Curve Fitting By Method Of Least Squares, Appendix I
REFERENCES[1] FRANCO, A. dosSantos., 1964. -Harmonic Analysis of Tides for 7 Days of Hourly observations. Int. Hyd. Rev. Vol. XLI, No. 2.[2] HORN, W . . 1960 - Some Recent Approaches to Tidal Problems. Int. Hyd. Rev. Vol.XXXVII, No. 2.[3] MUNK, W . , and HASSELMANN. K. , 1964 - Super-Resolution of Tides, Studies on Oceanography 1964, pp. 339-344. Washington. D.C.APPENDIX ICURVE FITTING BY M K T H U D OF LEAST SQUARESSuppose we have a function g(x) defined at the n point Xp x, . . . x,, and which to fit afunction f ( x ) dependent on the m parameters ai, a . . . a, such that the sum of the errors[g(X() - f ( x ] is least, ie :e [ g ( x . ) - f(x»)]is minimali -iThis is true if all derivatives of e with respect to the parameters are zero :3e a,i. e.3 a (1)3e 0' " 3a,(2)J 1. 2i [g(x») - f(x»)] - 0da ti.i-11 g(x,) v JiT-i1-11-1(x.)af(x,)m (3)j . 1, 2 . m (4) To be able to solve these equations we chose a particularly simple relation for f ( X j ) :f(x,) a i f , ( x i ) a,f,(x a,f,(xi)f(x,) ajf,(Xt)J ii. e. 'af(x,)(6)fj(x )3aj(5)(7)ring (4), which reads in matrix notation :c)f(Xi)aa,3f(xi)aa,3f(x,)aa 3f(xa)aa, (x,)aaiaf(x.)3f(x,)g(x )gT aa,af(Xt)3Saja,af(xj3f(x,)3a, -3aT3f(x,)3a, f(xi)f(x7)af(x 3a,(8) f(Xi)aa.3f(x,) a.af(xj3a.af(xi)aa,g(Xn)663f(x,) a, 3f(x.)3a.f(x.)
From (6) :fi(xi)f/x,)f(x,)f(x,)f,(xi) . f.(xi)f,(x,) . f.(x,)ai(9)f(x,)fi(x,) . . .f.(x,And part of (8) becomesfi(x,)fi(x,) . fi(x,)t2(xi)fstx,) . f.(xi) f.(x,)3f(x,)3a,af(xa)aa, f(x,) aiaf(xi)(10)3a,? f(x,)f.(xjaa.3f(x,)3a,Term the first matrix above F or just F, the vector of coefficients A, and the vectormeasured points G, then equation (8) becomesFG -- F ( F ' A )FG (FF')Aof(11)(12)Now FF' is quare and will usually be non-singular :A ( (FF 1 )" 1 F) . G(13)FF' is termed the normal matrix and is m-square and symmetrical. The m coefficients canthus be found from the n measured points.Now consider the special casef ( x i ) Ag A i cos(SiXi) A a c o s ( s , x ) . . . A, cos(s»xi) B, s i n t S j X j ) B, sin (s,x . . . B, sin(s,x )Also x. 0Xi irThe matrix F now is :11(14)1(n' n - 1).1 cos(SiT)cos(s,2r) .cos(S n'T)1 cos(s t)cos(s,2T) .cos(s,n'T)(15)1 cos(s,T)cos(s,2T) .cos(s,n'T)0 sin (SiT)sin (si2r) .sin (sin't)0 sin(s«t)sin(s 2r) .sin(s,n'T)i. e. it contains n columns 2m 1 rows. The terms in the product FF' will beLet s, 067
X., co .,A,. co.,s,«,),0.1 , 0 '; : ;I I"'.in,.,.,)L0: ', : .{ "' "8',,, (.in l,,,.) .„.„„fnaT;//,.':!2'"",„ , V,.„„„.«„ „„(„.„, ; ,' ; ; ; 'K 0j'It f o l l n w y that FF' is symrnetrical as it ought to be. Instead of calculating (2m 1 )(2m l ) ( 2 m 2 ) / 2Wf r e q u i r e only ———————n———————— Consider ( 1 6 ) :?1J -- COs(SiHT) "p-l1/2 COS(SjHT)COS(H(S, Sj)T)"' VCOS(H(S,-(20)-IS )-t)\x ». -0 '(21)JNow :n-1n-1(j -- V — T)(22) COS(MOL)T) -- Re V e """"X--1 0n rReDU rnUpT1 - e ""7e 2e 2- e i . e oT - He - ——— ———— ——e'Te i e 2(23)"c . /'n(4 T \J,— — ,.-.,( n - l ) Sin *. —»—v 3 )/Ree.(24) oT\sin \- -)cos v( 2 /0) Ti- -1 —7-rTsin —— )( 25)- (V) (--:), sin ( ( 2sin ( 2nn--ll), " 2). oT sm ;rsin ((2n-l) y)y; - -j1 l —————————22 Lgins / L)-*-' if,sin ((2n-l) (si Sj) T/2) 2 — — s i n ( ( s i s j ) T/2)Unless &)„„ Si68 sin ((2n-l) ( S j - S t ) T / 2 ) 1——sin((sj-s,)T/2)J(26)(27),--,(28)term
