04 Dec

least square approximation pdf

17 0 obj To test /Type/Encoding %PDF-1.2 There are no solutions to Ax Db. 13.1. 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 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 Approximation problems on other intervals [a,b] can be accomplished using a lin-ear change of variable. /Type/Font << /Type/Font 2 Chapter 5. /FontDescriptor 33 0 R A Better Approach: Orthogonal Polynomials. 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 /Name/F1 /BaseFont/DKEPNY+CMR8 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 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. 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 1062.5 826.4] /LastChar 196 The matrix A and vector b of the normal equation (7) are: A = 2 6 6 6 6 4 3 = 6. x. /Name/F2 Vocabulary words: least-squares solution. 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. 1000 1000 1055.6 1055.6 1055.6 777.8 666.7 666.7 450 450 450 450 777.8 777.8 0 0 endobj endobj /LastChar 196 /LastChar 196 275 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 A multiple-input-multiple-output (MIMO) is a communication system withntransmit antennas and mreceive anten-nas. endobj 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 >> Recipe: find a least-squares solution (two ways). endobj /Subtype/Type1 Usually the function φ(x)does not go through points [x i 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 Then the discrete least-square ap-proximation problem has a unique solution. These points are illustrated in the next example. Example 4.1 324.7 531.3 590.3 295.1 324.7 560.8 295.1 885.4 590.3 531.3 590.3 560.8 414.1 419.1 708.3 795.8 767.4 826.4 767.4 826.4 0 0 767.4 619.8 590.3 590.3 885.4 885.4 295.1 /Filter[/FlateDecode] Section 6.5 The Method of Least Squares ¶ permalink Objectives. • Least squares approximation — Data: A function, f(x). 11 0 obj Instead of Ax Db we solve Abx Dp. 8.1 - Discrete Least Squares Approximation. 500 500 500 500 500 500 500 500 500 500 500 277.8 277.8 277.8 777.8 472.2 472.2 777.8 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 … 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 /Subtype/Type1 /Name/F8 875 531.3 531.3 875 849.5 799.8 812.5 862.3 738.4 707.2 884.3 879.6 419 581 880.8 /Subtype/Type1 277.8 500] /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 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. /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 /Encoding 11 0 R 734 761.6 666.2 761.6 720.6 544 707.2 734 734 1006 734 734 598.4 272 489.6 272 489.6 /BaseFont/ZXBOAY+CMR10 Least Squares Approximations 221 Figure 4.7: The projection p DAbx is closest to b,sobxminimizes E Dkb Axk2. 34 0 obj /Subtype/Type1 24 0 obj 277.8 305.6 500 500 500 500 500 750 444.4 500 722.2 777.8 500 902.8 1013.9 777.8 /Widths[272 489.6 816 489.6 816 761.6 272 380.8 380.8 489.6 761.6 272 326.4 272 489.6 Learn to turn a best-fit problem into a least-squares problem. 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 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. 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 21 0 obj Problem: Given these measurements of the two quantities x and y, find y 7: x 1 = 2. x. 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. 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. /FontDescriptor 16 0 R /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 /BaseFont/GTEUSJ+CMSY10 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 endobj Two such data- tting techniques are polynomial interpolation and piecewise polynomial interpolation. Linear systems with more equations than unknowns typically do not have solutions. << 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 0 625 833.3 756.4 705.8 763.6 708.3 708.3 708.3 708.3 708.3 649.3 649.3 472.2 472.2 472.2 472.2 10 0 obj /Type/Encoding /FontDescriptor 13 0 R /FirstChar 33 << 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. 161/minus/periodcentered/multiply/asteriskmath/divide/diamondmath/plusminus/minusplus/circleplus/circleminus Figure 2: The continuous least squares approximation of order 2 for f(x) = cos(πx) on [-1,1]. /Encoding 28 0 R /BaseFont/GYIZGA+CMCSC10 In this example, let m = 1, n = 2, A = £ 1 1 ⁄, and b = £ 2 ⁄. 