla/integral/unionsq/intersectionsq/subsetsqequal/supersetsqequal/section/dagger/daggerdbl/paragraph/club/diamond/heart/spade/arrowleft Title: Abdi-LeastSquares-pretty.dvi Created Date: 9/23/2003 5:46:46 PM Approximation problems on other intervals [a,b] can be accomplished using a lin-ear change of variable. Here we describe continuous least-square approximations of a function f(x) by using polynomials. >> << 495.7 376.2 612.3 619.8 639.2 522.3 467 610.1 544.1 607.2 471.5 576.4 631.6 659.7 Interpolation techniques, of any 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 826.4 295.1 826.4 531.3 826.4 /Type/Font << /LastChar 196 /Subtype/Type1 160/space/Gamma/Delta/Theta/Lambda/Xi/Pi/Sigma/Upsilon/Phi/Psi 173/Omega/alpha/beta/gamma/delta/epsilon1/zeta/eta/theta/iota/kappa/lambda/mu/nu/xi/pi/rho/sigma/tau/upsilon/phi/chi/psi/tie] /Encoding 21 0 R 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. 500 500 500 500 500 500 500 500 500 500 500 277.8 277.8 777.8 500 777.8 500 530.9 /Length 2358 680.6 777.8 736.1 555.6 722.2 750 750 1027.8 750 750 611.1 277.8 500 277.8 500 277.8 x��YKs���W�HU 1ă 9�M���l���ڷL�L۬�H��lO��ӍH��TZo*�������[Q��4z���[zL?���K-?U�K�FI�D����,�i����2�m�6@b8�뿿y��G+ttsI&�(�e&���?�m����IT����q{w�u�liL�SϘ�����y�4џn~�"P�E����)�E�j{�_��p�*�O��Jf�0[6�]�኉���C�l���@< ��l`r��Ҫb)ab�Q"2�ٳ?5�Ё���U*�{��W}��R�W����Q�F�,��v��&�Ӫ�~��ߗ�"�C����]�?���΋��rx�W;"�X�v��Ջހ>���!�����R@����h�$����1c��if i,Y��tv�h�fHe�qc�*�I�ꃣ�(�"�� x�P`��z�t������e?����eW�n��h7�^ /Type/Font 500 500 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 625 833.3 >> /Type/Font 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 272 272 272 761.6 462.4 endobj /FontDescriptor 19 0 R 777.8 777.8 1000 500 500 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 3 0 obj << 277.8 500] These points are illustrated in the next example. >> A Better Approach: Orthogonal Polynomials. The method of least squares calculates the line of best fit by minimising the sum of the squares of the vertical distances of the points to th e line. 324.7 531.3 531.3 531.3 531.3 531.3 795.8 472.2 531.3 767.4 826.4 531.3 958.7 1076.8 /Subtype/Type1 least squares problem, this problem is known to be NP hard. 1002.4 873.9 615.8 720 413.2 413.2 413.2 1062.5 1062.5 434 564.4 454.5 460.2 546.7 295.1 826.4 501.7 501.7 826.4 795.8 752.1 767.4 811.1 722.6 693.1 833.5 795.8 382.6 Example 4.1 Picture: geometry of a least-squares solution. 8.1 - Discrete Least Squares Approximation. Solution Let P 2(x) = a 0 +a 1x+a 2x2. 