Then : - -f2 (2n-l) sin((2n-l) ( s s . ) T / 2 ) - 1J4[ sin((si Sj) T / 2 ) ) JUnless s Sj 0Then : tj 4- [2 ( 2 n - l ) (2n-,l)] nE0(30)EQ(29) EQ(28)EQ(35)EQ(38)BY SYMMETRYEQ(39)Figure Next consider the terms in the right hand top rectanglei tj . C O S ( S ( H T )K "0,r » '»-i - {\ sin'(n(si Sj.,)T) n(Sin(Sj.HT)n-l sin(n(sj.,-'si) T)'-I x-oNow similarly to (25) :D- I sin(H T). /nco-T \/"o T / oT\. sin(-3-)(n-l)X'"0sln.{ )sinrcos(y)-cos((2n-l)y)-[ Sirusin( )-I j rcos ((St Sj.) T / 2 ) - c o s ((2n-l) (St Sj.,) T/2)' 4[sin ((s s J T/2)69
Unless scos ((s- s ) t / 2 ) ) - cos ((2n-l )(s, ,-s ) T/2)—————————sm((Sj., -s, T/2))——————— s,Then : 'J f cos ((s s J T/2) - cos((2n-l) (s, s, ,) T/2)) 14 — — — — — — s i n ( t s s ) T/2)——————————JFinally the lower right triangle : n-i sin(si.,HT) sin(s,.,HT)x.o (36)— 1' "i1 cos(n(s, ,-S,.)T) - i'cos(H(Si. s,.,)T) "L X-oK-o(37)JBy using method applied to arrive at (28) :ll"sin((2n-l) \ j.ii gj-«;( s - s .) T/2)sin »"'((2n-l)1 sin \\fii-11' /" i""' / (Sj. Si.) T/2) 14 [[ ss ii nn ((Sj.-s )((Sj.-s .) T/2T / )2 ) "'sm ((s,. Sj.,) T / 2 ) J4sin((s,j" ::Unless s , s,Thens(a n 1i\ \\4 'r——I l n - i )sin ((2n-l) (Sj. s.) T/2) ']sin (Tsi., Sj.,) T / 2 ) J- ————.———7-7—————— ——r———j-f—t—————— I,,\3''lIn addition one can use :s i n ( ( s S j ) T / 2 ) sin(S T / 2 ) cos(s r / 2 ) sin(Sj i / 2 ) cos(s, -[/2)(40)s i n ( ( S i - S j ) t / 2 ) . sin(S( T / 2 ) cos(s T / 2 ) - sin(Sj T/2) cos(s T/2)(41)cos ((s s ) T / 2 ) -- cos(s, T / 2 ) cos(sT / 2 ) - sin(s T / 2 ) sin(s T / 2 )(42)cos ((s -s ) i / 2 ) - cos(s t / 2 ) cos(s T / 2 ) sin(s T / 2 ) s i n ( S j T/2)(43)All terms can thus be derived from four tables :sin(sT / 2 ) , cos(sT/2)sin ((2n-l)s, T/2), cos ( ( 2 n - l ) s T/2)APPENDIX IITIDAL ANALYSIS PROGRAMMEThe programme given below is written in Manchester Auto-Code (MAC), which is compatible with Extended Mercury Auto-Code (EMA). This programme is therefore acceptableto computers such as ATLAS and ORION II which have EMA compilers.A few notes on the peculiarities of MAC should make it easier to follow the programme.70
As is usual in programming languages, a distinction is made between integers and nonintegers. In MAC the former are held in the machine in fixed point form, and are referredto by any of the 12 symbolsI J K L M N O P Q R S TThese are termed indices, and can be used as subscripts in one dimensional arrays.The remaining letters of the alphabet A - H and U — Z are reserved for numbers heldin floating point form to a precision of eight significant figures. It is also permissible todenote floating point numbers by primed letters of the alphabet in the range A' - H' a n d U ' Z'.Finally the location with the symbolic address it permanently contains3.14159267.thenumberBrackets