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 0 761.6 679.6 652.8 734 707.2 761.6 707.2 761.6 0 0 707.2 571.2 544 544 816 816 272 /FirstChar 33 Least-squares applications • least-squares data fitting • growing sets of regressors • system identification • growing sets of measurements and recursive least-squares 6–1. Picture: geometry of a least-squares solution. The sample times are assumed to be increasing: s 0 < s 1 < ::: < s m.A B-spline curve that ts the data is parameterized /Type/Font /Name/F6 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. Title: Abdi-LeastSquares-pretty.dvi Created Date: 9/23/2003 5:46:46 PM >> /BaseFont/WMUXAW+CMSY8 Interpolation techniques, of any << /BaseFont/EENXFQ+CMMI10 462.4 761.6 734 693.4 707.2 747.8 666.2 639 768.3 734 353.2 503 761.2 611.8 897.2 Here we describe continuous least-square approximations of a function f(x) by using polynomials. >> << 20 0 obj 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. 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] 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 >> << 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 10.1.1 Least-Squares Approximation ofa Function We have described least-squares approximation to fit a set of discrete data. 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 /Filter /FlateDecode 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 0 777.8 277.8 500 555.6 444.4 555.6 444.4 305.6 500 555.6 277.8 305.6 527.8 277.8 833.3 555.6 >> Least Squares Regression Imagine you … Approximation and Interpolation We will now apply our minimization results to the interpolation and least squares fitting of data and functions. 750 708.3 722.2 763.9 680.6 652.8 784.7 750 361.1 513.9 777.8 625 916.7 750 777.8 531.3 826.4 826.4 826.4 826.4 0 0 826.4 826.4 826.4 1062.5 531.3 531.3 826.4 826.4 This paper concentrates on the MIMO application. 28 0 obj /Widths[1000 500 500 1000 1000 1000 777.8 1000 1000 611.1 611.1 1000 1000 1000 777.8 P. Sam Johnson (NIT Karnataka) Curve Fitting Using Least-Square Principle February 6, 2020 11/32. >> 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 stream 597.2 736.1 736.1 527.8 527.8 583.3 583.3 583.3 583.3 750 750 750 750 1044.4 1044.4 544 516.8 380.8 386.2 380.8 544 516.8 707.2 516.8 516.8 435.2 489.6 979.2 489.6 489.6 /FirstChar 33 /Type/Font >> /Subtype/Type1 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 0 683.3 902.8 844.4 755.5 /FontDescriptor 9 0 R It turns out that although the above method is relatively straightforward, the resulting linear systems are often 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 27 0 obj 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 endobj Least-squares data fitting we are given: /FontDescriptor 30 0 R /FontDescriptor 39 0 R 826.4 295.1 531.3] Suppose you have a large number n of experimentally determined points, through which you want to pass a curve. /Encoding 21 0 R >> 570 517 571.4 437.2 540.3 595.8 625.7 651.4 277.8] The answer agrees with what we had earlier but it is put on a systematic footing. 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 1200.9 299.2 489.6 489.6 489.6 489.6 489.6 734 435.2 489.6 707.2 761.6 489.6 883.8 992.6 Let’s illustrate with a simple example. ���,y'�,�WҐ0���0U�"y�Ұ�PNK�Tah 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). 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 /FirstChar 33 But normally one CURVE FITTING { LEAST SQUARES APPROXIMATION Data analysis and curve tting: Imagine that we are studying a physical system involving two quantities: x and y. 