37 0 obj 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.. << In this section the situation is just the opposite. << 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 /Type/Font Interpolation techniques, of any 299.2 489.6 489.6 489.6 489.6 489.6 734 435.2 489.6 707.2 761.6 489.6 883.8 992.6 343.8 593.8 312.5 937.5 625 562.5 625 593.8 459.5 443.8 437.5 625 593.8 812.5 593.8 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 0 642.9 885.4 806.2 736.8 Figure 4.3 shows the big picture for least squares. 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 /LastChar 196 /Subtype/Type1 31 0 obj /Type/Encoding The matrix A and vector b of the normal equation (7) are: A = 2 6 6 6 6 4 /Name/F3 7 0 obj /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 endobj Example 4.1 But normally one 531.3 531.3 413.2 413.2 295.1 531.3 531.3 649.3 531.3 295.1 885.4 795.8 885.4 443.6 277.8 500] endobj 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 /LastChar 196 /Type/Font 535.6 641.1 613.3 302.2 424.4 635.6 513.3 746.7 613.3 635.6 557.8 635.6 602.2 457.8 28 0 obj /Widths[791.7 583.3 583.3 638.9 638.9 638.9 638.9 805.6 805.6 805.6 805.6 1277.8 >> x��YKs���W�HU
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x�P`��z�t������e?����eW�n��h7�^ 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 << 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. Fit the data in the table using quadratic polynomial least squares method. ���,y'�,�WҐ0���0U�"y�Ұ�PNK�Tah 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 /Widths[1000 500 500 1000 1000 1000 777.8 1000 1000 611.1 611.1 1000 1000 1000 777.8 777.8 777.8 1000 1000 777.8 777.8 1000 777.8] 5 = 10. x. This paper concentrates on the MIMO application. 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 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). /Widths[622.5 466.3 591.4 828.1 517 362.8 654.2 1000 1000 1000 1000 277.8 277.8 500 << Recipe: find a least-squares solution (two ways). /Encoding 28 0 R /Encoding 21 0 R 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 /FontDescriptor 19 0 R 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 1200.9 666.7 666.7 666.7 666.7 611.1 611.1 444.4 444.4 444.4 444.4 500 500 388.9 388.9 277.8 • Least squares approximation — Data: A function, f(x). << 761.6 679.6 652.8 734 707.2 761.6 707.2 761.6 0 0 707.2 571.2 544 544 816 816 272 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. 34 0 obj /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 stream /BaseFont/KDMDUP+CMBX12 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 Let’s illustrate with a simple example. Solution Let P 2(x) = a 0 +a 1x+a 2x2. Least-squares data ﬁtting we are given: /Name/F2 /LastChar 196 /Type/Font 13.1. 500 500 500 500 500 500 500 500 500 500 500 277.8 277.8 777.8 500 777.8 500 530.9 489.6 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 /Encoding 11 0 R /Encoding 7 0 R P. Sam Johnson (NIT Karnataka) Curve Fitting Using Least-Square Principle February 6, 2020 11/32. /Subtype/Type1 4 = 8. x. 491.3 383.7 615.2 517.4 762.5 598.1 525.2 494.2 349.5 400.2 673.4 531.3 295.1 0 0 /LastChar 196 /Type/Encoding endobj 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 endobj Figure 2: The continuous least squares approximation of order 2 for f(x) = cos(πx) on [-1,1]. /Length 2358 >> /FontDescriptor 39 0 R 13. %PDF-1.4 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 /BaseFont/EENXFQ+CMMI10 /Name/F5 42 0 obj /Type/Font In this section, we answer the following important question: Approximation problems on other intervals [a,b] can be accomplished using a lin-ear change of variable. 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 /Encoding 11 0 R << /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 endobj 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 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. 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] 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] the approximation. 17 0 obj << 40 0 obj 2 Chapter 5. 8 >< >: a 0 R 1 0 1dx+a 1 R 1 … 750 758.5 714.7 827.9 738.2 643.1 786.2 831.3 439.6 554.5 849.3 680.6 970.1 803.5 /FirstChar 33 444.4 611.1 777.8 777.8 777.8 777.8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 /FirstChar 33 720.1 807.4 730.7 1264.5 869.1 841.6 743.3 867.7 906.9 643.4 586.3 662.8 656.2 1054.6 Instead of Ax Db we solve Abx Dp. << >> /FontDescriptor 36 0 R << 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 A multiple-input-multiple-output (MIMO) is a communication system withntransmit antennas and mreceive anten-nas. /Widths[295.1 531.3 885.4 531.3 885.4 826.4 295.1 413.2 413.2 531.3 826.4 295.1 354.2 Usually the function φ(x)does not go through points [x i Least Squares Regression Imagine you … This system is overdetermined and inconsistent. 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 Suppose you have a large number n of experimentally determined points, through which you want to pass a curve. /Filter[/FlateDecode] << The answer agrees with what we had earlier but it is put on a systematic footing. 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]���% )�g�������Q�y��6;�/d��R��� ��B^��ʋ��6����+�9�LK�"8�6�� ~�#8w��'��F��eH&�O�E