may not be used to group multiplications, but can be used to enclose a subscript. Examples areA3means AAPmeans AA(P 3) means Abut PA means the integer P multiplied by the floating point number A. The cummutative property of multiplication between integers and non-integers is thus not preserved.Behind the Immediate Access Core Store is a backing store consisting of one or moredrums. Access to and from drum store is by means of the cp6 andcp? instructions as indicatedbelow :cp6 (X) AO, NThis transfers the N values A o , A I . . . . A n - 1, from consecutive drum store addressesstarting with drum store address X to consecutive Immediate Access Store addresses starting with AO. Similarly the instructionscp7 (X) AO, Nreads N consecutive I.A.S. locations and writes them to drum.At the head of the programme it is necessary to place dimension statements (directives )specifying the maximum size of the various arrays used, i.e. the directiveA —— 103reserves 104 storage locations for the subscripted array A , A,. A g.Finally, as in general a programme is too large to be held in its entirely in the Immediate Access Store, it is divided up into chapters. The programme as a whole is held onthe drum, and each chapter is brought down into the core store as required. The chaptersare labelledCHAPTER 0, CHAPTER 1, CHAPTER 2, etc.CHAPTER 0 is the first chapter to be obeyed, but the last one to be read in.Instructions are punched one to a card, and comment can be added, provided it is separated by one or more blank columns from the instruction.Since tide gauge readings are normally rounded off to the nearest tenth of a foot, it isour custom to punch sea heights as an integral number of tenths of a foot, and to restorethe decimal point by programme.71
MAC PROGRAMME FOR TIDAL ANALYSIS.CHAPTER 1A—— 102B—— 102C —— 102BO 0READ (Q)READ (K)K K - 1M -- 2QT Q 1P - 1(1) QR E A D (BP)REPEAT4"7 ( A ' ) B l , QP -. 1 ( 1 ) QBP - GBPREPEATP -- 0(1) KREAD ( A )A -- 0. 1 AAO - AO AJ -- 1 ( 1 ) QB - 'pCO.S(PBJ)A.I ;- A.J ABREPEATJ - T(l) MI -- J - QB V SIN (FBI)A.I AJ ABREPEATREPEATCLOSECHAPTER 2VARIABLES 1P -- 0(1) MCP -- 0N 0(1) MREAD(BN)CP -- CP ANBNREPEATREPEATCLOSECHAPTER 3VARIABLES 1P 1(1) QR E A D (BP)REPEATcp7(D') B l , QP 1(1) QREAD (BP)REPEATc ?7(E') Bl, QR E A D (X)TIDAL ANALYSIS FOR UP TO 51 CONSTITUENTSSPEED FOR MEAN SEA LEVELNO. OF CONSTITUENTSNO. OF HOURLY OBSERVATIONSSPEEDS ( D E G . / M E A N SOLAR HOUR)SPEEDS TO DRUMSPEEDS TO R A D I A N SHOURLY HEIGHTSRESTORE DECIMAL POINTROW OF I N V E R S E MATRIXNODE FACTORSNODE FACTORS TO DRUMEQUILIBRIUM ARGUMENTSEQUILIBRIUM ARGUMENTS TO DRUMHOURS FROM START OF YEAR72
