Experimental Errors and Error Analysis

Transcription

1 Expermental Errors and Error Analss Rajee Prabhakar Unerst of Texas at Astn, Astn, TX Freeman Research Grop Meetng March 10, 004

2 Topcs coered Tpes of expermental errors Redcng errors Descrbng errors qanttatel Measred arables Qanttes calclated from measred arables (error propagaton) Least-sqares ft straght lne polnomal arbtrar fncton Ref: Bengton and Robnson, Data Redcton and Error Analss for the Phscal Scences, nd edton, McGraw Hll, NY 199

3 Qalt of Experments and Expermental Error Accrac Measre of correctness of reslt Precson Measre of reprodcblt of reslt Tpes of errors Mstakes Wrong membrane sed Nmber recorded ncorrectl, 1.9 wrtten as 1.9 Sstematc error Change n membrane castng condtons Incorrect calbraton Random error Flctatons n temperatre, pressre etc.

4 Redcng Errors Plannng experments carefll Objecte s to nderstand and redce sorces of sstematc error Don t ge n to the t s alwas been done ths wa mentalt Use more accrate nstrments Repeat the measrement, f possble

5 Uncertantes n Measred Varables Flctatons Often ndependent of actal ale Readng an analog nstrment ½ of smallest scale dson/least cont Uncertant estmate standard deaton, ½ of least cont Repeat measrement f possble and take standard deaton of the data

6 Parameters calclated from measred arables Propagaton of Errors V e.g. Volme measrement = LW H o o o o If the measred ales are L, W and H, V V V V( L, W, H) V + L W H o + + L W H WoHo LoHo LoWo Error, V = V V o N N ( ) ( ) 1 1 = V o = N = 1 = 1 Varance, V V lm V V N N

7 Propagaton of errors general formla ( ) 1 ( ) ( ) ( ( ) ( ) ( (, ) ) 1 lm 1 lm ) N N N f N N = = = + = = + + = +, Neglectng the co - arance + + (1) () co-arance

8 Volme example smplfed eqaton + V V V + + V L W H L W H SnceV = LWH, ( WH ) + ( HL) + ( LW ) V L W H V V V + + V L W H L W H V L W H V + + L W H Smplfed form of eqaton ald when the dependent arable s a prodct of ndependent arables to the 1 or -1 power onl

9 Example - Lmtatons of smplfed eqaton If V = π R H, V V V R + H R H ( RH ) + ( R ) π π V R H V V V R + H R H V R H V + R H

10 Least-sqares ft to a straght lne = a + b*x Objecte s to fnd the ales of a and b that prode the best predcton of for a gen x Goodness-of-ft parameter, χ 1 χ = ( a bx ) (3) To mnmze the error, χ χ = 0 = a b Resltng eqatons for a and b pgs. 104 and 113 of Bengton Errors n a and b - Eqatons on pgs. 109 & 114 of Bengton

11 Least-sqares ft to an eqaton lnear n the coeffcents Example, Polnomal: = a + b*x + c*x +.. Other fnctons: = a + b*e x Smlar mathematcal treatment as for straght lne. Howeer, determnants get bgger and more nmeros. See pg 117 (last 3 lnes) and 118 for example of qadratc eqaton. Alternate method Matrx solton (pg. 11)

12 Matrx Method for eqaton lnear n coeffcents x ( ) = af( x) k k k = 1 The coeffcent matrx a s gen b, a = βα 1 m where 1 β = f ( x ) ; row matrx k k α 1 = f ( x ) f ( x) ; smmetrc matrx lk l k Error n the coeffcents are elements of the nerse α matrx - Dagonal elements are arances - Off-dagonal elements are co-arances = α 1 ak kk = α 1 aa l k lk

13 Example Qadratc ft to Permeablt data P= b+ c p+ d( p) Compare wth, m x ( ) = af( x) = af( x) + af( x) + af( x) k k k = 1 Therefore, a = b; a = c; a = d; f = 1; f = p; f = ( p) Sole for a, b and c sng matrx method Yo wll also obtan arances and coarances (preos slde) Then, calclate error n P sng eqs. (1) or (). Example n Excel fle: Error analss demo.xls; sheet: P For more detals, ncldng an example, Bengton: pgs

14 Least-sqares ft to an fncton Best-ft for fnctons that are not lnear n the coeffcents Tral-and-error methods (Ch. 8, Bengton) Fnd coeffcent ales b sng SOLVER fncton n Excel To get errors, ar one coeffcent at a tme to ncrease b 1 χ The change n the coeffcent s t s standard deaton

15 Example S ft to dal mode eqaton f * g S = e+ 1 + g* p Determne coeffcents b mnmzng χ, N 1 S S calc expt χ = N = 1 b arng the coeffcents e, f and g. Change ntal gess ales of the coeffcents to confrm best - ft ales. Then determne for each coeffcent, b the method descrbed on the preos slde. Example n Excel fle: Error analss demo.xls; sheet: S

16 Calclatng D eff from best ft eqatons of P and S ( ) dp dp = + D C eff P p d p p dc p + ( ) + 3 ( ) = = fg e + (1 + gp) b c p d p NUMR DENR Calclate NUMR & DENR and ther errors. Calclate D eff. Error n D eff, D eff NUMR DENR = + D NUMR DENR eff

17 References Bengton Pgs. 1-6: Accrac, precson and ncertantes Pgs : Propagaton of errors wth specfc examples Pgs : Least-sqares ft to a straght lne Pgs : Least-sqares ft to a fncton lnear n the coeffcents, ncldng an example for a qadratc ft Mcrosoft Excel Help MINVERSE worksheet fncton MMULT worksheet fncton

18 Qestons