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 Here we describe continuous least-square approximations of a function f(x) by using polynomials. >> /FontDescriptor 26 0 R 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 endobj /Widths[660.7 490.6 632.1 882.1 544.1 388.9 692.4 1062.5 1062.5 1062.5 1062.5 295.1 319.4 552.8 552.8 552.8 552.8 552.8 552.8 552.8 552.8 552.8 552.8 552.8 319.4 319.4 /Widths[622.5 466.3 591.4 828.1 517 362.8 654.2 1000 1000 1000 1000 277.8 277.8 500 /Encoding 28 0 R /BaseFont/XCEACZ+CMR12 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 In such situations, the least squares solution to a linear system is one means of getting as Least-Squares Approximation of a Function We have described least-squares approximation to t a set of discrete data. We can rewrite this linear system as a matrix system Ax = b where: A = −1 1 2 1 1 −2 and b = 10 5 20 /Encoding 7 0 R 762.8 642 790.6 759.3 613.2 584.4 682.8 583.3 944.4 828.5 580.6 682.6 388.9 388.9 /Subtype/Type1 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 40 0 obj 2 = 4. x. 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]���% /Widths[342.6 581 937.5 562.5 937.5 875 312.5 437.5 437.5 562.5 875 312.5 375 312.5 /Subtype/Type1 << /Name/F9 /BaseFont/LLQVLW+CMMI8 37 0 obj 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] 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). /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 /BaseFont/YYYEYA+CMEX10 14 0 obj /Encoding 21 0 R /Type/Font 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 << Figure 4.3 shows the big picture for least squares. /Type/Encoding 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 0 458.3 458.3 416.7 416.7 500 500 500 500 500 500 500 500 500 500 500 277.8 277.8 777.8 500 777.8 500 530.9 160/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] 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 36 0 R >> 777.8 777.8 1000 1000 777.8 777.8 1000 777.8] ��y�� ��&�u���7�`��m����f�� 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 /Length 2358 Learn examples of best-fit problems. /Name/F5 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 �"7?q�p\� Least Squares. Least Squares Interpolation 1. 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. 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. FINDING THE LEAST SQUARES APPROXIMATION Here we discuss the least squares approximation problem on only the interval [ 1;1]. 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 /Subtype/Type1 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 0 826.4 295.1 826.4 531.3 826.4 833.3 1444.4 1277.8 555.6 1111.1 1111.1 1111.1 1111.1 1111.1 944.4 1277.8 555.6 1000 The least-squares line. Figure 1: Least squares polynomial approximation. 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 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. 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 /Length 2566 694.5 295.1] /Type/Font /Encoding 11 0 R the approximation. /Type/Encoding /Encoding 11 0 R 343.8 593.8 312.5 937.5 625 562.5 625 593.8 459.5 443.8 437.5 625 593.8 812.5 593.8 666.7 666.7 666.7 666.7 611.1 611.1 444.4 444.4 444.4 444.4 500 500 388.9 388.9 277.8 /FirstChar 33 /Name/F7 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 We will do this using orthogonal projections and a general approximation theorem … << 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 6 /LastChar 196 4.3. /LastChar 196 Approximation problems on other intervals [a;b] can be accomplished using a linear change of variable. 4 = 8. x. 13. 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�. 3 0 obj << 1277.8 811.1 811.1 875 875 666.7 666.7 666.7 666.7 666.7 666.7 888.9 888.9 888.9 324.7 531.3 531.3 531.3 531.3 531.3 795.8 472.2 531.3 767.4 826.4 531.3 958.7 1076.8 It should be noted, however, that the proposed method is general and can be applied to any integer linear least-square problem. /Name/F4 >> /FirstChar 33 << We would like to find the least squares approximation to b and the least squares solution xˆ to this system. 8 >< >: a 0 R 1 0 1dx+a 1 R 1 … endobj /LastChar 196 /Encoding 11 0 R 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 Fit the data in the table using quadratic polynomial least squares method. /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 0 0 0 0 0 0 0 615.3 833.3 762.8 694.4 742.4 831.3 779.9 583.3 666.7 612.2 0 0 772.4 /Type/Font /LastChar 196 >> 7 0 obj 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 endobj /LastChar 196 /FontDescriptor 23 0 R 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. << stream 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 If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. Then the discrete least-square approximation problem has a unique solution. Example. 