777.8 777.8 1000 1000 777.8 777.8 1000 777.8] /FontDescriptor 13 0 R 173/circlemultiply/circledivide/circledot/circlecopyrt/openbullet/bullet/equivasymptotic/equivalence/reflexsubset/reflexsuperset/lessequal/greaterequal/precedesequal/followsequal/similar/approxequal/propersubset/propersuperset/lessmuch/greatermuch/precedes/follows/arrowleft/spade] /Length 2566 /LastChar 196 756.4 705.8 763.6 708.3 708.3 708.3 708.3 708.3 649.3 649.3 472.2 472.2 472.2 472.2 /FontDescriptor 36 0 R /Name/F1 7-2 Least Squares Estimation Version 1.3 Solving for the βˆ i yields the least squares parameter estimates: βˆ 0 = P x2 i P y i− P x P x y n P x2 i − ( P x i)2 βˆ 1 = n P x iy − x y n P x 2 i − ( P x i) (5) where the P ’s are implicitly taken to be from i = 1 to n in each case. 27 0 obj 272 272 489.6 544 435.2 544 435.2 299.2 489.6 544 272 299.2 516.8 272 816 544 489.6 28 0 obj There is a formula (the Lagrange interpolation formula) producing a polynomial curve of degree n −1 which goes through the points exactly. 545.5 825.4 663.6 972.9 795.8 826.4 722.6 826.4 781.6 590.3 767.4 795.8 795.8 1091 656.3 625 625 937.5 937.5 312.5 343.8 562.5 562.5 562.5 562.5 562.5 849.5 500 574.1 826.4 295.1 531.3] 424.4 552.8 552.8 552.8 552.8 552.8 813.9 494.4 915.6 735.6 824.4 635.6 975 1091.7 /Type/Font 813.9 813.9 669.4 319.4 552.8 319.4 552.8 319.4 319.4 613.3 580 591.1 624.4 557.8 /Widths[319.4 552.8 902.8 552.8 902.8 844.4 319.4 436.1 436.1 552.8 844.4 319.4 377.8 Then the discrete least-square approximation problem has a unique solution. 42 0 obj Approximation and Interpolation We will now apply our minimization results to the interpolation and least squares fitting of data and functions. In this example, let m = 1, n = 2, A = £ 1 1 ⁄, and b = £ 2 ⁄. 708.3 795.8 767.4 826.4 767.4 826.4 0 0 767.4 619.8 590.3 590.3 885.4 885.4 295.1 14 0 obj View Regression Equation using Least Square Approximation with Example.pdf from DM 101 at SASTRA University, School of Law, Thanjavur. 460.7 580.4 896 722.6 1020.4 843.3 806.2 673.6 835.7 800.2 646.2 618.6 718.8 618.8 minimize the sum of the square of the distances between the approximation and the data, is referred to as the method of least squares • There are other ways … �"7?q�p\� endobj 275 1000 666.7 666.7 888.9 888.9 0 0 555.6 555.6 666.7 500 722.2 722.2 777.8 777.8 783.4 872.8 823.4 619.8 708.3 654.8 0 0 816.7 682.4 596.2 547.3 470.1 429.5 467 533.2 << The proposed method b orrows the idea of the least squares approximation (LSA, W ang and Leng, 2007) and can be used to handle a large class of parametric regression models on a distributed system. 413.2 590.3 560.8 767.4 560.8 560.8 472.2 531.3 1062.5 531.3 531.3 531.3 0 0 0 0 /BaseFont/KDMDUP+CMBX12 Least-squares applications • least-squares data fitting • growing sets of regressors • system identification • growing sets of measurements and recursive least-squares 6–1. 1111.1 1511.1 1111.1 1511.1 1111.1 1511.1 1055.6 944.4 472.2 833.3 833.3 833.3 833.3 /FirstChar 33 Learn examples of best-fit problems. In this section, we answer the following important question: where p(t) is a polynomial, e.g., p(t) = a 0 + a 1 t+ a 2 t2: The problem can be viewed as solving the overdetermined system of equa-tions, 2 6 6 6 6 4 y 1 y 2::: y N 3 7 7 7 7 5 761.6 272 489.6] 597.2 736.1 736.1 527.8 527.8 583.3 583.3 583.3 583.3 750 750 750 750 1044.4 1044.4 492.9 510.4 