2103104105106107108109110111PRINT 'TIDAL CONSTITUENTS'PRINT 'HOURS FROM START OF YEAR'PRINT (X) 4, 0NEWLINE 2PRINT 'MEAN SEA LEVEL'PRINT (C0)2. 2NEWLINE 2SPACE 8PRINT 'SPEED AMPLITUDE G NODE FACTOR EQ. A R G . 'NEWLINE 2CLOSECHAPTER 4VARIABLES 1P 1(1) QS P QE RADIUS (CP. CS)F PARCTAN (CP. - CS) p6(D' P - 1)Z. 1CALL DOWN P TH. NODE FACTORE E/ZF F/GCONVERT FROM RADIANS TO DEGREESDOWN 2 / 5PRINT (P) 2. 0SPACEcp6(A' P - 1)Z, 1CALL DOWN P TH. SPEEDPRINT (Z)2, 6 PRINT P TH. SPEEDPRINT (E)2, 4 PRINT P TH. AMPLITUDESPACEPRINT ;F)3, 2 PRINT P TH. LAGSPACE 3cp6(D' P - 1)Z, 1CALL DOWN P TH. NODE FACTORPRINT ( Z ) l , 3 PRINT P TH. NODE FACTORSPACE 3cp6(E' P - 1)Z. 1CALL DOWN P TH. EQ. ARG.PRINT (Z)3. 1 PRINT P TH. EQ. ARG.NEWLINE 2REPEATENDCLOSECHAPTER 5VARIABLES 12) f6(A' P - 1)V, 1 CALL DOWN P TH. SPEEDq)6(E' P - 1)Z, 1CALL DOWN P TH. EQ. ARG.F Z XV - F3) JUMP 4. F 0 . 0F F 360JUMP 34) JUMP 5. 360.0 FF F - 360JUMP 45) UPCLOSECHAPTER 0VARIABLES 1G 71/180TO CONVERT DEG. TO RADIANSA' 0D' 1000E' 2000CLOSE73
DISCUSSION (Chairman : M. GOUGENHEIM)LE PRESIDENT remercie Mr. Shipley de son interessant expose qui va sans doute soulever un certainnombre de questions de la part des participants.L'amiral FRANCO fait remarquer qu'une premiere question se pose qui est liee a la communication queva faire M. Van Ette.Mr. Shippley a parle de la conformation des matrices pour la solution du probleme de la determinationdes constantes harmoniques. et 1'Amiral Franco pense qu'il serait interesaant d'entendre 1'opinion duProfesseur Schoemaker a ce sujet.D'autre part, Mr. Shipley a dit qu'il employait le systeme complet, c'est-a-dire non divis6 par1'heure centrale, ce qui double le nombre d'inconnues.II a dit egalement que, malgre cela, le temps employe pour obtenir les resultats est le memeque si 1'on employait 1'heure centrale. Or deux personnes sont presentes qui peuvent egalement intervenir A ce sujet, ce sont MM. Cartwright qui a employe pour sa methode de Fourier dans I'analyse desmarees la formule de Watt pour determiner les termes connus des equations normales, et M. Horn(un autre M. Horn) qui emploie 1'heure centrale et qui pourrait dire.aussi quelques mots & ce sujet.Pr. SCHOEMAKER (Netherlands) said it was rather difficult to dive straight into this matter and suggested they should hold a simultaneous discussion on Mr Shipley's paper and the paper to be presentedby Mr van Ette and himself, because he and Mr van Ette had something to add on the special subjectof matrices and taking central points of observations. He wondered whether other delegates would prefer to start the discussion now or to wait until his and Mr van Ette's paper had also been presented.LE PRESIDENT indique que Mr. Horn prefere attendre et demande la position de Mr. Cartwright surce point.Mr. CARTWRIGHT (United Kingdom) had not come prepared to offer any contribution on this subject,partly because he had not himself done any research on harmonic tidal analysis for some years. However ,he saw that it was really a question of comparison between the least squares analysis and theFourier method. There was really little difference because the least squares method was that whichgave a minimum to the expected error or expected variance of error, and since it was a minimum onehad to differ considerably from the method of least squares to obtain an error which was much greater .It was simply a question of