is the calculated value of the dependent variable at point i. The best parameters have values that minimize the squares of the errors

Transcription

1 Multple Lnear and Polynomal Regresson wth Statstcal Analyss Gven a set of data of measured (or observed) values of a dependent varable: y versus n ndependent varables x 1, x, x n, multple lnear regresson attempts to fnd the best values of the parameters a 0, a 1, a n for the equaton ŷ ˆ + y = a0 + a1x1, + a x, + K an xn, s the calculated value of the dependent varable at pont. The best parameters have values that mnmze the squares of the errors S N = ( y yˆ ) = 1 In polynomal regresson there s only one ndependent varable, thus y ˆ = a + a x + a x + K+ a 0 1 n x n Multple Lnear and Polynomal Regresson wth Statstcal Analyss Typcal examples of multple lnear and polynomal regressons nclude correlaton of temperature dependent physcal propertes, correlaton of heat transfer data usng dmensonless groups, correlaton of non-deal phase equlbrum data and correlaton of reacton rate data. The software packages enable hgh precson correlaton of the data, however statstcal analyss s essental to determne the qualty of the ft (how well the regresson model fts the data) and the stablty of the model (the level of dependence of the model parameters on the partcular set of data). The most mportant ndcators for such studes are the resdual plot (qualty of the ft) and 95% confdence ntervals (stablty of the model) 1

2 Correlaton of Heat Capacty Data for Ethane A polynomal has to be ftted to heat capacty data provded by Ingham et al*. Ths data set ncludes 41 data ponts n the temperature range of 100 K 400 K. The degree of the polynomal: n C = a + a T + a T + K+ a T p 0 1 where C p s the heat capacty n J/kg-mol K, T s the temperature n K, and a 0, a 1,... are the regresson model parameters, whch best represents the data, has to be found. The goodness of ft should be determned based on the varance, the correlaton coeffcent (R ), the confdence ntervals of the parameters, and the resdual plot. Ingham, H.; Frend, D.G.; Ely, J.F.; "Thermophyscal Propertes of Ethane"; J. Phys. Ref. Data 1991, 0, 75 n Heat Capacty Data for Ethane, Fttng a 3 rd Degree Polynomal Model type

3 Heat Capacty Data for Ethane, Fttng a 3rd Degree Polynomal All the parameters ndcate satsfactory model. Note, however, the dfferences of orders of magntude between the parameter values. Ths may lmt the hghest degree of polynomal to be ftted Rmsd and varance values used for comparson between dfferent models Heat Capacty Data for Ethane, Calculated (3rd Degree Polynomal) and Expermental Values On the scale of the entre range of the C p data the ft seems to be excellent 3

4 Heat Capacty Data for Ethane, Resdual Plot for the 3rd Degree Polynomal Maxmal Error ~ 1% Hgh resoluton resdual plot shows oscllatory behavor whch s not explaned by the 3 rd degree polynomal Heat Capacty Data for Ethane, Defnng Standardzed Temperature Values for Hgh Order Polynomal Fttng 4

5 Heat Capacty Data for Ethane, Fttng a 5rd Degree Polynomal Usng standardzed values yelds model parameters of smlar magntude, enables fttng hgher order polynomals and mproves consderably all the statstcal ndcators Heat Capacty Data for Ethane, Resdual Plot of a 5 th Degree Polynomal Maxmal Error ~ 0.03% Usng standardzed ndependent varable values enables fttng polynomals wth precson hgher than justfed by the expermental error. 5

6 Modelng Vapor Pressure Data for Ethane A vapor pressure data set provded by Ingham et al* ncludes 107 data ponts n the temperature range of 9 K 304 K. Ths temperature range covers almost completely the range between the trpe pont temperature (= K) and the crtcal temperature (T C = K). The temperature dependence of the vapor pressure should be modeled by the Clapeyron, Antone and Wagner equatons The Clapeyron equaton s a two parameter equaton: B ln P = A+ where P s the vapor pressure (Pa), T T temperature (K), A and B are parameters *Ingham, H.; Frend, D.G.; Ely, J.F.; "Thermophyscal Propertes of Ethane"; J. Phys. Ref. Data 1991, 0, 75 Modelng Vapor Pressure Data for Ethane by the Clapeyron Equaton usng Lnear Regresson = ln(p_pa) = 1/T_K 6

7 Modelng Vapor Pressure Data for Ethane by the Clapeyron Equaton usng Lnear Regresson All the ndcators show good, acceptable ft! Modelng Vapor Pressure Data for Ethane by the Clapeyron Equaton usng Lnear Regresson Maxmal error n ln(p) ~80% n P ~ 5% The resdual plot reveals large unexplaned curvature n the data 7

8 Modelng Vapor Pressure Data for Ethane by the Antone Equaton usng Non-lnear Regresson B ln P= A+ T + C Model type and soluton algorthm Intal guess from Clapeyron eqn. Modelng Vapor Pressure Data for Ethane by the Antone Equaton usng Non-lnear Regresson Varance smaller by orders of magntude than Clapeyron Expermental and calculated values cannot be dstngushed. 8

9 Modelng Vapor Pressure Data for Ethane by the Antone Equaton usng Non-lnear Regresson Maxmal error n ln(p) ~1% n P ~ 5% Random resdual dstrbuton n the low pressure range, unexplaned curvature n the hgh pressure range Modelng Vapor Pressure Data for Ethane wth the Wagner Equaton ln P R aτ + bτ + cτ + dτ = T Where T R = T/T C s the reduced temperature P R = P/P C s the reduced pressure and τ = 1 - T R. For ethane T C = K, P C =4.870E+06 Pa In order to obtan the model parameters usng lnear regresson the followng varables are defned: Tr = T_K / lnpr = ln(p_pa / ) t = (1 - Tr) / Tr t15 = (1 - Tr) ^ 1.5 / Tr t3 = (1 - Tr) ^ 3 / Tr t6 = (1 - Tr) ^ 6 / Tr R 9

10 Modelng Vapor Pressure Data for Ethane wth the Wagner Equaton usng Multple Lnear Regresson Tr = T_K / lnpr = ln(p_pa / ) t = (1 - Tr) / Tr t15 = (1 - Tr) ^ 1.5 / Tr t3 = (1 - Tr) ^ 3 / Tr t6 = (1 - Tr) ^ 6 / Tr Modelng Vapor Pressure Data for Ethane wth the Wagner Equaton usng Multple Lnear Regresson Maxmal Error n ln(pr) ~ 0.6% Note random resduals dstrbuton n the entre data range 10