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Discrete Least Squares Approximations One of the most fundamental problems in science and engineering is data tting{constructing a function that, in some sense, conforms to given data points. 694.5 295.1] 24 0 obj Figure 1: Least squares polynomial approximation. 14 0 obj Example Find the least squares approximating polynomial of degree 2 for f(x) = sinˇxon [0;1]. /Subtype/Type1 1062.5 826.4] /FirstChar 33 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. The problem can be stated as follows: endobj 21 0 obj endobj Problem: Given these measurements of the two quantities x and y, find y 7: x 1 = 2. x. The sample times are assumed to be increasing: s 0 < s 1 < ::: < s m.A B-spline curve that ts the data is parameterized /FontDescriptor 26 0 R We would like to ﬁnd the least squares approximation to b and the least squares solution xˆ to this system. /Type/Font 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). 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 /BaseFont/LLQVLW+CMMI8 /BaseFont/GYIZGA+CMCSC10 /Name/F8 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 << 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 /Name/F4 Here we describe continuous least-square approximations of a function f(x) by using polynomials. /Name/F1 /Encoding 11 0 R /FirstChar 33 �"7?q�p\� 820.5 796.1 695.6 816.7 847.5 605.6 544.6 625.8 612.8 987.8 713.3 668.3 724.7 666.7 /FontDescriptor 33 0 R /Name/F10 /Subtype/Type1 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 … 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 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 >> 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 << 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] /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 570 517 571.4 437.2 540.3 595.8 625.7 651.4 277.8] 813.9 813.9 669.4 319.4 552.8 319.4 552.8 319.4 319.4 613.3 580 591.1 624.4 557.8 761.6 272 489.6] /Type/Font 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 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 0 458.3 458.3 416.7 416.7 500 555.6 527.8 391.7 394.4 388.9 555.6 527.8 722.2 527.8 527.8 444.4 500 1000 500 In this example, let m = 1, n = 2, A = £ 1 1 ⁄, and b = £ 2 ⁄. Here we describe continuous least-square approximations of a function f(x) by using polynomials. 826.4 295.1 531.3] 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] endobj 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 least squares problem, this problem is known to be NP hard. /LastChar 196 — 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. 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. /Differences[0/Gamma/Delta/Theta/Lambda/Xi/Pi/Sigma/Upsilon/Phi/Psi/Omega/alpha/beta/gamma/delta/epsilon1/zeta/eta/theta/iota/kappa/lambda/mu/nu/xi/pi/rho/sigma/tau/upsilon/phi/chi/psi/omega/epsilon/theta1/pi1/rho1/sigma1/phi1/arrowlefttophalf/arrowleftbothalf/arrowrighttophalf/arrowrightbothalf/arrowhookleft/arrowhookright/triangleright/triangleleft/zerooldstyle/oneoldstyle/twooldstyle/threeoldstyle/fouroldstyle/fiveoldstyle/sixoldstyle/sevenoldstyle/eightoldstyle/nineoldstyle/period/comma/less/slash/greater/star/partialdiff/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/flat/natural/sharp/slurbelow/slurabove/lscript/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/dotlessi/dotlessj/weierstrass/vector/tie/psi 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�. << 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 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. 495.7 376.2 612.3 619.8 639.2 522.3 467 610.1 544.1 607.2 471.5 576.4 631.6 659.7 826.4 826.4 826.4 826.4 826.4 826.4 826.4 826.4 826.4 826.4 1062.5 1062.5 826.4 826.4 >> 161/minus/periodcentered/multiply/asteriskmath/divide/diamondmath/plusminus/minusplus/circleplus/circleminus /LastChar 196 566.7 843 683.3 988.9 813.9 844.4 741.7 844.4 800 611.1 786.1 813.9 813.9 1105.5 >> Two such data- tting techniques are polynomial interpolation and piecewise polynomial interpolation. We will do this using orthogonal projections and a general approximation theorem … /LastChar 196 277.8 500 555.6 444.4 555.6 444.4 305.6 500 555.6 277.8 305.6 527.8 277.8 833.3 555.6 /Subtype/Type1 /Subtype/Type1 >> 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 8.1 - Discrete Least Squares Approximation. 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. /BaseFont/DKEPNY+CMR8 /FirstChar 33 FINDING THE LEAST SQUARES APPROXIMATION Here we discuss the least squares approximation problem on only the interval [ 1;1]. 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 4 Least-Squares Approximation by QR Factorization 4.1 Formulation of Least-Squares Approximation Problems Least-squares problems arise, for instance, when one seeks to determine the relation between an independent variable, say time, and a measured dependent variable, say position or velocity of an object. /FontDescriptor 16 0 R 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 0 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 272 272 272 761.6 462.4 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 /BaseFont/YYYEYA+CMEX10 FINDING THE LEAST SQUARES APPROXIMATION We solve the least squares approximation problem on only the interval [−1,1]. 20 0 obj /FirstChar 33 << >> 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 /Encoding 21 0 R /BaseFont/XCEACZ+CMR12 Vocabulary words: least-squares solution. 4.3. To test << Section 6.5 The Method of Least Squares ¶ permalink Objectives. Title: Abdi-LeastSquares-pretty.dvi Created Date: 9/23/2003 5:46:46 PM 27 0 obj 324.7 531.3 590.3 295.1 324.7 560.8 295.1 885.4 590.3 531.3 590.3 560.8 414.1 419.1 460.7 580.4 896 722.6 1020.4 843.3 806.2 673.6 835.7 800.2 646.2 618.6 718.8 618.8 f��\0W(I�D��fNI5�-�T*zL��"Eux��T�$'�àU[d}��}|��#-��������y�Y���}�7�����+И�U��U��R�W��K�w���Ɠߧ���Y�Ȩ���k���2�&+tFp�(�"�$8�]���3ol��1%8g+�JR���_�%뇤_�I ���wI20TF�%�i�/�G�Y�3����z78���������h�o�E/�m�&`�����
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