CURVE FITTING AND DATA REGRESSION

Transcription

1 Numercl Methods Process Sstems Engneerng CURVE FIING AND DAA REGRESSION Numercl methods n chemcl engneerng Dr. Edwn Zondervn

2 Numercl Methods Process Sstems Engneerng Dngerous curves!!! hs s not ectl wht I ment

3 Numercl Methods Process Sstems Engneerng OVERVIEW We re gong to ft mesurements to models tod. You wll lso lern wht R ctull mens

4 Numercl Methods Process Sstems Engneerng FIING MODELS O DAA s the mesured vrble s the controlled vrble

5 5 Numercl Methods Process Sstems Engneerng HOW O FI A MODEL O HE DAA We would lke to ft the followng model to the dt: Frst step: If we hve N dt ponts, we could wrte the model s the product of mtr nd vector: N N N N M M M M M ŷ s clled the desgn mtr nd re the ft prmeters

6 6 Numercl Methods Process Sstems Engneerng RESIDUALS Second step: work out the resduls for ech dt pont: hrd step: Work out the sum of squres of the resduls: d d d d d d d

7 Numercl Methods Process Sstems Engneerng MINIMIZING HE SUM OF SQUARES Choose the prmeter vector such tht the sum of squres of the resduls s mnmzed; the prtl dervtve wth respect to ech prmeter should be set to zero: [ ] 9-7 7

8 8 Numercl Methods Process Sstems Engneerng MINIMIZING HE SUM OF SQUARES e e e e e e e e e e e ] [ ] [ ] [ ] [

9 9 Numercl Methods Process Sstems Engneerng USING MALAB FOR LLSQ If we hve the sme number of dt ponts s ft prmeters, we cn solve the sstem: As = \ 9-

10 Numercl Methods Process Sstems Engneerng USING MALAB FPR LLSQ If there re more dt ponts N>, we cn wrte n nlogue, but mbe consstent soluton does not est the sstem s over specfed. However, mtlb wll fnd vlues for the vector whch mnmze -, so.e. lest squres ft.

11 Numercl Methods Process Sstems Engneerng FI O OUR PROBLEM N=length; :, = onesn,; :, = ; :, =.^; :, =.^; mesurements ftted: = A =\;

12 resdul Numercl Methods Process Sstems Engneerng HOW GOOD IS HE MODEL? For model to mke sense the dt ponts should be scttered rndoml round the model predctons, the men of the resduls d should be zero. d It s lws good to check f the resduls re not correlted wth the mesured vlues, f tht s the cse, t cn ndcte tht our model s wrong.

13 Numercl Methods Process Sstems Engneerng REGRESSION COEFFICIENS Vrnce mesured n the dt s: Vrnce of the resduls s: Vrnce n the model s: N error d N 9-9- model N 9-

14 Numercl Methods Process Sstems Engneerng REGRESSION COEFFICIENS Gven tht the error s uncorrelted we cn stte tht: error model 9- R model error 9-5 R 9-6 SSE SS SSE: Sum of errors squred SSR: Sum of squres dt SS: otl sum of squres model

15 Numercl Methods Process Sstems Engneerng SAISICAL ANALYSIS An uncorrellted error men wll be zero SSE, SS nd SSR wll hve -dstrbutons nd the rtos wll hve n F-dstrbuton. If SSR/SSE s lrge, the model s good! here s chnge tht the model s rubbsh, but tht SSR/SSE wll eld good vlue, Anlss of Vrnce ANOVA wll be good tool to clculte the probblt of such thng hppenng! 5

16 Numercl Methods Process Sstems Engneerng BACK O HE EAMPLE Stts: N = SSE = 8. SS =.59 SSR =.59 R =.965 Source Deg. Of freedom Sum of squres Regresson K = SSR = F = Resdul N-K-= SSE = F-vlue otl 99 SS =.55 F>657 mens: ver unlkel!!! F = SSR//SSE/95 6

17 Numercl Methods Process Sstems Engneerng CONFIDENCE LIMIS FOR HE FI PARAMEERS Usng the t-dstrbuton, the confdence lmts for the ft prmeters cn be set s: t error [ ], t error [ ] 9-7, Degree of freedom Must be looked up from sttstcl tble Is the -th dgonl element of the smmetrc mtr 7

18 Numercl Methods Process Sstems Engneerng CONFIDENCE LIMIS FOR HE PREDICED POINS A confdence ntervl for ech predcted vlue s gven b: t error [ ], t error [ ], 9-8 8

19 Numercl Methods Process Sstems Engneerng SUMMARY We hve seen how ft prmeters of model cn ftted to dt set, usng the lner lest squres method. We found out how to clculte the regresson coeffcents nd how to perform sttstcl nlss of the model usng ANOVA. We lso postulted epressons for the confdence lmts for the ft prmeters nd the predcted ponts 9