choosing one's filter and deciding whether one had corrected for all the sideband effects of that filter. As far as he could see, there was no essential difference betq'een the twomethods. Of course, the least squares method had the advantage that it could be very readily adaptedto any set of data, even if it had large gaps in it, whereas the Fourier method could only be conveniently applied to a continuous series of data.LE PRESIDENT pense qu'A vrai dire on ne peut pas parlor d'erreur ; on peut parler d'incertitude surle resultat. La methode des moindres carr s donne la solution la plus probable ; les methodes qui endifferent legerement donnent des solutions de probability un peu moindres, c'est-S-dire une incertitudel ?gerement plus grande ; mats le resultat peut etre a u s s i exact en soi-meme. C'est simplement 1'incertitude qui 1'affecte qui peut etre plus grande.LE PRESIDENT estime, avant d'ouvrir la discussion sur la methode de Mr. Shipley, qu'il serait preferable d'entendre les exposes de MM. Van Ette et Schoemaker. Comme sur ce point vient en n u mero 3, le President suggere de suivre le programme et de demander a Mr. Zetler de bien vouloir presenter sa communication,74
RECENT DEVELOPMENTS IN TIDAL ANALYSIS IN SOUTH AFRICAA.M. SHIPLEYC.S.I.R. Oceanographic Research Unit, University of Cape TownINTRODUCTIONIdeally a method of Tidal Analysis should be able to do the following :1 / Deal with a record of any length, up to a year or more.2/ Handle a record which has gaps in it.3/ Use any number of constituents within reason (say up to 64).The above scheme is rather ambitious unless one has a very large and fast computerand up to now we have refrained from attempting the analysis of records with large gaps .A year's record with a gap of say a week or so can be handled by using a portion of therecord, of 3 or 6 months' duration immediately before or after the gap. and from the resulting analysis interpolating the missing hourly heights. One then starts from scratch againand uses the now complete year's record for a major analysis.MATHEMATICAL MODELThe mathematical model used was the usual one, i.e. the sea level at any instant wasassumed to be given byZi -- An tr-lf,H,cos(S,t u, - g,)(1)where n is the number of constituents used and the rest of the symbols have their usualmeanings. By means of the addition formula for the cosine function equation (1) can be written as"t - friI- A cos S t , B sin S t(2)whereA, -f,H,cos(u, - g,) \(3)B, - - f,H,sin(u, - g,) Hence if the A, and B,. are known the tidal constants are given byH, /A ——B?-r(4)g, . u, - ARCTAN j A ./The quadrant in which the inverse tangent lies must be ;hosen so that the numeratorhas the same sign as sin g, and the denominator the same sign as cos g . On our I.C.T.1301 machine this is automatically taken care of by the compiler, (i.e. ARCTAN (X/Y) isregarded as a function of two arguments.)59r?R.*»ce D» s 9 . - sfMfoSt M. o»« T(»es"TE »JATlo»)AtK P o M hn. yjlLC u - A IL iu?—KofsjAcoC t ».") W(, e\ XH.