505.6 612.3 361.7 429.7 553.2 317.1 939.8 644.7 513.5 534.8 474.4 479.5 stream We will do this using orthogonal projections and a general approximation theorem … /BaseFont/ZXBOAY+CMR10 endobj endobj /FontDescriptor 39 0 R 388.9 1000 1000 416.7 528.6 429.2 432.8 520.5 465.6 489.6 477 576.2 344.5 411.8 520.6 /Name/F10 /Widths[277.8 500 833.3 500 833.3 777.8 277.8 388.9 388.9 500 777.8 277.8 333.3 277.8 465 322.5 384 636.5 500 277.8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 the approximation. Also suppose that we expect a linear relationship between these two quantities, that is, we expect y = ax+b, for some constants a and b. 720.1 807.4 730.7 1264.5 869.1 841.6 743.3 867.7 906.9 643.4 586.3 662.8 656.2 1054.6 /Differences[0/Gamma/Delta/Theta/Lambda/Xi/Pi/Sigma/Upsilon/Phi/Psi/Omega/ff/fi/fl/ffi/ffl/dotlessi/dotlessj/grave/acute/caron/breve/macron/ring/cedilla/germandbls/ae/oe/oslash/AE/OE/Oslash/suppress/exclam/quotedblright/numbersign/dollar/percent/ampersand/quoteright/parenleft/parenright/asterisk/plus/comma/hyphen/period/slash/zero/one/two/three/four/five/six/seven/eight/nine/colon/semicolon/exclamdown/equal/questiondown/question/at/A/B/C/D/E/F/G/H/I/J/K/L/M/N/O/P/Q/R/S/T/U/V/W/X/Y/Z/bracketleft/quotedblleft/bracketright/circumflex/dotaccent/quoteleft/a/b/c/d/e/f/g/h/i/j/k/l/m/n/o/p/q/r/s/t/u/v/w/x/y/z/endash/emdash/hungarumlaut/tilde/dieresis/suppress 833.3 1444.4 1277.8 555.6 1111.1 1111.1 1111.1 1111.1 1111.1 944.4 1277.8 555.6 1000 << 777.8 694.4 666.7 750 722.2 777.8 722.2 777.8 0 0 722.2 583.3 555.6 555.6 833.3 833.3 10 0 obj 3 = 6. x. 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 /BaseFont/WMUXAW+CMSY8 /FirstChar 33 727.8 813.9 786.1 844.4 786.1 844.4 0 0 786.1 552.8 552.8 319.4 319.4 523.6 302.2 Least squares method, also called least squares approximation, in statistics, a method for estimating the true value of some quantity based on a consideration of errors in observations or measurements. 1444.4 555.6 1000 1444.4 472.2 472.2 527.8 527.8 527.8 527.8 666.7 666.7 1000 1000 >> 2 = 4. x. Least Squares. 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 458.3 458.3 416.7 416.7 4.3. The matrix A and vector b of the normal equation (7) are: A = 2 6 6 6 6 4 << Example Find the least squares approximating polynomial of degree 2 for f(x) = sinˇxon [0;1]. /Differences[0/Gamma/Delta/Theta/Lambda/Xi/Pi/Sigma/Upsilon/Phi/Psi/Omega/arrowup/arrowdown/quotesingle/exclamdown/questiondown/dotlessi/dotlessj/grave/acute/caron/breve/macron/ring/cedilla/germandbls/ae/oe/oslash/AE/OE/Oslash/suppress/exclam/quotedblright/numbersign/dollar/percent/ampersand/quoteright/parenleft/parenright/asterisk/plus/comma/hyphen/period/slash/zero/one/two/three/four/five/six/seven/eight/nine/colon/semicolon/less/equal/greater/question/at/A/B/C/D/E/F/G/H/I/J/K/L/M/N/O/P/Q/R/S/T/U/V/W/X/Y/Z/bracketleft/quotedblleft/bracketright/circumflex/dotaccent/quoteleft/a/b/c/d/e/f/g/h/i/j/k/l/m/n/o/p/q/r/s/t/u/v/w/x/y/z/endash/emdash/hungarumlaut/tilde/dieresis/suppress /Encoding 7 0 R /Subtype/Type1 /LastChar 196 /Encoding 11 0 R 5 = 10. x. /LastChar 196 1277.8 811.1 811.1 875 875 666.7 666.7 666.7 666.7 666.7 666.7 888.9 888.9 888.9 500 500 611.1 500 277.8 833.3 750 833.3 416.7 666.7 666.7 777.8 777.8 444.4 444.4 /Type/Font endobj 34 0 obj 24 0 obj 295.1 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 295.1 /Type/Encoding 2 Least-Squares Fitting The data points are f(s k;P k)gm k=0, where s k are the sample times and P k are the sample data. 489.6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 611.8 816 /Name/F6 277.8 500 555.6 444.4 555.6 444.4 305.6 500 555.6 277.8 305.6 527.8 277.8 833.3 555.6 /Subtype/Type1 << 0 0 0 0 0 0 0 0 0 0 777.8 277.8 777.8 500 777.8 500 777.8 777.8 777.8 777.8 0 0 777.8 11 0 obj /Name/F2 591.1 613.3 613.3 835.6 613.3 613.3 502.2 552.8 1105.5 552.8 552.8 552.8 0 0 0 0 Let’s illustrate with a simple example. 295.1 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 295.1 295.1 x��Z�o���_!�B���ޥ��@�\� m���偖(��$:�8��}gf�4)�d����@_��rwfvv�7��W�+�DV#'W����i���ͤ�vr5�9�K9�~9͕t����?r�K�e����t�Z��>q���\}�]�����Or�Z�6H|������8����E����>��`��C�k���ww۩��C��?��rj]���% But normally one 444.4 611.1 777.8 777.8 777.8 777.8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 << /Widths[1000 500 500 1000 1000 1000 777.8 1000 1000 611.1 611.1 1000 1000 1000 777.8 /Type/Font 531.3 826.4 826.4 826.4 826.4 0 0 826.4 826.4 826.4 1062.5 531.3 531.3 826.4 826.4 160/space/Gamma/Delta/Theta/Lambda/Xi/Pi/Sigma/Upsilon/Phi/Psi 173/Omega/ff/fi/fl/ffi/ffl/dotlessi/dotlessj/grave/acute/caron/breve/macron/ring/cedilla/germandbls/ae/oe/oslash/AE/OE/Oslash/suppress/dieresis] 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 663.6 885.4 826.4 736.8 708.3 708.3 826.4 826.4 472.2 472.2 472.2 649.3 826.4 826.4 826.4 826.4 0 0 0 0 0 21 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). << << /Name/F5 /FirstChar 33 /FirstChar 33 562.5 562.5 562.5 562.5 562.5 562.5 562.5 562.5 562.5 562.5 562.5 312.5 312.5 342.6 /FontDescriptor 23 0 R 299.2 489.6 489.6 489.6 489.6 489.6 734 435.2 489.6 707.2 761.6 489.6 883.8 992.6 /Name/F8 0 0 0 0 0 0 0 0 0 0 0 0 675.9 937.5 875 787 750 879.6 812.5 875 812.5 875 0 0 812.5 611.1 798.5 656.8 526.5 771.4 527.8 718.7 594.9 844.5 544.5 677.8 762 689.7 1200.9 /LastChar 196 844.4 319.4 552.8] stream Notes on least squares approximation Given n data points (x 1,y 1),...,(x n,y n), we would like to find the line L, with an equation of the form y = mx + b, which is the “best fit” for the given data points. 8 >< >: a 0 R 1 0 1dx+a 1 R 1 … << 844.4 844.4 844.4 523.6 844.4 813.9 770.8 786.1 829.2 741.7 712.5 851.4 813.9 405.6 6 >> %PDF-1.4 767.4 767.4 826.4 826.4 649.3 849.5 694.7 562.6 821.7 560.8 758.3 631 904.2 585.5 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. 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2020 least square approximation pdf