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 /FirstChar 33 844.4 844.4 844.4 523.6 844.4 813.9 770.8 786.1 829.2 741.7 712.5 851.4 813.9 405.6 761.6 272 489.6] 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 /LastChar 196 endobj endobj 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 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�^ >> 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.. 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 /Name/F3 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 Orthogonal polynomials • General orthogonal polynomials — Space: polynomials over domain D — Weighting function: w(x) > 0 View Regression Equation using Least Square Approximation with Example.pdf from DM 101 at SASTRA University, School of Law, Thanjavur. /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 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 << In this section, we answer the following important question: /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 )�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��. /Widths[1062.5 531.3 531.3 1062.5 1062.5 1062.5 826.4 1062.5 1062.5 649.3 649.3 1062.5 531.3 531.3 413.2 413.2 295.1 531.3 531.3 649.3 531.3 295.1 885.4 795.8 885.4 443.6 The problem can be stated as follows: 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 << 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 /LastChar 196 FINDING THE LEAST SQUARES APPROXIMATION We solve the least squares approximation problem on only the interval [−1,1]. %PDF-1.4 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 272 272 272 761.6 462.4 545.5 825.4 663.6 972.9 795.8 826.4 722.6 826.4 781.6 590.3 767.4 795.8 795.8 1091 31 0 obj >> Approximation and interpolation Definition:Least-square method is such approximation, in which φ(x)is „interlaced“ between given points[x i,y i]in such a way, that the „distance“ between functions f and φ is in some sense minimal. endobj /FirstChar 33 << /Type/Font /Type/Font 795.8 795.8 649.3 295.1 531.3 295.1 531.3 295.1 295.1 531.3 590.3 472.2 590.3 472.2 791.7 777.8] 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 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 /Subtype/Type1 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 /Subtype/Type1 42 0 obj Example Find the least squares approximating polynomial of degree 2 for f(x) = sinˇxon [0;1]. /Type/Font Solution Let P 2(x) = a 0 +a 1x+a 2x2. There is a formula (the Lagrange interpolation formula) producing a polynomial curve of degree n −1 which goes through the points exactly. 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޺�(�"�$8�]���3ol��1%8g+�JR���_�%뇤_�I ���wI20TF�%�i�/�G�Y�3����z78���������h�o�E/�m�&`����� /���#8��C|��`v�K����#�Ң�AZ��͛0C��2��aWon�l��� \.�YE>�)�jntvK�=��G��4J4J�庁o�$Bv ��Ã�#Y�aJ����x��m���D/��sA� S劸��51��W����ӆd�/�jQ�KP'��h�8�*��� �!M���\�d�lHu�@�� r+�[��S��Qu0h�+� �4S%��z�G�I� >�N�6�J�x��0*���l���d��h �z�ڧ\�C�����/ͼ�0#�; �I��}��f�z^��R�U���a�*�c��BX�/���. /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 This system is overdetermined and inconsistent. /BaseFont/KDMDUP+CMBX12 Instead of splitting up x we are splitting up b. endobj 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. /Name/F10 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 — 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 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. 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] 812.5 875 562.5 1018.5 1143.5 875 312.5 562.5] least squares problem, this problem is known to be NP hard. /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 /FirstChar 33 5 = 10. x. 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 0 675.9 1067.1 879.6 844.9 768.5 844.9 839.1 625 782.4 864.6 849.5 1162 849.5 849.5 endobj >> 295.1 826.4 531.3 826.4 531.3 559.7 795.8 801.4 757.3 871.7 778.7 672.4 827.9 872.8 535.6 641.1 613.3 302.2 424.4 635.6 513.3 746.7 613.3 635.6 557.8 635.6 602.2 457.8 844.4 319.4 552.8] .Kastatic.Org and *.kasandbox.org are unblocked x and y, find y 7 x... 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