Consider now the case where we have m consecutive hourly observations of sea level,taken at hours 0, 1, 2, . (m-1).Then taking t 0, 1,(m-1)equations (2) become m equations of condition.Write them in matrix form as CX Z(5)Here C is a matrix with m rows and (2n 1) columns. It is in fact the matrix whose(p l)th row is1cos S,p . cos S p sin S p . sin S pXis the column vectorX(6)A,And Z is the column vector A(7)Z,i.e. Z is the column vector of hourly sea levels.Note that counting starts at the zero-ith sea level (Zg).Let C' be the transpose of C. To obtain the normal equations we must pre-multiplyequations (5) by C'C'CX C ' Z(8)The matrix CC' is square and in general non-singular.Denote its inverse by BB (C'C)'(9)Then the solution to the normal equations (8) isX - BC'Zwhere X is the solution vector whose transpose is (Ag . A , B . . B This in conjunction with equations (4) enables us to find the tidal constants and thussolve the problem.60
PRACTICAL CONSIDERATIONSThe above theory is quite straightforward, but as is to be expected a number of practical difficulties arose when trying to implement it. One of these is the enormous amount ofmachine time to generate and invert the normal matrix. For this reason it was decided tochoose certain fixed record lengths each with its corresponding set of constituents, and ineach case to generate and invert the matrix on a once for all time basis. The inverse matrix could then be kept on punched cards or magnetic tape and used over and over again.As a beginning we used the set of twelve constituents listed in TABLE N . 1. and arecord length of 697 hourly observations, (i.e. 29 Days). The reason for this choice of constituents is that they are the ones used on the Doodson-L6ge machine which the South AfricanHydrographic Office has used in the past for doing predictions.Table No. 1Tidal Constituents Used in 29 day Analysis.No.NAMESPEED00M.S.L.00.000 000 000 001M,28.984 104 237 302S,30.000 000 000 003N,28.439 729 541 504K,30.082 137 278 605 27.968 208 474 606M,57.968 208 474 607MS,58.984 104 237 308J,15.585 443 335 109Q,13.398 660 902 210P,14.958 931 360 7110,13.943 035 598 012K,15.041 068 639 3On using analyses obtained by this method it was found that the results were not verygood, in the sense that the agreement between computed and observed sea level left muchto be desired. The author must thank Vice-Admiral A. dos Santos Franco for pointing outthe reason for this. The noise level in a tidal record is such that the signal to noise ratioprohibits the separation of close constituents such as K, and S, or P and K , unless a record of length considerably longer than 29 days is analysed, or unless certain correctionsdependent on equilibrium tide theory are applied. The method of applying these correctionsis very lucidly explained in an article by dos Santos Franco in the I.H. Review for July1964.On thinking matters over it was decided that the first of the two alternatives would bethe better one, for the reasons set out below.I / I n the case of every harbour for which we have records, these records have an unbroken length far exceeding 29 days. It was felt that if one had a record of say tour months'duration one would get better results by analysing the record as a whole rather than doingfour separate monthly analyses and taking vectorial means.61
2/ It was felt that with a record of a length of three months or longer, it would berealistic to look for far more than 12 constituents in the analysis. Accordingly for 3, 4 and6 months' records, inverse normal matrices for 25 constituents were prepared. These constituents are listed in TABLE N 2. For a record extending to 369 days. we felt we could doeven better, and in consequence an inverse normal matrix for the 51 constituents, listed inTABLE N" 3., was computed.Table No. 2Constituents Used in 3, 4 and 6 month Analysis.No.NAMESPEED00M.S.L.00.000 000 000 001Mm00.544 374 695 802MSf01.015 895 762 703Mf01.098 033 041 3042Q,12.854 286 206 505Q,13.39B 660 902 2060,13.943 035 598 007P 14.958 931 360 708S,15.000 000 000 009K,15.041 068 638 310J,15.585 443 335 1112N,2 7 . 8 9 5 354 845 812t-l,27.968 208 474 613N,28.439 729 541 514v,28.512 583 170 415M,28.984 104 237 316X,29.455 625 304 217L,2 9 . 5 2 8 478 933 118T,29.958 933 322 419S,30.000 000 000 020K,30.082 137 278 621M,43 476 156 356 022SK,45.041 068 639 323M,57.968 208 474 624MS 58.984 104 237 325S,60.000 000 000 0The University of Cape Town has an I.C.T. 1301 computer. This is a fixed wordlength, binary coded decimal machine, which operates with an 8-digit mantissa when doingarithmetic in the floating point mode. Normal matrices have the reputation of being illconditioned, so that it was felt that double precision arithmetic should be used for their generation and inversion. For this reason the Computing Centre of the University of the Witwatersrand (who have a variable word length machine) was commissioned to do the "once forall time" generation and inversion. In the case of a 369 days' record with 51 constituents ,an eighteen decimal digit mantissa was used, and in the other cases 16 digits. The final results were in all cases rounded oft to eight significant figures.62
Table No. 3Constituents Used in 369 Days Analysis.No.NAMESPEED00M.S.L.00.000 000 000 001Sa00.041 068 639 302Ssa00.082 137 278 603Mm00. 544 374 695 804MSf01.015 895 762 705Mf01.098 033 041 3062Qi12.854 286 206 5070,12.927 139 835 308Q 13.398 660 902 209P,13.471 514 531 1100,13.943 035 598 011M,14.496 693 943 612TI,14.917 864 683 113P,14.958 931 360 714Si15.000 000 000 015Ki15.041 068 639 316*YI15.082 135 300 017(pi15.123 205 918 018*9i15.512 589 700 019Ji15.585 443 335 120SOi16.056 964 402 02100»16.139 101 680 622*OQt27.341 696 400 023MNS,27.423 833 778 9242N,27.895 354 845 825(1,2 7 . 9 6 8 208 474 626N,2 8 . 4 3 9 729 541 527v,28.512 583 170 428OP,28.901 966 958 729M,28.984 104 237 330MSKi29.066 241 516 031X,29.455 625 304 232L,29.528 478 933 133T,29.958 933 322 434S,30.000 000 000 035*R,30.041 066 700 036K,30.082 137 278 663
No.NAMESPEED37MSN,30.544 374 695 838KJ,30.626 511 974 4392SM»31.015 895 762 740M343.476 156 356 041MKs44.025 172 876 642SKa45.041 068 639 343MN«57.423 833 778 944M«57.968 208 474 645MS,58.984 104 237 346MK,59.066 241 516 047S,60.000 000 000 048SK 60.082 137 278 649MSN,87.423 833 778 9502MSe87.968 208 474 6512MK,88.050 345 753 3(*) Those constituents marked with asterisks are the ones for which wehave been unable to obtain the speedsto 12 significant figures. The lastthree or four unknown significant figures have been filled in with nonsignificant zeroes.The I.C.T. 1301 machine will not accept programmes written in FORTRAN, but usesa source language called "Manchester Auto-Code" (MAC). Moreover, the punching format for"MAC" is not compatible with that for FORTRAN. However, the analysis programme has nowbeen rewritten in FORTRAN IV and the matrices repunched in a FORTRAN format. This hasbeen done so as to run the programme on an IBM 360.Horn (1960) has pointed out that if an odd number of observations is used and the timeorigin is taken at the central observation, the matrix can be partitioned into two sub-matrices,one of order (m 1) and the other of order m. This would save a certain amount of machinetime in the generation and inversion of the matrices, but we did not do this for the followingtwo reasons :I / The calculation has to be done once only.2/ Mr. B.K.P. Horn, of the Witwatersrand Computing Centre (unrelated to the Hornmentioned above), found numerous short cuts in the generation of the normal matrix. M r .Horn's method is given in detail in appendix I. The time origin is zero hour on the firstday of the record.The actual analysis programme written in "MAC" is given in appendix II. Programmes and matrices for both "MAC" and FORTRAN compilers can be supplied on punched card*.Without entering into the pros and cons of one source language, as opposed to another,it is worth pointing out that, as will be seen from appendix II, in "MAC" it is permissibleto use zero as a subscript. This is not the case in any of the "dialects" of FORTRAN, andmust be borne in mind if the programme is translated into FORTRAN.64
Table No. 4Inverse normal matrix for 4,177 consecutive hourly .26052400-0.28183409-0.22769069-0. 41094046-0.31879032-0.44171545-0.49957147-0. 15172311-0.62050428-0. 28467-0.11886801-0.40859698-0.77195263-0. -05/-06/-06/-06/-06/-06/-06/-060. 0.46599105-0.412104430.34970862-0.11874929-0. 561169-0.11054498-0.55386493-0. 12348530-0.11821442-0. 11615885-0. 5-0.11317983-0. 10665987-0.87373759-0.14591290-0. 82602-0.22101678-0.16866295-0.95385359-0. 12-0.11665763-0.30199684-0.74429265-0.39350277-0. 321-0.31033438-0.28322517-0.51759686-0. 87-0.26370B40-0.43115157-0. -06/-U6/-06/-08/-05/-U5/-07/-07/-08/-06/-06Table N 4 is put in purely for illustrative purposes to show the format in which thematrices are punched on cards. This is a floating point format, retaining eight significantfigures irrespective of how small the number is.For example0.73618984 / - 09means0.73618984 x 10"*The first six columns of the inverse matrix for a six months' record are shown.In conclusion I should like to thank the Hydrographic Office of the South African Navyfor their help and collaboration, and for providing the funds necessary for producing the matrices.65
curve fitting by mkthud of least squares Suppose we have a function g(x) defined at the n point Xp x, . x,, and which to fit a function f(x) dependent on the m parameters ai, a . . . a, such that the sum of the errors
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Part 5 - CURVE FITTING Describes techniques to fit curves (curve fitting) to discrete data to obtain intermediate estimates. There are two general approaches for curve fitting: Least Squares regression: Data exhibit a significant degree of scatter. The strategy is to derive a single curve that represents the general trend of the data .
I. METHODS OF POLYNOMIAL CURVE-FITTING 1 By Use of Linear Equations By the Formula of Lagrange By Newton's Formula Curve Fitting by Spiine Functions I I. METHOD OF LEAST SQUARES 24 Polynomials of Least Squares Least Squares Polynomial Approximation with Restra i nts III. A METHOD OF SURFACE FITTING 37 Bicubic Spline Functions
Keywords Curve fitting · Surface fitting · Discrete polynomial curve · Discrete polynomial surface · Local optimal · Outliers 1 Introduction . The method of least squares is most commonly used for model fitting. This method estimates model parameters by minimizing the sum of squared residuals from all data, where
Curve fitting by method of least squares for parabola Y aX2 bX c ƩY i aƩX i 2 bƩX i nc ƩX i Y i aƩX i 3 bƩX i 2 cƩX i ƩX i 2Y i aƩX i 4 bƩX i 3 cƩX i 2 P.P.Krishnaraj RSET. Curve fitting by method of least squares for exponential curve Y aebX Taking log on both sides log 10 Y log 10 a bXlog 10 e Y A BX ƩY i nA BƩX i ƩX i Y i AƩX
polynomial curve fitting. Polynomials are one of the most The Polynomial Curve Fitting uses the method of least squares when fitting data. The fitting process requires a model that relates the response data to the predictor data with one or more coefficients. The result of the fitting process is an estimate of
EPA Test Method 1: EPA Test Method 2 EPA Test Method 3A. EPA Test Method 4 . Method 3A Oxygen & Carbon Dioxide . EPA Test Method 3A. Method 6C SO. 2. EPA Test Method 6C . Method 7E NOx . EPA Test Method 7E. Method 10 CO . EPA Test Method 10 . Method 25A Hydrocarbons (THC) EPA Test Method 25A. Method 30B Mercury (sorbent trap) EPA Test Method .
For best fitting theory curve (red curve) P(y1,.yN;a) becomes maximum! Use logarithm of product, get a sum and maximize sum: ln 2 ( ; ) 2 1 ln ( ,., ; ) 1 1 2 1 i N N i i i N y f x a P y y a OR minimize χ2with: Principle of least squares!!! Curve fitting - Least squares Principle of least squares!!! (Χ2 minimization)
